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"62f78d205c303365b156144e260330ee3fd0798bb85f15e76eb44215d7759979", "global_argument_positions": null, "global_definition": null, "id": "PD0004", "kernel_signature_unchanged": true, "name": "Prime", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "p" ], "reviewed_definition_id": "PD0004", "shared_definition_identity": "PD0004", "signature": "Prime(p)", "summary": "p is nonunit and every factorization of p has a unit factor.", "topological_layer": 0, "transitive_dependencies": [] }, { "arity": 2, "defined_template": "∀ d. Dvd(d,a) → Dvd(d,b) → d = 1", "defined_template_parts": [ { "kind": "text", "text": "∀ d. " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,b)" }, { "kind": "text", "text": " → d = 1" } ], "dependencies": [ "PD0003" ], "dependency_names": [ "Dvd" ], "exact_ast_verified": true, "expanded_template": "forall d. (exists x. a = d * x) -> (exists y. b = d * y) -> d = 1", "expansion_sha256": "9309813a1797d5b48ddd1596997e568cefb9260ed22f0486c4675d6a70b58f8b", "global_argument_positions": null, "global_definition": null, "id": "PD0005", "kernel_signature_unchanged": true, "name": "Coprime", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "a", "b" ], "reviewed_definition_id": "PD0005", "shared_definition_identity": "PD0005", "signature": "Coprime(a,b)", "summary": "Every common divisor of a and b is one.", "topological_layer": 1, "transitive_dependencies": [ "PD0003" ] }, { "arity": 4, "defined_template": "n = d · q + r ∧ Lt(r,d)", "defined_template_parts": [ { "kind": "text", "text": "n = d · q + r ∧ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(r,d)" } ], "dependencies": [ "PD0002" ], "dependency_names": [ "Lt" ], "exact_ast_verified": true, "expanded_template": "n = d * q + r /\\ exists h. h + S r = d", "expansion_sha256": "16c8dd61da6b99cec76efc87e4597c37f5f3a82e6ff4c9f6271b1ccf8bd869cd", "global_argument_positions": null, "global_definition": null, "id": "PD0007", "kernel_signature_unchanged": true, "name": "DivRem", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "n", "d", "q", "r" ], "reviewed_definition_id": "PD0007", "shared_definition_identity": "PD0007", "signature": "DivRem(n,d,q,r)", "summary": "q and r are a quotient and a strict remainder for n by d.", "topological_layer": 1, "transitive_dependencies": [ "PD0002" ] }, { "arity": 3, "defined_template": "∃ u. ∃ v. a + m · u = b + m · v", "defined_template_parts": [ { "kind": "text", "text": "∃ u. ∃ v. a + m · u = b + m · v" } ], "dependencies": [], "dependency_names": [], "exact_ast_verified": true, "expanded_template": "exists u v. a + m * u = b + m * v", "expansion_sha256": "14b22c6894ec6a5f7d1a8a1cb62865dc5effcb102a2fdae0d7b9a87820a44fb6", "global_argument_positions": null, "global_definition": null, "id": "PD0008", "kernel_signature_unchanged": true, "name": "ModEq", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "m", "a", "b" ], "reviewed_definition_id": "PD0008", "shared_definition_identity": "PD0008", "signature": "ModEq(m,a,b)", "summary": "Balanced-natural congruence modulo m.", "topological_layer": 0, "transitive_dependencies": [] }, { "arity": 4, "defined_template": "S x ≤ S (S i · c) ∧ (∃ y. b = y · S (S i · c) + x)", "defined_template_parts": [ { "kind": "text", "text": "S x ≤ S (S i · c) ∧ (∃ y. b = y · S (S i · c) + x)" } ], "dependencies": [], "dependency_names": [], "exact_ast_verified": true, "expanded_template": "((exists ff_h_defined_beta_at. ff_h_defined_beta_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_defined_beta_at. b = ff_q_defined_beta_at * S ((S (i)) * c) + (x))", "expansion_sha256": "469d8f255ea8dd94ea2088cf49a8c80a8bccf69fd14a841394717e7e449e73e9", "global_argument_positions": null, "global_definition": null, "id": "PD0013", "kernel_signature_unchanged": true, "name": "BetaAt", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "i", "x" ], "reviewed_definition_id": "PD0013", "shared_definition_identity": "PD0013", "signature": "BetaAt(b,c,i,x)", "summary": "x is the bounded beta-decoded value at index i.", "topological_layer": 0, "transitive_dependencies": [] }, { "arity": 4, "defined_template": "∃ ff_u_defined_product. ∃ ff_v_defined_product. BetaAt(ff_u_defined_product,ff_v_defined_product,0,1) ∧ (BetaAt(ff_u_defined_product,ff_v_defined_product,l,z) ∧ (∀ x. Lt(x,l) → ∃ y. ∃ n. ∃ m. BetaAt(b,c,x,y) ∧ (BetaAt(ff_u_defined_product,ff_v_defined_product,x,n) ∧ (BetaAt(ff_u_defined_product,ff_v_defined_product,S x,m) ∧ m = n · y))))", "defined_template_parts": [ { "kind": "text", "text": "∃ ff_u_defined_product. ∃ ff_v_defined_product. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(ff_u_defined_product,ff_v_defined_product,0,1)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(ff_u_defined_product,ff_v_defined_product,l,z)" }, { "kind": "text", "text": " ∧ (∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,l)" }, { "kind": "text", "text": " → ∃ y. ∃ n. ∃ m. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,x,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(ff_u_defined_product,ff_v_defined_product,x,n)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(ff_u_defined_product,ff_v_defined_product,S x,m)" }, { "kind": "text", "text": " ∧ m = n · y))))" } ], "dependencies": [ "PD0002", "PD0013" ], "dependency_names": [ "Lt", "BetaAt" ], "exact_ast_verified": true, "expanded_template": "exists ff_u_defined_product ff_v_defined_product. ((((exists ff_h_defined_product_start. ff_h_defined_product_start + S (1) = S ((S (0)) * ff_v_defined_product)) /\\ exists ff_q_defined_product_start. ff_u_defined_product = ff_q_defined_product_start * S ((S (0)) * ff_v_defined_product) + (1))) /\\ ((((exists ff_h_defined_product_terminal. ff_h_defined_product_terminal + S (z) = S ((S (l)) * ff_v_defined_product)) /\\ exists ff_q_defined_product_terminal. ff_u_defined_product = ff_q_defined_product_terminal * S ((S (l)) * ff_v_defined_product) + (z))) /\\ forall ff_i_defined_product. (exists ff_lt_defined_product_bound. ff_lt_defined_product_bound + S ff_i_defined_product = l) -> exists ff_p_defined_product ff_r_defined_product ff_s_defined_product. ((((exists ff_h_defined_product_factor. ff_h_defined_product_factor + S (ff_p_defined_product) = S ((S (ff_i_defined_product)) * c)) /\\ exists ff_q_defined_product_factor. b = ff_q_defined_product_factor * S ((S (ff_i_defined_product)) * c) + (ff_p_defined_product))) /\\ ((((exists ff_h_defined_product_partial. ff_h_defined_product_partial + S (ff_r_defined_product) = S ((S (ff_i_defined_product)) * ff_v_defined_product)) /\\ exists ff_q_defined_product_partial. ff_u_defined_product = ff_q_defined_product_partial * S ((S (ff_i_defined_product)) * ff_v_defined_product) + (ff_r_defined_product))) /\\ ((((exists ff_h_defined_product_successor. ff_h_defined_product_successor + S (ff_s_defined_product) = S ((S (S ff_i_defined_product)) * ff_v_defined_product)) /\\ exists ff_q_defined_product_successor. ff_u_defined_product = ff_q_defined_product_successor * S ((S (S ff_i_defined_product)) * ff_v_defined_product) + (ff_s_defined_product))) /\\ ff_s_defined_product = ff_r_defined_product * ff_p_defined_product)))))", "expansion_sha256": "f39381e906a4c89c28a06167c706e2e14937eaa40a879fc5658d85f9892c1653", "global_argument_positions": null, "global_definition": null, "id": "PD0014", "kernel_signature_unchanged": true, "name": "Product", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "l", "z" ], "reviewed_definition_id": "PD0014", "shared_definition_identity": "PD0014", "signature": "Product(b,c,l,z)", "summary": "z is the product of a beta-coded prefix of length l.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0013" ] }, { "arity": 4, "defined_template": "∀ ff_i_defined_repeat. Lt(ff_i_defined_repeat,l) → BetaAt(b,c,ff_i_defined_repeat,a)", "defined_template_parts": [ { "kind": "text", "text": "∀ ff_i_defined_repeat. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(ff_i_defined_repeat,l)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,ff_i_defined_repeat,a)" } ], "dependencies": [ "PD0002", "PD0013" ], "dependency_names": [ "Lt", "BetaAt" ], "exact_ast_verified": true, "expanded_template": "forall ff_i_defined_repeat. (exists ff_lt_defined_repeat_bound. ff_lt_defined_repeat_bound + S ff_i_defined_repeat = l) -> (((exists ff_h_defined_repeat_decoded. ff_h_defined_repeat_decoded + S (a) = S ((S (ff_i_defined_repeat)) * c)) /\\ exists ff_q_defined_repeat_decoded. b = ff_q_defined_repeat_decoded * S ((S (ff_i_defined_repeat)) * c) + (a)))", "expansion_sha256": "5ddd6737391a6ae01b8d7fad7a817a96233301d196787fc9c22d602290da819d", "global_argument_positions": null, "global_definition": null, "id": "PD0019", "kernel_signature_unchanged": true, "name": "Repeat", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "a", "l" ], "reviewed_definition_id": "PD0019", "shared_definition_identity": "PD0019", "signature": "Repeat(b,c,a,l)", "summary": "The decoded prefix repeats a for l positions.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0013" ] }, { "arity": 3, "defined_template": "∃ ff_b_defined_power. ∃ ff_c_defined_power. Repeat(ff_b_defined_power,ff_c_defined_power,a,e) ∧ Product(ff_b_defined_power,ff_c_defined_power,e,z)", "defined_template_parts": [ { "kind": "text", "text": "∃ ff_b_defined_power. ∃ ff_c_defined_power. " }, { "definition": "PD0019", "kind": "definition", "text": "Repeat(ff_b_defined_power,ff_c_defined_power,a,e)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0014", "kind": "definition", "text": "Product(ff_b_defined_power,ff_c_defined_power,e,z)" } ], "dependencies": [ "PD0014", "PD0019" ], "dependency_names": [ "Product", "Repeat" ], "exact_ast_verified": true, "expanded_template": "exists ff_b_defined_power ff_c_defined_power. ((forall ff_i_defined_power_repeat. (exists ff_lt_defined_power_repeat_bound. ff_lt_defined_power_repeat_bound + S ff_i_defined_power_repeat = e) -> (((exists ff_h_defined_power_repeat_decoded. ff_h_defined_power_repeat_decoded + S (a) = S ((S (ff_i_defined_power_repeat)) * ff_c_defined_power)) /\\ exists ff_q_defined_power_repeat_decoded. ff_b_defined_power = ff_q_defined_power_repeat_decoded * S ((S (ff_i_defined_power_repeat)) * ff_c_defined_power) + (a)))) /\\ (exists ff_u_defined_power_product ff_v_defined_power_product. ((((exists ff_h_defined_power_product_start. ff_h_defined_power_product_start + S (1) = S ((S (0)) * ff_v_defined_power_product)) /\\ exists ff_q_defined_power_product_start. ff_u_defined_power_product = ff_q_defined_power_product_start * S ((S (0)) * ff_v_defined_power_product) + (1))) /\\ ((((exists ff_h_defined_power_product_terminal. ff_h_defined_power_product_terminal + S (z) = S ((S (e)) * ff_v_defined_power_product)) /\\ exists ff_q_defined_power_product_terminal. ff_u_defined_power_product = ff_q_defined_power_product_terminal * S ((S (e)) * ff_v_defined_power_product) + (z))) /\\ forall ff_i_defined_power_product. (exists ff_lt_defined_power_product_bound. ff_lt_defined_power_product_bound + S ff_i_defined_power_product = e) -> exists ff_p_defined_power_product ff_r_defined_power_product ff_s_defined_power_product. ((((exists ff_h_defined_power_product_factor. ff_h_defined_power_product_factor + S (ff_p_defined_power_product) = S ((S (ff_i_defined_power_product)) * ff_c_defined_power)) /\\ exists ff_q_defined_power_product_factor. ff_b_defined_power = ff_q_defined_power_product_factor * S ((S (ff_i_defined_power_product)) * ff_c_defined_power) + (ff_p_defined_power_product))) /\\ ((((exists ff_h_defined_power_product_partial. ff_h_defined_power_product_partial + S (ff_r_defined_power_product) = S ((S (ff_i_defined_power_product)) * ff_v_defined_power_product)) /\\ exists ff_q_defined_power_product_partial. ff_u_defined_power_product = ff_q_defined_power_product_partial * S ((S (ff_i_defined_power_product)) * ff_v_defined_power_product) + (ff_r_defined_power_product))) /\\ ((((exists ff_h_defined_power_product_successor. ff_h_defined_power_product_successor + S (ff_s_defined_power_product) = S ((S (S ff_i_defined_power_product)) * ff_v_defined_power_product)) /\\ exists ff_q_defined_power_product_successor. ff_u_defined_power_product = ff_q_defined_power_product_successor * S ((S (S ff_i_defined_power_product)) * ff_v_defined_power_product) + (ff_s_defined_power_product))) /\\ ff_s_defined_power_product = ff_r_defined_power_product * ff_p_defined_power_product)))))))", "expansion_sha256": "7bf8dc5933e3f5226c6d1c81ba902c46bb7688d42f73313aa8d8deeda0b2ba81", "global_argument_positions": null, "global_definition": null, "id": "PD0020", "kernel_signature_unchanged": true, "name": "Pow", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "a", "e", "z" ], "reviewed_definition_id": "PD0020", "shared_definition_identity": "PD0020", "signature": "Pow(a,e,z)", "summary": "z is the relational e-th power of a.", "topological_layer": 2, "transitive_dependencies": [ "PD0002", "PD0013", "PD0014", "PD0019" ] }, { "arity": 4, "defined_template": "S x ≤ S (S i · c) ∧ (∃ y. b = y · S (S i · c) + x)", "defined_template_parts": [ { "kind": "text", "text": "S x ≤ S (S i · c) ∧ (∃ y. b = y · S (S i · c) + x)" } ], "dependencies": [], "dependency_names": [], "exact_ast_verified": true, "expanded_template": "((exists ff_h_defined_beta_at. ff_h_defined_beta_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_defined_beta_at. b = ff_q_defined_beta_at * S ((S (i)) * c) + (x))", "expansion_sha256": "469d8f255ea8dd94ea2088cf49a8c80a8bccf69fd14a841394717e7e449e73e9", "global_argument_positions": null, "global_definition": null, "id": "ND0001", "kernel_signature_unchanged": true, "name": "Beta", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "i", "x" ], "reviewed_definition_id": "ND0001", "shared_definition_identity": "ND0001", "signature": "Beta(b,c,i,x)", "summary": "Exact hygienic Gödel-beta extraction; a signature-identical alias of checked BetaAt.", "topological_layer": 0, "transitive_dependencies": [] }, { "arity": 6, "defined_template": "Beta(b,c,i · w + j,z)", "defined_template_parts": [ { "definition": "ND0001", "kind": "definition", "text": "Beta(b,c,i · w + j,z)" } ], "dependencies": [ "ND0001" ], "dependency_names": [ "Beta" ], "exact_ast_verified": true, "expanded_template": "((exists fs_h_matrix_explorer. fs_h_matrix_explorer + S (z) = S ((S (i * w + j)) * c)) /\\ exists fs_q_matrix_explorer. b = fs_q_matrix_explorer * S ((S (i * w + j)) * c) + (z))", "expansion_sha256": "2ed14165f516f8f0272e7f05cfe7f130fefff3c0234403c63235d86f4f77d45b", "global_argument_positions": null, "global_definition": null, "id": "ND0003", "kernel_signature_unchanged": true, "name": "MatrixAt", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "w", "i", "j", "z" ], "reviewed_definition_id": "ND0003", "shared_definition_identity": "ND0003", "signature": "MatrixAt(b,c,w,i,j,z)", "summary": "The exact natural matrix entry stored at flattened beta index i*w+j.", "topological_layer": 1, "transitive_dependencies": [ "ND0001" ] }, { "arity": 3, "defined_template": "Lt(r,m) ∧ ModEq(m,a,r)", "defined_template_parts": [ { "definition": "PD0002", "kind": "definition", "text": "Lt(r,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,a,r)" } ], "dependencies": [ "PD0002", "PD0008" ], "dependency_names": [ "Lt", "ModEq" ], "exact_ast_verified": true, "expanded_template": "((exists ff_gap_binary_advanced. ff_gap_binary_advanced + S (r) = m) /\\ (exists ff_left_binary_advanced_congruence ff_right_binary_advanced_congruence. (a) + m * ff_left_binary_advanced_congruence = (r) + m * ff_right_binary_advanced_congruence))", "expansion_sha256": "b5def46e1332ad938c788d97ce9b1a5e9f66cb1981cdf5dfd62d8b9a1b6410e0", "global_argument_positions": null, "global_definition": null, "id": "ND0023", "kernel_signature_unchanged": true, "name": "CanonicalModularResidue", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "m", "a", "r" ], "reviewed_definition_id": "ND0023", "shared_definition_identity": "ND0023", "signature": "CanonicalModularResidue(m,a,r)", "summary": "A strictly bounded canonical natural residue r exists fom_value_mrf_secondwavesource. ((((exists fom_beta_height_mrf_secondwavesource_entry. fom_beta_height_mrf_secondwavesource_entry + S (fom_value_mrf_secondwavesource) = S ((S (fom_index_mrf_secondwavesource)) * mdr_e_secondwave)) /\\ exists fom_beta_quotient_mrf_secondwavesource_entry. mdr_b_secondwave = fom_beta_quotient_mrf_secondwavesource_entry * S ((S (fom_index_mrf_secondwavesource)) * mdr_e_secondwave) + (fom_value_mrf_secondwavesource))) /\\ (exists fom_gap_mrf_secondwavesource_value_bound. fom_gap_mrf_secondwavesource_value_bound + S (fom_value_mrf_secondwavesource) = B))) -> exists mdr_z_secondwave. (((exists mdr_gap_secondwavebound. mdr_gap_secondwavebound + S (mdr_z_secondwave) = (T)) /\\ (forall mdr_i_secondwaveprefix mdr_a_secondwaveprefix. (exists mdr_gap_secondwaveprefixb. mdr_gap_secondwaveprefixb + S (mdr_i_secondwaveprefix) = (l)) -> (((exists ff_h_mdr_secondwaveprefixo. ff_h_mdr_secondwaveprefixo + S (mdr_a_secondwaveprefix) = S ((S (mdr_i_secondwaveprefix)) * mdr_e_secondwave)) /\\ exists ff_q_mdr_secondwaveprefixo. mdr_b_secondwave = ff_q_mdr_secondwaveprefixo * S ((S (mdr_i_secondwaveprefix)) * mdr_e_secondwave) + (mdr_a_secondwaveprefix))) -> (((exists ff_h_mdr_secondwaveprefixn. ff_h_mdr_secondwaveprefixn + S (mdr_a_secondwaveprefix) = S ((S (mdr_i_secondwaveprefix)) * c)) /\\ exists ff_q_mdr_secondwaveprefixn. mdr_z_secondwave = ff_q_mdr_secondwaveprefixn * S ((S (mdr_i_secondwaveprefix)) * c) + (mdr_a_secondwaveprefix))))))))", "expansion_sha256": "1ba61efb69895e675f9c7915f6bf093858f2b7887494a5a3457005dd2c215d43", "global_argument_positions": null, "global_definition": null, "id": "ND0112", "kernel_signature_unchanged": true, "name": "UniformBetaPrefixBox", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "c", "T", "l", "B" ], "reviewed_definition_id": "ND0112", "shared_definition_identity": "ND0112", "signature": "UniformBetaPrefixBox(c,T,l,B)", "summary": "One fixed scale c and positive finite code bound T recode every actual length-l prefix with values below B; completeness is part of the relation.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0013" ] }, { "arity": 4, "defined_template": "(∀ x. Lt(x,l) → ∃ y. BetaAt(b,c,x,y) ∧ Lt(y,B)) ∧ (∀ x. ∀ y. ∀ z. Lt(x,l) → Lt(y,l) → BetaAt(b,c,x,z) → BetaAt(b,c,y,z) → x = y)", "defined_template_parts": [ { "kind": "text", "text": "(∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,l)" }, { "kind": "text", "text": " → ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,x,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,B)" }, { "kind": "text", "text": ") ∧ (∀ x. ∀ y. ∀ z. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,l)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,l)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,x,z)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,y,z)" }, { "kind": "text", "text": " → x = y)" } ], "dependencies": [ "PD0013", "PD0002" ], "dependency_names": [ "BetaAt", "Lt" ], "exact_ast_verified": true, "expanded_template": "((forall fom_index_mrf_secondwavebound. (exists fom_gap_mrf_secondwavebound_index_bound. fom_gap_mrf_secondwavebound_index_bound + S (fom_index_mrf_secondwavebound) = l) -> exists fom_value_mrf_secondwavebound. ((((exists fom_beta_height_mrf_secondwavebound_entry. fom_beta_height_mrf_secondwavebound_entry + S (fom_value_mrf_secondwavebound) = S ((S (fom_index_mrf_secondwavebound)) * c)) /\\ exists fom_beta_quotient_mrf_secondwavebound_entry. b = fom_beta_quotient_mrf_secondwavebound_entry * S ((S (fom_index_mrf_secondwavebound)) * c) + (fom_value_mrf_secondwavebound))) /\\ (exists fom_gap_mrf_secondwavebound_value_bound. fom_gap_mrf_secondwavebound_value_bound + S (fom_value_mrf_secondwavebound) = B))) /\\ (forall mdr_i_secondwavedistinct mdr_j_secondwavedistinct mdr_a_secondwavedistinct. (exists mdr_gap_secondwavedistincti. mdr_gap_secondwavedistincti + S (mdr_i_secondwavedistinct) = (l)) -> (exists mdr_gap_secondwavedistinctj. mdr_gap_secondwavedistinctj + S (mdr_j_secondwavedistinct) = (l)) -> (((exists ff_h_mdr_secondwavedistinctfirst. ff_h_mdr_secondwavedistinctfirst + S (mdr_a_secondwavedistinct) = S ((S (mdr_i_secondwavedistinct)) * c)) /\\ exists ff_q_mdr_secondwavedistinctfirst. b = ff_q_mdr_secondwavedistinctfirst * S ((S (mdr_i_secondwavedistinct)) * c) + (mdr_a_secondwavedistinct))) -> (((exists ff_h_mdr_secondwavedistinctsecond. ff_h_mdr_secondwavedistinctsecond + S (mdr_a_secondwavedistinct) = S ((S (mdr_j_secondwavedistinct)) * c)) /\\ exists ff_q_mdr_secondwavedistinctsecond. b = ff_q_mdr_secondwavedistinctsecond * S ((S (mdr_j_secondwavedistinct)) * c) + (mdr_a_secondwavedistinct))) -> mdr_i_secondwavedistinct = mdr_j_secondwavedistinct))", "expansion_sha256": "68fd24790f5e6d7535551afcc8226f6f0c47e6b2b001fc60f56c757f7c2b7807", "global_argument_positions": null, "global_definition": null, "id": "ND0113", "kernel_signature_unchanged": true, "name": "FiniteMatrixSelector", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "l", "B" ], "reviewed_definition_id": "ND0113", "shared_definition_identity": "ND0113", "signature": "FiniteMatrixSelector(b,c,l,B)", "summary": "Actual beta-decoded matrix coordinates are all below B and pairwise distinct; the list may be empty.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0013" ] }, { "arity": 5, "defined_template": "∀ ics_index_secondwave. ∀ ics_value0_secondwave. ∀ ics_value1_secondwave. Lt(ics_index_secondwave,l) → BetaAt(ab,ac,ics_index_secondwave,ics_value0_secondwave) → BetaAt(db,dc,ics_index_secondwave,ics_value1_secondwave) → ics_value0_secondwave = ics_value1_secondwave", "defined_template_parts": [ { "kind": "text", "text": "∀ ics_index_secondwave. ∀ ics_value0_secondwave. ∀ ics_value1_secondwave. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(ics_index_secondwave,l)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(ab,ac,ics_index_secondwave,ics_value0_secondwave)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(db,dc,ics_index_secondwave,ics_value1_secondwave)" }, { "kind": "text", "text": " → ics_value0_secondwave = ics_value1_secondwave" } ], "dependencies": [ "PD0013", "PD0002" ], "dependency_names": [ "BetaAt", "Lt" ], "exact_ast_verified": true, "expanded_template": "forall ics_index_secondwave ics_value0_secondwave ics_value1_secondwave. (exists ics_gap_secondwave_bound. ics_gap_secondwave_bound + S (ics_index_secondwave) = (l)) -> (((exists fs_h_ics_secondwave_at0. fs_h_ics_secondwave_at0 + S (ics_value0_secondwave) = S ((S (ics_index_secondwave)) * ac)) /\\ exists fs_q_ics_secondwave_at0. ab = fs_q_ics_secondwave_at0 * S ((S (ics_index_secondwave)) * ac) + (ics_value0_secondwave))) -> (((exists fs_h_ics_secondwave_at1. fs_h_ics_secondwave_at1 + S (ics_value1_secondwave) = S ((S (ics_index_secondwave)) * dc)) /\\ exists fs_q_ics_secondwave_at1. db = fs_q_ics_secondwave_at1 * S ((S (ics_index_secondwave)) * dc) + (ics_value1_secondwave))) -> ics_value0_secondwave = ics_value1_secondwave", "expansion_sha256": "2be65a06be66dea4605cd6d2a005862b33b0383c1aeeefb0addac79029c58752", "global_argument_positions": null, "global_definition": null, "id": "ND0121", "kernel_signature_unchanged": true, "name": "IntegerVectorZero", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "ab", "ac", "db", "dc", "l" ], "reviewed_definition_id": "ND0121", "shared_definition_identity": "ND0121", "signature": "IntegerVectorZero(ab,ac,db,dc,l)", "summary": "Every actual positive component equals its negative component, so each represented integer is zero.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0013" ] }, { "arity": 4, "defined_template": "∀ fom_index_pfp_lowertier. Lt(fom_index_pfp_lowertier,l) → ∃ x. BetaAt(b,c,fom_index_pfp_lowertier,x) ∧ Lt(x,B)", "defined_template_parts": [ { "kind": "text", "text": "∀ fom_index_pfp_lowertier. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(fom_index_pfp_lowertier,l)" }, { "kind": "text", "text": " → ∃ x. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,fom_index_pfp_lowertier,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,B)" } ], "dependencies": [ "PD0002", "PD0013" ], "dependency_names": [ "Lt", "BetaAt" ], "exact_ast_verified": true, "expanded_template": "forall fom_index_pfp_lowertier. (exists fom_gap_pfp_lowertier_index_bound. fom_gap_pfp_lowertier_index_bound + S (fom_index_pfp_lowertier) = (l)) -> exists fom_value_pfp_lowertier. ((((exists fom_beta_height_pfp_lowertier_entry. fom_beta_height_pfp_lowertier_entry + S (fom_value_pfp_lowertier) = S ((S (fom_index_pfp_lowertier)) * (c))) /\\ exists fom_beta_quotient_pfp_lowertier_entry. (b) = fom_beta_quotient_pfp_lowertier_entry * S ((S (fom_index_pfp_lowertier)) * (c)) + (fom_value_pfp_lowertier))) /\\ (exists fom_gap_pfp_lowertier_value_bound. fom_gap_pfp_lowertier_value_bound + S (fom_value_pfp_lowertier) = (B)))", "expansion_sha256": "8761d21550e7982c1c21c17dbccefafaee5466ac658344d943b5c1fe7b26dea4", "global_argument_positions": null, "global_definition": null, "id": "ND0262", "kernel_signature_unchanged": true, "name": "BetaPrefixInto", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "l", "B" ], "reviewed_definition_id": "ND0262", "shared_definition_identity": "ND0262", "signature": "BetaPrefixInto(b,c,l,B)", "summary": "Every actual beta entry below the strict length l has a witnessed value below B. The same generic graph describes canonical polynomial coefficients when B is the modulus; empty prefixes are allowed.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0013" ] }, { "arity": 5, "defined_template": "∀ mdr_i_pfp_lowertier. ∀ mdr_a_pfp_lowertier. Lt(mdr_i_pfp_lowertier,l) → BetaAt(b,c,mdr_i_pfp_lowertier,mdr_a_pfp_lowertier) → BetaAt(d,e,mdr_i_pfp_lowertier,mdr_a_pfp_lowertier)", "defined_template_parts": [ { "kind": "text", "text": "∀ mdr_i_pfp_lowertier. ∀ mdr_a_pfp_lowertier. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(mdr_i_pfp_lowertier,l)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,mdr_i_pfp_lowertier,mdr_a_pfp_lowertier)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,mdr_i_pfp_lowertier,mdr_a_pfp_lowertier)" } ], "dependencies": [ "PD0002", "PD0013" ], "dependency_names": [ "Lt", "BetaAt" ], "exact_ast_verified": true, "expanded_template": "forall mdr_i_pfp_lowertier mdr_a_pfp_lowertier. (exists mdr_gap_pfp_lowertierb. mdr_gap_pfp_lowertierb + S (mdr_i_pfp_lowertier) = ((l))) -> (((exists ff_h_mdr_pfp_lowertiero. ff_h_mdr_pfp_lowertiero + S (mdr_a_pfp_lowertier) = S ((S (mdr_i_pfp_lowertier)) * (c))) /\\ exists ff_q_mdr_pfp_lowertiero. (b) = ff_q_mdr_pfp_lowertiero * S ((S (mdr_i_pfp_lowertier)) * (c)) + (mdr_a_pfp_lowertier))) -> (((exists ff_h_mdr_pfp_lowertiern. ff_h_mdr_pfp_lowertiern + S (mdr_a_pfp_lowertier) = S ((S (mdr_i_pfp_lowertier)) * (e))) /\\ exists ff_q_mdr_pfp_lowertiern. (d) = ff_q_mdr_pfp_lowertiern * S ((S (mdr_i_pfp_lowertier)) * (e)) + (mdr_a_pfp_lowertier)))", "expansion_sha256": "57aad2596b9538685346d56869bebc6ec32fa4c404c5e14035aa98e32c535823", "global_argument_positions": null, "global_definition": null, "id": "ND0263", "kernel_signature_unchanged": true, "name": "BetaPrefixEqual", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "d", "e", "l" ], "reviewed_definition_id": "ND0263", "shared_definition_identity": "ND0263", "signature": "BetaPrefixEqual(b,c,d,e,l)", "summary": "Every decoded source entry at i exists pfp_source_lowertier pfp_residue_lowertier. ((((exists ff_h_pfp_lowertiersource. ff_h_pfp_lowertiersource + S (pfp_source_lowertier) = S ((S (pfp_index_lowertier)) * (c))) /\\ exists ff_q_pfp_lowertiersource. (b) = ff_q_pfp_lowertiersource * S ((S (pfp_index_lowertier)) * (c)) + (pfp_source_lowertier))) /\\ (((((exists ff_h_pfp_lowertiertarget. ff_h_pfp_lowertiertarget + S (pfp_residue_lowertier) = S ((S (pfp_index_lowertier)) * (e))) /\\ exists ff_q_pfp_lowertiertarget. (d) = ff_q_pfp_lowertiertarget * S ((S (pfp_index_lowertier)) * (e)) + (pfp_residue_lowertier))) /\\ ((((exists pfa_gap_lowertierresiduebound. pfa_gap_lowertierresiduebound + S (pfp_residue_lowertier) = ((p))) /\\ ((exists pfa_offset_left_lowertierresiduecongruence pfa_offset_right_lowertierresiduecongruence. (pfp_source_lowertier) + ((p)) * pfa_offset_left_lowertierresiduecongruence = (pfp_residue_lowertier) + ((p)) * pfa_offset_right_lowertierresiduecongruence))))))))", "expansion_sha256": "441c40dfe8f79274574b49c4789286c576ac9cc52f5c5198b596ea362e3fac53", "global_argument_positions": null, "global_definition": null, "id": "ND0269", "kernel_signature_unchanged": true, "name": "FpCoefficientReduction", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "p", "b", "c", "d", "e", "l" ], "reviewed_definition_id": "ND0269", "shared_definition_identity": "ND0269", "signature": "FpCoefficientReduction(p,b,c,d,e,l)", "summary": "Actual source and target coefficient entries are related by the existing canonical-residue graph at each i (((exists fs_h_jt_definition_jordanat. fs_h_jt_definition_jordanat + S (jt_value_definition_jordan) = S ((S (jt_index_definition_jordan)) * c)) /\\ exists fs_q_jt_definition_jordanat. b = fs_q_jt_definition_jordanat * S ((S (jt_index_definition_jordan)) * c) + (jt_value_definition_jordan))) -> (exists jt_factor_definition_jordandivides. (jt_value_definition_jordan)=(d)*jt_factor_definition_jordandivides)", "expansion_sha256": "18ca730fb939fd309c35bf37e0eee36ac6678fca7d4b672ece0c5594788f8cb7", "global_argument_positions": null, "global_definition": null, "id": "ND0371", "kernel_signature_unchanged": true, "name": "JordanTupleAllDivisible", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "d", "b", "c", "k" ], "reviewed_definition_id": "ND0371", "shared_definition_identity": "ND0371", "signature": "JordanTupleAllDivisible(d,b,c,k)", "summary": "The natural d divides every actual decoded coordinate in the finite prefix.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0003", "PD0013" ] }, { "arity": 4, "defined_template": "∀ jt_divisor_definition_jordan. Dvd(jt_divisor_definition_jordan,n) → JordanTupleAllDivisible(jt_divisor_definition_jordan,b,c,k) → jt_divisor_definition_jordan = 1", "defined_template_parts": [ { "kind": "text", "text": "∀ jt_divisor_definition_jordan. " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(jt_divisor_definition_jordan,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(jt_divisor_definition_jordan,b,c,k)" }, { "kind": "text", "text": " → jt_divisor_definition_jordan = 1" } ], "dependencies": [ "PD0003", "ND0371" ], "dependency_names": [ "Dvd", "JordanTupleAllDivisible" ], "exact_ast_verified": true, "expanded_template": "forall jt_divisor_definition_jordan. (exists jt_factor_definition_jordanmodulus. (n)=(jt_divisor_definition_jordan)*jt_factor_definition_jordanmodulus) -> (forall jt_index_definition_jordancoordinates jt_value_definition_jordancoordinates. (exists jt_gap_definition_jordancoordinatesindex. jt_gap_definition_jordancoordinatesindex+S (jt_index_definition_jordancoordinates)=(k)) -> (((exists fs_h_jt_definition_jordancoordinatesat. fs_h_jt_definition_jordancoordinatesat + S (jt_value_definition_jordancoordinates) = S ((S (jt_index_definition_jordancoordinates)) * c)) /\\ exists fs_q_jt_definition_jordancoordinatesat. b = fs_q_jt_definition_jordancoordinatesat * S ((S (jt_index_definition_jordancoordinates)) * c) + (jt_value_definition_jordancoordinates))) -> (exists jt_factor_definition_jordancoordinatesdivides. (jt_value_definition_jordancoordinates)=(jt_divisor_definition_jordan)*jt_factor_definition_jordancoordinatesdivides)) -> jt_divisor_definition_jordan=1", "expansion_sha256": "85f8c4a5884350b48a68844fec169ac9bf20f2a2e9127020d9cb06f87f3d8cf1", "global_argument_positions": null, "global_definition": null, "id": "ND0372", "kernel_signature_unchanged": true, "name": "JordanPrimitiveTuple", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "n", "b", "c", "k" ], "reviewed_definition_id": "ND0372", "shared_definition_identity": "ND0372", "signature": "JordanPrimitiveTuple(n,b,c,k)", "summary": "Every common divisor of n and all decoded coordinates equals one; boundedness is separate.", "topological_layer": 2, "transitive_dependencies": [ "ND0371", "PD0002", "PD0003", "PD0013" ] }, { "arity": 6, "defined_template": "∀ jt_index_definition_jordan. ∀ jt_left_definition_jordan. ∀ jt_right_definition_jordan. Lt(jt_index_definition_jordan,k) → BetaAt(b,c,jt_index_definition_jordan,jt_left_definition_jordan) → BetaAt(d,e,jt_index_definition_jordan,jt_right_definition_jordan) → ModEq(n,jt_left_definition_jordan,jt_right_definition_jordan)", "defined_template_parts": [ { "kind": "text", "text": "∀ jt_index_definition_jordan. ∀ jt_left_definition_jordan. ∀ jt_right_definition_jordan. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(jt_index_definition_jordan,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,jt_index_definition_jordan,jt_left_definition_jordan)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,jt_index_definition_jordan,jt_right_definition_jordan)" }, { "kind": "text", "text": " → " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,jt_left_definition_jordan,jt_right_definition_jordan)" } ], "dependencies": [ "PD0002", "PD0013", "PD0008" ], "dependency_names": [ "Lt", "BetaAt", "ModEq" ], "exact_ast_verified": true, "expanded_template": "forall jt_index_definition_jordan jt_left_definition_jordan jt_right_definition_jordan. (exists jt_gap_definition_jordanindex. jt_gap_definition_jordanindex+S (jt_index_definition_jordan)=(k)) -> (((exists fs_h_jt_definition_jordanleft. fs_h_jt_definition_jordanleft + S (jt_left_definition_jordan) = S ((S (jt_index_definition_jordan)) * c)) /\\ exists fs_q_jt_definition_jordanleft. b = fs_q_jt_definition_jordanleft * S ((S (jt_index_definition_jordan)) * c) + (jt_left_definition_jordan))) -> (((exists fs_h_jt_definition_jordanright. fs_h_jt_definition_jordanright + S (jt_right_definition_jordan) = S ((S (jt_index_definition_jordan)) * e)) /\\ exists fs_q_jt_definition_jordanright. d = fs_q_jt_definition_jordanright * S ((S (jt_index_definition_jordan)) * e) + (jt_right_definition_jordan))) -> (exists jt_left_definition_jordanmod jt_right_definition_jordanmod. (jt_left_definition_jordan)+(n)*jt_left_definition_jordanmod=(jt_right_definition_jordan)+(n)*jt_right_definition_jordanmod)", "expansion_sha256": "0f4f927f0eb1e9590d6920e832afca7863becdbb1f74c2a835fa4db534c15630", "global_argument_positions": null, "global_definition": null, "id": "ND0373", "kernel_signature_unchanged": true, "name": "JordanTupleCongruence", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "n", "b", "c", "d", "e", "k" ], "reviewed_definition_id": "ND0373", "shared_definition_identity": "ND0373", "signature": "JordanTupleCongruence(n,b,c,d,e,k)", "summary": "All paired actual decoded coordinates are congruent modulo n, without imposing a canonical range.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0008", "PD0013" ] }, { "arity": 7, "defined_template": "(∀ x. Lt(x,j) → ∃ y. ∃ z. BetaAt(B,C,x,y) ∧ BetaAt(D,E,x,z) ∧ (BetaPrefixInto(y,z,k,n) ∧ JordanPrimitiveTuple(n,y,z,k))) ∧ ((∀ x. ∀ y. BetaPrefixInto(x,y,k,n) → JordanPrimitiveTuple(n,x,y,k) → ∃ z. ∃ m. ∃ i. Lt(z,j) ∧ (BetaAt(B,C,z,m) ∧ BetaAt(D,E,z,i) ∧ IntegerVectorZero(x,y,m,i,k))) ∧ (∀ x. ∀ y. ∀ z. ∀ m. ∀ i. ∀ u. Lt(x,j) → Lt(y,j) → BetaAt(B,C,x,z) ∧ BetaAt(D,E,x,m) → BetaAt(B,C,y,i) ∧ BetaAt(D,E,y,u) → IntegerVectorZero(z,m,i,u,k) → x = y))", "defined_template_parts": [ { "kind": "text", "text": "(∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,j)" }, { "kind": "text", "text": " → ∃ y. ∃ z. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,x,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,x,z)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(y,z,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,y,z,k)" }, { "kind": "text", "text": ")) ∧ ((∀ x. ∀ y. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x,y,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,x,y,k)" }, { "kind": "text", "text": " → ∃ z. ∃ m. ∃ i. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,z,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,z,i)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,m,i,k)" }, { "kind": "text", "text": ")) ∧ (∀ x. ∀ y. ∀ z. ∀ m. ∀ i. ∀ u. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,x,m)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,y,i)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,y,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(z,m,i,u,k)" }, { "kind": "text", "text": " → x = y))" } ], "dependencies": [ "PD0002", "PD0013", "ND0262", "ND0372", "ND0121" ], "dependency_names": [ "Lt", "BetaAt", "BetaPrefixInto", "JordanPrimitiveTuple", "IntegerVectorZero" ], "exact_ast_verified": true, "expanded_template": "((forall jt_i_definition_jordan. (exists jt_gap_definition_jordansoundindex. jt_gap_definition_jordansoundindex+S (jt_i_definition_jordan)=(j)) -> exists jt_b_definition_jordan jt_c_definition_jordan. ((((((exists fs_h_jt_definition_jordansoundcode. fs_h_jt_definition_jordansoundcode + S (jt_b_definition_jordan) = S ((S (jt_i_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordansoundcode. B = fs_q_jt_definition_jordansoundcode * S ((S (jt_i_definition_jordan)) * C) + (jt_b_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordansoundscale. fs_h_jt_definition_jordansoundscale + S (jt_c_definition_jordan) = S ((S (jt_i_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordansoundscale. D = fs_q_jt_definition_jordansoundscale * S ((S (jt_i_definition_jordan)) * E) + (jt_c_definition_jordan))))) /\\ (((forall jt_index_definition_jordanbound. (exists jt_gap_definition_jordanboundindex. jt_gap_definition_jordanboundindex+S (jt_index_definition_jordanbound)=(k)) -> exists jt_value_definition_jordanbound. ((((exists fs_h_jt_definition_jordanboundat. fs_h_jt_definition_jordanboundat + S (jt_value_definition_jordanbound) = S ((S (jt_index_definition_jordanbound)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordanboundat. jt_b_definition_jordan = fs_q_jt_definition_jordanboundat * S ((S (jt_index_definition_jordanbound)) * jt_c_definition_jordan) + (jt_value_definition_jordanbound))) /\\ (exists jt_gap_definition_jordanboundvalue. jt_gap_definition_jordanboundvalue+S (jt_value_definition_jordanbound)=(n)))) /\\ (forall jt_divisor_definition_jordanprimitive. (exists jt_factor_definition_jordanprimitivemodulus. (n)=(jt_divisor_definition_jordanprimitive)*jt_factor_definition_jordanprimitivemodulus) -> (forall jt_index_definition_jordanprimitivecoordinates jt_value_definition_jordanprimitivecoordinates. (exists jt_gap_definition_jordanprimitivecoordinatesindex. jt_gap_definition_jordanprimitivecoordinatesindex+S (jt_index_definition_jordanprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordanprimitivecoordinatesat. fs_h_jt_definition_jordanprimitivecoordinatesat + S (jt_value_definition_jordanprimitivecoordinates) = S ((S (jt_index_definition_jordanprimitivecoordinates)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordanprimitivecoordinatesat. jt_b_definition_jordan = fs_q_jt_definition_jordanprimitivecoordinatesat * S ((S (jt_index_definition_jordanprimitivecoordinates)) * jt_c_definition_jordan) + (jt_value_definition_jordanprimitivecoordinates))) -> (exists jt_factor_definition_jordanprimitivecoordinatesdivides. (jt_value_definition_jordanprimitivecoordinates)=(jt_divisor_definition_jordanprimitive)*jt_factor_definition_jordanprimitivecoordinatesdivides)) -> jt_divisor_definition_jordanprimitive=1))))) /\\ (((forall jt_b_definition_jordan jt_c_definition_jordan. (forall jt_index_definition_jordaninputbound. (exists jt_gap_definition_jordaninputboundindex. jt_gap_definition_jordaninputboundindex+S (jt_index_definition_jordaninputbound)=(k)) -> exists jt_value_definition_jordaninputbound. ((((exists fs_h_jt_definition_jordaninputboundat. fs_h_jt_definition_jordaninputboundat + S (jt_value_definition_jordaninputbound) = S ((S (jt_index_definition_jordaninputbound)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordaninputboundat. jt_b_definition_jordan = fs_q_jt_definition_jordaninputboundat * S ((S (jt_index_definition_jordaninputbound)) * jt_c_definition_jordan) + (jt_value_definition_jordaninputbound))) /\\ (exists jt_gap_definition_jordaninputboundvalue. jt_gap_definition_jordaninputboundvalue+S (jt_value_definition_jordaninputbound)=(n)))) -> (forall jt_divisor_definition_jordaninputprimitive. (exists jt_factor_definition_jordaninputprimitivemodulus. (n)=(jt_divisor_definition_jordaninputprimitive)*jt_factor_definition_jordaninputprimitivemodulus) -> (forall jt_index_definition_jordaninputprimitivecoordinates jt_value_definition_jordaninputprimitivecoordinates. (exists jt_gap_definition_jordaninputprimitivecoordinatesindex. jt_gap_definition_jordaninputprimitivecoordinatesindex+S (jt_index_definition_jordaninputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordaninputprimitivecoordinatesat. fs_h_jt_definition_jordaninputprimitivecoordinatesat + S (jt_value_definition_jordaninputprimitivecoordinates) = S ((S (jt_index_definition_jordaninputprimitivecoordinates)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordaninputprimitivecoordinatesat. jt_b_definition_jordan = fs_q_jt_definition_jordaninputprimitivecoordinatesat * S ((S (jt_index_definition_jordaninputprimitivecoordinates)) * jt_c_definition_jordan) + (jt_value_definition_jordaninputprimitivecoordinates))) -> (exists jt_factor_definition_jordaninputprimitivecoordinatesdivides. (jt_value_definition_jordaninputprimitivecoordinates)=(jt_divisor_definition_jordaninputprimitive)*jt_factor_definition_jordaninputprimitivecoordinatesdivides)) -> jt_divisor_definition_jordaninputprimitive=1) -> exists jt_i_definition_jordan jt_d_definition_jordan jt_e_definition_jordan. ((exists jt_gap_definition_jordancompleteindex. jt_gap_definition_jordancompleteindex+S (jt_i_definition_jordan)=(j)) /\\ (((((((exists fs_h_jt_definition_jordancompletecode. fs_h_jt_definition_jordancompletecode + S (jt_d_definition_jordan) = S ((S (jt_i_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordancompletecode. B = fs_q_jt_definition_jordancompletecode * S ((S (jt_i_definition_jordan)) * C) + (jt_d_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordancompletescale. fs_h_jt_definition_jordancompletescale + S (jt_e_definition_jordan) = S ((S (jt_i_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordancompletescale. D = fs_q_jt_definition_jordancompletescale * S ((S (jt_i_definition_jordan)) * E) + (jt_e_definition_jordan))))) /\\ (forall jt_index_definition_jordanrepresented jt_left_definition_jordanrepresented jt_right_definition_jordanrepresented. (exists jt_gap_definition_jordanrepresentedindex. jt_gap_definition_jordanrepresentedindex+S (jt_index_definition_jordanrepresented)=(k)) -> (((exists fs_h_jt_definition_jordanrepresentedleft. fs_h_jt_definition_jordanrepresentedleft + S (jt_left_definition_jordanrepresented) = S ((S (jt_index_definition_jordanrepresented)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordanrepresentedleft. jt_b_definition_jordan = fs_q_jt_definition_jordanrepresentedleft * S ((S (jt_index_definition_jordanrepresented)) * jt_c_definition_jordan) + (jt_left_definition_jordanrepresented))) -> (((exists fs_h_jt_definition_jordanrepresentedright. fs_h_jt_definition_jordanrepresentedright + S (jt_right_definition_jordanrepresented) = S ((S (jt_index_definition_jordanrepresented)) * jt_e_definition_jordan)) /\\ exists fs_q_jt_definition_jordanrepresentedright. jt_d_definition_jordan = fs_q_jt_definition_jordanrepresentedright * S ((S (jt_index_definition_jordanrepresented)) * jt_e_definition_jordan) + (jt_right_definition_jordanrepresented))) -> jt_left_definition_jordanrepresented=jt_right_definition_jordanrepresented))))) /\\ (forall jt_i_definition_jordan jt_h_definition_jordan jt_b_definition_jordan jt_c_definition_jordan jt_d_definition_jordan jt_e_definition_jordan. (exists jt_gap_definition_jordanfirstindex. jt_gap_definition_jordanfirstindex+S (jt_i_definition_jordan)=(j)) -> (exists jt_gap_definition_jordansecondindex. jt_gap_definition_jordansecondindex+S (jt_h_definition_jordan)=(j)) -> (((((exists fs_h_jt_definition_jordanfirstcode. fs_h_jt_definition_jordanfirstcode + S (jt_b_definition_jordan) = S ((S (jt_i_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordanfirstcode. B = fs_q_jt_definition_jordanfirstcode * S ((S (jt_i_definition_jordan)) * C) + (jt_b_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordanfirstscale. fs_h_jt_definition_jordanfirstscale + S (jt_c_definition_jordan) = S ((S (jt_i_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordanfirstscale. D = fs_q_jt_definition_jordanfirstscale * S ((S (jt_i_definition_jordan)) * E) + (jt_c_definition_jordan))))) -> (((((exists fs_h_jt_definition_jordansecondcode. fs_h_jt_definition_jordansecondcode + S (jt_d_definition_jordan) = S ((S (jt_h_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordansecondcode. B = fs_q_jt_definition_jordansecondcode * S ((S (jt_h_definition_jordan)) * C) + (jt_d_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordansecondscale. fs_h_jt_definition_jordansecondscale + S (jt_e_definition_jordan) = S ((S (jt_h_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordansecondscale. D = fs_q_jt_definition_jordansecondscale * S ((S (jt_h_definition_jordan)) * E) + (jt_e_definition_jordan))))) -> (forall jt_index_definition_jordansame jt_left_definition_jordansame jt_right_definition_jordansame. (exists jt_gap_definition_jordansameindex. jt_gap_definition_jordansameindex+S (jt_index_definition_jordansame)=(k)) -> (((exists fs_h_jt_definition_jordansameleft. fs_h_jt_definition_jordansameleft + S (jt_left_definition_jordansame) = S ((S (jt_index_definition_jordansame)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordansameleft. jt_b_definition_jordan = fs_q_jt_definition_jordansameleft * S ((S (jt_index_definition_jordansame)) * jt_c_definition_jordan) + (jt_left_definition_jordansame))) -> (((exists fs_h_jt_definition_jordansameright. fs_h_jt_definition_jordansameright + S (jt_right_definition_jordansame) = S ((S (jt_index_definition_jordansame)) * jt_e_definition_jordan)) /\\ exists fs_q_jt_definition_jordansameright. jt_d_definition_jordan = fs_q_jt_definition_jordansameright * S ((S (jt_index_definition_jordansame)) * jt_e_definition_jordan) + (jt_right_definition_jordansame))) -> jt_left_definition_jordansame=jt_right_definition_jordansame) -> jt_i_definition_jordan=jt_h_definition_jordan))))", "expansion_sha256": "c2c11cc8e62edd82a6a87a262358222ef02f869a8aa0fe6665c2e600009bad90", "global_argument_positions": null, "global_definition": null, "id": "ND0374", "kernel_signature_unchanged": true, "name": "JordanTupleEnumeration", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "k", "n", "B", "C", "D", "E", "j" ], "reviewed_definition_id": "ND0374", "shared_definition_identity": "ND0374", "signature": "JordanTupleEnumeration(k,n,B,C,D,E,j)", "summary": "Sound, complete and duplicate-free enumeration of canonical primitive tuples; both code and scale columns are actual beta data.", "topological_layer": 3, "transitive_dependencies": [ "ND0121", "ND0262", "ND0371", "ND0372", "PD0002", "PD0003", "PD0013" ] }, { "arity": 3, "defined_template": "¬k = 0 ∧ (¬n = 0 ∧ (∃ x. ∃ y. ∃ z. ∃ m. JordanTupleEnumeration(k,n,x,y,z,m,j)))", "defined_template_parts": [ { "kind": "text", "text": "¬k = 0 ∧ (¬n = 0 ∧ (∃ x. ∃ y. ∃ z. ∃ m. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,x,y,z,m,j)" }, { "kind": "text", "text": "))" } ], "dependencies": [ "ND0374" ], "dependency_names": [ "JordanTupleEnumeration" ], "exact_ast_verified": true, "expanded_template": "((~((k)=0)) /\\ (((~((n)=0)) /\\ (exists jt_codes_definition_jordan jt_code_scale_definition_jordan jt_scales_definition_jordan jt_scale_scale_definition_jordan. ((forall jt_i_definition_jordanenum. (exists jt_gap_definition_jordanenumsoundindex. jt_gap_definition_jordanenumsoundindex+S (jt_i_definition_jordanenum)=(j)) -> exists jt_b_definition_jordanenum jt_c_definition_jordanenum. ((((((exists fs_h_jt_definition_jordanenumsoundcode. fs_h_jt_definition_jordanenumsoundcode + S (jt_b_definition_jordanenum) = S ((S (jt_i_definition_jordanenum)) * jt_code_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumsoundcode. jt_codes_definition_jordan = fs_q_jt_definition_jordanenumsoundcode * S ((S (jt_i_definition_jordanenum)) * jt_code_scale_definition_jordan) + (jt_b_definition_jordanenum))) /\\ (((exists fs_h_jt_definition_jordanenumsoundscale. fs_h_jt_definition_jordanenumsoundscale + S (jt_c_definition_jordanenum) = S ((S (jt_i_definition_jordanenum)) * jt_scale_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumsoundscale. jt_scales_definition_jordan = fs_q_jt_definition_jordanenumsoundscale * S ((S (jt_i_definition_jordanenum)) * jt_scale_scale_definition_jordan) + (jt_c_definition_jordanenum))))) /\\ (((forall jt_index_definition_jordanenumbound. (exists jt_gap_definition_jordanenumboundindex. jt_gap_definition_jordanenumboundindex+S (jt_index_definition_jordanenumbound)=(k)) -> exists jt_value_definition_jordanenumbound. ((((exists fs_h_jt_definition_jordanenumboundat. fs_h_jt_definition_jordanenumboundat + S (jt_value_definition_jordanenumbound) = S ((S (jt_index_definition_jordanenumbound)) * jt_c_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenumboundat. jt_b_definition_jordanenum = fs_q_jt_definition_jordanenumboundat * S ((S (jt_index_definition_jordanenumbound)) * jt_c_definition_jordanenum) + (jt_value_definition_jordanenumbound))) /\\ (exists jt_gap_definition_jordanenumboundvalue. jt_gap_definition_jordanenumboundvalue+S (jt_value_definition_jordanenumbound)=(n)))) /\\ (forall jt_divisor_definition_jordanenumprimitive. (exists jt_factor_definition_jordanenumprimitivemodulus. (n)=(jt_divisor_definition_jordanenumprimitive)*jt_factor_definition_jordanenumprimitivemodulus) -> (forall jt_index_definition_jordanenumprimitivecoordinates jt_value_definition_jordanenumprimitivecoordinates. (exists jt_gap_definition_jordanenumprimitivecoordinatesindex. jt_gap_definition_jordanenumprimitivecoordinatesindex+S (jt_index_definition_jordanenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordanenumprimitivecoordinatesat. fs_h_jt_definition_jordanenumprimitivecoordinatesat + S (jt_value_definition_jordanenumprimitivecoordinates) = S ((S (jt_index_definition_jordanenumprimitivecoordinates)) * jt_c_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenumprimitivecoordinatesat. jt_b_definition_jordanenum = fs_q_jt_definition_jordanenumprimitivecoordinatesat * S ((S (jt_index_definition_jordanenumprimitivecoordinates)) * jt_c_definition_jordanenum) + (jt_value_definition_jordanenumprimitivecoordinates))) -> (exists jt_factor_definition_jordanenumprimitivecoordinatesdivides. (jt_value_definition_jordanenumprimitivecoordinates)=(jt_divisor_definition_jordanenumprimitive)*jt_factor_definition_jordanenumprimitivecoordinatesdivides)) -> jt_divisor_definition_jordanenumprimitive=1))))) /\\ (((forall jt_b_definition_jordanenum jt_c_definition_jordanenum. (forall jt_index_definition_jordanenuminputbound. (exists jt_gap_definition_jordanenuminputboundindex. jt_gap_definition_jordanenuminputboundindex+S (jt_index_definition_jordanenuminputbound)=(k)) -> exists jt_value_definition_jordanenuminputbound. ((((exists fs_h_jt_definition_jordanenuminputboundat. fs_h_jt_definition_jordanenuminputboundat + S (jt_value_definition_jordanenuminputbound) = S ((S (jt_index_definition_jordanenuminputbound)) * jt_c_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenuminputboundat. jt_b_definition_jordanenum = fs_q_jt_definition_jordanenuminputboundat * S ((S (jt_index_definition_jordanenuminputbound)) * jt_c_definition_jordanenum) + (jt_value_definition_jordanenuminputbound))) /\\ (exists jt_gap_definition_jordanenuminputboundvalue. jt_gap_definition_jordanenuminputboundvalue+S (jt_value_definition_jordanenuminputbound)=(n)))) -> (forall jt_divisor_definition_jordanenuminputprimitive. (exists jt_factor_definition_jordanenuminputprimitivemodulus. (n)=(jt_divisor_definition_jordanenuminputprimitive)*jt_factor_definition_jordanenuminputprimitivemodulus) -> (forall jt_index_definition_jordanenuminputprimitivecoordinates jt_value_definition_jordanenuminputprimitivecoordinates. (exists jt_gap_definition_jordanenuminputprimitivecoordinatesindex. jt_gap_definition_jordanenuminputprimitivecoordinatesindex+S (jt_index_definition_jordanenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordanenuminputprimitivecoordinatesat. fs_h_jt_definition_jordanenuminputprimitivecoordinatesat + S (jt_value_definition_jordanenuminputprimitivecoordinates) = S ((S (jt_index_definition_jordanenuminputprimitivecoordinates)) * jt_c_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenuminputprimitivecoordinatesat. jt_b_definition_jordanenum = fs_q_jt_definition_jordanenuminputprimitivecoordinatesat * S ((S (jt_index_definition_jordanenuminputprimitivecoordinates)) * jt_c_definition_jordanenum) + (jt_value_definition_jordanenuminputprimitivecoordinates))) -> (exists jt_factor_definition_jordanenuminputprimitivecoordinatesdivides. (jt_value_definition_jordanenuminputprimitivecoordinates)=(jt_divisor_definition_jordanenuminputprimitive)*jt_factor_definition_jordanenuminputprimitivecoordinatesdivides)) -> jt_divisor_definition_jordanenuminputprimitive=1) -> exists jt_i_definition_jordanenum jt_d_definition_jordanenum jt_e_definition_jordanenum. ((exists jt_gap_definition_jordanenumcompleteindex. jt_gap_definition_jordanenumcompleteindex+S (jt_i_definition_jordanenum)=(j)) /\\ (((((((exists fs_h_jt_definition_jordanenumcompletecode. fs_h_jt_definition_jordanenumcompletecode + S (jt_d_definition_jordanenum) = S ((S (jt_i_definition_jordanenum)) * jt_code_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumcompletecode. jt_codes_definition_jordan = fs_q_jt_definition_jordanenumcompletecode * S ((S (jt_i_definition_jordanenum)) * jt_code_scale_definition_jordan) + (jt_d_definition_jordanenum))) /\\ (((exists fs_h_jt_definition_jordanenumcompletescale. fs_h_jt_definition_jordanenumcompletescale + S (jt_e_definition_jordanenum) = S ((S (jt_i_definition_jordanenum)) * jt_scale_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumcompletescale. jt_scales_definition_jordan = fs_q_jt_definition_jordanenumcompletescale * S ((S (jt_i_definition_jordanenum)) * jt_scale_scale_definition_jordan) + (jt_e_definition_jordanenum))))) /\\ (forall jt_index_definition_jordanenumrepresented jt_left_definition_jordanenumrepresented jt_right_definition_jordanenumrepresented. (exists jt_gap_definition_jordanenumrepresentedindex. jt_gap_definition_jordanenumrepresentedindex+S (jt_index_definition_jordanenumrepresented)=(k)) -> (((exists fs_h_jt_definition_jordanenumrepresentedleft. fs_h_jt_definition_jordanenumrepresentedleft + S (jt_left_definition_jordanenumrepresented) = S ((S (jt_index_definition_jordanenumrepresented)) * jt_c_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenumrepresentedleft. jt_b_definition_jordanenum = fs_q_jt_definition_jordanenumrepresentedleft * S ((S (jt_index_definition_jordanenumrepresented)) * jt_c_definition_jordanenum) + (jt_left_definition_jordanenumrepresented))) -> (((exists fs_h_jt_definition_jordanenumrepresentedright. fs_h_jt_definition_jordanenumrepresentedright + S (jt_right_definition_jordanenumrepresented) = S ((S (jt_index_definition_jordanenumrepresented)) * jt_e_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenumrepresentedright. jt_d_definition_jordanenum = fs_q_jt_definition_jordanenumrepresentedright * S ((S (jt_index_definition_jordanenumrepresented)) * jt_e_definition_jordanenum) + (jt_right_definition_jordanenumrepresented))) -> jt_left_definition_jordanenumrepresented=jt_right_definition_jordanenumrepresented))))) /\\ (forall jt_i_definition_jordanenum jt_h_definition_jordanenum jt_b_definition_jordanenum jt_c_definition_jordanenum jt_d_definition_jordanenum jt_e_definition_jordanenum. (exists jt_gap_definition_jordanenumfirstindex. jt_gap_definition_jordanenumfirstindex+S (jt_i_definition_jordanenum)=(j)) -> (exists jt_gap_definition_jordanenumsecondindex. jt_gap_definition_jordanenumsecondindex+S (jt_h_definition_jordanenum)=(j)) -> (((((exists fs_h_jt_definition_jordanenumfirstcode. fs_h_jt_definition_jordanenumfirstcode + S (jt_b_definition_jordanenum) = S ((S (jt_i_definition_jordanenum)) * jt_code_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumfirstcode. jt_codes_definition_jordan = fs_q_jt_definition_jordanenumfirstcode * S ((S (jt_i_definition_jordanenum)) * jt_code_scale_definition_jordan) + (jt_b_definition_jordanenum))) /\\ (((exists fs_h_jt_definition_jordanenumfirstscale. fs_h_jt_definition_jordanenumfirstscale + S (jt_c_definition_jordanenum) = S ((S (jt_i_definition_jordanenum)) * jt_scale_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumfirstscale. jt_scales_definition_jordan = fs_q_jt_definition_jordanenumfirstscale * S ((S (jt_i_definition_jordanenum)) * jt_scale_scale_definition_jordan) + (jt_c_definition_jordanenum))))) -> (((((exists fs_h_jt_definition_jordanenumsecondcode. fs_h_jt_definition_jordanenumsecondcode + S (jt_d_definition_jordanenum) = S ((S (jt_h_definition_jordanenum)) * jt_code_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumsecondcode. jt_codes_definition_jordan = fs_q_jt_definition_jordanenumsecondcode * S ((S (jt_h_definition_jordanenum)) * jt_code_scale_definition_jordan) + (jt_d_definition_jordanenum))) /\\ (((exists fs_h_jt_definition_jordanenumsecondscale. fs_h_jt_definition_jordanenumsecondscale + S (jt_e_definition_jordanenum) = S ((S (jt_h_definition_jordanenum)) * jt_scale_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanenumsecondscale. jt_scales_definition_jordan = fs_q_jt_definition_jordanenumsecondscale * S ((S (jt_h_definition_jordanenum)) * jt_scale_scale_definition_jordan) + (jt_e_definition_jordanenum))))) -> (forall jt_index_definition_jordanenumsame jt_left_definition_jordanenumsame jt_right_definition_jordanenumsame. (exists jt_gap_definition_jordanenumsameindex. jt_gap_definition_jordanenumsameindex+S (jt_index_definition_jordanenumsame)=(k)) -> (((exists fs_h_jt_definition_jordanenumsameleft. fs_h_jt_definition_jordanenumsameleft + S (jt_left_definition_jordanenumsame) = S ((S (jt_index_definition_jordanenumsame)) * jt_c_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenumsameleft. jt_b_definition_jordanenum = fs_q_jt_definition_jordanenumsameleft * S ((S (jt_index_definition_jordanenumsame)) * jt_c_definition_jordanenum) + (jt_left_definition_jordanenumsame))) -> (((exists fs_h_jt_definition_jordanenumsameright. fs_h_jt_definition_jordanenumsameright + S (jt_right_definition_jordanenumsame) = S ((S (jt_index_definition_jordanenumsame)) * jt_e_definition_jordanenum)) /\\ exists fs_q_jt_definition_jordanenumsameright. jt_d_definition_jordanenum = fs_q_jt_definition_jordanenumsameright * S ((S (jt_index_definition_jordanenumsame)) * jt_e_definition_jordanenum) + (jt_right_definition_jordanenumsame))) -> jt_left_definition_jordanenumsame=jt_right_definition_jordanenumsame) -> jt_i_definition_jordanenum=jt_h_definition_jordanenum))))))))", "expansion_sha256": "aac9a9dc7e0fb4118c4675a3842f3c0d2c954a93e5a318bc8894774ce8ca6b0b", "global_argument_positions": null, "global_definition": null, "id": "ND0375", "kernel_signature_unchanged": true, "name": "JordanTotient", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "k", "n", "j" ], "reviewed_definition_id": "ND0375", "shared_definition_identity": "ND0375", "signature": "JordanTotient(k,n,j)", "summary": "Positive order and modulus with an actual tuple enumeration of length j. No product identity or existence theorem is a clause.", "topological_layer": 4, "transitive_dependencies": [ "ND0121", "ND0262", "ND0371", "ND0372", "ND0374", "PD0002", "PD0003", "PD0013" ] }, { "arity": 8, "defined_template": "∃ jt_index_definition_jordan. ∃ jt_code_definition_jordan. ∃ jt_scale_definition_jordan. Lt(jt_index_definition_jordan,j) ∧ (BetaAt(B,C,jt_index_definition_jordan,jt_code_definition_jordan) ∧ BetaAt(D,E,jt_index_definition_jordan,jt_scale_definition_jordan) ∧ IntegerVectorZero(b,c,jt_code_definition_jordan,jt_scale_definition_jordan,k))", "defined_template_parts": [ { "kind": "text", "text": "∃ jt_index_definition_jordan. ∃ jt_code_definition_jordan. ∃ jt_scale_definition_jordan. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(jt_index_definition_jordan,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,jt_index_definition_jordan,jt_code_definition_jordan)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,jt_index_definition_jordan,jt_scale_definition_jordan)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,jt_code_definition_jordan,jt_scale_definition_jordan,k)" }, { "kind": "text", "text": ")" } ], "dependencies": [ "PD0002", "PD0013", "ND0121" ], "dependency_names": [ "Lt", "BetaAt", "IntegerVectorZero" ], "exact_ast_verified": true, "expanded_template": "exists jt_index_definition_jordan jt_code_definition_jordan jt_scale_definition_jordan. ((exists jt_gap_definition_jordanindex. jt_gap_definition_jordanindex+S (jt_index_definition_jordan)=(j)) /\\ (((((((exists fs_h_jt_definition_jordancode. fs_h_jt_definition_jordancode + S (jt_code_definition_jordan) = S ((S (jt_index_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordancode. B = fs_q_jt_definition_jordancode * S ((S (jt_index_definition_jordan)) * C) + (jt_code_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordanscale. fs_h_jt_definition_jordanscale + S (jt_scale_definition_jordan) = S ((S (jt_index_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordanscale. D = fs_q_jt_definition_jordanscale * S ((S (jt_index_definition_jordan)) * E) + (jt_scale_definition_jordan))))) /\\ (forall jt_index_definition_jordanequal jt_left_definition_jordanequal jt_right_definition_jordanequal. (exists jt_gap_definition_jordanequalindex. jt_gap_definition_jordanequalindex+S (jt_index_definition_jordanequal)=(k)) -> (((exists fs_h_jt_definition_jordanequalleft. fs_h_jt_definition_jordanequalleft + S (jt_left_definition_jordanequal) = S ((S (jt_index_definition_jordanequal)) * c)) /\\ exists fs_q_jt_definition_jordanequalleft. b = fs_q_jt_definition_jordanequalleft * S ((S (jt_index_definition_jordanequal)) * c) + (jt_left_definition_jordanequal))) -> (((exists fs_h_jt_definition_jordanequalright. fs_h_jt_definition_jordanequalright + S (jt_right_definition_jordanequal) = S ((S (jt_index_definition_jordanequal)) * jt_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanequalright. jt_code_definition_jordan = fs_q_jt_definition_jordanequalright * S ((S (jt_index_definition_jordanequal)) * jt_scale_definition_jordan) + (jt_right_definition_jordanequal))) -> jt_left_definition_jordanequal=jt_right_definition_jordanequal))))", "expansion_sha256": "f0cd44a82fc41ee2b69119c84d015bd2305e405322dbee07e14b7fbebc126106", "global_argument_positions": null, "global_definition": null, "id": "ND0376", "kernel_signature_unchanged": true, "name": "JordanTupleListed", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "b", "c", "k", "B", "C", "D", "E", "j" ], "reviewed_definition_id": "ND0376", "shared_definition_identity": "ND0376", "signature": "JordanTupleListed(b,c,k,B,C,D,E,j)", "summary": "An actual decoded tuple occurs, up to coordinate equality, at an actual position in the two-column enumeration.", "topological_layer": 2, "transitive_dependencies": [ "ND0121", "PD0002", "PD0013" ] }, { "arity": 9, "defined_template": "(∀ x. Lt(x,j) → ∃ y. ∃ z. BetaAt(B,C,x,y) ∧ BetaAt(D,E,x,z) ∧ (BetaPrefixInto(y,z,k,n) ∧ JordanPrimitiveTuple(n,y,z,k))) ∧ ((∀ x. ∀ y. ∀ z. ∀ m. ∀ i. ∀ u. Lt(x,j) → Lt(y,j) → BetaAt(B,C,x,z) ∧ BetaAt(D,E,x,m) → BetaAt(B,C,y,i) ∧ BetaAt(D,E,y,u) → IntegerVectorZero(z,m,i,u,k) → x = y) ∧ (∀ x. Lt(x,limit) → BetaPrefixInto(x,c,k,n) → JordanPrimitiveTuple(n,x,c,k) → JordanTupleListed(x,c,k,B,C,D,E,j)))", "defined_template_parts": [ { "kind": "text", "text": "(∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,j)" }, { "kind": "text", "text": " → ∃ y. ∃ z. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,x,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,x,z)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(y,z,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,y,z,k)" }, { "kind": "text", "text": ")) ∧ ((∀ x. ∀ y. ∀ z. ∀ m. ∀ i. ∀ u. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,x,m)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,y,i)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,y,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(z,m,i,u,k)" }, { "kind": "text", "text": " → x = y) ∧ (∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,limit)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,x,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(x,c,k,B,C,D,E,j)" }, { "kind": "text", "text": "))" } ], "dependencies": [ "PD0002", "PD0013", "ND0262", "ND0372", "ND0121", "ND0376" ], "dependency_names": [ "Lt", "BetaAt", "BetaPrefixInto", "JordanPrimitiveTuple", "IntegerVectorZero", "JordanTupleListed" ], "exact_ast_verified": true, "expanded_template": "((forall jt_i_definition_jordan. (exists jt_gap_definition_jordansoundindex. jt_gap_definition_jordansoundindex+S (jt_i_definition_jordan)=(j)) -> exists jt_b_definition_jordan jt_e_definition_jordan. ((((((exists fs_h_jt_definition_jordansoundcode. fs_h_jt_definition_jordansoundcode + S (jt_b_definition_jordan) = S ((S (jt_i_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordansoundcode. B = fs_q_jt_definition_jordansoundcode * S ((S (jt_i_definition_jordan)) * C) + (jt_b_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordansoundscale. fs_h_jt_definition_jordansoundscale + S (jt_e_definition_jordan) = S ((S (jt_i_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordansoundscale. D = fs_q_jt_definition_jordansoundscale * S ((S (jt_i_definition_jordan)) * E) + (jt_e_definition_jordan))))) /\\ (((forall jt_index_definition_jordanbound. (exists jt_gap_definition_jordanboundindex. jt_gap_definition_jordanboundindex+S (jt_index_definition_jordanbound)=(k)) -> exists jt_value_definition_jordanbound. ((((exists fs_h_jt_definition_jordanboundat. fs_h_jt_definition_jordanboundat + S (jt_value_definition_jordanbound) = S ((S (jt_index_definition_jordanbound)) * jt_e_definition_jordan)) /\\ exists fs_q_jt_definition_jordanboundat. jt_b_definition_jordan = fs_q_jt_definition_jordanboundat * S ((S (jt_index_definition_jordanbound)) * jt_e_definition_jordan) + (jt_value_definition_jordanbound))) /\\ (exists jt_gap_definition_jordanboundvalue. jt_gap_definition_jordanboundvalue+S (jt_value_definition_jordanbound)=(n)))) /\\ (forall jt_divisor_definition_jordanprimitive. (exists jt_factor_definition_jordanprimitivemodulus. (n)=(jt_divisor_definition_jordanprimitive)*jt_factor_definition_jordanprimitivemodulus) -> (forall jt_index_definition_jordanprimitivecoordinates jt_value_definition_jordanprimitivecoordinates. (exists jt_gap_definition_jordanprimitivecoordinatesindex. jt_gap_definition_jordanprimitivecoordinatesindex+S (jt_index_definition_jordanprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordanprimitivecoordinatesat. fs_h_jt_definition_jordanprimitivecoordinatesat + S (jt_value_definition_jordanprimitivecoordinates) = S ((S (jt_index_definition_jordanprimitivecoordinates)) * jt_e_definition_jordan)) /\\ exists fs_q_jt_definition_jordanprimitivecoordinatesat. jt_b_definition_jordan = fs_q_jt_definition_jordanprimitivecoordinatesat * S ((S (jt_index_definition_jordanprimitivecoordinates)) * jt_e_definition_jordan) + (jt_value_definition_jordanprimitivecoordinates))) -> (exists jt_factor_definition_jordanprimitivecoordinatesdivides. (jt_value_definition_jordanprimitivecoordinates)=(jt_divisor_definition_jordanprimitive)*jt_factor_definition_jordanprimitivecoordinatesdivides)) -> jt_divisor_definition_jordanprimitive=1))))) /\\ (((forall jt_i_definition_jordan jt_h_definition_jordan jt_b_definition_jordan jt_e_definition_jordan jt_d_definition_jordan jt_f_definition_jordan. (exists jt_gap_definition_jordanfirstindex. jt_gap_definition_jordanfirstindex+S (jt_i_definition_jordan)=(j)) -> (exists jt_gap_definition_jordansecondindex. jt_gap_definition_jordansecondindex+S (jt_h_definition_jordan)=(j)) -> (((((exists fs_h_jt_definition_jordanfirstcode. fs_h_jt_definition_jordanfirstcode + S (jt_b_definition_jordan) = S ((S (jt_i_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordanfirstcode. B = fs_q_jt_definition_jordanfirstcode * S ((S (jt_i_definition_jordan)) * C) + (jt_b_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordanfirstscale. fs_h_jt_definition_jordanfirstscale + S (jt_e_definition_jordan) = S ((S (jt_i_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordanfirstscale. D = fs_q_jt_definition_jordanfirstscale * S ((S (jt_i_definition_jordan)) * E) + (jt_e_definition_jordan))))) -> (((((exists fs_h_jt_definition_jordansecondcode. fs_h_jt_definition_jordansecondcode + S (jt_d_definition_jordan) = S ((S (jt_h_definition_jordan)) * C)) /\\ exists fs_q_jt_definition_jordansecondcode. B = fs_q_jt_definition_jordansecondcode * S ((S (jt_h_definition_jordan)) * C) + (jt_d_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordansecondscale. fs_h_jt_definition_jordansecondscale + S (jt_f_definition_jordan) = S ((S (jt_h_definition_jordan)) * E)) /\\ exists fs_q_jt_definition_jordansecondscale. D = fs_q_jt_definition_jordansecondscale * S ((S (jt_h_definition_jordan)) * E) + (jt_f_definition_jordan))))) -> (forall jt_index_definition_jordansame jt_left_definition_jordansame jt_right_definition_jordansame. (exists jt_gap_definition_jordansameindex. jt_gap_definition_jordansameindex+S (jt_index_definition_jordansame)=(k)) -> (((exists fs_h_jt_definition_jordansameleft. fs_h_jt_definition_jordansameleft + S (jt_left_definition_jordansame) = S ((S (jt_index_definition_jordansame)) * jt_e_definition_jordan)) /\\ exists fs_q_jt_definition_jordansameleft. jt_b_definition_jordan = fs_q_jt_definition_jordansameleft * S ((S (jt_index_definition_jordansame)) * jt_e_definition_jordan) + (jt_left_definition_jordansame))) -> (((exists fs_h_jt_definition_jordansameright. fs_h_jt_definition_jordansameright + S (jt_right_definition_jordansame) = S ((S (jt_index_definition_jordansame)) * jt_f_definition_jordan)) /\\ exists fs_q_jt_definition_jordansameright. jt_d_definition_jordan = fs_q_jt_definition_jordansameright * S ((S (jt_index_definition_jordansame)) * jt_f_definition_jordan) + (jt_right_definition_jordansame))) -> jt_left_definition_jordansame=jt_right_definition_jordansame) -> jt_i_definition_jordan=jt_h_definition_jordan) /\\ (forall jt_z_definition_jordan. (exists jt_gap_definition_jordancodeindex. jt_gap_definition_jordancodeindex+S (jt_z_definition_jordan)=(limit)) -> (forall jt_index_definition_jordaninputbound. (exists jt_gap_definition_jordaninputboundindex. jt_gap_definition_jordaninputboundindex+S (jt_index_definition_jordaninputbound)=(k)) -> exists jt_value_definition_jordaninputbound. ((((exists fs_h_jt_definition_jordaninputboundat. fs_h_jt_definition_jordaninputboundat + S (jt_value_definition_jordaninputbound) = S ((S (jt_index_definition_jordaninputbound)) * c)) /\\ exists fs_q_jt_definition_jordaninputboundat. jt_z_definition_jordan = fs_q_jt_definition_jordaninputboundat * S ((S (jt_index_definition_jordaninputbound)) * c) + (jt_value_definition_jordaninputbound))) /\\ (exists jt_gap_definition_jordaninputboundvalue. jt_gap_definition_jordaninputboundvalue+S (jt_value_definition_jordaninputbound)=(n)))) -> (forall jt_divisor_definition_jordaninputprimitive. (exists jt_factor_definition_jordaninputprimitivemodulus. (n)=(jt_divisor_definition_jordaninputprimitive)*jt_factor_definition_jordaninputprimitivemodulus) -> (forall jt_index_definition_jordaninputprimitivecoordinates jt_value_definition_jordaninputprimitivecoordinates. (exists jt_gap_definition_jordaninputprimitivecoordinatesindex. jt_gap_definition_jordaninputprimitivecoordinatesindex+S (jt_index_definition_jordaninputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordaninputprimitivecoordinatesat. fs_h_jt_definition_jordaninputprimitivecoordinatesat + S (jt_value_definition_jordaninputprimitivecoordinates) = S ((S (jt_index_definition_jordaninputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_definition_jordaninputprimitivecoordinatesat. jt_z_definition_jordan = fs_q_jt_definition_jordaninputprimitivecoordinatesat * S ((S (jt_index_definition_jordaninputprimitivecoordinates)) * c) + (jt_value_definition_jordaninputprimitivecoordinates))) -> (exists jt_factor_definition_jordaninputprimitivecoordinatesdivides. (jt_value_definition_jordaninputprimitivecoordinates)=(jt_divisor_definition_jordaninputprimitive)*jt_factor_definition_jordaninputprimitivecoordinatesdivides)) -> jt_divisor_definition_jordaninputprimitive=1) -> (exists jt_index_definition_jordanlisted jt_code_definition_jordanlisted jt_scale_definition_jordanlisted. ((exists jt_gap_definition_jordanlistedindex. jt_gap_definition_jordanlistedindex+S (jt_index_definition_jordanlisted)=(j)) /\\ (((((((exists fs_h_jt_definition_jordanlistedcode. fs_h_jt_definition_jordanlistedcode + S (jt_code_definition_jordanlisted) = S ((S (jt_index_definition_jordanlisted)) * C)) /\\ exists fs_q_jt_definition_jordanlistedcode. B = fs_q_jt_definition_jordanlistedcode * S ((S (jt_index_definition_jordanlisted)) * C) + (jt_code_definition_jordanlisted))) /\\ (((exists fs_h_jt_definition_jordanlistedscale. fs_h_jt_definition_jordanlistedscale + S (jt_scale_definition_jordanlisted) = S ((S (jt_index_definition_jordanlisted)) * E)) /\\ exists fs_q_jt_definition_jordanlistedscale. D = fs_q_jt_definition_jordanlistedscale * S ((S (jt_index_definition_jordanlisted)) * E) + (jt_scale_definition_jordanlisted))))) /\\ (forall jt_index_definition_jordanlistedequal jt_left_definition_jordanlistedequal jt_right_definition_jordanlistedequal. (exists jt_gap_definition_jordanlistedequalindex. jt_gap_definition_jordanlistedequalindex+S (jt_index_definition_jordanlistedequal)=(k)) -> (((exists fs_h_jt_definition_jordanlistedequalleft. fs_h_jt_definition_jordanlistedequalleft + S (jt_left_definition_jordanlistedequal) = S ((S (jt_index_definition_jordanlistedequal)) * c)) /\\ exists fs_q_jt_definition_jordanlistedequalleft. jt_z_definition_jordan = fs_q_jt_definition_jordanlistedequalleft * S ((S (jt_index_definition_jordanlistedequal)) * c) + (jt_left_definition_jordanlistedequal))) -> (((exists fs_h_jt_definition_jordanlistedequalright. fs_h_jt_definition_jordanlistedequalright + S (jt_right_definition_jordanlistedequal) = S ((S (jt_index_definition_jordanlistedequal)) * jt_scale_definition_jordanlisted)) /\\ exists fs_q_jt_definition_jordanlistedequalright. jt_code_definition_jordanlisted = fs_q_jt_definition_jordanlistedequalright * S ((S (jt_index_definition_jordanlistedequal)) * jt_scale_definition_jordanlisted) + (jt_right_definition_jordanlistedequal))) -> jt_left_definition_jordanlistedequal=jt_right_definition_jordanlistedequal)))))))))", "expansion_sha256": "af269db2f4c407fe47ee8defcbb4dbc12e2339b7c0ff000989d9d7277007d175", "global_argument_positions": null, "global_definition": null, "id": "ND0377", "kernel_signature_unchanged": true, "name": "JordanTupleScan", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "k", "n", "c", "limit", "B", "C", "D", "E", "j" ], "reviewed_definition_id": "ND0377", "shared_definition_identity": "ND0377", "signature": "JordanTupleScan(k,n,c,limit,B,C,D,E,j)", "summary": "A duplicate-free partial enumeration covers precisely the primitive bounded tuples tested by the finite code scan.", "topological_layer": 3, "transitive_dependencies": [ "ND0121", "ND0262", "ND0371", "ND0372", "ND0376", "PD0002", "PD0003", "PD0013" ] }, { "arity": 4, "defined_template": "∀ jt_code_definition_jordan. ∀ jt_scale_definition_jordan. BetaPrefixInto(jt_code_definition_jordan,jt_scale_definition_jordan,k,n) → ∃ x. Lt(x,T) ∧ IntegerVectorZero(jt_code_definition_jordan,jt_scale_definition_jordan,x,c,k)", "defined_template_parts": [ { "kind": "text", "text": "∀ jt_code_definition_jordan. ∀ jt_scale_definition_jordan. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(jt_code_definition_jordan,jt_scale_definition_jordan,k,n)" }, { "kind": "text", "text": " → ∃ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,T)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(jt_code_definition_jordan,jt_scale_definition_jordan,x,c,k)" } ], "dependencies": [ "PD0002", "ND0262", "ND0121" ], "dependency_names": [ "Lt", "BetaPrefixInto", "IntegerVectorZero" ], "exact_ast_verified": true, "expanded_template": "forall jt_code_definition_jordan jt_scale_definition_jordan. (forall jt_index_definition_jordanbound. (exists jt_gap_definition_jordanboundindex. jt_gap_definition_jordanboundindex+S (jt_index_definition_jordanbound)=(k)) -> exists jt_value_definition_jordanbound. ((((exists fs_h_jt_definition_jordanboundat. fs_h_jt_definition_jordanboundat + S (jt_value_definition_jordanbound) = S ((S (jt_index_definition_jordanbound)) * jt_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanboundat. jt_code_definition_jordan = fs_q_jt_definition_jordanboundat * S ((S (jt_index_definition_jordanbound)) * jt_scale_definition_jordan) + (jt_value_definition_jordanbound))) /\\ (exists jt_gap_definition_jordanboundvalue. jt_gap_definition_jordanboundvalue+S (jt_value_definition_jordanbound)=(n)))) -> exists jt_representative_definition_jordan. ((exists jt_gap_definition_jordanindex. jt_gap_definition_jordanindex+S (jt_representative_definition_jordan)=(T)) /\\ (forall jt_index_definition_jordanequal jt_left_definition_jordanequal jt_right_definition_jordanequal. (exists jt_gap_definition_jordanequalindex. jt_gap_definition_jordanequalindex+S (jt_index_definition_jordanequal)=(k)) -> (((exists fs_h_jt_definition_jordanequalleft. fs_h_jt_definition_jordanequalleft + S (jt_left_definition_jordanequal) = S ((S (jt_index_definition_jordanequal)) * jt_scale_definition_jordan)) /\\ exists fs_q_jt_definition_jordanequalleft. jt_code_definition_jordan = fs_q_jt_definition_jordanequalleft * S ((S (jt_index_definition_jordanequal)) * jt_scale_definition_jordan) + (jt_left_definition_jordanequal))) -> (((exists fs_h_jt_definition_jordanequalright. fs_h_jt_definition_jordanequalright + S (jt_right_definition_jordanequal) = S ((S (jt_index_definition_jordanequal)) * c)) /\\ exists fs_q_jt_definition_jordanequalright. jt_representative_definition_jordan = fs_q_jt_definition_jordanequalright * S ((S (jt_index_definition_jordanequal)) * c) + (jt_right_definition_jordanequal))) -> jt_left_definition_jordanequal=jt_right_definition_jordanequal))", "expansion_sha256": "20edc07362a9066a50fcc7c241c008edd9a5129387aeb2ca9bfe7d7e7f4e64b8", "global_argument_positions": null, "global_definition": null, "id": "ND0378", "kernel_signature_unchanged": true, "name": "JordanTupleRepresentatives", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "k", "n", "c", "T" ], "reviewed_definition_id": "ND0378", "shared_definition_identity": "ND0378", "signature": "JordanTupleRepresentatives(k,n,c,T)", "summary": "Every bounded tuple has a coordinate-equal representative with common scale c and code below T.", "topological_layer": 2, "transitive_dependencies": [ "ND0121", "ND0262", "PD0002", "PD0013" ] }, { "arity": 9, "defined_template": "∀ jt_index_definition_jordan. Lt(jt_index_definition_jordan,k) → ∃ x. ∃ y. ∃ z. BetaAt(b,c,jt_index_definition_jordan,x) ∧ (BetaAt(d,e,jt_index_definition_jordan,y) ∧ (BetaAt(f,g,jt_index_definition_jordan,z) ∧ (ModEq(m,z,x) ∧ ModEq(n,z,y))))", "defined_template_parts": [ { "kind": "text", "text": "∀ jt_index_definition_jordan. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(jt_index_definition_jordan,k)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,jt_index_definition_jordan,x)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,jt_index_definition_jordan,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(f,g,jt_index_definition_jordan,z)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,z,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,z,y)" }, { "kind": "text", "text": ")))" } ], "dependencies": [ "PD0002", "PD0013", "PD0008" ], "dependency_names": [ "Lt", "BetaAt", "ModEq" ], "exact_ast_verified": true, "expanded_template": "forall jt_index_definition_jordan. (exists jt_gap_definition_jordanindex. jt_gap_definition_jordanindex+S (jt_index_definition_jordan)=(k)) -> exists jt_left_definition_jordan jt_right_definition_jordan jt_output_definition_jordan. ((((exists fs_h_jt_definition_jordanleft. fs_h_jt_definition_jordanleft + S (jt_left_definition_jordan) = S ((S (jt_index_definition_jordan)) * c)) /\\ exists fs_q_jt_definition_jordanleft. b = fs_q_jt_definition_jordanleft * S ((S (jt_index_definition_jordan)) * c) + (jt_left_definition_jordan))) /\\ (((((exists fs_h_jt_definition_jordanright. fs_h_jt_definition_jordanright + S (jt_right_definition_jordan) = S ((S (jt_index_definition_jordan)) * e)) /\\ exists fs_q_jt_definition_jordanright. d = fs_q_jt_definition_jordanright * S ((S (jt_index_definition_jordan)) * e) + (jt_right_definition_jordan))) /\\ (((((exists fs_h_jt_definition_jordanoutput. fs_h_jt_definition_jordanoutput + S (jt_output_definition_jordan) = S ((S (jt_index_definition_jordan)) * g)) /\\ exists fs_q_jt_definition_jordanoutput. f = fs_q_jt_definition_jordanoutput * S ((S (jt_index_definition_jordan)) * g) + (jt_output_definition_jordan))) /\\ (((exists jt_left_definition_jordanmodleft jt_right_definition_jordanmodleft. (jt_output_definition_jordan)+(m)*jt_left_definition_jordanmodleft=(jt_left_definition_jordan)+(m)*jt_right_definition_jordanmodleft) /\\ (exists jt_left_definition_jordanmodright jt_right_definition_jordanmodright. (jt_output_definition_jordan)+(n)*jt_left_definition_jordanmodright=(jt_right_definition_jordan)+(n)*jt_right_definition_jordanmodright))))))))", "expansion_sha256": "31203a4107b942e018725d4c94b463da2429754e4980664ce0a3496e2416f5a8", "global_argument_positions": null, "global_definition": null, "id": "ND0379", "kernel_signature_unchanged": true, "name": "JordanTupleCRT", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "m", "n", "b", "c", "d", "e", "f", "g", "k" ], "reviewed_definition_id": "ND0379", "shared_definition_identity": "ND0379", "signature": "JordanTupleCRT(m,n,b,c,d,e,f,g,k)", "summary": "Actual output coordinates satisfy both component congruences. Neither coprimality nor canonicality is implicit.", "topological_layer": 1, "transitive_dependencies": [ "PD0002", "PD0008", "PD0013" ] }, { "arity": 9, "defined_template": "BetaPrefixInto(f,g,k,m · n) ∧ (JordanTupleCongruence(m,f,g,b,c,k) ∧ JordanTupleCongruence(n,f,g,d,e,k))", "defined_template_parts": [ { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(f,g,k,m · n)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m,f,g,b,c,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,f,g,d,e,k)" }, { "kind": "text", "text": ")" } ], "dependencies": [ "ND0262", "ND0373" ], "dependency_names": [ "BetaPrefixInto", "JordanTupleCongruence" ], "exact_ast_verified": true, "expanded_template": "((forall jt_index_definition_jordanbound. (exists jt_gap_definition_jordanboundindex. jt_gap_definition_jordanboundindex+S (jt_index_definition_jordanbound)=(k)) -> exists jt_value_definition_jordanbound. ((((exists fs_h_jt_definition_jordanboundat. fs_h_jt_definition_jordanboundat + S (jt_value_definition_jordanbound) = S ((S (jt_index_definition_jordanbound)) * g)) /\\ exists fs_q_jt_definition_jordanboundat. f = fs_q_jt_definition_jordanboundat * S ((S (jt_index_definition_jordanbound)) * g) + (jt_value_definition_jordanbound))) /\\ (exists jt_gap_definition_jordanboundvalue. jt_gap_definition_jordanboundvalue+S (jt_value_definition_jordanbound)=(m*n)))) /\\ (((forall jt_index_definition_jordanleft jt_left_definition_jordanleft jt_right_definition_jordanleft. (exists jt_gap_definition_jordanleftindex. jt_gap_definition_jordanleftindex+S (jt_index_definition_jordanleft)=(k)) -> (((exists fs_h_jt_definition_jordanleftleft. fs_h_jt_definition_jordanleftleft + S (jt_left_definition_jordanleft) = S ((S (jt_index_definition_jordanleft)) * g)) /\\ exists fs_q_jt_definition_jordanleftleft. f = fs_q_jt_definition_jordanleftleft * S ((S (jt_index_definition_jordanleft)) * g) + (jt_left_definition_jordanleft))) -> (((exists fs_h_jt_definition_jordanleftright. fs_h_jt_definition_jordanleftright + S (jt_right_definition_jordanleft) = S ((S (jt_index_definition_jordanleft)) * c)) /\\ exists fs_q_jt_definition_jordanleftright. b = fs_q_jt_definition_jordanleftright * S ((S (jt_index_definition_jordanleft)) * c) + (jt_right_definition_jordanleft))) -> (exists jt_left_definition_jordanleftmod jt_right_definition_jordanleftmod. (jt_left_definition_jordanleft)+(m)*jt_left_definition_jordanleftmod=(jt_right_definition_jordanleft)+(m)*jt_right_definition_jordanleftmod)) /\\ (forall jt_index_definition_jordanright jt_left_definition_jordanright jt_right_definition_jordanright. (exists jt_gap_definition_jordanrightindex. jt_gap_definition_jordanrightindex+S (jt_index_definition_jordanright)=(k)) -> (((exists fs_h_jt_definition_jordanrightleft. fs_h_jt_definition_jordanrightleft + S (jt_left_definition_jordanright) = S ((S (jt_index_definition_jordanright)) * g)) /\\ exists fs_q_jt_definition_jordanrightleft. f = fs_q_jt_definition_jordanrightleft * S ((S (jt_index_definition_jordanright)) * g) + (jt_left_definition_jordanright))) -> (((exists fs_h_jt_definition_jordanrightright. fs_h_jt_definition_jordanrightright + S (jt_right_definition_jordanright) = S ((S (jt_index_definition_jordanright)) * e)) /\\ exists fs_q_jt_definition_jordanrightright. d = fs_q_jt_definition_jordanrightright * S ((S (jt_index_definition_jordanright)) * e) + (jt_right_definition_jordanright))) -> (exists jt_left_definition_jordanrightmod jt_right_definition_jordanrightmod. (jt_left_definition_jordanright)+(n)*jt_left_definition_jordanrightmod=(jt_right_definition_jordanright)+(n)*jt_right_definition_jordanrightmod)))))", "expansion_sha256": "ca05ef70c026d117130a65e2f3e6654bffd7ac9000edd78259a271b5b91fede5", "global_argument_positions": null, "global_definition": null, "id": "ND0380", "kernel_signature_unchanged": true, "name": "JordanCanonicalTupleCRT", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "m", "n", "b", "c", "d", "e", "f", "g", "k" ], "reviewed_definition_id": "ND0380", "shared_definition_identity": "ND0380", "signature": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)", "summary": "The output tuple is bounded by m*n and congruent to each actual input tuple at its component modulus.", "topological_layer": 2, "transitive_dependencies": [ "ND0262", "ND0373", "PD0002", "PD0008", "PD0013" ] }, { "arity": 18, "defined_template": "∀ jt_index_definition_jordan. Lt(jt_index_definition_jordan,q) → ∃ x. ∃ y. ∃ z. ∃ i. ∃ j. ∃ w. ∃ x0. ∃ x1. Lt(x,u) ∧ (Lt(y,v) ∧ (jt_index_definition_jordan = v · x + y ∧ (BetaAt(A,B,x,z) ∧ BetaAt(C,D,x,i) ∧ (BetaAt(E,F,y,j) ∧ BetaAt(G,H,y,w) ∧ (BetaAt(P,Q,jt_index_definition_jordan,x0) ∧ BetaAt(R,T,jt_index_definition_jordan,x1) ∧ (JordanCanonicalTupleCRT(m,n,z,i,j,w,x0,x1,k) ∧ JordanPrimitiveTuple(m · n,x0,x1,k)))))))", "defined_template_parts": [ { "kind": "text", "text": "∀ jt_index_definition_jordan. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(jt_index_definition_jordan,q)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ i. ∃ j. ∃ w. ∃ x0. ∃ x1. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,v)" }, { "kind": "text", "text": " ∧ (jt_index_definition_jordan = v · x + y ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,i)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,y,j)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,y,w)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,jt_index_definition_jordan,x0)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,jt_index_definition_jordan,x1)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,z,i,j,w,x0,x1,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,x0,x1,k)" }, { "kind": "text", "text": "))))))" } ], "dependencies": [ "PD0002", "PD0013", "ND0380", "ND0372" ], "dependency_names": [ "Lt", "BetaAt", "JordanCanonicalTupleCRT", "JordanPrimitiveTuple" ], "exact_ast_verified": true, "expanded_template": "forall jt_index_definition_jordan. (exists jt_gap_definition_jordanindex. jt_gap_definition_jordanindex+S (jt_index_definition_jordan)=(q)) -> exists jt_row_definition_jordan jt_column_definition_jordan jt_b_definition_jordan jt_c_definition_jordan jt_d_definition_jordan jt_e_definition_jordan jt_f_definition_jordan jt_g_definition_jordan. ((exists jt_gap_definition_jordanrow. jt_gap_definition_jordanrow+S (jt_row_definition_jordan)=(u)) /\\ (((exists jt_gap_definition_jordancolumn. jt_gap_definition_jordancolumn+S (jt_column_definition_jordan)=(v)) /\\ (((jt_index_definition_jordan=(v)*jt_row_definition_jordan+jt_column_definition_jordan) /\\ (((((((exists fs_h_jt_definition_jordanleftcode. fs_h_jt_definition_jordanleftcode + S (jt_b_definition_jordan) = S ((S (jt_row_definition_jordan)) * B)) /\\ exists fs_q_jt_definition_jordanleftcode. A = fs_q_jt_definition_jordanleftcode * S ((S (jt_row_definition_jordan)) * B) + (jt_b_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordanleftscale. fs_h_jt_definition_jordanleftscale + S (jt_c_definition_jordan) = S ((S (jt_row_definition_jordan)) * D)) /\\ exists fs_q_jt_definition_jordanleftscale. C = fs_q_jt_definition_jordanleftscale * S ((S (jt_row_definition_jordan)) * D) + (jt_c_definition_jordan))))) /\\ (((((((exists fs_h_jt_definition_jordanrightcode. fs_h_jt_definition_jordanrightcode + S (jt_d_definition_jordan) = S ((S (jt_column_definition_jordan)) * F)) /\\ exists fs_q_jt_definition_jordanrightcode. E = fs_q_jt_definition_jordanrightcode * S ((S (jt_column_definition_jordan)) * F) + (jt_d_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordanrightscale. fs_h_jt_definition_jordanrightscale + S (jt_e_definition_jordan) = S ((S (jt_column_definition_jordan)) * H)) /\\ exists fs_q_jt_definition_jordanrightscale. G = fs_q_jt_definition_jordanrightscale * S ((S (jt_column_definition_jordan)) * H) + (jt_e_definition_jordan))))) /\\ (((((((exists fs_h_jt_definition_jordanoutputcode. fs_h_jt_definition_jordanoutputcode + S (jt_f_definition_jordan) = S ((S (jt_index_definition_jordan)) * Q)) /\\ exists fs_q_jt_definition_jordanoutputcode. P = fs_q_jt_definition_jordanoutputcode * S ((S (jt_index_definition_jordan)) * Q) + (jt_f_definition_jordan))) /\\ (((exists fs_h_jt_definition_jordanoutputscale. fs_h_jt_definition_jordanoutputscale + S (jt_g_definition_jordan) = S ((S (jt_index_definition_jordan)) * T)) /\\ exists fs_q_jt_definition_jordanoutputscale. R = fs_q_jt_definition_jordanoutputscale * S ((S (jt_index_definition_jordan)) * T) + (jt_g_definition_jordan))))) /\\ (((((forall jt_index_definition_jordancrtbound. (exists jt_gap_definition_jordancrtboundindex. jt_gap_definition_jordancrtboundindex+S (jt_index_definition_jordancrtbound)=(k)) -> exists jt_value_definition_jordancrtbound. ((((exists fs_h_jt_definition_jordancrtboundat. fs_h_jt_definition_jordancrtboundat + S (jt_value_definition_jordancrtbound) = S ((S (jt_index_definition_jordancrtbound)) * jt_g_definition_jordan)) /\\ exists fs_q_jt_definition_jordancrtboundat. jt_f_definition_jordan = fs_q_jt_definition_jordancrtboundat * S ((S (jt_index_definition_jordancrtbound)) * jt_g_definition_jordan) + (jt_value_definition_jordancrtbound))) /\\ (exists jt_gap_definition_jordancrtboundvalue. jt_gap_definition_jordancrtboundvalue+S (jt_value_definition_jordancrtbound)=(m*n)))) /\\ (((forall jt_index_definition_jordancrtleft jt_left_definition_jordancrtleft jt_right_definition_jordancrtleft. (exists jt_gap_definition_jordancrtleftindex. jt_gap_definition_jordancrtleftindex+S (jt_index_definition_jordancrtleft)=(k)) -> (((exists fs_h_jt_definition_jordancrtleftleft. fs_h_jt_definition_jordancrtleftleft + S (jt_left_definition_jordancrtleft) = S ((S (jt_index_definition_jordancrtleft)) * jt_g_definition_jordan)) /\\ exists fs_q_jt_definition_jordancrtleftleft. jt_f_definition_jordan = fs_q_jt_definition_jordancrtleftleft * S ((S (jt_index_definition_jordancrtleft)) * jt_g_definition_jordan) + (jt_left_definition_jordancrtleft))) -> (((exists fs_h_jt_definition_jordancrtleftright. fs_h_jt_definition_jordancrtleftright + S (jt_right_definition_jordancrtleft) = S ((S (jt_index_definition_jordancrtleft)) * jt_c_definition_jordan)) /\\ exists fs_q_jt_definition_jordancrtleftright. jt_b_definition_jordan = fs_q_jt_definition_jordancrtleftright * S ((S (jt_index_definition_jordancrtleft)) * jt_c_definition_jordan) + (jt_right_definition_jordancrtleft))) -> (exists jt_left_definition_jordancrtleftmod jt_right_definition_jordancrtleftmod. (jt_left_definition_jordancrtleft)+(m)*jt_left_definition_jordancrtleftmod=(jt_right_definition_jordancrtleft)+(m)*jt_right_definition_jordancrtleftmod)) /\\ (forall jt_index_definition_jordancrtright jt_left_definition_jordancrtright jt_right_definition_jordancrtright. (exists jt_gap_definition_jordancrtrightindex. jt_gap_definition_jordancrtrightindex+S (jt_index_definition_jordancrtright)=(k)) -> (((exists fs_h_jt_definition_jordancrtrightleft. fs_h_jt_definition_jordancrtrightleft + S (jt_left_definition_jordancrtright) = S ((S (jt_index_definition_jordancrtright)) * jt_g_definition_jordan)) /\\ exists fs_q_jt_definition_jordancrtrightleft. jt_f_definition_jordan = fs_q_jt_definition_jordancrtrightleft * S ((S (jt_index_definition_jordancrtright)) * jt_g_definition_jordan) + (jt_left_definition_jordancrtright))) -> (((exists fs_h_jt_definition_jordancrtrightright. fs_h_jt_definition_jordancrtrightright + S (jt_right_definition_jordancrtright) = S ((S (jt_index_definition_jordancrtright)) * jt_e_definition_jordan)) /\\ exists fs_q_jt_definition_jordancrtrightright. jt_d_definition_jordan = fs_q_jt_definition_jordancrtrightright * S ((S (jt_index_definition_jordancrtright)) * jt_e_definition_jordan) + (jt_right_definition_jordancrtright))) -> (exists jt_left_definition_jordancrtrightmod jt_right_definition_jordancrtrightmod. (jt_left_definition_jordancrtright)+(n)*jt_left_definition_jordancrtrightmod=(jt_right_definition_jordancrtright)+(n)*jt_right_definition_jordancrtrightmod)))))) /\\ (forall jt_divisor_definition_jordanprimitive. (exists jt_factor_definition_jordanprimitivemodulus. (m*n)=(jt_divisor_definition_jordanprimitive)*jt_factor_definition_jordanprimitivemodulus) -> (forall jt_index_definition_jordanprimitivecoordinates jt_value_definition_jordanprimitivecoordinates. (exists jt_gap_definition_jordanprimitivecoordinatesindex. jt_gap_definition_jordanprimitivecoordinatesindex+S (jt_index_definition_jordanprimitivecoordinates)=(k)) -> (((exists fs_h_jt_definition_jordanprimitivecoordinatesat. fs_h_jt_definition_jordanprimitivecoordinatesat + S (jt_value_definition_jordanprimitivecoordinates) = S ((S (jt_index_definition_jordanprimitivecoordinates)) * jt_g_definition_jordan)) /\\ exists fs_q_jt_definition_jordanprimitivecoordinatesat. jt_f_definition_jordan = fs_q_jt_definition_jordanprimitivecoordinatesat * S ((S (jt_index_definition_jordanprimitivecoordinates)) * jt_g_definition_jordan) + (jt_value_definition_jordanprimitivecoordinates))) -> (exists jt_factor_definition_jordanprimitivecoordinatesdivides. (jt_value_definition_jordanprimitivecoordinates)=(jt_divisor_definition_jordanprimitive)*jt_factor_definition_jordanprimitivecoordinatesdivides)) -> jt_divisor_definition_jordanprimitive=1))))))))))))))", "expansion_sha256": "bdfd5eca3fd237a3e5927aa6094b3e70b6de825c4439186914a405d1da1dbf95", "global_argument_positions": null, "global_definition": null, "id": "ND0381", "kernel_signature_unchanged": true, "name": "JordanRectangleCRT", "origin": "shared-hygienic-conservative-definition-not-proof-authority", "parameters": [ "m", "n", "k", "A", "B", "C", "D", "u", "E", "F", "G", "H", "v", "P", "Q", "R", "T", "q" ], "reviewed_definition_id": "ND0381", "shared_definition_identity": "ND0381", "signature": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q)", "summary": "At each flat rectangle index below q the output has actual entries giving a canonical primitive CRT tuple of the actual source entries.", "topological_layer": 3, "transitive_dependencies": [ "ND0262", "ND0371", "ND0372", "ND0373", "ND0380", "PD0002", "PD0003", "PD0008", "PD0013" ] } ], "edge_count": 254, "edges": [ { "kind": "proof_dependency", "source": "JT0004", "target": "JT0005" }, { "kind": "proof_dependency", "source": "JT0002", "target": "JT0005" }, { "kind": "proof_dependency", "source": "JT0007", "target": "JT000B" }, { "kind": "proof_dependency", "source": "JT0006", "target": "JT000C" }, { "kind": "proof_dependency", "source": "JT000D", "target": "JT000F" }, { "kind": "proof_dependency", "source": "JT0010", "target": "JT0012" }, { "kind": "proof_dependency", "source": "JT0011", "target": "JT0012" }, { "kind": "proof_dependency", "source": "JT0012", "target": "JT0013" }, { "kind": "proof_dependency", "source": "JT0013", "target": "JT0014" }, { "kind": "proof_dependency", "source": "JT0014", "target": "JT0015" }, { "kind": "proof_dependency", "source": "JT0016", "target": "JT0019" }, { "kind": "proof_dependency", "source": "JT0018", "target": "JT0019" }, { "kind": "proof_dependency", "source": "JT0017", "target": "JT0019" }, { "kind": "proof_dependency", "source": "JT001A", "target": "JT001C" }, { "kind": "proof_dependency", "source": "JT001B", "target": "JT001C" }, { "kind": "proof_dependency", "source": "JT0019", "target": "JT001C" }, { "kind": "proof_dependency", "source": "JT001F", "target": "JT0020" }, { "kind": "proof_dependency", "source": "JT001E", "target": "JT0021" }, { "kind": "proof_dependency", "source": "JT0003", "target": "JT0022" }, { "kind": "proof_dependency", "source": "JT0022", "target": "JT0024" }, { "kind": "proof_dependency", "source": "JT0020", "target": "JT0024" }, { "kind": "proof_dependency", "source": "JT0005", "target": "JT0024" }, { "kind": "proof_dependency", "source": "JT0024", "target": "JT0025" }, { "kind": "proof_dependency", "source": "JT0021", "target": "JT0026" }, { "kind": "proof_dependency", "source": "JT0002", "target": "JT0026" }, { "kind": "proof_dependency", "source": "JT001F", "target": "JT0026" }, { "kind": "proof_dependency", "source": "JT001D", "target": "JT0027" }, { "kind": "proof_dependency", "source": "JT0015", "target": "JT0027" }, { "kind": "proof_dependency", "source": "JT001C", "target": "JT0027" }, { "kind": "proof_dependency", "source": "JT0023", "target": "JT0027" }, { "kind": "proof_dependency", "source": "JT0026", "target": "JT0027" }, { "kind": "proof_dependency", "source": "JT001E", "target": "JT0028" }, { "kind": "proof_dependency", "source": "JT0028", "target": "JT0029" }, { "kind": "proof_dependency", "source": "JT0027", "target": "JT0029" }, { "kind": "proof_dependency", "source": "JT0025", "target": "JT0029" }, { "kind": "proof_dependency", "source": "JT002A", "target": "JT002C" }, { "kind": "proof_dependency", "source": "JT002B", "target": "JT002C" }, { "kind": "proof_dependency", "source": "JT002C", "target": "JT0032" }, { "kind": "proof_dependency", "source": "JT002F", "target": "JT0032" }, { "kind": "proof_dependency", "source": "JT0031", "target": "JT0032" }, { "kind": 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"document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=bezout_mod_left]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "e5ae5490cb5cb67b7fbad0dfd01dd27a17eb07bd5aa06a7e5cf90532da5aef4c", "membership": "stable", "name": "bezout_mod_left", "proof_tag": "PA001W", "provenance": [ "stable" ], "script": [ "intro m", "intro n", "intro xp", "intro yp", "intro xn", "intro yn", "intro h", "exists xp", "exists xn", "trans m * xp + n * yp", "apply add_comm", "trans 1 + (m * xn + n * yn)", "exact h", "trans 1 + (n * yn + m * xn)", "congr", "refl", "apply add_comm", "symm", "apply add_assoc" ], "script_sha256": "671069595acda0bd92eab5e5c6be80775a4471d0374c93a9f93f7ef9335dc8cc", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * v", "statement_sha256": "7c933e89b5e783e0fe57654937ee8202b8b0a917184b23463fccbda85de308b0", "summary": "A balanced Bezout identity selects the right coefficient modulo the left modulus.", "summary_sha256": "623459c55c27addeaa8a67a3b05a025eec601b61e36612e9ff02cd0a2865910f" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "add_assoc", "add_comm" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=bezout_mod_left]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "bezout_mod_left", "parent_alpha_version": "v34", "proof_bundle_node_id": 119, "reference_route": "jordan-totient/checkpoint.html#theorem-bezout_mod_left", "script": [ "intro m", "intro n", "intro xp", "intro yp", "intro xn", "intro yn", "intro h", "exists xp", "exists xn", "trans m * xp + n * yp", "apply add_comm", "trans 1 + (m * xn + n * yn)", "exact h", "trans 1 + (n * yn + m * xn)", "congr", "refl", "apply add_comm", "symm", "apply add_assoc" ], "script_sha256": "671069595acda0bd92eab5e5c6be80775a4471d0374c93a9f93f7ef9335dc8cc", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. n * yp + m * u = (1 + n * yn) + m * v", "statement_sha256": 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"stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=bezout_mod_right]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "6ccb293c0092a76550b64a950ae7afd928fff0fce27051e7d9c1cd1b6a10f356", "membership": "stable", "name": "bezout_mod_right", "proof_tag": "PA001X", "provenance": [ "stable" ], "script": [ "intro m", "intro n", "intro xp", "intro yp", "intro xn", "intro yn", "intro h", "exists yp", "exists yn", "trans 1 + (m * xn + n * yn)", "exact h", "symm", "apply add_assoc" ], "script_sha256": "fa59bb212574931c5bc4d3c48a092318bd78e844ab12aa648f55b7315f3a0898", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. m * xp + n * u = (1 + m * xn) + n * v", "statement_sha256": "2b9829ff7d6d6582927a7b85c058b531717e3d91259f9bcc574a76f687eec068", "summary": "A balanced Bezout identity selects the left coefficient modulo the right modulus.", "summary_sha256": "4da853cb1345df8137f32f820ba84f76c8cfea5486a4473abbf3dbd0313ba112" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "add_assoc" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=bezout_mod_right]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "bezout_mod_right", "parent_alpha_version": "v34", "proof_bundle_node_id": 120, "reference_route": "jordan-totient/checkpoint.html#theorem-bezout_mod_right", "script": [ "intro m", "intro n", "intro xp", "intro yp", "intro xn", "intro yn", "intro h", "exists yp", "exists yn", "trans 1 + (m * xn + n * yn)", "exact h", "symm", "apply add_assoc" ], "script_sha256": "fa59bb212574931c5bc4d3c48a092318bd78e844ab12aa648f55b7315f3a0898", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall m n xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn) -> exists u v. m * xp + n * u = (1 + m * xn) + n * v", "statement_sha256": "2b9829ff7d6d6582927a7b85c058b531717e3d91259f9bcc574a76f687eec068" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "mod_eq_predecessor_cancel", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "add_assoc", "add_comm", "mul_succ_left" ], "dependencies_sha256": "1a53d87d5eda46d621b2411e0f71a65d3f99bc989d4becadba0c2ac7a242e694", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "35e5bcc4216481e35319981c1f0536cda87b5ec2a6c2215b49d5b686cea7686f", "cut_nodes": 9, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 25, "proof_edges": 210, "proof_nodes": 315, "proof_objects": 192, "reused_objects": 19, "status": "checked" }, "enrollment_index": 154, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=mod_eq_predecessor_cancel]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "9e7108b19406a3b98f1405f8570ae00f4faf2a00bceca6e4df9b0ff65164bb21", "membership": "stable", "name": "mod_eq_predecessor_cancel", "proof_tag": "PA0025", "provenance": [ "stable" ], "script": [ "intro k", "intro a", "intro z", "exists 0", "exists z", "rewrite PA5", "rewrite PA3", "specialize mul_succ_left k", "specialize mul_succ_left z", "rewrite mul_succ_left", "trans a + (z + k * z)", "apply add_assoc", "congr", "refl", "apply add_comm" ], "script_sha256": "0ae073a1f56464b5ea93c3d1997f7f807c2bbfbdbdd39da767ca280673a7e1a5", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall k a z. exists u v. ((a + z) + k * z) + S k * u = a + S k * v", "statement_sha256": "ff9e6397219fcf402e7a0e46e5d1ce0cee284f43b12ebdd124e21082c0170db4", "summary": "The predecessor of a successor acts as minus one in balanced congruence.", "summary_sha256": "293a4ec4026986473ad6de66d2c763af9eaa3f0e332ce0f11fc818c861929aa5" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "add_assoc", "add_comm", "mul_succ_left" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=mod_eq_predecessor_cancel]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "mod_eq_predecessor_cancel", "parent_alpha_version": "v34", "proof_bundle_node_id": 121, "reference_route": "jordan-totient/checkpoint.html#theorem-mod_eq_predecessor_cancel", "script": [ "intro k", "intro a", "intro z", "exists 0", "exists z", "rewrite PA5", "rewrite PA3", "specialize mul_succ_left k", "specialize mul_succ_left z", "rewrite mul_succ_left", "trans a + (z + k * z)", "apply add_assoc", "congr", "refl", "apply add_comm" ], "script_sha256": "0ae073a1f56464b5ea93c3d1997f7f807c2bbfbdbdd39da767ca280673a7e1a5", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall k a z. exists u v. ((a + z) + k * z) + S k * u = a + S k * v", "statement_sha256": "ff9e6397219fcf402e7a0e46e5d1ce0cee284f43b12ebdd124e21082c0170db4" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "binary_crt", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "nonzero_is_succ", "coprime_balanced_bezout", "bezout_mod_left", "bezout_mod_right", "mod_eq_mul_left", "mul_add", "mul_one", "dvd_to_mod_zero", "mul_assoc", "mul_comm", "mod_eq_add", "mod_eq_refl", "mod_eq_trans", "mod_eq_predecessor_cancel", "zero_add" ], "dependencies_sha256": "4d0a90befe5020de60529fafbd35888ae2e9245371de29fcf6b731fd637756f9", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "fd5384ba933f194fe0229c01f97f3261d12dfe8586196f1fdd390349aefad2df", "cut_nodes": 144, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 51, "proof_edges": 2037, "proof_nodes": 5044, "proof_objects": 1929, "reused_objects": 109, "status": "checked" }, "enrollment_index": 155, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=binary_crt]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "0e6e2b144b6f4f6eae048e90d708c98b49c53309d0ed7db50017dc0e482e79db", "membership": "stable", "name": "binary_crt", "proof_tag": "PA0026", "provenance": [ "stable" ], "script": [ "intro m", "intro n", "intro a", "intro b", "intro hm", "intro hn", "intro hcop", "have hms : exists k. m = S k", "specialize nonzero_is_succ m", "apply nonzero_is_succ", "exact hm", "have hns : exists k. n = S k", "specialize nonzero_is_succ n", "apply nonzero_is_succ", "exact hn", "have hbez : exists xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn)", "specialize coprime_balanced_bezout m", "specialize coprime_balanced_bezout n", "apply coprime_balanced_bezout", "exact hcop", "cases hms", "cases hns", "cases hbez", "cases hbez_witness", "cases hbez_witness_witness", "cases hbez_witness_witness_witness", "have hbl : exists u v. n * x3 + m * u = (1 + n * x5) + m * v", "specialize bezout_mod_left m", "specialize bezout_mod_left n", "specialize bezout_mod_left x2", "specialize bezout_mod_left x3", "specialize bezout_mod_left x4", "specialize bezout_mod_left x5", "apply bezout_mod_left", "exact hbez_witness_witness_witness_witness", "have hbr : exists u v. m * x2 + n * u = (1 + m * x4) + n * v", "specialize bezout_mod_right m", "specialize bezout_mod_right n", "specialize bezout_mod_right x2", "specialize bezout_mod_right x3", "specialize bezout_mod_right x4", "specialize bezout_mod_right x5", "apply bezout_mod_right", "exact hbez_witness_witness_witness_witness", "have hal0 : exists u v. (a * (n * x3)) + m * u = (a * (1 + n * x5)) + m * v", "specialize mod_eq_mul_left m", "specialize mod_eq_mul_left (n * x3)", "specialize mod_eq_mul_left (1 + n * x5)", "specialize mod_eq_mul_left a", "apply mod_eq_mul_left", "exact hbl", "have hal : exists u v. (a * (n * x3)) + m * u = (a + a * (n * x5)) + m * v", "have haexpand : a * (1 + n * x5) = a + a * (n * x5)", "trans a * 1 + a * (n * x5)", "apply mul_add", "congr", "apply mul_one", "refl", "rewrite <- haexpand", "exact hal0", "have hbm : exists u v. (b * (m * x2)) + m * u = 0 + m * v", "apply dvd_to_mod_zero", "exists b * x2", "trans (b * m) * x2", "symm", "apply mul_assoc", "trans (m * b) * x2", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hym : exists u v. ((a * (n * x3)) + (b * (m * x2))) + m * u = ((a + a * (n * x5)) + 0) + m * v", "specialize mod_eq_add m", "specialize mod_eq_add (a * (n * x3))", "specialize mod_eq_add (a + a * (n * x5))", "specialize mod_eq_add (b * (m * x2))", "specialize mod_eq_add 0", "apply mod_eq_add", "exact hal", "exact hbm", "have hym_norm : exists u v. ((a * (n * x3)) + (b * (m * x2))) + m * u = (a + a * (n * x5)) + m * v", "have hym_zero : (a + a * (n * x5)) + 0 = a + a * (n * x5)", "rewrite PA3", "refl", "rewrite <- hym_zero", "exact hym", "have hkm : exists u v. (x * (a * (n * x5))) + m * u = (x * (a * (n * x5))) + m * v", "specialize mod_eq_refl m", "specialize mod_eq_refl (x * (a * (n * x5)))", "apply mod_eq_refl", "have hymk : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + m * u = ((a + a * (n * x5)) + (x * (a * (n * x5)))) + m * v", "specialize mod_eq_add m", "specialize mod_eq_add ((a * (n * x3)) + (b * (m * x2)))", "specialize mod_eq_add (a + a * (n * x5))", "specialize mod_eq_add (x * (a * (n * x5)))", "specialize mod_eq_add (x * (a * (n * x5)))", "apply mod_eq_add", "exact hym_norm", "exact hkm", "have hcancelm : exists u v. ((a + a * (n * x5)) + x * (a * (n * x5))) + S x * u = a + S x * v", "specialize mod_eq_predecessor_cancel x", "specialize mod_eq_predecessor_cancel a", "specialize mod_eq_predecessor_cancel (a * (n * x5))", "apply mod_eq_predecessor_cancel", "rewrite <- hms_witness at hcancelm", "rewrite <- hms_witness at hcancelm", "have hbasem : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + m * u = a + m * v", "specialize mod_eq_trans m", "specialize mod_eq_trans (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))", "specialize mod_eq_trans ((a + a * (n * x5)) + (x * (a * (n * x5))))", "specialize mod_eq_trans a", "apply mod_eq_trans", "exact hymk", "exact hcancelm", "have hknznm : exists u v. (x1 * (b * (m * x4))) + m * u = 0 + m * v", "apply dvd_to_mod_zero", "exists x1 * (b * x4)", "trans x1 * ((b * m) * x4)", "congr", "refl", "symm", "apply mul_assoc", "trans x1 * ((m * b) * x4)", "congr", "refl", "congr", "apply mul_comm", "refl", "trans x1 * (m * (b * x4))", "congr", "refl", "apply mul_assoc", "trans (x1 * m) * (b * x4)", "symm", "apply mul_assoc", "trans (m * x1) * (b * x4)", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hfinalm0 : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + m * u = (a + 0) + m * v", "specialize mod_eq_add m", "specialize mod_eq_add (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))", "specialize mod_eq_add a", "specialize mod_eq_add (x1 * (b * (m * x4)))", "specialize mod_eq_add 0", "apply mod_eq_add", "exact hbasem", "exact hknznm", "have hazerom : exists u v. (a + 0) + m * u = a + m * v", "exists 0", "exists 0", "simp", "have hfinalm : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + m * u = a + m * v", "specialize mod_eq_trans m", "specialize mod_eq_trans ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4))))", "specialize mod_eq_trans (a + 0)", "specialize mod_eq_trans a", "apply mod_eq_trans", "exact hfinalm0", "exact hazerom", "have hbn0 : exists u v. (b * (m * x2)) + n * u = (b * (1 + m * x4)) + n * v", "specialize mod_eq_mul_left n", "specialize mod_eq_mul_left (m * x2)", "specialize mod_eq_mul_left (1 + m * x4)", "specialize mod_eq_mul_left b", "apply mod_eq_mul_left", "exact hbr", "have hbn : exists u v. (b * (m * x2)) + n * u = (b + b * (m * x4)) + n * v", "have hbexpand : b * (1 + m * x4) = b + b * (m * x4)", "trans b * 1 + b * (m * x4)", "apply mul_add", "congr", "apply mul_one", "refl", "rewrite <- hbexpand", "exact hbn0", "have han : exists u v. (a * (n * x3)) + n * u = 0 + n * v", "apply dvd_to_mod_zero", "exists a * x3", "trans (a * n) * x3", "symm", "apply mul_assoc", "trans (n * a) * x3", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hyn0 : exists u v. ((a * (n * x3)) + (b * (m * x2))) + n * u = (0 + (b + b * (m * x4))) + n * v", "specialize mod_eq_add n", "specialize mod_eq_add (a * (n * x3))", "specialize mod_eq_add 0", "specialize mod_eq_add (b * (m * x2))", "specialize mod_eq_add (b + b * (m * x4))", "apply mod_eq_add", "exact han", "exact hbn", "have hyn_norm : exists u v. ((a * (n * x3)) + (b * (m * x2))) + n * u = (b + b * (m * x4)) + n * v", "have hyn_zero : 0 + (b + b * (m * x4)) = b + b * (m * x4)", "specialize zero_add (b + b * (m * x4))", "exact zero_add", "rewrite <- hyn_zero", "exact hyn0", "have hkmz : exists u v. (x * (a * (n * x5))) + n * u = 0 + n * v", "apply dvd_to_mod_zero", "exists x * (a * x5)", "trans x * ((a * n) * x5)", "congr", "refl", "symm", "apply mul_assoc", "trans x * ((n * a) * x5)", "congr", "refl", "congr", "apply mul_comm", "refl", "trans x * (n * (a * x5))", "congr", "refl", "apply mul_assoc", "trans (x * n) * (a * x5)", "symm", "apply mul_assoc", "trans (n * x) * (a * x5)", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hyn1 : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + n * u = ((b + b * (m * x4)) + 0) + n * v", "specialize mod_eq_add n", "specialize mod_eq_add ((a * (n * x3)) + (b * (m * x2)))", "specialize mod_eq_add (b + b * (m * x4))", "specialize mod_eq_add (x * (a * (n * x5)))", "specialize mod_eq_add 0", "apply mod_eq_add", "exact hyn_norm", "exact hkmz", "have hyn1_norm : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + n * u = (b + b * (m * x4)) + n * v", "have hyn1_zero : (b + b * (m * x4)) + 0 = b + b * (m * x4)", "rewrite PA3", "refl", "rewrite <- hyn1_zero", "exact hyn1", "have hkn : exists u v. (x1 * (b * (m * x4))) + n * u = (x1 * (b * (m * x4))) + n * v", "specialize mod_eq_refl n", "specialize mod_eq_refl (x1 * (b * (m * x4)))", "apply mod_eq_refl", "have hynk : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + n * u = ((b + b * (m * x4)) + (x1 * (b * (m * x4)))) + n * v", "specialize mod_eq_add n", "specialize mod_eq_add (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))", "specialize mod_eq_add (b + b * (m * x4))", "specialize mod_eq_add (x1 * (b * (m * x4)))", "specialize mod_eq_add (x1 * (b * (m * x4)))", "apply mod_eq_add", "exact hyn1_norm", "exact hkn", "have hcanceln : exists u v. ((b + b * (m * x4)) + x1 * (b * (m * x4))) + S x1 * u = b + S x1 * v", "specialize mod_eq_predecessor_cancel x1", "specialize mod_eq_predecessor_cancel b", "specialize mod_eq_predecessor_cancel (b * (m * x4))", "apply mod_eq_predecessor_cancel", "rewrite <- hns_witness at hcanceln", "rewrite <- hns_witness at hcanceln", "have hfinaln : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + n * u = b + n * v", "specialize mod_eq_trans n", "specialize mod_eq_trans ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4))))", "specialize mod_eq_trans ((b + b * (m * x4)) + (x1 * (b * (m * x4))))", "specialize mod_eq_trans b", "apply mod_eq_trans", "exact hynk", "exact hcanceln", "exists (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))", "split", "exact hfinalm", "exact hfinaln" ], "script_sha256": "8db963d420b0ce840011a723d4905fbac814e1af3bc8678bf51658d0a163bd64", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> exists x. (exists u v. x + m * u = a + m * v) /\\ (exists r s. x + n * r = b + n * s)", "statement_sha256": "25e1f5213a9a5b04f3a20077b936ba8814f6ae8951f4f72e5f5e2135d7c9148f", "summary": "Constructive binary CRT for positive coprime natural moduli using balanced congruence.", "summary_sha256": "6fdbe6d82f278b76d4f9960e8951be2039a5cb670e0494f6ae813f531705f237" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "nonzero_is_succ", "coprime_balanced_bezout", "bezout_mod_left", "bezout_mod_right", "mod_eq_mul_left", "mul_add", "mul_one", "dvd_to_mod_zero", "mul_assoc", "mul_comm", "mod_eq_add", "mod_eq_refl", "mod_eq_trans", "mod_eq_predecessor_cancel", "zero_add" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=binary_crt]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "binary_crt", "parent_alpha_version": "v34", "proof_bundle_node_id": 122, "reference_route": "jordan-totient/checkpoint.html#theorem-binary_crt", "script": [ "intro m", "intro n", "intro a", "intro b", "intro hm", "intro hn", "intro hcop", "have hms : exists k. m = S k", "specialize nonzero_is_succ m", "apply nonzero_is_succ", "exact hm", "have hns : exists k. n = S k", "specialize nonzero_is_succ n", "apply nonzero_is_succ", "exact hn", "have hbez : exists xp yp xn yn. m * xp + n * yp = 1 + (m * xn + n * yn)", "specialize coprime_balanced_bezout m", "specialize coprime_balanced_bezout n", "apply coprime_balanced_bezout", "exact hcop", "cases hms", "cases hns", "cases hbez", "cases hbez_witness", "cases hbez_witness_witness", "cases hbez_witness_witness_witness", "have hbl : exists u v. n * x3 + m * u = (1 + n * x5) + m * v", "specialize bezout_mod_left m", "specialize bezout_mod_left n", "specialize bezout_mod_left x2", "specialize bezout_mod_left x3", "specialize bezout_mod_left x4", "specialize bezout_mod_left x5", "apply bezout_mod_left", "exact hbez_witness_witness_witness_witness", "have hbr : exists u v. m * x2 + n * u = (1 + m * x4) + n * v", "specialize bezout_mod_right m", "specialize bezout_mod_right n", "specialize bezout_mod_right x2", "specialize bezout_mod_right x3", "specialize bezout_mod_right x4", "specialize bezout_mod_right x5", "apply bezout_mod_right", "exact hbez_witness_witness_witness_witness", "have hal0 : exists u v. (a * (n * x3)) + m * u = (a * (1 + n * x5)) + m * v", "specialize mod_eq_mul_left m", "specialize mod_eq_mul_left (n * x3)", "specialize mod_eq_mul_left (1 + n * x5)", "specialize mod_eq_mul_left a", "apply mod_eq_mul_left", "exact hbl", "have hal : exists u v. (a * (n * x3)) + m * u = (a + a * (n * x5)) + m * v", "have haexpand : a * (1 + n * x5) = a + a * (n * x5)", "trans a * 1 + a * (n * x5)", "apply mul_add", "congr", "apply mul_one", "refl", "rewrite <- haexpand", "exact hal0", "have hbm : exists u v. (b * (m * x2)) + m * u = 0 + m * v", "apply dvd_to_mod_zero", "exists b * x2", "trans (b * m) * x2", "symm", "apply mul_assoc", "trans (m * b) * x2", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hym : exists u v. ((a * (n * x3)) + (b * (m * x2))) + m * u = ((a + a * (n * x5)) + 0) + m * v", "specialize mod_eq_add m", "specialize mod_eq_add (a * (n * x3))", "specialize mod_eq_add (a + a * (n * x5))", "specialize mod_eq_add (b * (m * x2))", "specialize mod_eq_add 0", "apply mod_eq_add", "exact hal", "exact hbm", "have hym_norm : exists u v. ((a * (n * x3)) + (b * (m * x2))) + m * u = (a + a * (n * x5)) + m * v", "have hym_zero : (a + a * (n * x5)) + 0 = a + a * (n * x5)", "rewrite PA3", "refl", "rewrite <- hym_zero", "exact hym", "have hkm : exists u v. (x * (a * (n * x5))) + m * u = (x * (a * (n * x5))) + m * v", "specialize mod_eq_refl m", "specialize mod_eq_refl (x * (a * (n * x5)))", "apply mod_eq_refl", "have hymk : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + m * u = ((a + a * (n * x5)) + (x * (a * (n * x5)))) + m * v", "specialize mod_eq_add m", "specialize mod_eq_add ((a * (n * x3)) + (b * (m * x2)))", "specialize mod_eq_add (a + a * (n * x5))", "specialize mod_eq_add (x * (a * (n * x5)))", "specialize mod_eq_add (x * (a * (n * x5)))", "apply mod_eq_add", "exact hym_norm", "exact hkm", "have hcancelm : exists u v. ((a + a * (n * x5)) + x * (a * (n * x5))) + S x * u = a + S x * v", "specialize mod_eq_predecessor_cancel x", "specialize mod_eq_predecessor_cancel a", "specialize mod_eq_predecessor_cancel (a * (n * x5))", "apply mod_eq_predecessor_cancel", "rewrite <- hms_witness at hcancelm", "rewrite <- hms_witness at hcancelm", "have hbasem : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + m * u = a + m * v", "specialize mod_eq_trans m", "specialize mod_eq_trans (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))", "specialize mod_eq_trans ((a + a * (n * x5)) + (x * (a * (n * x5))))", "specialize mod_eq_trans a", "apply mod_eq_trans", "exact hymk", "exact hcancelm", "have hknznm : exists u v. (x1 * (b * (m * x4))) + m * u = 0 + m * v", "apply dvd_to_mod_zero", "exists x1 * (b * x4)", "trans x1 * ((b * m) * x4)", "congr", "refl", "symm", "apply mul_assoc", "trans x1 * ((m * b) * x4)", "congr", "refl", "congr", "apply mul_comm", "refl", "trans x1 * (m * (b * x4))", "congr", "refl", "apply mul_assoc", "trans (x1 * m) * (b * x4)", "symm", "apply mul_assoc", "trans (m * x1) * (b * x4)", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hfinalm0 : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + m * u = (a + 0) + m * v", "specialize mod_eq_add m", "specialize mod_eq_add (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))", "specialize mod_eq_add a", "specialize mod_eq_add (x1 * (b * (m * x4)))", "specialize mod_eq_add 0", "apply mod_eq_add", "exact hbasem", "exact hknznm", "have hazerom : exists u v. (a + 0) + m * u = a + m * v", "exists 0", "exists 0", "simp", "have hfinalm : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + m * u = a + m * v", "specialize mod_eq_trans m", "specialize mod_eq_trans ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4))))", "specialize mod_eq_trans (a + 0)", "specialize mod_eq_trans a", "apply mod_eq_trans", "exact hfinalm0", "exact hazerom", "have hbn0 : exists u v. (b * (m * x2)) + n * u = (b * (1 + m * x4)) + n * v", "specialize mod_eq_mul_left n", "specialize mod_eq_mul_left (m * x2)", "specialize mod_eq_mul_left (1 + m * x4)", "specialize mod_eq_mul_left b", "apply mod_eq_mul_left", "exact hbr", "have hbn : exists u v. (b * (m * x2)) + n * u = (b + b * (m * x4)) + n * v", "have hbexpand : b * (1 + m * x4) = b + b * (m * x4)", "trans b * 1 + b * (m * x4)", "apply mul_add", "congr", "apply mul_one", "refl", "rewrite <- hbexpand", "exact hbn0", "have han : exists u v. (a * (n * x3)) + n * u = 0 + n * v", "apply dvd_to_mod_zero", "exists a * x3", "trans (a * n) * x3", "symm", "apply mul_assoc", "trans (n * a) * x3", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hyn0 : exists u v. ((a * (n * x3)) + (b * (m * x2))) + n * u = (0 + (b + b * (m * x4))) + n * v", "specialize mod_eq_add n", "specialize mod_eq_add (a * (n * x3))", "specialize mod_eq_add 0", "specialize mod_eq_add (b * (m * x2))", "specialize mod_eq_add (b + b * (m * x4))", "apply mod_eq_add", "exact han", "exact hbn", "have hyn_norm : exists u v. ((a * (n * x3)) + (b * (m * x2))) + n * u = (b + b * (m * x4)) + n * v", "have hyn_zero : 0 + (b + b * (m * x4)) = b + b * (m * x4)", "specialize zero_add (b + b * (m * x4))", "exact zero_add", "rewrite <- hyn_zero", "exact hyn0", "have hkmz : exists u v. (x * (a * (n * x5))) + n * u = 0 + n * v", "apply dvd_to_mod_zero", "exists x * (a * x5)", "trans x * ((a * n) * x5)", "congr", "refl", "symm", "apply mul_assoc", "trans x * ((n * a) * x5)", "congr", "refl", "congr", "apply mul_comm", "refl", "trans x * (n * (a * x5))", "congr", "refl", "apply mul_assoc", "trans (x * n) * (a * x5)", "symm", "apply mul_assoc", "trans (n * x) * (a * x5)", "congr", "apply mul_comm", "refl", "apply mul_assoc", "have hyn1 : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + n * u = ((b + b * (m * x4)) + 0) + n * v", "specialize mod_eq_add n", "specialize mod_eq_add ((a * (n * x3)) + (b * (m * x2)))", "specialize mod_eq_add (b + b * (m * x4))", "specialize mod_eq_add (x * (a * (n * x5)))", "specialize mod_eq_add 0", "apply mod_eq_add", "exact hyn_norm", "exact hkmz", "have hyn1_norm : exists u v. (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + n * u = (b + b * (m * x4)) + n * v", "have hyn1_zero : (b + b * (m * x4)) + 0 = b + b * (m * x4)", "rewrite PA3", "refl", "rewrite <- hyn1_zero", "exact hyn1", "have hkn : exists u v. (x1 * (b * (m * x4))) + n * u = (x1 * (b * (m * x4))) + n * v", "specialize mod_eq_refl n", "specialize mod_eq_refl (x1 * (b * (m * x4)))", "apply mod_eq_refl", "have hynk : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + n * u = ((b + b * (m * x4)) + (x1 * (b * (m * x4)))) + n * v", "specialize mod_eq_add n", "specialize mod_eq_add (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5))))", "specialize mod_eq_add (b + b * (m * x4))", "specialize mod_eq_add (x1 * (b * (m * x4)))", "specialize mod_eq_add (x1 * (b * (m * x4)))", "apply mod_eq_add", "exact hyn1_norm", "exact hkn", "have hcanceln : exists u v. ((b + b * (m * x4)) + x1 * (b * (m * x4))) + S x1 * u = b + S x1 * v", "specialize mod_eq_predecessor_cancel x1", "specialize mod_eq_predecessor_cancel b", "specialize mod_eq_predecessor_cancel (b * (m * x4))", "apply mod_eq_predecessor_cancel", "rewrite <- hns_witness at hcanceln", "rewrite <- hns_witness at hcanceln", "have hfinaln : exists u v. ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))) + n * u = b + n * v", "specialize mod_eq_trans n", "specialize mod_eq_trans ((((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4))))", "specialize mod_eq_trans ((b + b * (m * x4)) + (x1 * (b * (m * x4))))", "specialize mod_eq_trans b", "apply mod_eq_trans", "exact hynk", "exact hcanceln", "exists (((a * (n * x3)) + (b * (m * x2))) + (x * (a * (n * x5)))) + (x1 * (b * (m * x4)))", "split", "exact hfinalm", "exact hfinaln" ], "script_sha256": "8db963d420b0ce840011a723d4905fbac814e1af3bc8678bf51658d0a163bd64", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall m n a b. ~(m = 0) -> ~(n = 0) -> (forall d. (exists u. m = d * u) -> (exists v. n = d * v) -> d = 1) -> exists x. (exists u v. x + m * u = a + m * v) /\\ (exists r s. x + n * r = b + n * s)", "statement_sha256": "25e1f5213a9a5b04f3a20077b936ba8814f6ae8951f4f72e5f5e2135d7c9148f" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_modulus_coprime_base", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "divides_remainder", "divisor_one", "mul_comm" ], "dependencies_sha256": "c8cda3defb9d15e5d00ae7b28d453cd7d68ae9b2b434453cbe6dc782f3b8df87", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "f7e3c800ecbe062e709ee0d115860418d316389e361f5b26c5e460340ed6ce42", "cut_nodes": 24, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 30, "proof_edges": 683, "proof_nodes": 874, "proof_objects": 630, "reused_objects": 54, "status": "checked" }, "enrollment_index": 158, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_modulus_coprime_base]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "387def9eab4333dcc4a105953f9a506c9c9afc51f3039fa4c7bbfc0225b5bfe1", "membership": "stable", "name": "beta_modulus_coprime_base", "proof_tag": "PA0015", "provenance": [ "stable" ], "script": [ "intro c", "intro k", "intro d", "intro hm", "intro hc", "have hstep : S (k * c) = c * k + 1", "simp [mul_comm]", "have h1 : exists w. 1 = d * w", "specialize divides_remainder d", "specialize divides_remainder (S (k * c))", "specialize divides_remainder c", "specialize divides_remainder k", "specialize divides_remainder 1", "apply divides_remainder", "exact hm", "exact hc", "exact hstep", "specialize divisor_one d", "apply divisor_one", "exact h1" ], "script_sha256": "c06c2d8fae1638fd3d937cdb3a50ac1fce3b9b323c40077beae8ba12b29638f0", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall c k d. (exists u. S (k * c) = d * u) -> (exists v. c = d * v) -> d = 1", "statement_sha256": "e3888d0932249958689f3ec9e8d191cf866212c101dabfe5b62c198c39c4d373", "summary": "Every beta-shaped successor modulus is coprime to its base c.", "summary_sha256": "e413e55e37f6e9bdf11382f314b2dbf49788f79f22c8898730012da9ec8c22fb" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "divides_remainder", "divisor_one", "mul_comm" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_modulus_coprime_base]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_modulus_coprime_base", "parent_alpha_version": "v34", "proof_bundle_node_id": 123, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_modulus_coprime_base", "script": [ "intro c", "intro k", "intro d", "intro hm", "intro hc", "have hstep : S (k * c) = c * k + 1", "simp [mul_comm]", "have h1 : exists w. 1 = d * w", "specialize divides_remainder d", "specialize divides_remainder (S (k * c))", "specialize divides_remainder c", "specialize divides_remainder k", "specialize divides_remainder 1", "apply divides_remainder", "exact hm", "exact hc", "exact hstep", "specialize divisor_one d", "apply divisor_one", "exact h1" ], "script_sha256": "c06c2d8fae1638fd3d937cdb3a50ac1fce3b9b323c40077beae8ba12b29638f0", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall c k d. (exists u. S (k * c) = d * u) -> (exists v. c = d * v) -> d = 1", "statement_sha256": "e3888d0932249958689f3ec9e8d191cf866212c101dabfe5b62c198c39c4d373" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "common_divisor_beta_moduli_divides_gap_times_c", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "divides_remainder", "add_succ_left", "add_mul", "zero_add" ], "dependencies_sha256": "d656b601e271d5cfaafff54a5e37becd9db2d1e69acd28926fc68ba2656f2733", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "42a60bc7d8feda4822b114ce9a25a29d27d03a2ab6b9cb2e8a078f4942a8862a", "cut_nodes": 24, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 30, "proof_edges": 584, "proof_nodes": 855, "proof_objects": 540, "reused_objects": 45, "status": "checked" }, "enrollment_index": 159, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=common_divisor_beta_moduli_divides_gap_times_c]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "012a4fed3e23ebf05d389a3c9bf12721f0a37190352a56e58f9988e43a23d296", "membership": "stable", "name": "common_divisor_beta_moduli_divides_gap_times_c", "proof_tag": "PA0017", "provenance": [ "stable" ], "script": [ "intro c", "intro i", "intro j", "intro gap", "intro d", "intro hij", "intro hmi", "intro hmj", "have hstep : S ((S j) * c) = S ((S i) * c) * 1 + gap * c", "rewrite hij", "specialize add_succ_left i", "specialize add_succ_left gap", "rewrite <- add_succ_left", "simp [add_mul, zero_add]", "symm", "specialize add_succ_left_before (S i * c)", "specialize add_succ_left_before (gap * c)", "exact add_succ_left_before", "specialize divides_remainder d", "specialize divides_remainder (S ((S j) * c))", "specialize divides_remainder (S ((S i) * c))", "specialize divides_remainder 1", "specialize divides_remainder (gap * c)", "apply divides_remainder", "exact hmj", "exact hmi", "exact hstep" ], "script_sha256": "3785a14aeb21f7acb5863a027c6d61593462e88604049e613f78e241e060fa95", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall c i j gap d. j = i + gap -> (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> exists w. gap * c = d * w", "statement_sha256": "5e42d3c08bd8cc965eac53bfd7e6abfdf30d1a27ee43f0d38ab3febc24389b0f", "summary": "A common divisor of two ordered beta moduli divides the index gap times c.", "summary_sha256": "31f426295aeda56866468a91f331cf33ce816eac98e3555e6740a509c7a61ee7" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "divides_remainder", "add_succ_left", "add_mul", "zero_add" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=common_divisor_beta_moduli_divides_gap_times_c]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "common_divisor_beta_moduli_divides_gap_times_c", "parent_alpha_version": "v34", "proof_bundle_node_id": 124, "reference_route": "jordan-totient/checkpoint.html#theorem-common_divisor_beta_moduli_divides_gap_times_c", "script": [ "intro c", "intro i", "intro j", "intro gap", "intro d", "intro hij", "intro hmi", "intro hmj", "have hstep : S ((S j) * c) = S ((S i) * c) * 1 + gap * c", "rewrite hij", "specialize add_succ_left i", "specialize add_succ_left gap", "rewrite <- add_succ_left", "simp [add_mul, zero_add]", "symm", "specialize add_succ_left_before (S i * c)", "specialize add_succ_left_before (gap * c)", "exact add_succ_left_before", "specialize divides_remainder d", "specialize divides_remainder (S ((S j) * c))", "specialize divides_remainder (S ((S i) * c))", "specialize divides_remainder 1", "specialize divides_remainder (gap * c)", "apply divides_remainder", "exact hmj", "exact hmi", "exact hstep" ], "script_sha256": "3785a14aeb21f7acb5863a027c6d61593462e88604049e613f78e241e060fa95", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall c i j gap d. j = i + gap -> (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> exists w. gap * c = d * w", "statement_sha256": "5e42d3c08bd8cc965eac53bfd7e6abfdf30d1a27ee43f0d38ab3febc24389b0f" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_moduli_coprime_of_gap_dvd", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_modulus_coprime_base", "common_divisor_beta_moduli_divides_gap_times_c", "multiple_trans", "multiple_refl", "gauss_coprime_cancel", "mul_comm" ], "dependencies_sha256": "c09f4780969e5986ce0525f5f28640640082bab769b8828a149461a920c2f2f2", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "e31f87a1af34325c3566489d1e3ffe0220e721312a89d0f9a62ea1ea88cf3fe8", "cut_nodes": 175, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 56, "proof_edges": 1921, "proof_nodes": 6007, "proof_objects": 1803, "reused_objects": 119, "status": "checked" }, "enrollment_index": 160, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_moduli_coprime_of_gap_dvd]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "efa7e97212f702717f4e327a9c19e19df60ad2a500749676e2b88f3fd0488304", "membership": "stable", "name": "beta_moduli_coprime_of_gap_dvd", "proof_tag": "PA001Q", "provenance": [ "stable" ], "script": [ "intro c", "intro i", "intro j", "intro gap", "intro hij", "intro hgapc", "intro d", "intro hmi", "intro hmj", "have hcopdc : forall e. (exists u. d = e * u) -> (exists v. c = e * v) -> e = 1", "intro e", "intro hed", "intro hec", "have hmei : exists u. S ((S i) * c) = e * u", "specialize multiple_trans d", "specialize multiple_trans e", "specialize multiple_trans (S ((S i) * c))", "apply multiple_trans", "exact hmi", "exact hed", "specialize beta_modulus_coprime_base c", "specialize beta_modulus_coprime_base (S i)", "specialize beta_modulus_coprime_base e", "apply beta_modulus_coprime_base", "exact hmei", "exact hec", "have hgapprod : exists w. gap * c = d * w", "specialize common_divisor_beta_moduli_divides_gap_times_c c", "specialize common_divisor_beta_moduli_divides_gap_times_c i", "specialize common_divisor_beta_moduli_divides_gap_times_c j", "specialize common_divisor_beta_moduli_divides_gap_times_c gap", "specialize common_divisor_beta_moduli_divides_gap_times_c d", "apply common_divisor_beta_moduli_divides_gap_times_c", "exact hij", "exact hmi", "exact hmj", "cases hgapprod", "have hdivgap : exists w. gap = d * w", "specialize gauss_coprime_cancel d", "specialize gauss_coprime_cancel c", "specialize gauss_coprime_cancel gap", "apply gauss_coprime_cancel", "exact hcopdc", "exists x", "trans gap * c", "apply mul_comm", "exact hgapprod_witness", "have hdc : exists w. c = d * w", "specialize multiple_trans gap", "specialize multiple_trans d", "specialize multiple_trans c", "apply multiple_trans", "exact hgapc", "exact hdivgap", "specialize hcopdc d", "apply hcopdc", "specialize multiple_refl d", "exact multiple_refl", "exact hdc" ], "script_sha256": "915515f9d65c5b2736e44064366b3e6092c89038b21c83327bbeec9a44791022", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall c i j gap. j = i + gap -> (exists k. c = gap * k) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "statement_sha256": "df6c3daf567bc06a5ca361d2b1cfc976d2ec1bca03f3b8dd5e24dba8b0070602", "summary": "Beta moduli at an additive index gap dividing c are coprime.", "summary_sha256": "5c91cd2379b68f54cac837583bc3f1a19883116d920e6d56a1d23c027ce57396" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_modulus_coprime_base", "common_divisor_beta_moduli_divides_gap_times_c", "multiple_trans", "multiple_refl", "gauss_coprime_cancel", "mul_comm" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_moduli_coprime_of_gap_dvd]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_moduli_coprime_of_gap_dvd", "parent_alpha_version": "v34", "proof_bundle_node_id": 125, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_moduli_coprime_of_gap_dvd", "script": [ "intro c", "intro i", "intro j", "intro gap", "intro hij", "intro hgapc", "intro d", "intro hmi", "intro hmj", "have hcopdc : forall e. (exists u. d = e * u) -> (exists v. c = e * v) -> e = 1", "intro e", "intro hed", "intro hec", "have hmei : exists u. S ((S i) * c) = e * u", "specialize multiple_trans d", "specialize multiple_trans e", "specialize multiple_trans (S ((S i) * c))", "apply multiple_trans", "exact hmi", "exact hed", "specialize beta_modulus_coprime_base c", "specialize beta_modulus_coprime_base (S i)", "specialize beta_modulus_coprime_base e", "apply beta_modulus_coprime_base", "exact hmei", "exact hec", "have hgapprod : exists w. gap * c = d * w", "specialize common_divisor_beta_moduli_divides_gap_times_c c", "specialize common_divisor_beta_moduli_divides_gap_times_c i", "specialize common_divisor_beta_moduli_divides_gap_times_c j", "specialize common_divisor_beta_moduli_divides_gap_times_c gap", "specialize common_divisor_beta_moduli_divides_gap_times_c d", "apply common_divisor_beta_moduli_divides_gap_times_c", "exact hij", "exact hmi", "exact hmj", "cases hgapprod", "have hdivgap : exists w. gap = d * w", "specialize gauss_coprime_cancel d", "specialize gauss_coprime_cancel c", "specialize gauss_coprime_cancel gap", "apply gauss_coprime_cancel", "exact hcopdc", "exists x", "trans gap * c", "apply mul_comm", "exact hgapprod_witness", "have hdc : exists w. c = d * w", "specialize multiple_trans gap", "specialize multiple_trans d", "specialize multiple_trans c", "apply multiple_trans", "exact hgapc", "exact hdivgap", "specialize hcopdc d", "apply hcopdc", "specialize multiple_refl d", "exact multiple_refl", "exact hdc" ], "script_sha256": "915515f9d65c5b2736e44064366b3e6092c89038b21c83327bbeec9a44791022", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall c i j gap. j = i + gap -> (exists k. c = gap * k) -> forall d. 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S t + S h = S B) -> exists k. c = S t * k) -> (exists g. g + S i = j) -> (exists h. h + j = B) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. 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S ((S j) * c) = d * v) -> d = 1", "statement_sha256": "eb5bc880e1349d365cef2c73bd1b3412bac10615d2c14180fddbbb2f54dbcc2c", "summary": "Distinct indices in a bounded prefix have pairwise coprime beta moduli under a bounded common-multiple invariant.", "summary_sha256": "a3c85f2a33b3b1d000a4aef9e536c0941577132ec99f899dfbebe24daea99288" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "lt_trichotomy", "beta_moduli_coprime_of_lt_bounded_common_multiple" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_moduli_pairwise_coprime_bounded]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_moduli_pairwise_coprime_bounded", "parent_alpha_version": "v34", "proof_bundle_node_id": 129, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_moduli_pairwise_coprime_bounded", "script": [ "intro B", "intro c", "intro hcm", "intro i", "intro j", "intro hne", "intro hiB", "intro hjB", "intro d", "intro hdi", "intro hdj", "specialize lt_trichotomy i", "specialize lt_trichotomy j", "cases lt_trichotomy", "exfalso", "apply hne", "exact lt_trichotomy_left", "cases lt_trichotomy_right", "have hcopij : forall e. 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(exists u. d = e * u) -> (exists v. a = e * v) -> e = 1", "intro e", "intro hed", "intro hea", "have hen : exists q. n = e * q", "specialize multiple_trans d", "specialize multiple_trans e", "specialize multiple_trans n", "apply multiple_trans", "exact hdn", "exact hed", "specialize han e", "apply han", "exact hea", "exact hen", "have hdb : exists w. b = d * w", "specialize gauss_coprime_cancel d", "specialize gauss_coprime_cancel a", "specialize gauss_coprime_cancel b", "apply gauss_coprime_cancel", "exact hda", "exact hab", "specialize hbn d", "apply hbn", "exact hdb", "exact hdn" ], "script_sha256": "ecbffa2857d908c70ee2f439f0e15fb4484ffc4b8369ff4d56ff3c6b379a172f", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a b n. (forall d. (exists x. a = d * x) -> (exists y. n = d * y) -> d = 1) -> (forall d. (exists x. b = d * x) -> (exists y. n = d * y) -> d = 1) -> forall d. 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(exists k. P = m * k) -> (exists u v. x + P * u = a + P * v) -> exists r s. x + m * r = a + m * s", "statement_sha256": "246654070ff9a4577e17991d9189766a93fe7c11629ec2303a8ad111c86451e3", "summary": "Balanced congruence descends from a multiple modulus to every divisor modulus.", "summary_sha256": "fb3d86c67eb303083fbbf32b3bba31204db99473ff3fa21c944dd03e3b154b6a" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "mul_assoc" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=mod_eq_of_mod_eq_multiple]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "mod_eq_of_mod_eq_multiple", "parent_alpha_version": "v34", "proof_bundle_node_id": 131, "reference_route": "jordan-totient/checkpoint.html#theorem-mod_eq_of_mod_eq_multiple", "script": [ "intro m", "intro P", "intro x", "intro a", "intro hdiv", "intro hmod", "cases hdiv", "cases hmod", "cases hmod_witness", "rewrite hdiv_witness at hmod_witness_witness", "rewrite hdiv_witness at hmod_witness_witness", "exists x1 * x2", "exists x1 * x3", "trans x + (m * x1) * x2", "congr", "refl", "symm", "apply mul_assoc", "trans a + (m * x1) * x3", "exact hmod_witness_witness", "congr", "refl", "apply mul_assoc" ], "script_sha256": "1bdf5200eb5b8a88bc4374ac4122d5bc0476b0f1eb43d5042df80825aeac4304", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall m P x a. 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C * B = S t * q", "statement_sha256": "a514ad09db5eae76cb17193a87016a905895f8450a3f9bb35dc77ef00b299945" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_value_lt_scaled_base", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_value_le_code", "le_add_right", "succ_le_succ", "le_trans", "le_scaled_nonzero", "base_le_beta_modulus" ], "dependencies_sha256": "149c2f8ceab378b1685fb8841216faad3f768e765bf8bbe66d41c9a725e9c5e7", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "05aeb5bdd1f80ada1023ab728c03df39a8064475d8bfd5bdcdfff30bf6a8fb27", "cut_nodes": 31, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 33, "proof_edges": 529, "proof_nodes": 863, "proof_objects": 489, "reused_objects": 41, "status": "checked" }, "enrollment_index": 182, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_value_lt_scaled_base]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "4a5a82db6aeb668fc8a3a9a9b29e31e396e99cca1481a4cca6fd1de28358c920", "membership": "stable", "name": "beta_value_lt_scaled_base", "proof_tag": "PA002T", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro i", "intro x", "intro C", "intro s", "intro j", "intro hat", "intro hC", "have hxb : exists h. h + x = b", "specialize beta_value_le_code b", "specialize beta_value_le_code c", "specialize beta_value_le_code i", "specialize beta_value_le_code x", "apply beta_value_le_code", "exact hat", "have hbs : exists h. h + b = b + s", "specialize le_add_right b", "specialize le_add_right s", "exact le_add_right", "have hxs : exists h. h + x = b + s", "specialize le_trans x", "specialize le_trans b", "specialize le_trans (b + s)", "apply le_trans", "exact hxb", "exact hbs", "have hsx : exists h. h + S x = S (b + s)", "specialize succ_le_succ x", "specialize succ_le_succ (b + s)", "apply succ_le_succ", "exact hxs", "have hscale : exists h. h + S (b + s) = C * S (b + s)", "specialize le_scaled_nonzero C", "specialize le_scaled_nonzero (S (b + s))", "apply le_scaled_nonzero", "exact hC", "have hxbase : exists h. h + S x = C * S (b + s)", "specialize le_trans (S x)", "specialize le_trans (S (b + s))", "specialize le_trans (C * S (b + s))", "apply le_trans", "exact hsx", "exact hscale", "have hmod : exists h. h + C * S (b + s) = S ((S j) * (C * S (b + s)))", "specialize base_le_beta_modulus (C * S (b + s))", "specialize base_le_beta_modulus j", "exact base_le_beta_modulus", "specialize le_trans (S x)", "specialize le_trans (C * S (b + s))", "specialize le_trans (S ((S j) * (C * S (b + s))))", "apply le_trans", "exact hxbase", "exact hmod" ], "script_sha256": "a9b306e419d340966210ecb9bb07c2899f3f347d0a9fe95093d57d8b657a0db5", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c i x C s j. ((exists h. h + S x = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + x) -> ~(C = 0) -> exists h. h + S x = S ((S j) * (C * S (b + s)))", "statement_sha256": "723e34f7303079ad1d5510dcde76756e36f758aed266f6bf8bd225aa8b568268", "summary": "An old beta value fits every modulus after a constructive scaled-base rebase.", "summary_sha256": "2e243b2a2c08af6dfee1cc0940d4316dfa977b279f9babb1af0de7e0c1cac861" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_value_le_code", "le_add_right", "succ_le_succ", "le_trans", "le_scaled_nonzero", "base_le_beta_modulus" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_value_lt_scaled_base]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_value_lt_scaled_base", "parent_alpha_version": "v34", "proof_bundle_node_id": 138, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_value_lt_scaled_base", "script": [ "intro b", "intro c", "intro i", "intro x", "intro C", "intro s", "intro j", "intro hat", "intro hC", "have hxb : exists h. h + x = b", "specialize beta_value_le_code b", "specialize beta_value_le_code c", "specialize beta_value_le_code i", "specialize beta_value_le_code x", "apply beta_value_le_code", "exact hat", "have hbs : exists h. h + b = b + s", "specialize le_add_right b", "specialize le_add_right s", "exact le_add_right", "have hxs : exists h. h + x = b + s", "specialize le_trans x", "specialize le_trans b", "specialize le_trans (b + s)", "apply le_trans", "exact hxb", "exact hbs", "have hsx : exists h. h + S x = S (b + s)", "specialize succ_le_succ x", "specialize succ_le_succ (b + s)", "apply succ_le_succ", "exact hxs", "have hscale : exists h. h + S (b + s) = C * S (b + s)", "specialize le_scaled_nonzero C", "specialize le_scaled_nonzero (S (b + s))", "apply le_scaled_nonzero", "exact hC", "have hxbase : exists h. h + S x = C * S (b + s)", "specialize le_trans (S x)", "specialize le_trans (S (b + s))", "specialize le_trans (C * S (b + s))", "apply le_trans", "exact hsx", "exact hscale", "have hmod : exists h. h + C * S (b + s) = S ((S j) * (C * S (b + s)))", "specialize base_le_beta_modulus (C * S (b + s))", "specialize base_le_beta_modulus j", "exact base_le_beta_modulus", "specialize le_trans (S x)", "specialize le_trans (C * S (b + s))", "specialize le_trans (S ((S j) * (C * S (b + s))))", "apply le_trans", "exact hxbase", "exact hmod" ], "script_sha256": "a9b306e419d340966210ecb9bb07c2899f3f347d0a9fe95093d57d8b657a0db5", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c i x C s j. ((exists h. h + S x = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + x) -> ~(C = 0) -> exists h. h + S x = S ((S j) * (C * S (b + s)))", "statement_sha256": "723e34f7303079ad1d5510dcde76756e36f758aed266f6bf8bd225aa8b568268" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "new_value_lt_scaled_base", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "le_add_left", "succ_le_succ", "le_scaled_nonzero", "le_trans", "base_le_beta_modulus" ], "dependencies_sha256": "68f9377def64b94dfb087ceac279f2eb554e9e4064070fc1e4209ce37140b3fa", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "08fcf3a5d804a316fce0d656a44e4eaeb13cc336a455e351628d2f31ca1387b5", "cut_nodes": 27, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 31, "proof_edges": 486, "proof_nodes": 751, "proof_objects": 446, "reused_objects": 41, "status": "checked" }, "enrollment_index": 183, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=new_value_lt_scaled_base]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "68901b04bbd78020a7769a26f0bb56297418b4350722535e7bfb2c46993099ac", "membership": "stable", "name": "new_value_lt_scaled_base", "proof_tag": "PA002Q", "provenance": [ "stable" ], "script": [ "intro b", "intro s", "intro C", "intro j", "intro hC", "have hsb : exists h. h + s = b + s", "specialize le_add_left s", "specialize le_add_left b", "exact le_add_left", "have hss : exists h. h + S s = S (b + s)", "specialize succ_le_succ s", "specialize succ_le_succ (b + s)", "apply succ_le_succ", "exact hsb", "have hscale : exists h. h + S (b + s) = C * S (b + s)", "specialize le_scaled_nonzero C", "specialize le_scaled_nonzero (S (b + s))", "apply le_scaled_nonzero", "exact hC", "have hsbase : exists h. h + S s = C * S (b + s)", "specialize le_trans (S s)", "specialize le_trans (S (b + s))", "specialize le_trans (C * S (b + s))", "apply le_trans", "exact hss", "exact hscale", "have hmod : exists h. h + C * S (b + s) = S ((S j) * (C * S (b + s)))", "specialize base_le_beta_modulus (C * S (b + s))", "specialize base_le_beta_modulus j", "exact base_le_beta_modulus", "specialize le_trans (S s)", "specialize le_trans (C * S (b + s))", "specialize le_trans (S ((S j) * (C * S (b + s))))", "apply le_trans", "exact hsbase", "exact hmod" ], "script_sha256": "48c5e547619038dcfeb6edcf94db7fc3a494b691ec24b8001586703aea561567", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b s C j. ~(C = 0) -> exists h. h + S s = S ((S j) * (C * S (b + s)))", "statement_sha256": "17864693ef24007d91e415865da8a8623b72bb3e2c4aae995e72665c9929fb8c", "summary": "The appended value fits every modulus after the same constructive scaled-base rebase.", "summary_sha256": "3dda7a2c956047a316a352a6c7203f83c76b7cdd57a6a119d74c33593fee1290" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "le_add_left", "succ_le_succ", "le_scaled_nonzero", "le_trans", "base_le_beta_modulus" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=new_value_lt_scaled_base]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "new_value_lt_scaled_base", "parent_alpha_version": "v34", "proof_bundle_node_id": 139, "reference_route": "jordan-totient/checkpoint.html#theorem-new_value_lt_scaled_base", "script": [ "intro b", "intro s", "intro C", "intro j", "intro hC", "have hsb : exists h. h + s = b + s", "specialize le_add_left s", "specialize le_add_left b", "exact le_add_left", "have hss : exists h. h + S s = S (b + s)", "specialize succ_le_succ s", "specialize succ_le_succ (b + s)", "apply succ_le_succ", "exact hsb", "have hscale : exists h. h + S (b + s) = C * S (b + s)", "specialize le_scaled_nonzero C", "specialize le_scaled_nonzero (S (b + s))", "apply le_scaled_nonzero", "exact hC", "have hsbase : exists h. h + S s = C * S (b + s)", "specialize le_trans (S s)", "specialize le_trans (S (b + s))", "specialize le_trans (C * S (b + s))", "apply le_trans", "exact hss", "exact hscale", "have hmod : exists h. h + C * S (b + s) = S ((S j) * (C * S (b + s)))", "specialize base_le_beta_modulus (C * S (b + s))", "specialize base_le_beta_modulus j", "exact base_le_beta_modulus", "specialize le_trans (S s)", "specialize le_trans (C * S (b + s))", "specialize le_trans (S ((S j) * (C * S (b + s))))", "apply le_trans", "exact hsbase", "exact hmod" ], "script_sha256": "48c5e547619038dcfeb6edcf94db7fc3a494b691ec24b8001586703aea561567", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b s C j. ~(C = 0) -> exists h. h + S s = S ((S j) * (C * S (b + s)))", "statement_sha256": "17864693ef24007d91e415865da8a8623b72bb3e2c4aae995e72665c9929fb8c" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_exclusive_accumulated_product_step", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "mul_ne_zero", "right_factor_divides_product", "beta_modulus_nonzero", "le_of_succ_le_succ", "le_eq_or_lt", "multiple_mul_right", "le_succ_self", "le_trans", "lt_to_le", "lt_irrefl_expanded", "beta_moduli_pairwise_coprime_bounded", "coprime_mul_left" ], "dependencies_sha256": "1b9e0ac757e006980e32dbb42262f7b2f0545b6128c6c63c294b6bda9fba905d", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "1833bd103225a199bd1eb410fe3869a3963f49335c33a6b2462f7157f156cd58", "cut_nodes": 332, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 70, "proof_edges": 2481, "proof_nodes": 11222, "proof_objects": 2345, "reused_objects": 137, "status": "checked" }, "enrollment_index": 184, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_exclusive_accumulated_product_step]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "d317c4fe6e0afad919787b5253c4264d7b11696dc31d9dcd6d4da18ba91ab4ac", "membership": "stable", "name": "beta_exclusive_accumulated_product_step", "proof_tag": "PA001U", "provenance": [ "stable" ], "script": [ "intro N", "intro c", "intro k", "intro P", "intro hcm", "intro hkN", "intro hP", "intro hdiv", "intro hfuture", "have hnew : ~(S ((S k) * c) = 0)", "specialize beta_modulus_nonzero c", "specialize beta_modulus_nonzero k", "exact beta_modulus_nonzero", "split", "specialize mul_ne_zero P", "specialize mul_ne_zero (S ((S k) * c))", "intro hzero", "apply mul_ne_zero", "exact hP", "exact hnew", "exact hzero", "split", "intro i", "intro hi", "have hik : exists r. r + i = k", "specialize le_of_succ_le_succ i", "specialize le_of_succ_le_succ k", "apply le_of_succ_le_succ", "exact hi", "have hsplit : i = k \\/ exists r. r + S i = k", "specialize le_eq_or_lt i", "specialize le_eq_or_lt k", "apply le_eq_or_lt", "exact hik", "cases hsplit", "rewrite hsplit_left", "specialize right_factor_divides_product P", "specialize right_factor_divides_product (S ((S k) * c))", "exact right_factor_divides_product", "have hiP : exists q. P = S ((S i) * c) * q", "specialize hdiv i", "apply hdiv", "exact hsplit_right", "specialize multiple_mul_right (S ((S i) * c))", "specialize multiple_mul_right P", "specialize multiple_mul_right (S ((S k) * c))", "apply multiple_mul_right", "exact hiP", "intro j", "intro hSkj", "intro hjN", "have hkj : exists r. r + k = j", "have hkSk : exists r. r + k = S k", "specialize le_succ_self k", "exact le_succ_self", "specialize le_trans k", "specialize le_trans (S k)", "specialize le_trans j", "apply le_trans", "exact hkSk", "exact hSkj", "have hPj : forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "specialize hfuture j", "apply hfuture", "exact hkj", "exact hjN", "have hneq : ~(k = j)", "intro heq", "rewrite <- heq at hSkj", "specialize lt_irrefl_expanded k", "apply lt_irrefl_expanded", "exact hSkj", "have hkbound : exists r. r + k = N", "specialize lt_to_le k", "specialize lt_to_le N", "apply lt_to_le", "exact hkN", "have hpairs : forall i j. ~(i = j) -> (exists hi. hi + i = N) -> (exists hj. hj + j = N) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "specialize beta_moduli_pairwise_coprime_bounded N", "specialize beta_moduli_pairwise_coprime_bounded c", "apply beta_moduli_pairwise_coprime_bounded", "exact hcm", "have hnewj : forall d. (exists u. S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "specialize hpairs k", "specialize hpairs j", "apply hpairs", "exact hneq", "exact hkbound", "exact hjN", "specialize coprime_mul_left P", "specialize coprime_mul_left (S ((S k) * c))", "specialize coprime_mul_left (S ((S j) * c))", "apply coprime_mul_left", "exact hPj", "exact hnewj" ], "script_sha256": "7c45f8dda3f0828c3a5dee3520c59fa387c8371a4fbf982f28d76ba215897059", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))", "statement_sha256": "623df71bb00a1df01ed2b113ce65eb8e8c1b5ae56fd4a92a05861cd2b914b2bb", "summary": "Extend the accumulated target-modulus product for an exclusive prefix.", "summary_sha256": "069b92f72c53ac828bba187cd25c863c06a380c81cfcbe46a27a85e9b20c50fc" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "mul_ne_zero", "right_factor_divides_product", "beta_modulus_nonzero", "le_of_succ_le_succ", "le_eq_or_lt", "multiple_mul_right", "le_succ_self", "le_trans", "lt_to_le", "lt_irrefl_expanded", "beta_moduli_pairwise_coprime_bounded", "coprime_mul_left" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_exclusive_accumulated_product_step]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_exclusive_accumulated_product_step", "parent_alpha_version": "v34", "proof_bundle_node_id": 140, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_exclusive_accumulated_product_step", "script": [ "intro N", "intro c", "intro k", "intro P", "intro hcm", "intro hkN", "intro hP", "intro hdiv", "intro hfuture", "have hnew : ~(S ((S k) * c) = 0)", "specialize beta_modulus_nonzero c", "specialize beta_modulus_nonzero k", "exact beta_modulus_nonzero", "split", "specialize mul_ne_zero P", "specialize mul_ne_zero (S ((S k) * c))", "intro hzero", "apply mul_ne_zero", "exact hP", "exact hnew", "exact hzero", "split", "intro i", "intro hi", "have hik : exists r. r + i = k", "specialize le_of_succ_le_succ i", "specialize le_of_succ_le_succ k", "apply le_of_succ_le_succ", "exact hi", "have hsplit : i = k \\/ exists r. r + S i = k", "specialize le_eq_or_lt i", "specialize le_eq_or_lt k", "apply le_eq_or_lt", "exact hik", "cases hsplit", "rewrite hsplit_left", "specialize right_factor_divides_product P", "specialize right_factor_divides_product (S ((S k) * c))", "exact right_factor_divides_product", "have hiP : exists q. P = S ((S i) * c) * q", "specialize hdiv i", "apply hdiv", "exact hsplit_right", "specialize multiple_mul_right (S ((S i) * c))", "specialize multiple_mul_right P", "specialize multiple_mul_right (S ((S k) * c))", "apply multiple_mul_right", "exact hiP", "intro j", "intro hSkj", "intro hjN", "have hkj : exists r. r + k = j", "have hkSk : exists r. r + k = S k", "specialize le_succ_self k", "exact le_succ_self", "specialize le_trans k", "specialize le_trans (S k)", "specialize le_trans j", "apply le_trans", "exact hkSk", "exact hSkj", "have hPj : forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "specialize hfuture j", "apply hfuture", "exact hkj", "exact hjN", "have hneq : ~(k = j)", "intro heq", "rewrite <- heq at hSkj", "specialize lt_irrefl_expanded k", "apply lt_irrefl_expanded", "exact hSkj", "have hkbound : exists r. r + k = N", "specialize lt_to_le k", "specialize lt_to_le N", "apply lt_to_le", "exact hkN", "have hpairs : forall i j. ~(i = j) -> (exists hi. hi + i = N) -> (exists hj. hj + j = N) -> forall d. (exists u. S ((S i) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "specialize beta_moduli_pairwise_coprime_bounded N", "specialize beta_moduli_pairwise_coprime_bounded c", "apply beta_moduli_pairwise_coprime_bounded", "exact hcm", "have hnewj : forall d. (exists u. S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1", "specialize hpairs k", "specialize hpairs j", "apply hpairs", "exact hneq", "exact hkbound", "exact hjN", "specialize coprime_mul_left P", "specialize coprime_mul_left (S ((S k) * c))", "specialize coprime_mul_left (S ((S j) * c))", "apply coprime_mul_left", "exact hPj", "exact hnewj" ], "script_sha256": "7c45f8dda3f0828c3a5dee3520c59fa387c8371a4fbf982f28d76ba215897059", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall N c k P. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> (~(P * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))", "statement_sha256": "623df71bb00a1df01ed2b113ce65eb8e8c1b5ae56fd4a92a05861cd2b914b2bb" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_exclusive_recode_congruence_step", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_modulus_nonzero", "le_refl", "lt_to_le", "binary_crt_fold_step", "beta_at_exists", "beta_at_unique", "le_of_succ_le_succ", "le_eq_or_lt" ], "dependencies_sha256": "89ed14214f9bc23ee1b5b7a7fb80e09370fefd17d93bb4c8ad5536ba5f499f2e", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "282847b6bbbacfe5a6ff683a06c017ee5e7f1b4d77d8d9ae8b28cb35c9d0abd6", "cut_nodes": 215, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 65, "proof_edges": 2712, "proof_nodes": 7398, "proof_objects": 2583, "reused_objects": 130, "status": "checked" }, "enrollment_index": 185, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_exclusive_recode_congruence_step]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "32263e26c21d463a3914ff6a06834fc99adeab04e3e6c9de084087a56d67ca60", "membership": "stable", "name": "beta_exclusive_recode_congruence_step", "proof_tag": "PA002G", "provenance": [ "stable" ], "script": [ "intro N", "intro c", "intro b", "intro e", "intro k", "intro P", "intro z", "intro hkN", "intro hP", "intro hdiv", "intro hcong", "intro hfuture", "have hnew : ~(S ((S k) * c) = 0)", "specialize beta_modulus_nonzero c", "specialize beta_modulus_nonzero k", "exact beta_modulus_nonzero", "have hkbound : exists h. h + k = N", "specialize lt_to_le k", "specialize lt_to_le N", "apply lt_to_le", "exact hkN", "have hcop : forall d. (exists u. P = d * u) -> (exists v. S ((S k) * c) = d * v) -> d = 1", "specialize hfuture k", "apply hfuture", "specialize le_refl k", "exact le_refl", "exact hkbound", "have hvalue : exists a. ((exists h. h + S a = S ((S k) * e)) /\\ exists q. b = q * S ((S k) * e) + a)", "specialize beta_at_exists b", "specialize beta_at_exists e", "specialize beta_at_exists k", "exact beta_at_exists", "cases hvalue", "have hfold : exists z2. ((forall m a. (exists q. P = m * q) -> (exists u v. z + m * u = a + m * v) -> exists r s. z2 + m * r = a + m * s) /\\ exists q r. z2 + S ((S k) * c) * q = x + S ((S k) * c) * r)", "specialize binary_crt_fold_step P", "specialize binary_crt_fold_step (S ((S k) * c))", "specialize binary_crt_fold_step z", "specialize binary_crt_fold_step x", "apply binary_crt_fold_step", "exact hP", "exact hnew", "exact hcop", "cases hfold", "cases hfold_witness", "exists x1", "intro i", "intro a", "intro hi", "intro hati", "have hik : exists r. r + i = k", "specialize le_of_succ_le_succ i", "specialize le_of_succ_le_succ k", "apply le_of_succ_le_succ", "exact hi", "have hsplit : i = k \\/ exists r. r + S i = k", "specialize le_eq_or_lt i", "specialize le_eq_or_lt k", "apply le_eq_or_lt", "exact hik", "cases hsplit", "have hati_new : ((exists h. h + S a = S ((S k) * e)) /\\ exists q. b = q * S ((S k) * e) + a)", "rewrite <- hsplit_left", "rewrite <- hsplit_left", "exact hati", "have haeq : a = x", "specialize beta_at_unique b", "specialize beta_at_unique e", "specialize beta_at_unique k", "specialize beta_at_unique a", "specialize beta_at_unique x", "apply beta_at_unique", "exact hati_new", "exact hvalue_witness", "rewrite hsplit_left", "rewrite hsplit_left", "rewrite haeq", "exact hfold_witness_right", "have hmiP : exists q. P = S ((S i) * c) * q", "specialize hdiv i", "apply hdiv", "exact hsplit_right", "have hzold : exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v", "specialize hcong i", "specialize hcong a", "apply hcong", "exact hsplit_right", "exact hati", "specialize hfold_witness_left (S ((S i) * c))", "specialize hfold_witness_left a", "apply hfold_witness_left", "exact hmiP", "exact hzold" ], "script_sha256": "62681f7104e20d1991a037541c88e771d2a255e2131b8022e1b2b62662975034", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall N c b e k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v", "statement_sha256": "29947a185cb5bd3e6d5f3addc026b0c148f4e5aca0428989ce28c64955dce949", "summary": "Add the next source value to a target-base CRT code for an exclusive prefix.", "summary_sha256": "9bb872d86fe39377f2c1373b08563de6bade0a3f60aec39355aa27017d01e1cd" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_modulus_nonzero", "le_refl", "lt_to_le", "binary_crt_fold_step", "beta_at_exists", "beta_at_unique", "le_of_succ_le_succ", "le_eq_or_lt" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_exclusive_recode_congruence_step]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_exclusive_recode_congruence_step", "parent_alpha_version": "v34", "proof_bundle_node_id": 141, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_exclusive_recode_congruence_step", "script": [ "intro N", "intro c", "intro b", "intro e", "intro k", "intro P", "intro z", "intro hkN", "intro hP", "intro hdiv", "intro hcong", "intro hfuture", "have hnew : ~(S ((S k) * c) = 0)", "specialize beta_modulus_nonzero c", "specialize beta_modulus_nonzero k", "exact beta_modulus_nonzero", "have hkbound : exists h. h + k = N", "specialize lt_to_le k", "specialize lt_to_le N", "apply lt_to_le", "exact hkN", "have hcop : forall d. (exists u. P = d * u) -> (exists v. S ((S k) * c) = d * v) -> d = 1", "specialize hfuture k", "apply hfuture", "specialize le_refl k", "exact le_refl", "exact hkbound", "have hvalue : exists a. ((exists h. h + S a = S ((S k) * e)) /\\ exists q. b = q * S ((S k) * e) + a)", "specialize beta_at_exists b", "specialize beta_at_exists e", "specialize beta_at_exists k", "exact beta_at_exists", "cases hvalue", "have hfold : exists z2. ((forall m a. (exists q. P = m * q) -> (exists u v. z + m * u = a + m * v) -> exists r s. z2 + m * r = a + m * s) /\\ exists q r. z2 + S ((S k) * c) * q = x + S ((S k) * c) * r)", "specialize binary_crt_fold_step P", "specialize binary_crt_fold_step (S ((S k) * c))", "specialize binary_crt_fold_step z", "specialize binary_crt_fold_step x", "apply binary_crt_fold_step", "exact hP", "exact hnew", "exact hcop", "cases hfold", "cases hfold_witness", "exists x1", "intro i", "intro a", "intro hi", "intro hati", "have hik : exists r. r + i = k", "specialize le_of_succ_le_succ i", "specialize le_of_succ_le_succ k", "apply le_of_succ_le_succ", "exact hi", "have hsplit : i = k \\/ exists r. r + S i = k", "specialize le_eq_or_lt i", "specialize le_eq_or_lt k", "apply le_eq_or_lt", "exact hik", "cases hsplit", "have hati_new : ((exists h. h + S a = S ((S k) * e)) /\\ exists q. b = q * S ((S k) * e) + a)", "rewrite <- hsplit_left", "rewrite <- hsplit_left", "exact hati", "have haeq : a = x", "specialize beta_at_unique b", "specialize beta_at_unique e", "specialize beta_at_unique k", "specialize beta_at_unique a", "specialize beta_at_unique x", "apply beta_at_unique", "exact hati_new", "exact hvalue_witness", "rewrite hsplit_left", "rewrite hsplit_left", "rewrite haeq", "exact hfold_witness_right", "have hmiP : exists q. P = S ((S i) * c) * q", "specialize hdiv i", "apply hdiv", "exact hsplit_right", "have hzold : exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v", "specialize hcong i", "specialize hcong a", "apply hcong", "exact hsplit_right", "exact hati", "specialize hfold_witness_left (S ((S i) * c))", "specialize hfold_witness_left a", "apply hfold_witness_left", "exact hmiP", "exact hzold" ], "script_sha256": "62681f7104e20d1991a037541c88e771d2a255e2131b8022e1b2b62662975034", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall N c b e k P z. (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v", "statement_sha256": "29947a185cb5bd3e6d5f3addc026b0c148f4e5aca0428989ce28c64955dce949" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_exclusive_recode_invariant_step", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_exclusive_accumulated_product_step", "beta_exclusive_recode_congruence_step" ], "dependencies_sha256": "7cb47aa8e47cf90b38d4545594304bc793ab6f3573e456935a424115173beb1b", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "1057ae7a6b5e5e24ee9976a188920c7a8f12c002c51a4d1db60c7eb932b97ef1", "cut_nodes": 549, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 71, "proof_edges": 3937, "proof_nodes": 18709, "proof_objects": 3750, "reused_objects": 188, "status": "checked" }, "enrollment_index": 186, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_exclusive_recode_invariant_step]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "e1fa4801933489f8d2b9c6274efb0bd33303a4ed06495274d5ff2d82b2a1a64c", "membership": "stable", "name": "beta_exclusive_recode_invariant_step", "proof_tag": "PA002H", "provenance": [ "stable" ], "script": [ "intro N", "intro c", "intro b", "intro e", "intro k", "intro P", "intro z", "intro hcm", "intro hkN", "intro hP", "intro hdiv", "intro hcong", "intro hfuture", "have hproduct : (~(P * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))", "specialize beta_exclusive_accumulated_product_step N", "specialize beta_exclusive_accumulated_product_step c", "specialize beta_exclusive_accumulated_product_step k", "specialize beta_exclusive_accumulated_product_step P", "apply beta_exclusive_accumulated_product_step", "exact hcm", "exact hkN", "exact hP", "exact hdiv", "exact hfuture", "have hcodes : exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v", "specialize beta_exclusive_recode_congruence_step N", "specialize beta_exclusive_recode_congruence_step c", "specialize beta_exclusive_recode_congruence_step b", "specialize beta_exclusive_recode_congruence_step e", "specialize beta_exclusive_recode_congruence_step k", "specialize beta_exclusive_recode_congruence_step P", "specialize beta_exclusive_recode_congruence_step z", "apply beta_exclusive_recode_congruence_step", "exact hkN", "exact hP", "exact hdiv", "exact hcong", "exact hfuture", "cases hcodes", "cases hproduct", "cases hproduct_right", "exists x", "split", "exact hproduct_left", "split", "exact hproduct_right_left", "split", "exact hcodes_witness", "exact hproduct_right_right" ], "script_sha256": "2bf1e26793a87261321f10da3bd6ed4c270a26492e9db1f53e676e0e7b632670", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall N c b e k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))", "statement_sha256": "9eaa14524e19b2f11771b44c6676033b3e31080ee96371c11bac0f48ca1c51f6", "summary": "Combine modulus-product and cross-base congruence updates for an exclusive prefix.", "summary_sha256": "5b02ba82c2a4e67f3f6cbc07a1c208b3df03f3c4cc80754aa459b76dd380c816" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_exclusive_accumulated_product_step", "beta_exclusive_recode_congruence_step" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_exclusive_recode_invariant_step]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_exclusive_recode_invariant_step", "parent_alpha_version": "v34", "proof_bundle_node_id": 142, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_exclusive_recode_invariant_step", "script": [ "intro N", "intro c", "intro b", "intro e", "intro k", "intro P", "intro z", "intro hcm", "intro hkN", "intro hP", "intro hdiv", "intro hcong", "intro hfuture", "have hproduct : (~(P * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1))", "specialize beta_exclusive_accumulated_product_step N", "specialize beta_exclusive_accumulated_product_step c", "specialize beta_exclusive_accumulated_product_step k", "specialize beta_exclusive_accumulated_product_step P", "apply beta_exclusive_accumulated_product_step", "exact hcm", "exact hkN", "exact hP", "exact hdiv", "exact hfuture", "have hcodes : exists z2. forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v", "specialize beta_exclusive_recode_congruence_step N", "specialize beta_exclusive_recode_congruence_step c", "specialize beta_exclusive_recode_congruence_step b", "specialize beta_exclusive_recode_congruence_step e", "specialize beta_exclusive_recode_congruence_step k", "specialize beta_exclusive_recode_congruence_step P", "specialize beta_exclusive_recode_congruence_step z", "apply beta_exclusive_recode_congruence_step", "exact hkN", "exact hP", "exact hdiv", "exact hcong", "exact hfuture", "cases hcodes", "cases hproduct", "cases hproduct_right", "exists x", "split", "exact hproduct_left", "split", "exact hproduct_right_left", "split", "exact hcodes_witness", "exact hproduct_right_right" ], "script_sha256": "2bf1e26793a87261321f10da3bd6ed4c270a26492e9db1f53e676e0e7b632670", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall N c b e k P z. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> (exists h. h + S k = N) -> ~(P = 0) -> (forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) -> (forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) -> (forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1) -> exists z2. (~(P * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. P * S ((S k) * c) = S ((S i) * c) * q) /\\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. P * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))", "statement_sha256": "9eaa14524e19b2f11771b44c6676033b3e31080ee96371c11bac0f48ca1c51f6" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "bounded_beta_exclusive_recode_invariant", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "succ_ne_zero", "add_eq_zero_right", "coprime_one_left", "le_succ_self", "le_trans", "beta_exclusive_recode_invariant_step" ], "dependencies_sha256": "bf68d33a3f23730911d4b776ce74a98625862dc943b26b391798d6cfc804d583", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "2c9541dd49fb1a6242210f3062931f5313d4632e445135bcad5e9b06fee01295", "cut_nodes": 563, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 77, "proof_edges": 4065, "proof_nodes": 19155, "proof_objects": 3873, "reused_objects": 193, "status": "checked" }, "enrollment_index": 187, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=bounded_beta_exclusive_recode_invariant]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "9d774b43fd860c5283dbc3887fa7f42456c42b2e2572f9687e8f53f1c0407d74", "membership": "stable", "name": "bounded_beta_exclusive_recode_invariant", "proof_tag": "PA002I", "provenance": [ "stable" ], "script": [ "intro N", "intro c", "intro b", "intro e", "intro hcm", "induction k", "intro hkN", "exists 1", "exists 0", "split", "specialize succ_ne_zero 0", "exact succ_ne_zero", "split", "intro i", "intro hi", "exfalso", "cases hi", "have hsi0 : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi0", "split", "intro i", "intro a", "intro hi", "intro hati", "exfalso", "cases hi", "have hsi0 : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi0", "intro j", "intro h0j", "intro hjN", "intro d", "intro h1", "intro hm", "specialize coprime_one_left (S ((S j) * c))", "specialize coprime_one_left d", "apply coprime_one_left", "exact h1", "exact hm", "intro hkN", "have hkprev : exists h. h + k = N", "have hkstep : exists h. h + k = S k", "specialize le_succ_self k", "exact le_succ_self", "specialize le_trans k", "specialize le_trans (S k)", "specialize le_trans N", "apply le_trans", "exact hkstep", "exact hkN", "have hprev : exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))", "apply IH", "exact hkprev", "cases hprev", "cases hprev_witness", "cases hprev_witness_witness", "cases hprev_witness_witness_right", "cases hprev_witness_witness_right_right", "have hnext : exists z2. (~(x * S ((S k) * c) = 0) /\\ ((forall i. (exists h. h + S i = S k) -> exists q. x * S ((S k) * c) = S ((S i) * c) * q) /\\ ((forall i a. (exists h. h + S i = S k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z2 + S ((S i) * c) * u = a + S ((S i) * c) * v) /\\ forall j. (exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. x * S ((S k) * c) = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))", "specialize beta_exclusive_recode_invariant_step N", "specialize beta_exclusive_recode_invariant_step c", "specialize beta_exclusive_recode_invariant_step b", "specialize beta_exclusive_recode_invariant_step e", "specialize beta_exclusive_recode_invariant_step k", "specialize beta_exclusive_recode_invariant_step x", "specialize beta_exclusive_recode_invariant_step x1", "apply beta_exclusive_recode_invariant_step", "exact hcm", "exact hkN", "exact hprev_witness_witness_left", "exact hprev_witness_witness_right_left", "exact hprev_witness_witness_right_right_left", "exact hprev_witness_witness_right_right_right", "cases hnext", "exists x * S ((S k) * c)", "exists x2", "exact hnext_witness" ], "script_sha256": "1044693ab57bb836ae8558882a8c74a978103eddea8688f6af6fae4652f1a214", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall N c b e. (forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))", "statement_sha256": "29564e9bae7783feeb631c02f82f5e9af720bdfd28c9b4c49aba47a41ff0d6c3", "summary": "Fold an empty-based, exclusive beta prefix into another base with append readiness.", "summary_sha256": "675e50a7c48725a7270ba40155a5559631440ed7e34666238ec4bd006c27ae22" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "succ_ne_zero", "add_eq_zero_right", "coprime_one_left", "le_succ_self", "le_trans", "beta_exclusive_recode_invariant_step" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=bounded_beta_exclusive_recode_invariant]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "bounded_beta_exclusive_recode_invariant", "parent_alpha_version": "v34", "proof_bundle_node_id": 143, "reference_route": "jordan-totient/checkpoint.html#theorem-bounded_beta_exclusive_recode_invariant", "script": [ "intro N", "intro c", "intro b", "intro e", "intro hcm", "induction k", "intro hkN", "exists 1", "exists 0", "split", "specialize succ_ne_zero 0", "exact succ_ne_zero", "split", "intro i", "intro hi", "exfalso", "cases hi", "have hsi0 : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi0", "split", "intro i", "intro a", "intro hi", "intro hati", "exfalso", "cases hi", "have hsi0 : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi0", "intro j", "intro h0j", "intro hjN", "intro d", "intro h1", "intro hm", "specialize coprime_one_left (S ((S j) * c))", "specialize coprime_one_left d", "apply coprime_one_left", "exact h1", "exact hm", "intro hkN", "have hkprev : exists h. h + k = N", "have hkstep : exists h. h + k = S k", "specialize le_succ_self k", "exact le_succ_self", "specialize le_trans k", "specialize le_trans (S k)", "specialize le_trans N", "apply le_trans", "exact hkstep", "exact hkN", "have hprev : exists P z. 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(exists g. g + S k = j) -> (exists h. h + j = N) -> forall d. (exists u. x * S ((S k) * c) = d * u) -> (exists v. 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(forall t. (exists h. S t + S h = S N) -> exists q. c = S t * q) -> forall k. (exists h. h + k = N) -> exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * c) * q) /\\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * c) * u = a + S ((S i) * c) * v) /\\ forall j. (exists g. g + k = j) -> (exists h. h + j = N) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * c) = d * v) -> d = 1)))", "statement_sha256": "29564e9bae7783feeb631c02f82f5e9af720bdfd28c9b4c49aba47a41ff0d6c3" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_prefix_extend", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "bounded_common_multiple_exists", "scaled_bounded_common_multiple", "bounded_beta_exclusive_recode_invariant", "le_refl", "beta_modulus_nonzero", "binary_crt_fold_step", "new_value_lt_scaled_base", "beta_value_lt_scaled_base", "beta_at_of_mod_eq_bound" ], "dependencies_sha256": "273d5a4667b52f2687b31fbfb63de8c970fc9adee35fba6afb45ddd2653ffd58", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "511d3bde3fc45d7ab2748c39baacb167b9eb05f71d6c43ad1d1eb03fbb23c7f6", "cut_nodes": 867, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 80, "proof_edges": 4732, "proof_nodes": 29057, "proof_objects": 4508, "reused_objects": 225, "status": "checked" }, "enrollment_index": 188, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_prefix_extend]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "12b1fc10c9e32da47518d4b9654f37aa98c9fe65a00dcdf3e37153401025c4fb", "membership": "stable", "name": "beta_prefix_extend", "proof_tag": "PA002X", "provenance": [ "stable" ], "script": [ "intro k", "intro b", "intro e", "intro s", "have hC : exists C. (~(C = 0) /\\ forall t. (exists h. S t + S h = S k) -> exists q. C = S t * q)", "specialize bounded_common_multiple_exists k", "exact bounded_common_multiple_exists", "cases hC", "cases hC_witness", "have hcm2 : forall t. (exists h. S t + S h = S k) -> exists q. x * S (b + s) = S t * q", "specialize scaled_bounded_common_multiple k", "specialize scaled_bounded_common_multiple x", "specialize scaled_bounded_common_multiple (S (b + s))", "apply scaled_bounded_common_multiple", "exact hC_witness_right", "have hall : forall n. (exists h. h + n = k) -> exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = n) -> exists q. P = S ((S i) * (x * S (b + s))) * q) /\\ ((forall i a. (exists h. h + S i = n) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v) /\\ forall j. (exists g. g + n = j) -> (exists h. h + j = k) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * (x * S (b + s))) = d * v) -> d = 1)))", "specialize bounded_beta_exclusive_recode_invariant k", "specialize bounded_beta_exclusive_recode_invariant (x * S (b + s))", "specialize bounded_beta_exclusive_recode_invariant b", "specialize bounded_beta_exclusive_recode_invariant e", "apply bounded_beta_exclusive_recode_invariant", "exact hcm2", "have hinv : exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * (x * S (b + s))) * q) /\\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v) /\\ forall j. (exists g. g + k = j) -> (exists h. h + j = k) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * (x * S (b + s))) = d * v) -> d = 1)))", "specialize hall k", "apply hall", "specialize le_refl k", "exact le_refl", "cases hinv", "cases hinv_witness", "cases hinv_witness_witness", "cases hinv_witness_witness_right", "cases hinv_witness_witness_right_right", "have hcop : forall d. (exists u. x1 = d * u) -> (exists v. S ((S k) * (x * S (b + s))) = d * v) -> d = 1", "specialize hinv_witness_witness_right_right_right k", "apply hinv_witness_witness_right_right_right", "specialize le_refl k", "exact le_refl", "specialize le_refl k", "exact le_refl", "have hnew0 : ~(S ((S k) * (x * S (b + s))) = 0)", "specialize beta_modulus_nonzero (x * S (b + s))", "specialize beta_modulus_nonzero k", "exact beta_modulus_nonzero", "have hfold : exists z2. ((forall m a. (exists q. x1 = m * q) -> (exists u v. x2 + m * u = a + m * v) -> exists r t. z2 + m * r = a + m * t) /\\ exists q r. z2 + S ((S k) * (x * S (b + s))) * q = s + S ((S k) * (x * S (b + s))) * r)", "specialize binary_crt_fold_step x1", "specialize binary_crt_fold_step (S ((S k) * (x * S (b + s))))", "specialize binary_crt_fold_step x2", "specialize binary_crt_fold_step s", "apply binary_crt_fold_step", "exact hinv_witness_witness_left", "exact hnew0", "exact hcop", "cases hfold", "cases hfold_witness", "exists x3", "exists x * S (b + s)", "split", "specialize beta_at_of_mod_eq_bound x3", "specialize beta_at_of_mod_eq_bound (x * S (b + s))", "specialize beta_at_of_mod_eq_bound k", "specialize beta_at_of_mod_eq_bound s", "apply beta_at_of_mod_eq_bound", "specialize new_value_lt_scaled_base b", "specialize new_value_lt_scaled_base s", "specialize new_value_lt_scaled_base x", "specialize new_value_lt_scaled_base k", "apply new_value_lt_scaled_base", "exact hC_witness_left", "exact hfold_witness_right", "intro i", "intro a", "intro hi", "intro hati", "have hmi : exists q. x1 = S ((S i) * (x * S (b + s))) * q", "specialize hinv_witness_witness_right_left i", "apply hinv_witness_witness_right_left", "exact hi", "have hzold : exists u v. x2 + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v", "specialize hinv_witness_witness_right_right_left i", "specialize hinv_witness_witness_right_right_left a", "apply hinv_witness_witness_right_right_left", "exact hi", "exact hati", "have hznew : exists u v. x3 + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v", "specialize hfold_witness_left (S ((S i) * (x * S (b + s))))", "specialize hfold_witness_left a", "apply hfold_witness_left", "exact hmi", "exact hzold", "specialize beta_at_of_mod_eq_bound x3", "specialize beta_at_of_mod_eq_bound (x * S (b + s))", "specialize beta_at_of_mod_eq_bound i", "specialize beta_at_of_mod_eq_bound a", "apply beta_at_of_mod_eq_bound", "specialize beta_value_lt_scaled_base b", "specialize beta_value_lt_scaled_base e", "specialize beta_value_lt_scaled_base i", "specialize beta_value_lt_scaled_base a", "specialize beta_value_lt_scaled_base x", "specialize beta_value_lt_scaled_base s", "specialize beta_value_lt_scaled_base i", "apply beta_value_lt_scaled_base", "exact hati", "exact hC_witness_left", "exact hznew" ], "script_sha256": "3fbc35b172643704dab6487f323e4fec09cec4c02f89bd0999295b8ea41781e4", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\\ exists q. z = q * S ((S k) * c) + s) /\\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\\ exists q. z = q * S ((S i) * c) + a))", "statement_sha256": "292c7a2a1abbf591e9a3cfd77ad0533f12761f691460fca7a20f413c6f66a1e0", "summary": "Rebase an arbitrary decoded prefix and append one exact natural value.", "summary_sha256": "d30ed8ed83ba7fb68e19c73878b906930df845e3072cdf7530d004418b3de456" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "bounded_common_multiple_exists", "scaled_bounded_common_multiple", "bounded_beta_exclusive_recode_invariant", "le_refl", "beta_modulus_nonzero", "binary_crt_fold_step", "new_value_lt_scaled_base", "beta_value_lt_scaled_base", "beta_at_of_mod_eq_bound" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_prefix_extend]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_prefix_extend", "parent_alpha_version": "v34", "proof_bundle_node_id": 144, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_prefix_extend", "script": [ "intro k", "intro b", "intro e", "intro s", "have hC : exists C. (~(C = 0) /\\ forall t. (exists h. S t + S h = S k) -> exists q. C = S t * q)", "specialize bounded_common_multiple_exists k", "exact bounded_common_multiple_exists", "cases hC", "cases hC_witness", "have hcm2 : forall t. (exists h. S t + S h = S k) -> exists q. x * S (b + s) = S t * q", "specialize scaled_bounded_common_multiple k", "specialize scaled_bounded_common_multiple x", "specialize scaled_bounded_common_multiple (S (b + s))", "apply scaled_bounded_common_multiple", "exact hC_witness_right", "have hall : forall n. (exists h. h + n = k) -> exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = n) -> exists q. P = S ((S i) * (x * S (b + s))) * q) /\\ ((forall i a. (exists h. h + S i = n) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v) /\\ forall j. (exists g. g + n = j) -> (exists h. h + j = k) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * (x * S (b + s))) = d * v) -> d = 1)))", "specialize bounded_beta_exclusive_recode_invariant k", "specialize bounded_beta_exclusive_recode_invariant (x * S (b + s))", "specialize bounded_beta_exclusive_recode_invariant b", "specialize bounded_beta_exclusive_recode_invariant e", "apply bounded_beta_exclusive_recode_invariant", "exact hcm2", "have hinv : exists P z. (~(P = 0) /\\ ((forall i. (exists h. h + S i = k) -> exists q. P = S ((S i) * (x * S (b + s))) * q) /\\ ((forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> exists u v. z + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v) /\\ forall j. (exists g. g + k = j) -> (exists h. h + j = k) -> forall d. (exists u. P = d * u) -> (exists v. S ((S j) * (x * S (b + s))) = d * v) -> d = 1)))", "specialize hall k", "apply hall", "specialize le_refl k", "exact le_refl", "cases hinv", "cases hinv_witness", "cases hinv_witness_witness", "cases hinv_witness_witness_right", "cases hinv_witness_witness_right_right", "have hcop : forall d. (exists u. x1 = d * u) -> (exists v. S ((S k) * (x * S (b + s))) = d * v) -> d = 1", "specialize hinv_witness_witness_right_right_right k", "apply hinv_witness_witness_right_right_right", "specialize le_refl k", "exact le_refl", "specialize le_refl k", "exact le_refl", "have hnew0 : ~(S ((S k) * (x * S (b + s))) = 0)", "specialize beta_modulus_nonzero (x * S (b + s))", "specialize beta_modulus_nonzero k", "exact beta_modulus_nonzero", "have hfold : exists z2. ((forall m a. (exists q. x1 = m * q) -> (exists u v. x2 + m * u = a + m * v) -> exists r t. z2 + m * r = a + m * t) /\\ exists q r. z2 + S ((S k) * (x * S (b + s))) * q = s + S ((S k) * (x * S (b + s))) * r)", "specialize binary_crt_fold_step x1", "specialize binary_crt_fold_step (S ((S k) * (x * S (b + s))))", "specialize binary_crt_fold_step x2", "specialize binary_crt_fold_step s", "apply binary_crt_fold_step", "exact hinv_witness_witness_left", "exact hnew0", "exact hcop", "cases hfold", "cases hfold_witness", "exists x3", "exists x * S (b + s)", "split", "specialize beta_at_of_mod_eq_bound x3", "specialize beta_at_of_mod_eq_bound (x * S (b + s))", "specialize beta_at_of_mod_eq_bound k", "specialize beta_at_of_mod_eq_bound s", "apply beta_at_of_mod_eq_bound", "specialize new_value_lt_scaled_base b", "specialize new_value_lt_scaled_base s", "specialize new_value_lt_scaled_base x", "specialize new_value_lt_scaled_base k", "apply new_value_lt_scaled_base", "exact hC_witness_left", "exact hfold_witness_right", "intro i", "intro a", "intro hi", "intro hati", "have hmi : exists q. x1 = S ((S i) * (x * S (b + s))) * q", "specialize hinv_witness_witness_right_left i", "apply hinv_witness_witness_right_left", "exact hi", "have hzold : exists u v. x2 + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v", "specialize hinv_witness_witness_right_right_left i", "specialize hinv_witness_witness_right_right_left a", "apply hinv_witness_witness_right_right_left", "exact hi", "exact hati", "have hznew : exists u v. x3 + S ((S i) * (x * S (b + s))) * u = a + S ((S i) * (x * S (b + s))) * v", "specialize hfold_witness_left (S ((S i) * (x * S (b + s))))", "specialize hfold_witness_left a", "apply hfold_witness_left", "exact hmi", "exact hzold", "specialize beta_at_of_mod_eq_bound x3", "specialize beta_at_of_mod_eq_bound (x * S (b + s))", "specialize beta_at_of_mod_eq_bound i", "specialize beta_at_of_mod_eq_bound a", "apply beta_at_of_mod_eq_bound", "specialize beta_value_lt_scaled_base b", "specialize beta_value_lt_scaled_base e", "specialize beta_value_lt_scaled_base i", "specialize beta_value_lt_scaled_base a", "specialize beta_value_lt_scaled_base x", "specialize beta_value_lt_scaled_base s", "specialize beta_value_lt_scaled_base i", "apply beta_value_lt_scaled_base", "exact hati", "exact hC_witness_left", "exact hznew" ], "script_sha256": "3fbc35b172643704dab6487f323e4fec09cec4c02f89bd0999295b8ea41781e4", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall k b e s. exists z c. (((exists h. h + S s = S ((S k) * c)) /\\ exists q. z = q * S ((S k) * c) + s) /\\ forall i a. (exists h. h + S i = k) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. b = q * S ((S i) * e) + a) -> ((exists h. h + S a = S ((S i) * c)) /\\ exists q. z = q * S ((S i) * c) + a))", "statement_sha256": "292c7a2a1abbf591e9a3cfd77ad0533f12761f691460fca7a20f413c6f66a1e0" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_prefix_product_trace_exists", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_at_self_of_bound", "add_eq_zero_right", "succ_ne_zero", "beta_at_exists", "beta_prefix_extend", "zero_le", "succ_le_succ", "le_refl", "le_of_succ_le_succ", "le_eq_or_lt", "one_mul" ], "dependencies_sha256": "954ab39afe70296bc66f9af4ef46555d4f5406ef6e6cdfe58bfbc88d9fb99356", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "0dd19ccc0b06503a99e50627eaed2b9bda46c7a5f2685916bde03e1a17ae1f41", "cut_nodes": 899, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 85, "proof_edges": 4951, "proof_nodes": 29981, "proof_objects": 4717, "reused_objects": 235, "status": "checked" }, "enrollment_index": 189, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_prefix_product_trace_exists]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "5e6b32df8955c2375fa71dbba32c909c91561298f27521f74d71fce3cfe9c9de", "membership": "stable", "name": "beta_prefix_product_trace_exists", "proof_tag": "PA003W", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "induction l", "exists 1", "exists 1", "split", "specialize beta_at_self_of_bound 1", "specialize beta_at_self_of_bound 0", "specialize beta_at_self_of_bound 1", "apply beta_at_self_of_bound", "specialize one_mul 1", "rewrite one_mul", "specialize le_refl 2", "exact le_refl", "intro i", "intro hi", "exfalso", "cases hi", "have hsi0 : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi0", "have htrace : exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S (S i)) * v)) /\\ exists q. u = q * S ((S (S i)) * v) + s) /\\ s = r * p))))", "apply IH", "cases htrace", "cases htrace_witness", "cases htrace_witness_witness", "have hfactor : exists p. ((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p)", "specialize beta_at_exists b", "specialize beta_at_exists c", "specialize beta_at_exists l", "exact beta_at_exists", "cases hfactor", "have hlast : exists r. ((exists h. h + S r = S ((S l) * x1)) /\\ exists q. x = q * S ((S l) * x1) + r)", "specialize beta_at_exists x", "specialize beta_at_exists x1", "specialize beta_at_exists l", "exact beta_at_exists", "cases hlast", "have hext : exists z v. (((exists h. h + S (x3 * x2) = S ((S (S l)) * v)) /\\ exists q. z = q * S ((S (S l)) * v) + (x3 * x2)) /\\ forall i a. (exists h. h + S i = S l) -> ((exists h. h + S a = S ((S i) * x1)) /\\ exists q. x = q * S ((S i) * x1) + a) -> ((exists h. h + S a = S ((S i) * v)) /\\ exists q. z = q * S ((S i) * v) + a))", "specialize beta_prefix_extend (S l)", "specialize beta_prefix_extend x", "specialize beta_prefix_extend x1", "specialize beta_prefix_extend (x3 * x2)", "exact beta_prefix_extend", "cases hext", "cases hext_witness", "cases hext_witness_witness", "exists x4", "exists x5", "split", "specialize hext_witness_witness_right 0", "specialize hext_witness_witness_right 1", "apply hext_witness_witness_right", "have h0 : exists h. h + S 0 = S l", "have hzero : exists h. h + 0 = l", "specialize zero_le l", "exact zero_le", "specialize succ_le_succ 0", "specialize succ_le_succ l", "apply succ_le_succ", "exact hzero", "exact h0", "exact htrace_witness_witness_left", "intro i", "intro hi", "have hil : exists h. h + i = l", "specialize le_of_succ_le_succ i", "specialize le_of_succ_le_succ l", "apply le_of_succ_le_succ", "exact hi", "have hsplit : i = l \\/ exists h. h + S i = l", "specialize le_eq_or_lt i", "specialize le_eq_or_lt l", "apply le_eq_or_lt", "exact hil", "cases hsplit", "exists x2", "exists x3", "exists x3 * x2", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hfactor_witness", "split", "rewrite hsplit_left", "rewrite hsplit_left", "specialize hext_witness_witness_right l", "specialize hext_witness_witness_right x3", "apply hext_witness_witness_right", "specialize le_refl (S l)", "exact le_refl", "exact hlast_witness", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hext_witness_witness_left", "refl", "have hold : exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * x1)) /\\ exists q. x = q * S ((S i) * x1) + r) /\\ (((exists h. h + S s = S ((S (S i)) * x1)) /\\ exists q. x = q * S ((S (S i)) * x1) + s) /\\ s = r * p)))", "specialize htrace_witness_witness_right i", "apply htrace_witness_witness_right", "exact hsplit_right", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "cases hold_witness_witness_witness_right", "cases hold_witness_witness_witness_right_right", "exists x6", "exists x7", "exists x8", "split", "exact hold_witness_witness_witness_left", "split", "specialize hext_witness_witness_right i", "specialize hext_witness_witness_right x7", "apply hext_witness_witness_right", "exact hi", "exact hold_witness_witness_witness_right_left", "split", "specialize hext_witness_witness_right (S i)", "specialize hext_witness_witness_right x8", "apply hext_witness_witness_right", "specialize succ_le_succ (S i)", "specialize succ_le_succ l", "apply succ_le_succ", "exact hsplit_right", "exact hold_witness_witness_witness_right_right_left", "exact hold_witness_witness_witness_right_right_right" ], "script_sha256": "06e52e6f7f4934c01ed7a3a12820f89e79649f3e8b86fe706edfca74af8fddcc", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c l. exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S (S i)) * v)) /\\ exists q. u = q * S ((S (S i)) * v) + s) /\\ s = r * p))))", "statement_sha256": "81d12aa55dbd4e5d3fef95cdca2ed566435b28fc9e3406a884c8b76b2f0af5d2", "summary": "Every decoded beta factor prefix admits a beta-coded exact prefix-product trace.", "summary_sha256": "1e96a781d0d39adde48461b360a2de8975688b0271d0c3ed0fd237c8d193e4ea" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_at_self_of_bound", "add_eq_zero_right", "succ_ne_zero", "beta_at_exists", "beta_prefix_extend", "zero_le", "succ_le_succ", "le_refl", "le_of_succ_le_succ", "le_eq_or_lt", "one_mul" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_prefix_product_trace_exists]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_prefix_product_trace_exists", "parent_alpha_version": "v34", "proof_bundle_node_id": 145, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_prefix_product_trace_exists", "script": [ "intro b", "intro c", "induction l", "exists 1", "exists 1", "split", "specialize beta_at_self_of_bound 1", "specialize beta_at_self_of_bound 0", "specialize beta_at_self_of_bound 1", "apply beta_at_self_of_bound", "specialize one_mul 1", "rewrite one_mul", "specialize le_refl 2", "exact le_refl", "intro i", "intro hi", "exfalso", "cases hi", "have hsi0 : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi0", "have htrace : exists u v. 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(((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S (S i)) * v)) /\\ exists q. u = q * S ((S (S i)) * v) + s) /\\ s = r * p))))", "statement_sha256": "81d12aa55dbd4e5d3fef95cdca2ed566435b28fc9e3406a884c8b76b2f0af5d2" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_product_exists", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_prefix_product_trace_exists", "beta_at_exists" ], "dependencies_sha256": "ec2c46e7705fefd7283a5e807d1a2d50dd6cedc89324152e4f0703cbeeb708ff", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "6656c8e6b7f3ea457a7353422f510f6a97b4910fc83b7cbb50bf6a6b5170516f", "cut_nodes": 916, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 86, "proof_edges": 4979, "proof_nodes": 30487, "proof_objects": 4744, "reused_objects": 236, "status": "checked" }, "enrollment_index": 190, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_exists]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "abd21182b7a22e96b45fa9b8b2db8807f52a480542ae52397c3bf1e85bffcb70", "membership": "stable", "name": "beta_product_exists", "proof_tag": "PA003X", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro l", "have htrace : exists u v. 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(((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S (S i)) * v)) /\\ exists q. u = q * S ((S (S i)) * v) + s) /\\ s = r * p)))))", "statement_sha256": "65955dade14a69f532b45f1541232206f1b80a684b8424f6b7b39f1d24df7bb3" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_product_functional", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_at_unique", "le_refl", "le_succ", "mul_congr" ], "dependencies_sha256": "d2ea4c1e069c32c43c125562fcc44666164c4c991c67502014ac82ac133fdac5", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "12a5727a4593b04b404e2c26eb8a444f4a705305504076c3759078e7fab116d0", "cut_nodes": 36, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 60, "proof_edges": 947, "proof_nodes": 1382, "proof_objects": 909, "reused_objects": 39, "status": "checked" }, "enrollment_index": 191, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_functional]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "45151961c2ca2fe7bbc723545aa91921025a5601319c02e49e6f130b83780696", "membership": "stable", "name": "beta_product_functional", "proof_tag": "PA0051", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "induction l", "intro n", "intro u", "intro v", "intro m", "intro w", "intro d", "intro h1", "intro h2", "cases h1", "cases h1_right", "cases h2", "cases h2_right", "have hn : n = 1", "specialize beta_at_unique u", "specialize beta_at_unique v", "specialize beta_at_unique 0", "specialize beta_at_unique n", "specialize beta_at_unique 1", "apply beta_at_unique", "exact h1_right_left", "exact h1_left", "have hm : m = 1", "specialize beta_at_unique w", "specialize beta_at_unique d", "specialize beta_at_unique 0", "specialize beta_at_unique m", "specialize beta_at_unique 1", "apply beta_at_unique", "exact h2_right_left", "exact h2_left", "trans 1", "exact hn", "symm", "exact hm", "intro n", "intro u", "intro v", "intro m", "intro w", "intro d", "intro h1", "intro h2", "cases h1", "cases h1_right", "cases h2", "cases h2_right", "have hstep1 : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ (((exists h. h + S r = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + r) /\\ (((exists h. h + S s = S ((S S l) * v)) /\\ exists q. u = q * S ((S S l) * v) + s) /\\ s = r * p)))", "specialize h1_right_right l", "apply h1_right_right", "specialize le_refl (S l)", "exact le_refl", "cases hstep1", "cases hstep1_witness", "cases hstep1_witness_witness", "cases hstep1_witness_witness_witness", "cases hstep1_witness_witness_witness_right", "cases hstep1_witness_witness_witness_right_right", "have hstep2 : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ (((exists h. h + S r = S ((S l) * d)) /\\ exists q. w = q * S ((S l) * d) + r) /\\ (((exists h. h + S s = S ((S S l) * d)) /\\ exists q. w = q * S ((S S l) * d) + s) /\\ s = r * p)))", "specialize h2_right_right l", "apply h2_right_right", "specialize le_refl (S l)", "exact le_refl", "cases hstep2", "cases hstep2_witness", "cases hstep2_witness_witness", "cases hstep2_witness_witness_witness", "cases hstep2_witness_witness_witness_right", "cases hstep2_witness_witness_witness_right_right", "have hn : n = x2", "specialize beta_at_unique u", "specialize beta_at_unique v", "specialize beta_at_unique (S l)", "specialize beta_at_unique n", "specialize beta_at_unique x2", "apply beta_at_unique", "exact h1_right_left", "exact hstep1_witness_witness_witness_right_right_left", "have hm : m = x5", "specialize beta_at_unique w", "specialize beta_at_unique d", "specialize beta_at_unique (S l)", "specialize beta_at_unique m", "specialize beta_at_unique x5", "apply beta_at_unique", "exact h2_right_left", "exact hstep2_witness_witness_witness_right_right_left", "have hp : x = x3", "specialize beta_at_unique b", "specialize beta_at_unique c", "specialize beta_at_unique l", "specialize beta_at_unique x", "specialize beta_at_unique x3", "apply beta_at_unique", "exact hstep1_witness_witness_witness_left", "exact hstep2_witness_witness_witness_left", "have hprod1 : (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S x1 = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + x1) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))", "split", "exact h1_left", "split", "exact hstep1_witness_witness_witness_right_left", "intro i", "intro hi", "specialize h1_right_right i", "apply h1_right_right", "specialize le_succ (S i)", "specialize le_succ l", "apply le_succ", "exact hi", "have hprod2 : (((exists h. h + S 1 = S ((S 0) * d)) /\\ exists q. w = q * S ((S 0) * d) + 1) /\\ (((exists h. h + S x4 = S ((S l) * d)) /\\ exists q. w = q * S ((S l) * d) + x4) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * d)) /\\ exists q. w = q * S ((S i) * d) + r) /\\ (((exists h. h + S s = S ((S S i) * d)) /\\ exists q. w = q * S ((S S i) * d) + s) /\\ s = r * p)))))", "split", "exact h2_left", "split", "exact hstep2_witness_witness_witness_right_left", "intro i", "intro hi", "specialize h2_right_right i", "apply h2_right_right", "specialize le_succ (S i)", "specialize le_succ l", "apply le_succ", "exact hi", "have hprev : x1 = x4", "specialize IH x1", "specialize IH u", "specialize IH v", "specialize IH x4", "specialize IH w", "specialize IH d", "apply IH", "exact hprod1", "exact hprod2", "have hmul : x1 * x = x4 * x3", "specialize mul_congr x1", "specialize mul_congr x4", "specialize mul_congr x", "specialize mul_congr x3", "apply mul_congr", "exact hprev", "exact hp", "trans x2", "exact hn", "trans x1 * x", "exact hstep1_witness_witness_witness_right_right_right", "trans x4 * x3", "exact hmul", "trans x5", "symm", "exact hstep2_witness_witness_witness_right_right_right", "symm", "exact hm" ], "script_sha256": "2397e4b233b19bf8ae22c59cb8af1610812713bc2e65893803e539a88ead31f7", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c l n u v m w d. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + n) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p))))) -> (((exists h. h + S 1 = S ((S 0) * d)) /\\ exists q. w = q * S ((S 0) * d) + 1) /\\ (((exists h. h + S m = S ((S l) * d)) /\\ exists q. w = q * S ((S l) * d) + m) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * d)) /\\ exists q. w = q * S ((S i) * d) + r) /\\ (((exists h. h + S s = S ((S S i) * d)) /\\ exists q. w = q * S ((S S i) * d) + s) /\\ s = r * p))))) -> n = m", "statement_sha256": "0d622682bc03f814ac15b4a40a55d8a3ef7c98c0b2047884df7c25da6ca1a0d8", "summary": "The fully expanded beta-coded Product relation is functional in its terminal product.", "summary_sha256": "c34a96b0a7089288b1ec469bc2d55e474dd589eef3b7eb62a810f5b6f6831b84" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_at_unique", "le_refl", "le_succ", "mul_congr" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_functional]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_product_functional", "parent_alpha_version": "v34", "proof_bundle_node_id": 147, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_product_functional", "script": [ "intro b", "intro c", "induction l", "intro n", "intro u", "intro v", "intro m", "intro w", "intro d", "intro h1", "intro h2", "cases h1", "cases h1_right", "cases h2", "cases h2_right", "have hn : n = 1", "specialize beta_at_unique u", "specialize beta_at_unique v", "specialize beta_at_unique 0", "specialize beta_at_unique n", "specialize beta_at_unique 1", "apply beta_at_unique", "exact h1_right_left", "exact h1_left", "have hm : m = 1", "specialize beta_at_unique w", "specialize beta_at_unique d", "specialize beta_at_unique 0", "specialize beta_at_unique m", "specialize beta_at_unique 1", "apply beta_at_unique", "exact h2_right_left", "exact h2_left", "trans 1", "exact hn", "symm", "exact hm", "intro n", "intro u", "intro v", "intro m", "intro w", "intro d", "intro h1", "intro h2", "cases h1", "cases h1_right", "cases h2", "cases h2_right", "have hstep1 : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ (((exists h. h + S r = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + r) /\\ (((exists h. h + S s = S ((S S l) * v)) /\\ exists q. u = q * S ((S S l) * v) + s) /\\ s = r * p)))", "specialize h1_right_right l", "apply h1_right_right", "specialize le_refl (S l)", "exact le_refl", "cases hstep1", "cases hstep1_witness", "cases hstep1_witness_witness", "cases hstep1_witness_witness_witness", "cases hstep1_witness_witness_witness_right", "cases hstep1_witness_witness_witness_right_right", "have hstep2 : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ (((exists h. h + S r = S ((S l) * d)) /\\ exists q. w = q * S ((S l) * d) + r) /\\ (((exists h. h + S s = S ((S S l) * d)) /\\ exists q. w = q * S ((S S l) * d) + s) /\\ s = r * p)))", "specialize h2_right_right l", "apply h2_right_right", "specialize le_refl (S l)", "exact le_refl", "cases hstep2", "cases hstep2_witness", "cases hstep2_witness_witness", "cases hstep2_witness_witness_witness", "cases hstep2_witness_witness_witness_right", "cases hstep2_witness_witness_witness_right_right", "have hn : n = x2", "specialize beta_at_unique u", "specialize beta_at_unique v", "specialize beta_at_unique (S l)", "specialize beta_at_unique n", "specialize beta_at_unique x2", "apply beta_at_unique", "exact h1_right_left", "exact hstep1_witness_witness_witness_right_right_left", "have hm : m = x5", "specialize beta_at_unique w", "specialize beta_at_unique d", "specialize beta_at_unique (S l)", "specialize beta_at_unique m", "specialize beta_at_unique x5", "apply beta_at_unique", "exact h2_right_left", "exact hstep2_witness_witness_witness_right_right_left", "have hp : x = x3", "specialize beta_at_unique b", "specialize beta_at_unique c", "specialize beta_at_unique l", "specialize beta_at_unique x", "specialize beta_at_unique x3", "apply beta_at_unique", "exact hstep1_witness_witness_witness_left", "exact hstep2_witness_witness_witness_left", "have hprod1 : (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S x1 = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + x1) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))", "split", "exact h1_left", "split", "exact hstep1_witness_witness_witness_right_left", "intro i", "intro hi", "specialize h1_right_right i", "apply h1_right_right", "specialize le_succ (S i)", "specialize le_succ l", "apply le_succ", "exact hi", "have hprod2 : (((exists h. h + S 1 = S ((S 0) * d)) /\\ exists q. w = q * S ((S 0) * d) + 1) /\\ (((exists h. h + S x4 = S ((S l) * d)) /\\ exists q. w = q * S ((S l) * d) + x4) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * d)) /\\ exists q. w = q * S ((S i) * d) + r) /\\ (((exists h. h + S s = S ((S S i) * d)) /\\ exists q. w = q * S ((S S i) * d) + s) /\\ s = r * p)))))", "split", "exact h2_left", "split", "exact hstep2_witness_witness_witness_right_left", "intro i", "intro hi", "specialize h2_right_right i", "apply h2_right_right", "specialize le_succ (S i)", "specialize le_succ l", "apply le_succ", "exact hi", "have hprev : x1 = x4", "specialize IH x1", "specialize IH u", "specialize IH v", "specialize IH x4", "specialize IH w", "specialize IH d", "apply IH", "exact hprod1", "exact hprod2", "have hmul : x1 * x = x4 * x3", "specialize mul_congr x1", "specialize mul_congr x4", "specialize mul_congr x", "specialize mul_congr x3", "apply mul_congr", "exact hprev", "exact hp", "trans x2", "exact hn", "trans x1 * x", "exact hstep1_witness_witness_witness_right_right_right", "trans x4 * x3", "exact hmul", "trans x5", "symm", "exact hstep2_witness_witness_witness_right_right_right", "symm", "exact hm" ], "script_sha256": "2397e4b233b19bf8ae22c59cb8af1610812713bc2e65893803e539a88ead31f7", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c l n u v m w d. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + n) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p))))) -> (((exists h. h + S 1 = S ((S 0) * d)) /\\ exists q. w = q * S ((S 0) * d) + 1) /\\ (((exists h. h + S m = S ((S l) * d)) /\\ exists q. w = q * S ((S l) * d) + m) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * d)) /\\ exists q. w = q * S ((S i) * d) + r) /\\ (((exists h. h + S s = S ((S S i) * d)) /\\ exists q. w = q * S ((S S i) * d) + s) /\\ s = r * p))))) -> n = m", "statement_sha256": "0d622682bc03f814ac15b4a40a55d8a3ef7c98c0b2047884df7c25da6ca1a0d8" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_product_zero", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_at_unique" ], "dependencies_sha256": "aee880e8fa2a31079a779e18aae39ebc46230ee14bf423dd743f20f3c6cfaad0", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "63088511aac2ef0c5aa853cedc17a39582c96785bc149a143e3a166732358033", "cut_nodes": 31, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 60, "proof_edges": 778, "proof_nodes": 1171, "proof_objects": 742, "reused_objects": 37, "status": "checked" }, "enrollment_index": 193, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_zero]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "2f42793f118d6a3f8dd591330f1143e460b071caba258af0a9319c3483560c73", "membership": "stable", "name": "beta_product_zero", "proof_tag": "PA0049", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro n", "intro hproduct", "cases hproduct", "cases hproduct_witness", "cases hproduct_witness_witness", "cases hproduct_witness_witness_right", "specialize beta_at_unique x", "specialize beta_at_unique x1", "specialize beta_at_unique 0", "specialize beta_at_unique n", "specialize beta_at_unique 1", "apply beta_at_unique", "exact hproduct_witness_witness_right_left", "exact hproduct_witness_witness_left" ], "script_sha256": "d3670c3b1b337e7f2cd41493f7fb43032cfb9a3f2cd5f6362913e9bcb06e596d", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + n) /\\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) -> n = 1", "statement_sha256": "311da073e3b279a1c8a11c70e0c320d5728c686f858c65882f380864943c904f", "summary": "The product of an empty decoded prefix is one.", "summary_sha256": "baed9c40a5198bb3016d60afec23a7e62211e9d431f29ed4a24128afa59bcf05" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_at_unique" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_zero]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_product_zero", "parent_alpha_version": "v34", "proof_bundle_node_id": 148, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_product_zero", "script": [ "intro b", "intro c", "intro n", "intro hproduct", "cases hproduct", "cases hproduct_witness", "cases hproduct_witness_witness", "cases hproduct_witness_witness_right", "specialize beta_at_unique x", "specialize beta_at_unique x1", "specialize beta_at_unique 0", "specialize beta_at_unique n", "specialize beta_at_unique 1", "apply beta_at_unique", "exact hproduct_witness_witness_right_left", "exact hproduct_witness_witness_left" ], "script_sha256": "d3670c3b1b337e7f2cd41493f7fb43032cfb9a3f2cd5f6362913e9bcb06e596d", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + n) /\\ forall i. (exists h. h + S i = 0) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) -> n = 1", "statement_sha256": "311da073e3b279a1c8a11c70e0c320d5728c686f858c65882f380864943c904f" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_product_succ_decompose", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "le_refl", "le_succ", "beta_at_unique" ], "dependencies_sha256": "9f221c2eb14df06a7fd1893734c0e9474e54bd1a1c03fbf98dea50d180a0cc90", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "2a5502bdcf7379de0e5682ad43a19c8a55ac4e11adf199ef4675d83f0c848adc", "cut_nodes": 35, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 62, "proof_edges": 822, "proof_nodes": 1257, "proof_objects": 784, "reused_objects": 39, "status": "checked" }, "enrollment_index": 194, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_succ_decompose]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "9ae7ddb7db46f6721e61feefac197c7cf478bfa4acf49619ab9b221b4e932a26", "membership": "stable", "name": "beta_product_succ_decompose", "proof_tag": "PA004A", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro l", "intro n", "intro hproduct", "cases hproduct", "cases hproduct_witness", "cases hproduct_witness_witness", "cases hproduct_witness_witness_right", "have hstep : exists p r s. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ (((exists h. h + S r = S ((S l) * x1)) /\\ exists q. x = q * S ((S l) * x1) + r) /\\ (((exists h. h + S s = S ((S S l) * x1)) /\\ exists q. x = q * S ((S S l) * x1) + s) /\\ s = r * p)))", "specialize hproduct_witness_witness_right_right l", "apply hproduct_witness_witness_right_right", "specialize le_refl (S l)", "exact le_refl", "cases hstep", "cases hstep_witness", "cases hstep_witness_witness", "cases hstep_witness_witness_witness", "cases hstep_witness_witness_witness_right", "cases hstep_witness_witness_witness_right_right", "have hn : n = x4", "specialize beta_at_unique x", "specialize beta_at_unique x1", "specialize beta_at_unique (S l)", "specialize beta_at_unique n", "specialize beta_at_unique x4", "apply beta_at_unique", "exact hproduct_witness_witness_right_left", "exact hstep_witness_witness_witness_right_right_left", "exists x2", "exists x3", "split", "exact hstep_witness_witness_witness_left", "split", "exists x", "exists x1", "split", "exact hproduct_witness_witness_left", "split", "exact hstep_witness_witness_witness_right_left", "intro i", "intro hi", "specialize hproduct_witness_witness_right_right i", "apply hproduct_witness_witness_right_right", "specialize le_succ (S i)", "specialize le_succ l", "apply le_succ", "exact hi", "trans x4", "exact hn", "exact hstep_witness_witness_witness_right_right_right" ], "script_sha256": "d80eed06c75bc2d8b698aae0f411b7ae4b0542f84999c8d087ecf5c35e302cac", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c l n. (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S S l) * v)) /\\ exists q. u = q * S ((S S l) * v) + n) /\\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) -> exists p r. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S r = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + r) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) /\\ n = r * p))", "statement_sha256": "03469f0a2a01256aebd276b930955e026afe285ea8143da27e5d07d19f70c6a2", "summary": "A successor product decomposes into its prefix product and final decoded factor.", "summary_sha256": "2d7ab7a472caf55e23bc24b914d6882e9c5c240764061267d070a29018ce322d" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "le_refl", "le_succ", "beta_at_unique" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_succ_decompose]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_product_succ_decompose", "parent_alpha_version": "v34", "proof_bundle_node_id": 149, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_product_succ_decompose", "script": [ "intro b", "intro c", "intro l", "intro n", "intro hproduct", "cases hproduct", "cases hproduct_witness", "cases hproduct_witness_witness", "cases hproduct_witness_witness_right", "have hstep : exists p r s. 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(exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S S l) * v)) /\\ exists q. u = q * S ((S S l) * v) + n) /\\ forall i. (exists h. h + S i = S l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) -> exists p r. (((exists h. h + S p = S ((S l) * c)) /\\ exists q. b = q * S ((S l) * c) + p) /\\ ((exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S r = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + r) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) /\\ n = r * p))", "statement_sha256": "03469f0a2a01256aebd276b930955e026afe285ea8143da27e5d07d19f70c6a2" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_product_transport_prefix", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [], "dependencies_sha256": "01ba4719c80b6fe911b091a7c05124b64eeece964e09c058ef8f9805daca546b", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "4c40f0a22c7a5d071f32faee47106b462be9ba530351493ab9f1a8e5f1a79299", "cut_nodes": 0, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 29, "proof_edges": 58, "proof_nodes": 59, "proof_objects": 59, "reused_objects": 0, "status": "checked" }, "enrollment_index": 196, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_transport_prefix]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "f1a6d0797d988dda06030d56b1d9df0f90a698a12fafd37cc01834ca1cb4e8fd", "membership": "stable", "name": "beta_product_transport_prefix", "proof_tag": "PA004Y", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro z", "intro e", "intro l", "intro n", "intro hproduct", "intro hpres", "cases hproduct", "cases hproduct_witness", "cases hproduct_witness_witness", "cases hproduct_witness_witness_right", "exists x", "exists x1", "split", "exact hproduct_witness_witness_left", "split", "exact hproduct_witness_witness_right_left", "intro i", "intro hi", "have hstep : exists p r s. 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(exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + n) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p)))))) -> (forall i a. (exists h. h + S i = l) -> ((exists h. h + S a = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + a) -> ((exists h. h + S a = S ((S i) * e)) /\\ exists q. z = q * S ((S i) * e) + a)) -> (exists u v. (((exists h. h + S 1 = S ((S 0) * v)) /\\ exists q. u = q * S ((S 0) * v) + 1) /\\ (((exists h. h + S n = S ((S l) * v)) /\\ exists q. u = q * S ((S l) * v) + n) /\\ forall i. (exists h. h + S i = l) -> exists p r s. (((exists h. h + S p = S ((S i) * e)) /\\ exists q. z = q * S ((S i) * e) + p) /\\ (((exists h. h + S r = S ((S i) * v)) /\\ exists q. u = q * S ((S i) * v) + r) /\\ (((exists h. h + S s = S ((S S i) * v)) /\\ exists q. u = q * S ((S S i) * v) + s) /\\ s = r * p))))))", "statement_sha256": "db9d6aad8466f739736000f4984e1bb872f63f5ed7481f6ccfbeb90512ae0c52", "summary": "One-way extensional factor-prefix preservation transports Product without changing its trace.", "summary_sha256": "c3c993092e9c0956f6a3972e62a69c27aef07354847aac20fd2384300b7e21b1" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_product_transport_prefix]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_product_transport_prefix", "parent_alpha_version": "v34", "proof_bundle_node_id": 150, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_product_transport_prefix", "script": [ "intro b", "intro c", "intro z", "intro e", "intro l", "intro n", "intro hproduct", "intro hpres", "cases hproduct", "cases hproduct_witness", "cases hproduct_witness_witness", "cases hproduct_witness_witness_right", "exists x", "exists x1", "split", "exact hproduct_witness_witness_left", "split", "exact hproduct_witness_witness_right_left", "intro i", "intro hi", "have hstep : exists p r s. 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((forall ff_i_x_repeat. (exists ff_lt_x_repeat_bound. ff_lt_x_repeat_bound + S ff_i_x_repeat = e) -> (((exists ff_h_x_repeat_decoded. ff_h_x_repeat_decoded + S (a) = S ((S (ff_i_x_repeat)) * ff_c_x)) /\\ exists ff_q_x_repeat_decoded. ff_b_x = ff_q_x_repeat_decoded * S ((S (ff_i_x_repeat)) * ff_c_x) + (a)))) /\\ (exists ff_u_x_product ff_v_x_product. ((((exists ff_h_x_product_start. ff_h_x_product_start + S (1) = S ((S (0)) * ff_v_x_product)) /\\ exists ff_q_x_product_start. ff_u_x_product = ff_q_x_product_start * S ((S (0)) * ff_v_x_product) + (1))) /\\ ((((exists ff_h_x_product_terminal. ff_h_x_product_terminal + S (n) = S ((S (e)) * ff_v_x_product)) /\\ exists ff_q_x_product_terminal. ff_u_x_product = ff_q_x_product_terminal * S ((S (e)) * ff_v_x_product) + (n))) /\\ forall ff_i_x_product. (exists ff_lt_x_product_bound. ff_lt_x_product_bound + S ff_i_x_product = e) -> exists ff_p_x_product ff_r_x_product ff_s_x_product. ((((exists ff_h_x_product_factor. ff_h_x_product_factor + S (ff_p_x_product) = S ((S (ff_i_x_product)) * ff_c_x)) /\\ exists ff_q_x_product_factor. ff_b_x = ff_q_x_product_factor * S ((S (ff_i_x_product)) * ff_c_x) + (ff_p_x_product))) /\\ ((((exists ff_h_x_product_partial. ff_h_x_product_partial + S (ff_r_x_product) = S ((S (ff_i_x_product)) * ff_v_x_product)) /\\ exists ff_q_x_product_partial. ff_u_x_product = ff_q_x_product_partial * S ((S (ff_i_x_product)) * ff_v_x_product) + (ff_r_x_product))) /\\ ((((exists ff_h_x_product_successor. ff_h_x_product_successor + S (ff_s_x_product) = S ((S (S ff_i_x_product)) * ff_v_x_product)) /\\ exists ff_q_x_product_successor. ff_u_x_product = ff_q_x_product_successor * S ((S (S ff_i_x_product)) * ff_v_x_product) + (ff_s_x_product))) /\\ ff_s_x_product = ff_r_x_product * ff_p_x_product))))))))", "statement_sha256": "b773cba57ba44f87431d84dcaeb85cdf6ee363ab259e593ad25aa0ba4ec81543" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_zero", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_product_zero" ], "dependencies_sha256": "a3feac605c04cec534a725ae66902c86e09aa318d35a0151f20ac22e385b233a", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "f3e5705226dd57f733a8ac1adcf61e8612e09f72152f250a0fbe5b90d591dcf7", "cut_nodes": 32, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 61, "proof_edges": 831, "proof_nodes": 1224, "proof_objects": 795, "reused_objects": 37, "status": "checked" }, "enrollment_index": 281, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_zero]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "6725aff1d9e2760b38044d0d5c67461785ec808f2fb3eee19f02d31467401804", "membership": "stable", "name": "pow_zero", "proof_tag": "PA004B", "provenance": [ "stable" ], "script": [ "intro a", "intro e", "intro n", "intro he", "intro hpow", "rewrite he at hpow", "rewrite he at hpow", "rewrite he at hpow", "rewrite he at hpow", "cases hpow", "cases hpow_witness", "cases hpow_witness_witness", "specialize beta_product_zero x", "specialize beta_product_zero x1", "specialize beta_product_zero n", "apply beta_product_zero", "exact hpow_witness_witness_right" ], "script_sha256": "e9ff83aeb61ba2415c0b4dd423c1214dda0f51ed17b4808b8347017d24e63f54", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a e n. e = 0 -> (exists ff_b_z ff_c_z. ((forall ff_i_z_repeat. (exists ff_lt_z_repeat_bound. ff_lt_z_repeat_bound + S ff_i_z_repeat = e) -> (((exists ff_h_z_repeat_decoded. ff_h_z_repeat_decoded + S (a) = S ((S (ff_i_z_repeat)) * ff_c_z)) /\\ exists ff_q_z_repeat_decoded. ff_b_z = ff_q_z_repeat_decoded * S ((S (ff_i_z_repeat)) * ff_c_z) + (a)))) /\\ (exists ff_u_z_product ff_v_z_product. ((((exists ff_h_z_product_start. ff_h_z_product_start + S (1) = S ((S (0)) * ff_v_z_product)) /\\ exists ff_q_z_product_start. ff_u_z_product = ff_q_z_product_start * S ((S (0)) * ff_v_z_product) + (1))) /\\ ((((exists ff_h_z_product_terminal. ff_h_z_product_terminal + S (n) = S ((S (e)) * ff_v_z_product)) /\\ exists ff_q_z_product_terminal. ff_u_z_product = ff_q_z_product_terminal * S ((S (e)) * ff_v_z_product) + (n))) /\\ forall ff_i_z_product. (exists ff_lt_z_product_bound. ff_lt_z_product_bound + S ff_i_z_product = e) -> exists ff_p_z_product ff_r_z_product ff_s_z_product. ((((exists ff_h_z_product_factor. ff_h_z_product_factor + S (ff_p_z_product) = S ((S (ff_i_z_product)) * ff_c_z)) /\\ exists ff_q_z_product_factor. ff_b_z = ff_q_z_product_factor * S ((S (ff_i_z_product)) * ff_c_z) + (ff_p_z_product))) /\\ ((((exists ff_h_z_product_partial. ff_h_z_product_partial + S (ff_r_z_product) = S ((S (ff_i_z_product)) * ff_v_z_product)) /\\ exists ff_q_z_product_partial. ff_u_z_product = ff_q_z_product_partial * S ((S (ff_i_z_product)) * ff_v_z_product) + (ff_r_z_product))) /\\ ((((exists ff_h_z_product_successor. ff_h_z_product_successor + S (ff_s_z_product) = S ((S (S ff_i_z_product)) * ff_v_z_product)) /\\ exists ff_q_z_product_successor. ff_u_z_product = ff_q_z_product_successor * S ((S (S ff_i_z_product)) * ff_v_z_product) + (ff_s_z_product))) /\\ ff_s_z_product = ff_r_z_product * ff_p_z_product)))))))) -> n = 1", "statement_sha256": "4ce6988b5b348c217d652275b9972c4ff3171def98ff83629fd3bd581ae23db4", "summary": "The relational zeroth power is one.", "summary_sha256": "8f9e2cb5a8805c803cd642fbe10d192e544c463523c25e913faa5b960bf08284" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_product_zero" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_zero]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_zero", "parent_alpha_version": "v34", "proof_bundle_node_id": 157, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_zero", "script": [ "intro a", "intro e", "intro n", "intro he", "intro hpow", "rewrite he at hpow", "rewrite he at hpow", "rewrite he at hpow", "rewrite he at hpow", "cases hpow", "cases hpow_witness", "cases hpow_witness_witness", "specialize beta_product_zero x", "specialize beta_product_zero x1", "specialize beta_product_zero n", "apply beta_product_zero", "exact hpow_witness_witness_right" ], "script_sha256": "e9ff83aeb61ba2415c0b4dd423c1214dda0f51ed17b4808b8347017d24e63f54", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a e n. e = 0 -> (exists ff_b_z ff_c_z. ((forall ff_i_z_repeat. (exists ff_lt_z_repeat_bound. ff_lt_z_repeat_bound + S ff_i_z_repeat = e) -> (((exists ff_h_z_repeat_decoded. ff_h_z_repeat_decoded + S (a) = S ((S (ff_i_z_repeat)) * ff_c_z)) /\\ exists ff_q_z_repeat_decoded. ff_b_z = ff_q_z_repeat_decoded * S ((S (ff_i_z_repeat)) * ff_c_z) + (a)))) /\\ (exists ff_u_z_product ff_v_z_product. ((((exists ff_h_z_product_start. ff_h_z_product_start + S (1) = S ((S (0)) * ff_v_z_product)) /\\ exists ff_q_z_product_start. ff_u_z_product = ff_q_z_product_start * S ((S (0)) * ff_v_z_product) + (1))) /\\ ((((exists ff_h_z_product_terminal. ff_h_z_product_terminal + S (n) = S ((S (e)) * ff_v_z_product)) /\\ exists ff_q_z_product_terminal. ff_u_z_product = ff_q_z_product_terminal * S ((S (e)) * ff_v_z_product) + (n))) /\\ forall ff_i_z_product. (exists ff_lt_z_product_bound. ff_lt_z_product_bound + S ff_i_z_product = e) -> exists ff_p_z_product ff_r_z_product ff_s_z_product. ((((exists ff_h_z_product_factor. ff_h_z_product_factor + S (ff_p_z_product) = S ((S (ff_i_z_product)) * ff_c_z)) /\\ exists ff_q_z_product_factor. ff_b_z = ff_q_z_product_factor * S ((S (ff_i_z_product)) * ff_c_z) + (ff_p_z_product))) /\\ ((((exists ff_h_z_product_partial. ff_h_z_product_partial + S (ff_r_z_product) = S ((S (ff_i_z_product)) * ff_v_z_product)) /\\ exists ff_q_z_product_partial. ff_u_z_product = ff_q_z_product_partial * S ((S (ff_i_z_product)) * ff_v_z_product) + (ff_r_z_product))) /\\ ((((exists ff_h_z_product_successor. ff_h_z_product_successor + S (ff_s_z_product) = S ((S (S ff_i_z_product)) * ff_v_z_product)) /\\ exists ff_q_z_product_successor. ff_u_z_product = ff_q_z_product_successor * S ((S (S ff_i_z_product)) * ff_v_z_product) + (ff_s_z_product))) /\\ ff_s_z_product = ff_r_z_product * ff_p_z_product)))))))) -> n = 1", "statement_sha256": "4ce6988b5b348c217d652275b9972c4ff3171def98ff83629fd3bd581ae23db4" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_functional", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_repeat_transport_entry", "beta_product_transport_prefix", "beta_product_functional" ], "dependencies_sha256": "fe97151f16770cac3e15bbe326b267e21d5e832f95a2f0aa2d226a48251861ae", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "93be6fa1bf4660de3047ea977bd40d1d2725bbb6805d50cbf783f150d4ab770d", "cut_nodes": 71, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 63, "proof_edges": 1150, "proof_nodes": 2705, "proof_objects": 1111, "reused_objects": 40, "status": "checked" }, "enrollment_index": 282, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_functional]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "bd04d9186de1f0319b401f83abf85be017b8c4b34bff4f66fe7069706b83c84a", "membership": "stable", "name": "pow_functional", "proof_tag": "PA005G", "provenance": [ "stable" ], "script": [ "intro a", "intro e", "intro n", "intro m", "intro hn", "intro hm", "cases hn", "cases hn_witness", "cases hn_witness_witness", "cases hm", "cases hm_witness", "cases hm_witness_witness", "have htransport : exists ff_u_transport ff_v_transport. ((((exists ff_h_transport_start. ff_h_transport_start + S (1) = S ((S (0)) * ff_v_transport)) /\\ exists ff_q_transport_start. ff_u_transport = ff_q_transport_start * S ((S (0)) * ff_v_transport) + (1))) /\\ ((((exists ff_h_transport_terminal. ff_h_transport_terminal + S (n) = S ((S (e)) * ff_v_transport)) /\\ exists ff_q_transport_terminal. ff_u_transport = ff_q_transport_terminal * S ((S (e)) * ff_v_transport) + (n))) /\\ forall ff_i_transport. (exists ff_lt_transport_bound. ff_lt_transport_bound + S ff_i_transport = e) -> exists ff_p_transport ff_r_transport ff_s_transport. ((((exists ff_h_transport_factor. ff_h_transport_factor + S (ff_p_transport) = S ((S (ff_i_transport)) * x3)) /\\ exists ff_q_transport_factor. x2 = ff_q_transport_factor * S ((S (ff_i_transport)) * x3) + (ff_p_transport))) /\\ ((((exists ff_h_transport_partial. ff_h_transport_partial + S (ff_r_transport) = S ((S (ff_i_transport)) * ff_v_transport)) /\\ exists ff_q_transport_partial. ff_u_transport = ff_q_transport_partial * S ((S (ff_i_transport)) * ff_v_transport) + (ff_r_transport))) /\\ ((((exists ff_h_transport_successor. ff_h_transport_successor + S (ff_s_transport) = S ((S (S ff_i_transport)) * ff_v_transport)) /\\ exists ff_q_transport_successor. ff_u_transport = ff_q_transport_successor * S ((S (S ff_i_transport)) * ff_v_transport) + (ff_s_transport))) /\\ ff_s_transport = ff_r_transport * ff_p_transport)))))", "specialize beta_product_transport_prefix x", "specialize beta_product_transport_prefix x1", "specialize beta_product_transport_prefix x2", "specialize beta_product_transport_prefix x3", "specialize beta_product_transport_prefix e", "specialize beta_product_transport_prefix n", "apply beta_product_transport_prefix", "exact hn_witness_witness_right", "intro i", "intro p", "intro hi", "intro hp", "specialize beta_repeat_transport_entry x", "specialize beta_repeat_transport_entry x1", "specialize beta_repeat_transport_entry x2", "specialize beta_repeat_transport_entry x3", "specialize beta_repeat_transport_entry a", "specialize beta_repeat_transport_entry e", "have hentries : forall i p. (exists h. h + S i = e) -> (((exists ff_h_pow_transport_l. ff_h_pow_transport_l + S (p) = S ((S (i)) * x1)) /\\ exists ff_q_pow_transport_l. x = ff_q_pow_transport_l * S ((S (i)) * x1) + (p))) -> (((exists ff_h_pow_transport_r. ff_h_pow_transport_r + S (p) = S ((S (i)) * x3)) /\\ exists ff_q_pow_transport_r. x2 = ff_q_pow_transport_r * S ((S (i)) * x3) + (p)))", "apply beta_repeat_transport_entry", "exact hn_witness_witness_left", "exact hm_witness_witness_left", "specialize hentries i", "specialize hentries p", "apply hentries", "exact hi", "exact hp", "cases htransport", "cases htransport_witness", "cases hm_witness_witness_right", "cases hm_witness_witness_right_witness", "specialize beta_product_functional x2", "specialize beta_product_functional x3", "specialize beta_product_functional e", "specialize beta_product_functional n", "specialize beta_product_functional x4", "specialize beta_product_functional x5", "specialize beta_product_functional m", "specialize beta_product_functional x6", "specialize beta_product_functional x7", "apply beta_product_functional", "exact htransport_witness_witness", "exact hm_witness_witness_right_witness_witness" ], "script_sha256": "51a79d17b30f564a5bea434a068dfff8b442f8a3a6ead2b0007bf4937b102914", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a e n m. (exists ff_b_l ff_c_l. ((forall ff_i_l_repeat. (exists ff_lt_l_repeat_bound. ff_lt_l_repeat_bound + S ff_i_l_repeat = e) -> (((exists ff_h_l_repeat_decoded. ff_h_l_repeat_decoded + S (a) = S ((S (ff_i_l_repeat)) * ff_c_l)) /\\ exists ff_q_l_repeat_decoded. ff_b_l = ff_q_l_repeat_decoded * S ((S (ff_i_l_repeat)) * ff_c_l) + (a)))) /\\ (exists ff_u_l_product ff_v_l_product. ((((exists ff_h_l_product_start. ff_h_l_product_start + S (1) = S ((S (0)) * ff_v_l_product)) /\\ exists ff_q_l_product_start. ff_u_l_product = ff_q_l_product_start * S ((S (0)) * ff_v_l_product) + (1))) /\\ ((((exists ff_h_l_product_terminal. ff_h_l_product_terminal + S (n) = S ((S (e)) * ff_v_l_product)) /\\ exists ff_q_l_product_terminal. ff_u_l_product = ff_q_l_product_terminal * S ((S (e)) * ff_v_l_product) + (n))) /\\ forall ff_i_l_product. (exists ff_lt_l_product_bound. ff_lt_l_product_bound + S ff_i_l_product = e) -> exists ff_p_l_product ff_r_l_product ff_s_l_product. ((((exists ff_h_l_product_factor. ff_h_l_product_factor + S (ff_p_l_product) = S ((S (ff_i_l_product)) * ff_c_l)) /\\ exists ff_q_l_product_factor. ff_b_l = ff_q_l_product_factor * S ((S (ff_i_l_product)) * ff_c_l) + (ff_p_l_product))) /\\ ((((exists ff_h_l_product_partial. ff_h_l_product_partial + S (ff_r_l_product) = S ((S (ff_i_l_product)) * ff_v_l_product)) /\\ exists ff_q_l_product_partial. ff_u_l_product = ff_q_l_product_partial * S ((S (ff_i_l_product)) * ff_v_l_product) + (ff_r_l_product))) /\\ ((((exists ff_h_l_product_successor. ff_h_l_product_successor + S (ff_s_l_product) = S ((S (S ff_i_l_product)) * ff_v_l_product)) /\\ exists ff_q_l_product_successor. ff_u_l_product = ff_q_l_product_successor * S ((S (S ff_i_l_product)) * ff_v_l_product) + (ff_s_l_product))) /\\ ff_s_l_product = ff_r_l_product * ff_p_l_product)))))))) -> (exists ff_b_r ff_c_r. ((forall ff_i_r_repeat. (exists ff_lt_r_repeat_bound. ff_lt_r_repeat_bound + S ff_i_r_repeat = e) -> (((exists ff_h_r_repeat_decoded. ff_h_r_repeat_decoded + S (a) = S ((S (ff_i_r_repeat)) * ff_c_r)) /\\ exists ff_q_r_repeat_decoded. ff_b_r = ff_q_r_repeat_decoded * S ((S (ff_i_r_repeat)) * ff_c_r) + (a)))) /\\ (exists ff_u_r_product ff_v_r_product. ((((exists ff_h_r_product_start. ff_h_r_product_start + S (1) = S ((S (0)) * ff_v_r_product)) /\\ exists ff_q_r_product_start. ff_u_r_product = ff_q_r_product_start * S ((S (0)) * ff_v_r_product) + (1))) /\\ ((((exists ff_h_r_product_terminal. ff_h_r_product_terminal + S (m) = S ((S (e)) * ff_v_r_product)) /\\ exists ff_q_r_product_terminal. ff_u_r_product = ff_q_r_product_terminal * S ((S (e)) * ff_v_r_product) + (m))) /\\ forall ff_i_r_product. (exists ff_lt_r_product_bound. ff_lt_r_product_bound + S ff_i_r_product = e) -> exists ff_p_r_product ff_r_r_product ff_s_r_product. ((((exists ff_h_r_product_factor. ff_h_r_product_factor + S (ff_p_r_product) = S ((S (ff_i_r_product)) * ff_c_r)) /\\ exists ff_q_r_product_factor. ff_b_r = ff_q_r_product_factor * S ((S (ff_i_r_product)) * ff_c_r) + (ff_p_r_product))) /\\ ((((exists ff_h_r_product_partial. ff_h_r_product_partial + S (ff_r_r_product) = S ((S (ff_i_r_product)) * ff_v_r_product)) /\\ exists ff_q_r_product_partial. ff_u_r_product = ff_q_r_product_partial * S ((S (ff_i_r_product)) * ff_v_r_product) + (ff_r_r_product))) /\\ ((((exists ff_h_r_product_successor. ff_h_r_product_successor + S (ff_s_r_product) = S ((S (S ff_i_r_product)) * ff_v_r_product)) /\\ exists ff_q_r_product_successor. ff_u_r_product = ff_q_r_product_successor * S ((S (S ff_i_r_product)) * ff_v_r_product) + (ff_s_r_product))) /\\ ff_s_r_product = ff_r_r_product * ff_p_r_product)))))))) -> n = m", "statement_sha256": "a165bbcd196d1d36843397b85e6b018527828d717d34f17691cd3330f97b751d", "summary": "Relational powers have a unique natural value.", "summary_sha256": "543a6a00b26ec602109df44aa430621e980c95f5d84babac9debb1d85e429d28" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_repeat_transport_entry", "beta_product_transport_prefix", "beta_product_functional" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_functional]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_functional", "parent_alpha_version": "v34", "proof_bundle_node_id": 158, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_functional", "script": [ "intro a", "intro e", "intro n", "intro m", "intro hn", "intro hm", "cases hn", "cases hn_witness", "cases hn_witness_witness", "cases hm", "cases hm_witness", "cases hm_witness_witness", "have htransport : exists ff_u_transport ff_v_transport. ((((exists ff_h_transport_start. ff_h_transport_start + S (1) = S ((S (0)) * ff_v_transport)) /\\ exists ff_q_transport_start. ff_u_transport = ff_q_transport_start * S ((S (0)) * ff_v_transport) + (1))) /\\ ((((exists ff_h_transport_terminal. ff_h_transport_terminal + S (n) = S ((S (e)) * ff_v_transport)) /\\ exists ff_q_transport_terminal. ff_u_transport = ff_q_transport_terminal * S ((S (e)) * ff_v_transport) + (n))) /\\ forall ff_i_transport. (exists ff_lt_transport_bound. ff_lt_transport_bound + S ff_i_transport = e) -> exists ff_p_transport ff_r_transport ff_s_transport. ((((exists ff_h_transport_factor. ff_h_transport_factor + S (ff_p_transport) = S ((S (ff_i_transport)) * x3)) /\\ exists ff_q_transport_factor. x2 = ff_q_transport_factor * S ((S (ff_i_transport)) * x3) + (ff_p_transport))) /\\ ((((exists ff_h_transport_partial. ff_h_transport_partial + S (ff_r_transport) = S ((S (ff_i_transport)) * ff_v_transport)) /\\ exists ff_q_transport_partial. ff_u_transport = ff_q_transport_partial * S ((S (ff_i_transport)) * ff_v_transport) + (ff_r_transport))) /\\ ((((exists ff_h_transport_successor. ff_h_transport_successor + S (ff_s_transport) = S ((S (S ff_i_transport)) * ff_v_transport)) /\\ exists ff_q_transport_successor. ff_u_transport = ff_q_transport_successor * S ((S (S ff_i_transport)) * ff_v_transport) + (ff_s_transport))) /\\ ff_s_transport = ff_r_transport * ff_p_transport)))))", "specialize beta_product_transport_prefix x", "specialize beta_product_transport_prefix x1", "specialize beta_product_transport_prefix x2", "specialize beta_product_transport_prefix x3", "specialize beta_product_transport_prefix e", "specialize beta_product_transport_prefix n", "apply beta_product_transport_prefix", "exact hn_witness_witness_right", "intro i", "intro p", "intro hi", "intro hp", "specialize beta_repeat_transport_entry x", "specialize beta_repeat_transport_entry x1", "specialize beta_repeat_transport_entry x2", "specialize beta_repeat_transport_entry x3", "specialize beta_repeat_transport_entry a", "specialize beta_repeat_transport_entry e", "have hentries : forall i p. (exists h. h + S i = e) -> (((exists ff_h_pow_transport_l. ff_h_pow_transport_l + S (p) = S ((S (i)) * x1)) /\\ exists ff_q_pow_transport_l. x = ff_q_pow_transport_l * S ((S (i)) * x1) + (p))) -> (((exists ff_h_pow_transport_r. ff_h_pow_transport_r + S (p) = S ((S (i)) * x3)) /\\ exists ff_q_pow_transport_r. x2 = ff_q_pow_transport_r * S ((S (i)) * x3) + (p)))", "apply beta_repeat_transport_entry", "exact hn_witness_witness_left", "exact hm_witness_witness_left", "specialize hentries i", "specialize hentries p", "apply hentries", "exact hi", "exact hp", "cases htransport", "cases htransport_witness", "cases hm_witness_witness_right", "cases hm_witness_witness_right_witness", "specialize beta_product_functional x2", "specialize beta_product_functional x3", "specialize beta_product_functional e", "specialize beta_product_functional n", "specialize beta_product_functional x4", "specialize beta_product_functional x5", "specialize beta_product_functional m", "specialize beta_product_functional x6", "specialize beta_product_functional x7", "apply beta_product_functional", "exact htransport_witness_witness", "exact hm_witness_witness_right_witness_witness" ], "script_sha256": "51a79d17b30f564a5bea434a068dfff8b442f8a3a6ead2b0007bf4937b102914", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a e n m. (exists ff_b_l ff_c_l. ((forall ff_i_l_repeat. (exists ff_lt_l_repeat_bound. ff_lt_l_repeat_bound + S ff_i_l_repeat = e) -> (((exists ff_h_l_repeat_decoded. ff_h_l_repeat_decoded + S (a) = S ((S (ff_i_l_repeat)) * ff_c_l)) /\\ exists ff_q_l_repeat_decoded. ff_b_l = ff_q_l_repeat_decoded * S ((S (ff_i_l_repeat)) * ff_c_l) + (a)))) /\\ (exists ff_u_l_product ff_v_l_product. ((((exists ff_h_l_product_start. ff_h_l_product_start + S (1) = S ((S (0)) * ff_v_l_product)) /\\ exists ff_q_l_product_start. ff_u_l_product = ff_q_l_product_start * S ((S (0)) * ff_v_l_product) + (1))) /\\ ((((exists ff_h_l_product_terminal. ff_h_l_product_terminal + S (n) = S ((S (e)) * ff_v_l_product)) /\\ exists ff_q_l_product_terminal. ff_u_l_product = ff_q_l_product_terminal * S ((S (e)) * ff_v_l_product) + (n))) /\\ forall ff_i_l_product. (exists ff_lt_l_product_bound. ff_lt_l_product_bound + S ff_i_l_product = e) -> exists ff_p_l_product ff_r_l_product ff_s_l_product. ((((exists ff_h_l_product_factor. ff_h_l_product_factor + S (ff_p_l_product) = S ((S (ff_i_l_product)) * ff_c_l)) /\\ exists ff_q_l_product_factor. ff_b_l = ff_q_l_product_factor * S ((S (ff_i_l_product)) * ff_c_l) + (ff_p_l_product))) /\\ ((((exists ff_h_l_product_partial. ff_h_l_product_partial + S (ff_r_l_product) = S ((S (ff_i_l_product)) * ff_v_l_product)) /\\ exists ff_q_l_product_partial. ff_u_l_product = ff_q_l_product_partial * S ((S (ff_i_l_product)) * ff_v_l_product) + (ff_r_l_product))) /\\ ((((exists ff_h_l_product_successor. ff_h_l_product_successor + S (ff_s_l_product) = S ((S (S ff_i_l_product)) * ff_v_l_product)) /\\ exists ff_q_l_product_successor. ff_u_l_product = ff_q_l_product_successor * S ((S (S ff_i_l_product)) * ff_v_l_product) + (ff_s_l_product))) /\\ ff_s_l_product = ff_r_l_product * ff_p_l_product)))))))) -> (exists ff_b_r ff_c_r. ((forall ff_i_r_repeat. (exists ff_lt_r_repeat_bound. ff_lt_r_repeat_bound + S ff_i_r_repeat = e) -> (((exists ff_h_r_repeat_decoded. ff_h_r_repeat_decoded + S (a) = S ((S (ff_i_r_repeat)) * ff_c_r)) /\\ exists ff_q_r_repeat_decoded. ff_b_r = ff_q_r_repeat_decoded * S ((S (ff_i_r_repeat)) * ff_c_r) + (a)))) /\\ (exists ff_u_r_product ff_v_r_product. ((((exists ff_h_r_product_start. ff_h_r_product_start + S (1) = S ((S (0)) * ff_v_r_product)) /\\ exists ff_q_r_product_start. ff_u_r_product = ff_q_r_product_start * S ((S (0)) * ff_v_r_product) + (1))) /\\ ((((exists ff_h_r_product_terminal. ff_h_r_product_terminal + S (m) = S ((S (e)) * ff_v_r_product)) /\\ exists ff_q_r_product_terminal. ff_u_r_product = ff_q_r_product_terminal * S ((S (e)) * ff_v_r_product) + (m))) /\\ forall ff_i_r_product. (exists ff_lt_r_product_bound. ff_lt_r_product_bound + S ff_i_r_product = e) -> exists ff_p_r_product ff_r_r_product ff_s_r_product. ((((exists ff_h_r_product_factor. ff_h_r_product_factor + S (ff_p_r_product) = S ((S (ff_i_r_product)) * ff_c_r)) /\\ exists ff_q_r_product_factor. ff_b_r = ff_q_r_product_factor * S ((S (ff_i_r_product)) * ff_c_r) + (ff_p_r_product))) /\\ ((((exists ff_h_r_product_partial. ff_h_r_product_partial + S (ff_r_r_product) = S ((S (ff_i_r_product)) * ff_v_r_product)) /\\ exists ff_q_r_product_partial. ff_u_r_product = ff_q_r_product_partial * S ((S (ff_i_r_product)) * ff_v_r_product) + (ff_r_r_product))) /\\ ((((exists ff_h_r_product_successor. ff_h_r_product_successor + S (ff_s_r_product) = S ((S (S ff_i_r_product)) * ff_v_r_product)) /\\ exists ff_q_r_product_successor. ff_u_r_product = ff_q_r_product_successor * S ((S (S ff_i_r_product)) * ff_v_r_product) + (ff_s_r_product))) /\\ ff_s_r_product = ff_r_r_product * ff_p_r_product)))))))) -> n = m", "statement_sha256": "a165bbcd196d1d36843397b85e6b018527828d717d34f17691cd3330f97b751d" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_successor_decompose", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_product_succ_decompose", "beta_repeat_entry_eq", "le_refl", "le_succ" ], "dependencies_sha256": "ddbff7b115478609d487bb1d7e0b3fa4fe0fccac0f6d7e29dab78cbfe87e8d43", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "70defcae97b2be790af783d9f5f6ca51a853b585ac28ab575dddf8be61fd51ef", "cut_nodes": 72, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 63, "proof_edges": 923, "proof_nodes": 2541, "proof_objects": 882, "reused_objects": 42, "status": "checked" }, "enrollment_index": 283, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_successor_decompose]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "69ddc3dab4150854c8fa504ed49046e0f54c924a1bf8afa83b23fefa2fed1c50", "membership": "stable", "name": "pow_successor_decompose", "proof_tag": "PA004D", "provenance": [ "stable" ], "script": [ "intro a", "intro e", "intro se", "intro n", "intro hse", "intro hpow", "rewrite hse at hpow", "rewrite hse at hpow", "rewrite hse at hpow", "rewrite hse at hpow", "cases hpow", "cases hpow_witness", "cases hpow_witness_witness", "have hdecomp : exists p r. (((exists ff_h_pow_succ_factor. ff_h_pow_succ_factor + S (p) = S ((S (e)) * x1)) /\\ exists ff_q_pow_succ_factor. x = ff_q_pow_succ_factor * S ((S (e)) * x1) + (p))) /\\ ((exists ff_u_pow_succ_prefix ff_v_pow_succ_prefix. ((((exists ff_h_pow_succ_prefix_start. ff_h_pow_succ_prefix_start + S (1) = S ((S (0)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_start. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_start * S ((S (0)) * ff_v_pow_succ_prefix) + (1))) /\\ ((((exists ff_h_pow_succ_prefix_terminal. ff_h_pow_succ_prefix_terminal + S (r) = S ((S (e)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_terminal. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_terminal * S ((S (e)) * ff_v_pow_succ_prefix) + (r))) /\\ forall ff_i_pow_succ_prefix. (exists ff_lt_pow_succ_prefix_bound. ff_lt_pow_succ_prefix_bound + S ff_i_pow_succ_prefix = e) -> exists ff_p_pow_succ_prefix ff_r_pow_succ_prefix ff_s_pow_succ_prefix. ((((exists ff_h_pow_succ_prefix_factor. ff_h_pow_succ_prefix_factor + S (ff_p_pow_succ_prefix) = S ((S (ff_i_pow_succ_prefix)) * x1)) /\\ exists ff_q_pow_succ_prefix_factor. x = ff_q_pow_succ_prefix_factor * S ((S (ff_i_pow_succ_prefix)) * x1) + (ff_p_pow_succ_prefix))) /\\ ((((exists ff_h_pow_succ_prefix_partial. ff_h_pow_succ_prefix_partial + S (ff_r_pow_succ_prefix) = S ((S (ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_partial. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_partial * S ((S (ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix) + (ff_r_pow_succ_prefix))) /\\ ((((exists ff_h_pow_succ_prefix_successor. ff_h_pow_succ_prefix_successor + S (ff_s_pow_succ_prefix) = S ((S (S ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_successor. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_successor * S ((S (S ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix) + (ff_s_pow_succ_prefix))) /\\ ff_s_pow_succ_prefix = ff_r_pow_succ_prefix * ff_p_pow_succ_prefix)))))) /\\ n = r * p)", "specialize beta_product_succ_decompose x", "specialize beta_product_succ_decompose x1", "specialize beta_product_succ_decompose e", "specialize beta_product_succ_decompose n", "apply beta_product_succ_decompose", "exact hpow_witness_witness_right", "cases hdecomp", "cases hdecomp_witness", "cases hdecomp_witness_witness", "cases hdecomp_witness_witness_right", "have hpa : x2 = a", "specialize beta_repeat_entry_eq x", "specialize beta_repeat_entry_eq x1", "specialize beta_repeat_entry_eq a", "specialize beta_repeat_entry_eq (S e)", "specialize beta_repeat_entry_eq e", "specialize beta_repeat_entry_eq x2", "apply beta_repeat_entry_eq", "exact hpow_witness_witness_left", "specialize le_refl (S e)", "exact le_refl", "exact hdecomp_witness_witness_left", "exists x3", "split", "exists x", "exists x1", "split", "intro i", "intro hi", "specialize hpow_witness_witness_left i", "apply hpow_witness_witness_left", "specialize le_succ (S i)", "specialize le_succ e", "apply le_succ", "exact hi", "exact hdecomp_witness_witness_right_left", "trans x3 * x2", "exact hdecomp_witness_witness_right_right", "rewrite hpa", "refl" ], "script_sha256": "0ab836135b1de98e16545d52989a7943b380980a458f3be10b977099b653ea13", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a e se n. se = S e -> (exists ff_b_s ff_c_s. ((forall ff_i_s_repeat. (exists ff_lt_s_repeat_bound. ff_lt_s_repeat_bound + S ff_i_s_repeat = se) -> (((exists ff_h_s_repeat_decoded. ff_h_s_repeat_decoded + S (a) = S ((S (ff_i_s_repeat)) * ff_c_s)) /\\ exists ff_q_s_repeat_decoded. ff_b_s = ff_q_s_repeat_decoded * S ((S (ff_i_s_repeat)) * ff_c_s) + (a)))) /\\ (exists ff_u_s_product ff_v_s_product. ((((exists ff_h_s_product_start. ff_h_s_product_start + S (1) = S ((S (0)) * ff_v_s_product)) /\\ exists ff_q_s_product_start. ff_u_s_product = ff_q_s_product_start * S ((S (0)) * ff_v_s_product) + (1))) /\\ ((((exists ff_h_s_product_terminal. ff_h_s_product_terminal + S (n) = S ((S (se)) * ff_v_s_product)) /\\ exists ff_q_s_product_terminal. ff_u_s_product = ff_q_s_product_terminal * S ((S (se)) * ff_v_s_product) + (n))) /\\ forall ff_i_s_product. (exists ff_lt_s_product_bound. ff_lt_s_product_bound + S ff_i_s_product = se) -> exists ff_p_s_product ff_r_s_product ff_s_s_product. ((((exists ff_h_s_product_factor. ff_h_s_product_factor + S (ff_p_s_product) = S ((S (ff_i_s_product)) * ff_c_s)) /\\ exists ff_q_s_product_factor. ff_b_s = ff_q_s_product_factor * S ((S (ff_i_s_product)) * ff_c_s) + (ff_p_s_product))) /\\ ((((exists ff_h_s_product_partial. ff_h_s_product_partial + S (ff_r_s_product) = S ((S (ff_i_s_product)) * ff_v_s_product)) /\\ exists ff_q_s_product_partial. ff_u_s_product = ff_q_s_product_partial * S ((S (ff_i_s_product)) * ff_v_s_product) + (ff_r_s_product))) /\\ ((((exists ff_h_s_product_successor. ff_h_s_product_successor + S (ff_s_s_product) = S ((S (S ff_i_s_product)) * ff_v_s_product)) /\\ exists ff_q_s_product_successor. ff_u_s_product = ff_q_s_product_successor * S ((S (S ff_i_s_product)) * ff_v_s_product) + (ff_s_s_product))) /\\ ff_s_s_product = ff_r_s_product * ff_p_s_product)))))))) -> exists r. (exists ff_b_p ff_c_p. ((forall ff_i_p_repeat. (exists ff_lt_p_repeat_bound. ff_lt_p_repeat_bound + S ff_i_p_repeat = e) -> (((exists ff_h_p_repeat_decoded. ff_h_p_repeat_decoded + S (a) = S ((S (ff_i_p_repeat)) * ff_c_p)) /\\ exists ff_q_p_repeat_decoded. ff_b_p = ff_q_p_repeat_decoded * S ((S (ff_i_p_repeat)) * ff_c_p) + (a)))) /\\ (exists ff_u_p_product ff_v_p_product. ((((exists ff_h_p_product_start. ff_h_p_product_start + S (1) = S ((S (0)) * ff_v_p_product)) /\\ exists ff_q_p_product_start. ff_u_p_product = ff_q_p_product_start * S ((S (0)) * ff_v_p_product) + (1))) /\\ ((((exists ff_h_p_product_terminal. ff_h_p_product_terminal + S (r) = S ((S (e)) * ff_v_p_product)) /\\ exists ff_q_p_product_terminal. ff_u_p_product = ff_q_p_product_terminal * S ((S (e)) * ff_v_p_product) + (r))) /\\ forall ff_i_p_product. (exists ff_lt_p_product_bound. ff_lt_p_product_bound + S ff_i_p_product = e) -> exists ff_p_p_product ff_r_p_product ff_s_p_product. ((((exists ff_h_p_product_factor. ff_h_p_product_factor + S (ff_p_p_product) = S ((S (ff_i_p_product)) * ff_c_p)) /\\ exists ff_q_p_product_factor. ff_b_p = ff_q_p_product_factor * S ((S (ff_i_p_product)) * ff_c_p) + (ff_p_p_product))) /\\ ((((exists ff_h_p_product_partial. ff_h_p_product_partial + S (ff_r_p_product) = S ((S (ff_i_p_product)) * ff_v_p_product)) /\\ exists ff_q_p_product_partial. ff_u_p_product = ff_q_p_product_partial * S ((S (ff_i_p_product)) * ff_v_p_product) + (ff_r_p_product))) /\\ ((((exists ff_h_p_product_successor. ff_h_p_product_successor + S (ff_s_p_product) = S ((S (S ff_i_p_product)) * ff_v_p_product)) /\\ exists ff_q_p_product_successor. ff_u_p_product = ff_q_p_product_successor * S ((S (S ff_i_p_product)) * ff_v_p_product) + (ff_s_p_product))) /\\ ff_s_p_product = ff_r_p_product * ff_p_p_product)))))))) /\\ n = r * a", "statement_sha256": "2da86da57081e7d2fe9316cd1f50ad46d586cf8fbba80c04d7a65930124861c5", "summary": "A successor relational power is its predecessor power times the base.", "summary_sha256": "3ccf7ddffea66e0cdd09780ff6647d8442d87e1bd410790360b0b73a9f0390ed" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_product_succ_decompose", "beta_repeat_entry_eq", "le_refl", "le_succ" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_successor_decompose]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_successor_decompose", "parent_alpha_version": "v34", "proof_bundle_node_id": 159, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_successor_decompose", "script": [ "intro a", "intro e", "intro se", "intro n", "intro hse", "intro hpow", "rewrite hse at hpow", "rewrite hse at hpow", "rewrite hse at hpow", "rewrite hse at hpow", "cases hpow", "cases hpow_witness", "cases hpow_witness_witness", "have hdecomp : exists p r. (((exists ff_h_pow_succ_factor. ff_h_pow_succ_factor + S (p) = S ((S (e)) * x1)) /\\ exists ff_q_pow_succ_factor. x = ff_q_pow_succ_factor * S ((S (e)) * x1) + (p))) /\\ ((exists ff_u_pow_succ_prefix ff_v_pow_succ_prefix. ((((exists ff_h_pow_succ_prefix_start. ff_h_pow_succ_prefix_start + S (1) = S ((S (0)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_start. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_start * S ((S (0)) * ff_v_pow_succ_prefix) + (1))) /\\ ((((exists ff_h_pow_succ_prefix_terminal. ff_h_pow_succ_prefix_terminal + S (r) = S ((S (e)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_terminal. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_terminal * S ((S (e)) * ff_v_pow_succ_prefix) + (r))) /\\ forall ff_i_pow_succ_prefix. (exists ff_lt_pow_succ_prefix_bound. ff_lt_pow_succ_prefix_bound + S ff_i_pow_succ_prefix = e) -> exists ff_p_pow_succ_prefix ff_r_pow_succ_prefix ff_s_pow_succ_prefix. ((((exists ff_h_pow_succ_prefix_factor. ff_h_pow_succ_prefix_factor + S (ff_p_pow_succ_prefix) = S ((S (ff_i_pow_succ_prefix)) * x1)) /\\ exists ff_q_pow_succ_prefix_factor. x = ff_q_pow_succ_prefix_factor * S ((S (ff_i_pow_succ_prefix)) * x1) + (ff_p_pow_succ_prefix))) /\\ ((((exists ff_h_pow_succ_prefix_partial. ff_h_pow_succ_prefix_partial + S (ff_r_pow_succ_prefix) = S ((S (ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_partial. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_partial * S ((S (ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix) + (ff_r_pow_succ_prefix))) /\\ ((((exists ff_h_pow_succ_prefix_successor. ff_h_pow_succ_prefix_successor + S (ff_s_pow_succ_prefix) = S ((S (S ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix)) /\\ exists ff_q_pow_succ_prefix_successor. ff_u_pow_succ_prefix = ff_q_pow_succ_prefix_successor * S ((S (S ff_i_pow_succ_prefix)) * ff_v_pow_succ_prefix) + (ff_s_pow_succ_prefix))) /\\ ff_s_pow_succ_prefix = ff_r_pow_succ_prefix * ff_p_pow_succ_prefix)))))) /\\ n = r * p)", "specialize beta_product_succ_decompose x", "specialize beta_product_succ_decompose x1", "specialize beta_product_succ_decompose e", "specialize beta_product_succ_decompose n", "apply beta_product_succ_decompose", "exact hpow_witness_witness_right", "cases hdecomp", "cases hdecomp_witness", "cases hdecomp_witness_witness", "cases hdecomp_witness_witness_right", "have hpa : x2 = a", "specialize beta_repeat_entry_eq x", "specialize beta_repeat_entry_eq x1", "specialize beta_repeat_entry_eq a", "specialize beta_repeat_entry_eq (S e)", "specialize beta_repeat_entry_eq e", "specialize beta_repeat_entry_eq x2", "apply beta_repeat_entry_eq", "exact hpow_witness_witness_left", "specialize le_refl (S e)", "exact le_refl", "exact hdecomp_witness_witness_left", "exists x3", "split", "exists x", "exists x1", "split", "intro i", "intro hi", "specialize hpow_witness_witness_left i", "apply hpow_witness_witness_left", "specialize le_succ (S i)", "specialize le_succ e", "apply le_succ", "exact hi", "exact hdecomp_witness_witness_right_left", "trans x3 * x2", "exact hdecomp_witness_witness_right_right", "rewrite hpa", "refl" ], "script_sha256": "0ab836135b1de98e16545d52989a7943b380980a458f3be10b977099b653ea13", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a e se n. se = S e -> (exists ff_b_s ff_c_s. ((forall ff_i_s_repeat. (exists ff_lt_s_repeat_bound. ff_lt_s_repeat_bound + S ff_i_s_repeat = se) -> (((exists ff_h_s_repeat_decoded. ff_h_s_repeat_decoded + S (a) = S ((S (ff_i_s_repeat)) * ff_c_s)) /\\ exists ff_q_s_repeat_decoded. ff_b_s = ff_q_s_repeat_decoded * S ((S (ff_i_s_repeat)) * ff_c_s) + (a)))) /\\ (exists ff_u_s_product ff_v_s_product. ((((exists ff_h_s_product_start. ff_h_s_product_start + S (1) = S ((S (0)) * ff_v_s_product)) /\\ exists ff_q_s_product_start. ff_u_s_product = ff_q_s_product_start * S ((S (0)) * ff_v_s_product) + (1))) /\\ ((((exists ff_h_s_product_terminal. ff_h_s_product_terminal + S (n) = S ((S (se)) * ff_v_s_product)) /\\ exists ff_q_s_product_terminal. ff_u_s_product = ff_q_s_product_terminal * S ((S (se)) * ff_v_s_product) + (n))) /\\ forall ff_i_s_product. (exists ff_lt_s_product_bound. ff_lt_s_product_bound + S ff_i_s_product = se) -> exists ff_p_s_product ff_r_s_product ff_s_s_product. ((((exists ff_h_s_product_factor. ff_h_s_product_factor + S (ff_p_s_product) = S ((S (ff_i_s_product)) * ff_c_s)) /\\ exists ff_q_s_product_factor. ff_b_s = ff_q_s_product_factor * S ((S (ff_i_s_product)) * ff_c_s) + (ff_p_s_product))) /\\ ((((exists ff_h_s_product_partial. ff_h_s_product_partial + S (ff_r_s_product) = S ((S (ff_i_s_product)) * ff_v_s_product)) /\\ exists ff_q_s_product_partial. ff_u_s_product = ff_q_s_product_partial * S ((S (ff_i_s_product)) * ff_v_s_product) + (ff_r_s_product))) /\\ ((((exists ff_h_s_product_successor. ff_h_s_product_successor + S (ff_s_s_product) = S ((S (S ff_i_s_product)) * ff_v_s_product)) /\\ exists ff_q_s_product_successor. ff_u_s_product = ff_q_s_product_successor * S ((S (S ff_i_s_product)) * ff_v_s_product) + (ff_s_s_product))) /\\ ff_s_s_product = ff_r_s_product * ff_p_s_product)))))))) -> exists r. (exists ff_b_p ff_c_p. ((forall ff_i_p_repeat. (exists ff_lt_p_repeat_bound. ff_lt_p_repeat_bound + S ff_i_p_repeat = e) -> (((exists ff_h_p_repeat_decoded. ff_h_p_repeat_decoded + S (a) = S ((S (ff_i_p_repeat)) * ff_c_p)) /\\ exists ff_q_p_repeat_decoded. ff_b_p = ff_q_p_repeat_decoded * S ((S (ff_i_p_repeat)) * ff_c_p) + (a)))) /\\ (exists ff_u_p_product ff_v_p_product. ((((exists ff_h_p_product_start. ff_h_p_product_start + S (1) = S ((S (0)) * ff_v_p_product)) /\\ exists ff_q_p_product_start. ff_u_p_product = ff_q_p_product_start * S ((S (0)) * ff_v_p_product) + (1))) /\\ ((((exists ff_h_p_product_terminal. ff_h_p_product_terminal + S (r) = S ((S (e)) * ff_v_p_product)) /\\ exists ff_q_p_product_terminal. ff_u_p_product = ff_q_p_product_terminal * S ((S (e)) * ff_v_p_product) + (r))) /\\ forall ff_i_p_product. (exists ff_lt_p_product_bound. ff_lt_p_product_bound + S ff_i_p_product = e) -> exists ff_p_p_product ff_r_p_product ff_s_p_product. ((((exists ff_h_p_product_factor. ff_h_p_product_factor + S (ff_p_p_product) = S ((S (ff_i_p_product)) * ff_c_p)) /\\ exists ff_q_p_product_factor. ff_b_p = ff_q_p_product_factor * S ((S (ff_i_p_product)) * ff_c_p) + (ff_p_p_product))) /\\ ((((exists ff_h_p_product_partial. ff_h_p_product_partial + S (ff_r_p_product) = S ((S (ff_i_p_product)) * ff_v_p_product)) /\\ exists ff_q_p_product_partial. ff_u_p_product = ff_q_p_product_partial * S ((S (ff_i_p_product)) * ff_v_p_product) + (ff_r_p_product))) /\\ ((((exists ff_h_p_product_successor. ff_h_p_product_successor + S (ff_s_p_product) = S ((S (S ff_i_p_product)) * ff_v_p_product)) /\\ exists ff_q_p_product_successor. ff_u_p_product = ff_q_p_product_successor * S ((S (S ff_i_p_product)) * ff_v_p_product) + (ff_s_p_product))) /\\ ff_s_p_product = ff_r_p_product * ff_p_p_product)))))))) /\\ n = r * a", "statement_sha256": "2da86da57081e7d2fe9316cd1f50ad46d586cf8fbba80c04d7a65930124861c5" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_one_from_zero_successor", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "pow_successor_decompose", "pow_zero", "one_mul" ], "dependencies_sha256": "b5a342bcb7f582300c6c790009d1516d45dce0cf9f34026ea62e9d677b2663db", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "d709c0bb9f53c78a4819f904db0332005a2b37386460ef91f15173d46a194791", "cut_nodes": 107, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 64, "proof_edges": 1089, "proof_nodes": 3827, "proof_objects": 1043, "reused_objects": 47, "status": "checked" }, "enrollment_index": 318, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_one_from_zero_successor]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "c380a5daf1a1b0b8c70f79bfd6ffb0cdd92e935c453ea335e25165111025ee6e", "membership": "stable", "name": "pow_one_from_zero_successor", "proof_tag": "PA005T", "provenance": [ "stable" ], "script": [ "intro a", "intro z", "intro e", "intro n", "intro hz", "intro he", "intro hpow", "have hstep : exists r. (exists ff_b_one_predecessor ff_c_one_predecessor. ((forall ff_i_one_predecessor_repeat. (exists ff_lt_one_predecessor_repeat_bound. ff_lt_one_predecessor_repeat_bound + S ff_i_one_predecessor_repeat = z) -> (((exists ff_h_one_predecessor_repeat_decoded. ff_h_one_predecessor_repeat_decoded + S (a) = S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor)) /\\ exists ff_q_one_predecessor_repeat_decoded. ff_b_one_predecessor = ff_q_one_predecessor_repeat_decoded * S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor) + (a)))) /\\ (exists ff_u_one_predecessor_product ff_v_one_predecessor_product. ((((exists ff_h_one_predecessor_product_start. ff_h_one_predecessor_product_start + S (1) = S ((S (0)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_start. ff_u_one_predecessor_product = ff_q_one_predecessor_product_start * S ((S (0)) * ff_v_one_predecessor_product) + (1))) /\\ ((((exists ff_h_one_predecessor_product_terminal. ff_h_one_predecessor_product_terminal + S (r) = S ((S (z)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_terminal. ff_u_one_predecessor_product = ff_q_one_predecessor_product_terminal * S ((S (z)) * ff_v_one_predecessor_product) + (r))) /\\ forall ff_i_one_predecessor_product. (exists ff_lt_one_predecessor_product_bound. ff_lt_one_predecessor_product_bound + S ff_i_one_predecessor_product = z) -> exists ff_p_one_predecessor_product ff_r_one_predecessor_product ff_s_one_predecessor_product. ((((exists ff_h_one_predecessor_product_factor. ff_h_one_predecessor_product_factor + S (ff_p_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor)) /\\ exists ff_q_one_predecessor_product_factor. ff_b_one_predecessor = ff_q_one_predecessor_product_factor * S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor) + (ff_p_one_predecessor_product))) /\\ ((((exists ff_h_one_predecessor_product_partial. ff_h_one_predecessor_product_partial + S (ff_r_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_partial. ff_u_one_predecessor_product = ff_q_one_predecessor_product_partial * S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_r_one_predecessor_product))) /\\ ((((exists ff_h_one_predecessor_product_successor. ff_h_one_predecessor_product_successor + S (ff_s_one_predecessor_product) = S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_successor. ff_u_one_predecessor_product = ff_q_one_predecessor_product_successor * S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_s_one_predecessor_product))) /\\ ff_s_one_predecessor_product = ff_r_one_predecessor_product * ff_p_one_predecessor_product)))))))) /\\ n = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose z", "specialize pow_successor_decompose e", "specialize pow_successor_decompose n", "apply pow_successor_decompose", "exact he", "exact hpow", "cases hstep", "cases hstep_witness", "have hr : x = 1", "specialize pow_zero a", "specialize pow_zero z", "specialize pow_zero x", "apply pow_zero", "exact hz", "exact hstep_witness_left", "trans x * a", "exact hstep_witness_right", "rewrite hr", "specialize one_mul a", "exact one_mul" ], "script_sha256": "b855edd418df1658727b9a8c7083004956e91908dd346d661b8c490e1e7ad2f8", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a z e n. z = 0 -> e = S z -> (exists ff_b_one_carrier ff_c_one_carrier. ((forall ff_i_one_carrier_repeat. (exists ff_lt_one_carrier_repeat_bound. ff_lt_one_carrier_repeat_bound + S ff_i_one_carrier_repeat = e) -> (((exists ff_h_one_carrier_repeat_decoded. ff_h_one_carrier_repeat_decoded + S (a) = S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier)) /\\ exists ff_q_one_carrier_repeat_decoded. ff_b_one_carrier = ff_q_one_carrier_repeat_decoded * S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier) + (a)))) /\\ (exists ff_u_one_carrier_product ff_v_one_carrier_product. ((((exists ff_h_one_carrier_product_start. ff_h_one_carrier_product_start + S (1) = S ((S (0)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_start. ff_u_one_carrier_product = ff_q_one_carrier_product_start * S ((S (0)) * ff_v_one_carrier_product) + (1))) /\\ ((((exists ff_h_one_carrier_product_terminal. ff_h_one_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_terminal. ff_u_one_carrier_product = ff_q_one_carrier_product_terminal * S ((S (e)) * ff_v_one_carrier_product) + (n))) /\\ forall ff_i_one_carrier_product. (exists ff_lt_one_carrier_product_bound. ff_lt_one_carrier_product_bound + S ff_i_one_carrier_product = e) -> exists ff_p_one_carrier_product ff_r_one_carrier_product ff_s_one_carrier_product. ((((exists ff_h_one_carrier_product_factor. ff_h_one_carrier_product_factor + S (ff_p_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier)) /\\ exists ff_q_one_carrier_product_factor. ff_b_one_carrier = ff_q_one_carrier_product_factor * S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier) + (ff_p_one_carrier_product))) /\\ ((((exists ff_h_one_carrier_product_partial. ff_h_one_carrier_product_partial + S (ff_r_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_partial. ff_u_one_carrier_product = ff_q_one_carrier_product_partial * S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_r_one_carrier_product))) /\\ ((((exists ff_h_one_carrier_product_successor. ff_h_one_carrier_product_successor + S (ff_s_one_carrier_product) = S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_successor. ff_u_one_carrier_product = ff_q_one_carrier_product_successor * S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_s_one_carrier_product))) /\\ ff_s_one_carrier_product = ff_r_one_carrier_product * ff_p_one_carrier_product)))))))) -> n = a", "statement_sha256": "1bf1d61df893893438afb852dfe152d959aeec426e7bc17d9df82d77296c45cb", "summary": "A successor of a zero exponent gives the relational first power.", "summary_sha256": "c82b1c7a3ae249409ca32c65286bab2d9019b5b1cc1896c27be14066a87e6be4" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_successor_decompose", "pow_zero", "one_mul" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_one_from_zero_successor]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_one_from_zero_successor", "parent_alpha_version": "v34", "proof_bundle_node_id": 160, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_one_from_zero_successor", "script": [ "intro a", "intro z", "intro e", "intro n", "intro hz", "intro he", "intro hpow", "have hstep : exists r. (exists ff_b_one_predecessor ff_c_one_predecessor. ((forall ff_i_one_predecessor_repeat. (exists ff_lt_one_predecessor_repeat_bound. ff_lt_one_predecessor_repeat_bound + S ff_i_one_predecessor_repeat = z) -> (((exists ff_h_one_predecessor_repeat_decoded. ff_h_one_predecessor_repeat_decoded + S (a) = S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor)) /\\ exists ff_q_one_predecessor_repeat_decoded. ff_b_one_predecessor = ff_q_one_predecessor_repeat_decoded * S ((S (ff_i_one_predecessor_repeat)) * ff_c_one_predecessor) + (a)))) /\\ (exists ff_u_one_predecessor_product ff_v_one_predecessor_product. ((((exists ff_h_one_predecessor_product_start. ff_h_one_predecessor_product_start + S (1) = S ((S (0)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_start. ff_u_one_predecessor_product = ff_q_one_predecessor_product_start * S ((S (0)) * ff_v_one_predecessor_product) + (1))) /\\ ((((exists ff_h_one_predecessor_product_terminal. ff_h_one_predecessor_product_terminal + S (r) = S ((S (z)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_terminal. ff_u_one_predecessor_product = ff_q_one_predecessor_product_terminal * S ((S (z)) * ff_v_one_predecessor_product) + (r))) /\\ forall ff_i_one_predecessor_product. (exists ff_lt_one_predecessor_product_bound. ff_lt_one_predecessor_product_bound + S ff_i_one_predecessor_product = z) -> exists ff_p_one_predecessor_product ff_r_one_predecessor_product ff_s_one_predecessor_product. ((((exists ff_h_one_predecessor_product_factor. ff_h_one_predecessor_product_factor + S (ff_p_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor)) /\\ exists ff_q_one_predecessor_product_factor. ff_b_one_predecessor = ff_q_one_predecessor_product_factor * S ((S (ff_i_one_predecessor_product)) * ff_c_one_predecessor) + (ff_p_one_predecessor_product))) /\\ ((((exists ff_h_one_predecessor_product_partial. ff_h_one_predecessor_product_partial + S (ff_r_one_predecessor_product) = S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_partial. ff_u_one_predecessor_product = ff_q_one_predecessor_product_partial * S ((S (ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_r_one_predecessor_product))) /\\ ((((exists ff_h_one_predecessor_product_successor. ff_h_one_predecessor_product_successor + S (ff_s_one_predecessor_product) = S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product)) /\\ exists ff_q_one_predecessor_product_successor. ff_u_one_predecessor_product = ff_q_one_predecessor_product_successor * S ((S (S ff_i_one_predecessor_product)) * ff_v_one_predecessor_product) + (ff_s_one_predecessor_product))) /\\ ff_s_one_predecessor_product = ff_r_one_predecessor_product * ff_p_one_predecessor_product)))))))) /\\ n = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose z", "specialize pow_successor_decompose e", "specialize pow_successor_decompose n", "apply pow_successor_decompose", "exact he", "exact hpow", "cases hstep", "cases hstep_witness", "have hr : x = 1", "specialize pow_zero a", "specialize pow_zero z", "specialize pow_zero x", "apply pow_zero", "exact hz", "exact hstep_witness_left", "trans x * a", "exact hstep_witness_right", "rewrite hr", "specialize one_mul a", "exact one_mul" ], "script_sha256": "b855edd418df1658727b9a8c7083004956e91908dd346d661b8c490e1e7ad2f8", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a z e n. z = 0 -> e = S z -> (exists ff_b_one_carrier ff_c_one_carrier. ((forall ff_i_one_carrier_repeat. (exists ff_lt_one_carrier_repeat_bound. ff_lt_one_carrier_repeat_bound + S ff_i_one_carrier_repeat = e) -> (((exists ff_h_one_carrier_repeat_decoded. ff_h_one_carrier_repeat_decoded + S (a) = S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier)) /\\ exists ff_q_one_carrier_repeat_decoded. ff_b_one_carrier = ff_q_one_carrier_repeat_decoded * S ((S (ff_i_one_carrier_repeat)) * ff_c_one_carrier) + (a)))) /\\ (exists ff_u_one_carrier_product ff_v_one_carrier_product. ((((exists ff_h_one_carrier_product_start. ff_h_one_carrier_product_start + S (1) = S ((S (0)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_start. ff_u_one_carrier_product = ff_q_one_carrier_product_start * S ((S (0)) * ff_v_one_carrier_product) + (1))) /\\ ((((exists ff_h_one_carrier_product_terminal. ff_h_one_carrier_product_terminal + S (n) = S ((S (e)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_terminal. ff_u_one_carrier_product = ff_q_one_carrier_product_terminal * S ((S (e)) * ff_v_one_carrier_product) + (n))) /\\ forall ff_i_one_carrier_product. (exists ff_lt_one_carrier_product_bound. ff_lt_one_carrier_product_bound + S ff_i_one_carrier_product = e) -> exists ff_p_one_carrier_product ff_r_one_carrier_product ff_s_one_carrier_product. ((((exists ff_h_one_carrier_product_factor. ff_h_one_carrier_product_factor + S (ff_p_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier)) /\\ exists ff_q_one_carrier_product_factor. ff_b_one_carrier = ff_q_one_carrier_product_factor * S ((S (ff_i_one_carrier_product)) * ff_c_one_carrier) + (ff_p_one_carrier_product))) /\\ ((((exists ff_h_one_carrier_product_partial. ff_h_one_carrier_product_partial + S (ff_r_one_carrier_product) = S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_partial. ff_u_one_carrier_product = ff_q_one_carrier_product_partial * S ((S (ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_r_one_carrier_product))) /\\ ((((exists ff_h_one_carrier_product_successor. ff_h_one_carrier_product_successor + S (ff_s_one_carrier_product) = S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product)) /\\ exists ff_q_one_carrier_product_successor. ff_u_one_carrier_product = ff_q_one_carrier_product_successor * S ((S (S ff_i_one_carrier_product)) * ff_v_one_carrier_product) + (ff_s_one_carrier_product))) /\\ ff_s_one_carrier_product = ff_r_one_carrier_product * ff_p_one_carrier_product)))))))) -> n = a", "statement_sha256": "1bf1d61df893893438afb852dfe152d959aeec426e7bc17d9df82d77296c45cb" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_one", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "pow_one_from_zero_successor" ], "dependencies_sha256": "fb13cc872397ad40fba4ce446d4c5e3c4e18ff21c10061353f0c563dd8cabe52", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "16491d90b740143f121ec9fdfbf6c2adcc1d150f55bbb0735877ace24a9d913a", "cut_nodes": 108, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 65, "proof_edges": 1118, "proof_nodes": 3856, "proof_objects": 1072, "reused_objects": 47, "status": "checked" }, "enrollment_index": 319, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_one]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "8aeb705f0ac954cde9d822571ea5aa6ee97c71536eb28189077668e938f6aca9", "membership": "stable", "name": "pow_one", "proof_tag": "PA005U", "provenance": [ "stable" ], "script": [ "intro a", "intro e", "intro n", "intro he", "intro hpow", "specialize pow_one_from_zero_successor a", "specialize pow_one_from_zero_successor 0", "specialize pow_one_from_zero_successor e", "specialize pow_one_from_zero_successor n", "apply pow_one_from_zero_successor", "refl", "exact he", "exact hpow" ], "script_sha256": "e5063f997fac90a8a731f093e40094795d3ba17972824a7ebfcebbb42d780973", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a e n. e = 1 -> (exists ff_b_one ff_c_one. ((forall ff_i_one_repeat. (exists ff_lt_one_repeat_bound. ff_lt_one_repeat_bound + S ff_i_one_repeat = e) -> (((exists ff_h_one_repeat_decoded. ff_h_one_repeat_decoded + S (a) = S ((S (ff_i_one_repeat)) * ff_c_one)) /\\ exists ff_q_one_repeat_decoded. ff_b_one = ff_q_one_repeat_decoded * S ((S (ff_i_one_repeat)) * ff_c_one) + (a)))) /\\ (exists ff_u_one_product ff_v_one_product. ((((exists ff_h_one_product_start. ff_h_one_product_start + S (1) = S ((S (0)) * ff_v_one_product)) /\\ exists ff_q_one_product_start. ff_u_one_product = ff_q_one_product_start * S ((S (0)) * ff_v_one_product) + (1))) /\\ ((((exists ff_h_one_product_terminal. ff_h_one_product_terminal + S (n) = S ((S (e)) * ff_v_one_product)) /\\ exists ff_q_one_product_terminal. ff_u_one_product = ff_q_one_product_terminal * S ((S (e)) * ff_v_one_product) + (n))) /\\ forall ff_i_one_product. (exists ff_lt_one_product_bound. ff_lt_one_product_bound + S ff_i_one_product = e) -> exists ff_p_one_product ff_r_one_product ff_s_one_product. ((((exists ff_h_one_product_factor. ff_h_one_product_factor + S (ff_p_one_product) = S ((S (ff_i_one_product)) * ff_c_one)) /\\ exists ff_q_one_product_factor. ff_b_one = ff_q_one_product_factor * S ((S (ff_i_one_product)) * ff_c_one) + (ff_p_one_product))) /\\ ((((exists ff_h_one_product_partial. ff_h_one_product_partial + S (ff_r_one_product) = S ((S (ff_i_one_product)) * ff_v_one_product)) /\\ exists ff_q_one_product_partial. ff_u_one_product = ff_q_one_product_partial * S ((S (ff_i_one_product)) * ff_v_one_product) + (ff_r_one_product))) /\\ ((((exists ff_h_one_product_successor. ff_h_one_product_successor + S (ff_s_one_product) = S ((S (S ff_i_one_product)) * ff_v_one_product)) /\\ exists ff_q_one_product_successor. ff_u_one_product = ff_q_one_product_successor * S ((S (S ff_i_one_product)) * ff_v_one_product) + (ff_s_one_product))) /\\ ff_s_one_product = ff_r_one_product * ff_p_one_product)))))))) -> n = a", "statement_sha256": "6bc41df69ec62998d0ca8a1ad02daaab265b51bc19fa8d4b8891aa717dfab44a", "summary": "The relational first power of a natural is the natural itself.", "summary_sha256": "2951a5878bf4d654e35f637ff8e3d93104e5bc554a384b8e799df2d475aad8c3" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_one_from_zero_successor" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_one]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_one", "parent_alpha_version": "v34", "proof_bundle_node_id": 161, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_one", "script": [ "intro a", "intro e", "intro n", "intro he", "intro hpow", "specialize pow_one_from_zero_successor a", "specialize pow_one_from_zero_successor 0", "specialize pow_one_from_zero_successor e", "specialize pow_one_from_zero_successor n", "apply pow_one_from_zero_successor", "refl", "exact he", "exact hpow" ], "script_sha256": "e5063f997fac90a8a731f093e40094795d3ba17972824a7ebfcebbb42d780973", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a e n. e = 1 -> (exists ff_b_one ff_c_one. ((forall ff_i_one_repeat. (exists ff_lt_one_repeat_bound. ff_lt_one_repeat_bound + S ff_i_one_repeat = e) -> (((exists ff_h_one_repeat_decoded. ff_h_one_repeat_decoded + S (a) = S ((S (ff_i_one_repeat)) * ff_c_one)) /\\ exists ff_q_one_repeat_decoded. ff_b_one = ff_q_one_repeat_decoded * S ((S (ff_i_one_repeat)) * ff_c_one) + (a)))) /\\ (exists ff_u_one_product ff_v_one_product. ((((exists ff_h_one_product_start. ff_h_one_product_start + S (1) = S ((S (0)) * ff_v_one_product)) /\\ exists ff_q_one_product_start. ff_u_one_product = ff_q_one_product_start * S ((S (0)) * ff_v_one_product) + (1))) /\\ ((((exists ff_h_one_product_terminal. ff_h_one_product_terminal + S (n) = S ((S (e)) * ff_v_one_product)) /\\ exists ff_q_one_product_terminal. ff_u_one_product = ff_q_one_product_terminal * S ((S (e)) * ff_v_one_product) + (n))) /\\ forall ff_i_one_product. (exists ff_lt_one_product_bound. ff_lt_one_product_bound + S ff_i_one_product = e) -> exists ff_p_one_product ff_r_one_product ff_s_one_product. ((((exists ff_h_one_product_factor. ff_h_one_product_factor + S (ff_p_one_product) = S ((S (ff_i_one_product)) * ff_c_one)) /\\ exists ff_q_one_product_factor. ff_b_one = ff_q_one_product_factor * S ((S (ff_i_one_product)) * ff_c_one) + (ff_p_one_product))) /\\ ((((exists ff_h_one_product_partial. ff_h_one_product_partial + S (ff_r_one_product) = S ((S (ff_i_one_product)) * ff_v_one_product)) /\\ exists ff_q_one_product_partial. ff_u_one_product = ff_q_one_product_partial * S ((S (ff_i_one_product)) * ff_v_one_product) + (ff_r_one_product))) /\\ ((((exists ff_h_one_product_successor. ff_h_one_product_successor + S (ff_s_one_product) = S ((S (S ff_i_one_product)) * ff_v_one_product)) /\\ exists ff_q_one_product_successor. ff_u_one_product = ff_q_one_product_successor * S ((S (S ff_i_one_product)) * ff_v_one_product) + (ff_s_one_product))) /\\ ff_s_one_product = ff_r_one_product * ff_p_one_product)))))))) -> n = a", "statement_sha256": "6bc41df69ec62998d0ca8a1ad02daaab265b51bc19fa8d4b8891aa717dfab44a" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_successor_pair_mul", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "pow_successor_decompose", "pow_functional" ], "dependencies_sha256": "6d64d095f386f040846d75bba226e49d9906c514a984f3a6ccdf11dc41472d8a", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "be60a3277b36945d017a52111763bbff87e995c0efa27254d7a006126fd7515d", "cut_nodes": 145, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 65, "proof_edges": 1338, "proof_nodes": 5282, "proof_objects": 1293, "reused_objects": 46, "status": "checked" }, "enrollment_index": 320, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_successor_pair_mul]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "e22ac770699baeb766d14165eb17036f45759b492fb28fc2176b4acf9b65e1be", "membership": "stable", "name": "pow_successor_pair_mul", "proof_tag": "PA005H", "provenance": [ "stable" ], "script": [ "intro a", "intro e", "intro se", "intro r", "intro n", "intro hse", "intro hprevious", "intro hsuccessor", "have hstep : exists z. (exists ff_b_pair_decomposed ff_c_pair_decomposed. ((forall ff_i_pair_decomposed_repeat. (exists ff_lt_pair_decomposed_repeat_bound. ff_lt_pair_decomposed_repeat_bound + S ff_i_pair_decomposed_repeat = e) -> (((exists ff_h_pair_decomposed_repeat_decoded. ff_h_pair_decomposed_repeat_decoded + S (a) = S ((S (ff_i_pair_decomposed_repeat)) * ff_c_pair_decomposed)) /\\ exists ff_q_pair_decomposed_repeat_decoded. ff_b_pair_decomposed = ff_q_pair_decomposed_repeat_decoded * S ((S (ff_i_pair_decomposed_repeat)) * ff_c_pair_decomposed) + (a)))) /\\ (exists ff_u_pair_decomposed_product ff_v_pair_decomposed_product. ((((exists ff_h_pair_decomposed_product_start. ff_h_pair_decomposed_product_start + S (1) = S ((S (0)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_start. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_start * S ((S (0)) * ff_v_pair_decomposed_product) + (1))) /\\ ((((exists ff_h_pair_decomposed_product_terminal. ff_h_pair_decomposed_product_terminal + S (z) = S ((S (e)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_terminal. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_terminal * S ((S (e)) * ff_v_pair_decomposed_product) + (z))) /\\ forall ff_i_pair_decomposed_product. (exists ff_lt_pair_decomposed_product_bound. ff_lt_pair_decomposed_product_bound + S ff_i_pair_decomposed_product = e) -> exists ff_p_pair_decomposed_product ff_r_pair_decomposed_product ff_s_pair_decomposed_product. ((((exists ff_h_pair_decomposed_product_factor. ff_h_pair_decomposed_product_factor + S (ff_p_pair_decomposed_product) = S ((S (ff_i_pair_decomposed_product)) * ff_c_pair_decomposed)) /\\ exists ff_q_pair_decomposed_product_factor. ff_b_pair_decomposed = ff_q_pair_decomposed_product_factor * S ((S (ff_i_pair_decomposed_product)) * ff_c_pair_decomposed) + (ff_p_pair_decomposed_product))) /\\ ((((exists ff_h_pair_decomposed_product_partial. ff_h_pair_decomposed_product_partial + S (ff_r_pair_decomposed_product) = S ((S (ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_partial. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_partial * S ((S (ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product) + (ff_r_pair_decomposed_product))) /\\ ((((exists ff_h_pair_decomposed_product_successor. ff_h_pair_decomposed_product_successor + S (ff_s_pair_decomposed_product) = S ((S (S ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_successor. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_successor * S ((S (S ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product) + (ff_s_pair_decomposed_product))) /\\ ff_s_pair_decomposed_product = ff_r_pair_decomposed_product * ff_p_pair_decomposed_product)))))))) /\\ n = z * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose e", "specialize pow_successor_decompose se", "specialize pow_successor_decompose n", "apply pow_successor_decompose", "exact hse", "exact hsuccessor", "cases hstep", "cases hstep_witness", "have hz : x = r", "specialize pow_functional a", "specialize pow_functional e", "specialize pow_functional x", "specialize pow_functional r", "apply pow_functional", "exact hstep_witness_left", "exact hprevious", "trans x * a", "exact hstep_witness_right", "rewrite hz", "refl" ], "script_sha256": "7eeb7a6261fef70dcd181edff7dea72f45923859efad055d470de5689ac95bee", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a e se r n. se = S e -> (exists ff_b_pair_predecessor ff_c_pair_predecessor. ((forall ff_i_pair_predecessor_repeat. (exists ff_lt_pair_predecessor_repeat_bound. ff_lt_pair_predecessor_repeat_bound + S ff_i_pair_predecessor_repeat = e) -> (((exists ff_h_pair_predecessor_repeat_decoded. ff_h_pair_predecessor_repeat_decoded + S (a) = S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor)) /\\ exists ff_q_pair_predecessor_repeat_decoded. ff_b_pair_predecessor = ff_q_pair_predecessor_repeat_decoded * S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor) + (a)))) /\\ (exists ff_u_pair_predecessor_product ff_v_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_start. ff_h_pair_predecessor_product_start + S (1) = S ((S (0)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_start. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_start * S ((S (0)) * ff_v_pair_predecessor_product) + (1))) /\\ ((((exists ff_h_pair_predecessor_product_terminal. ff_h_pair_predecessor_product_terminal + S (r) = S ((S (e)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_terminal. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_terminal * S ((S (e)) * ff_v_pair_predecessor_product) + (r))) /\\ forall ff_i_pair_predecessor_product. (exists ff_lt_pair_predecessor_product_bound. ff_lt_pair_predecessor_product_bound + S ff_i_pair_predecessor_product = e) -> exists ff_p_pair_predecessor_product ff_r_pair_predecessor_product ff_s_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_factor. ff_h_pair_predecessor_product_factor + S (ff_p_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor)) /\\ exists ff_q_pair_predecessor_product_factor. ff_b_pair_predecessor = ff_q_pair_predecessor_product_factor * S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor) + (ff_p_pair_predecessor_product))) /\\ ((((exists ff_h_pair_predecessor_product_partial. ff_h_pair_predecessor_product_partial + S (ff_r_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_partial. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_partial * S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_r_pair_predecessor_product))) /\\ ((((exists ff_h_pair_predecessor_product_successor. ff_h_pair_predecessor_product_successor + S (ff_s_pair_predecessor_product) = S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_successor. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_successor * S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_s_pair_predecessor_product))) /\\ ff_s_pair_predecessor_product = ff_r_pair_predecessor_product * ff_p_pair_predecessor_product)))))))) -> (exists ff_b_pair_successor ff_c_pair_successor. ((forall ff_i_pair_successor_repeat. (exists ff_lt_pair_successor_repeat_bound. ff_lt_pair_successor_repeat_bound + S ff_i_pair_successor_repeat = se) -> (((exists ff_h_pair_successor_repeat_decoded. ff_h_pair_successor_repeat_decoded + S (a) = S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor)) /\\ exists ff_q_pair_successor_repeat_decoded. ff_b_pair_successor = ff_q_pair_successor_repeat_decoded * S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor) + (a)))) /\\ (exists ff_u_pair_successor_product ff_v_pair_successor_product. ((((exists ff_h_pair_successor_product_start. ff_h_pair_successor_product_start + S (1) = S ((S (0)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_start. ff_u_pair_successor_product = ff_q_pair_successor_product_start * S ((S (0)) * ff_v_pair_successor_product) + (1))) /\\ ((((exists ff_h_pair_successor_product_terminal. ff_h_pair_successor_product_terminal + S (n) = S ((S (se)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_terminal. ff_u_pair_successor_product = ff_q_pair_successor_product_terminal * S ((S (se)) * ff_v_pair_successor_product) + (n))) /\\ forall ff_i_pair_successor_product. (exists ff_lt_pair_successor_product_bound. ff_lt_pair_successor_product_bound + S ff_i_pair_successor_product = se) -> exists ff_p_pair_successor_product ff_r_pair_successor_product ff_s_pair_successor_product. ((((exists ff_h_pair_successor_product_factor. ff_h_pair_successor_product_factor + S (ff_p_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor)) /\\ exists ff_q_pair_successor_product_factor. ff_b_pair_successor = ff_q_pair_successor_product_factor * S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor) + (ff_p_pair_successor_product))) /\\ ((((exists ff_h_pair_successor_product_partial. ff_h_pair_successor_product_partial + S (ff_r_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_partial. ff_u_pair_successor_product = ff_q_pair_successor_product_partial * S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_r_pair_successor_product))) /\\ ((((exists ff_h_pair_successor_product_successor. ff_h_pair_successor_product_successor + S (ff_s_pair_successor_product) = S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_successor. ff_u_pair_successor_product = ff_q_pair_successor_product_successor * S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_s_pair_successor_product))) /\\ ff_s_pair_successor_product = ff_r_pair_successor_product * ff_p_pair_successor_product)))))))) -> n = r * a", "statement_sha256": "0057e37220405bfa53138560953bc6b0fed8aefe6c3f9babdf2fb60351e7424d", "summary": "A successor power paired with its predecessor equals predecessor times base.", "summary_sha256": "5324951a8474020d4e3dbbe0375815653a7f7df5dcebef0d7471461ca85d59fa" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_successor_decompose", "pow_functional" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_successor_pair_mul]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_successor_pair_mul", "parent_alpha_version": "v34", "proof_bundle_node_id": 162, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_successor_pair_mul", "script": [ "intro a", "intro e", "intro se", "intro r", "intro n", "intro hse", "intro hprevious", "intro hsuccessor", "have hstep : exists z. (exists ff_b_pair_decomposed ff_c_pair_decomposed. ((forall ff_i_pair_decomposed_repeat. (exists ff_lt_pair_decomposed_repeat_bound. ff_lt_pair_decomposed_repeat_bound + S ff_i_pair_decomposed_repeat = e) -> (((exists ff_h_pair_decomposed_repeat_decoded. ff_h_pair_decomposed_repeat_decoded + S (a) = S ((S (ff_i_pair_decomposed_repeat)) * ff_c_pair_decomposed)) /\\ exists ff_q_pair_decomposed_repeat_decoded. ff_b_pair_decomposed = ff_q_pair_decomposed_repeat_decoded * S ((S (ff_i_pair_decomposed_repeat)) * ff_c_pair_decomposed) + (a)))) /\\ (exists ff_u_pair_decomposed_product ff_v_pair_decomposed_product. ((((exists ff_h_pair_decomposed_product_start. ff_h_pair_decomposed_product_start + S (1) = S ((S (0)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_start. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_start * S ((S (0)) * ff_v_pair_decomposed_product) + (1))) /\\ ((((exists ff_h_pair_decomposed_product_terminal. ff_h_pair_decomposed_product_terminal + S (z) = S ((S (e)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_terminal. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_terminal * S ((S (e)) * ff_v_pair_decomposed_product) + (z))) /\\ forall ff_i_pair_decomposed_product. (exists ff_lt_pair_decomposed_product_bound. ff_lt_pair_decomposed_product_bound + S ff_i_pair_decomposed_product = e) -> exists ff_p_pair_decomposed_product ff_r_pair_decomposed_product ff_s_pair_decomposed_product. ((((exists ff_h_pair_decomposed_product_factor. ff_h_pair_decomposed_product_factor + S (ff_p_pair_decomposed_product) = S ((S (ff_i_pair_decomposed_product)) * ff_c_pair_decomposed)) /\\ exists ff_q_pair_decomposed_product_factor. ff_b_pair_decomposed = ff_q_pair_decomposed_product_factor * S ((S (ff_i_pair_decomposed_product)) * ff_c_pair_decomposed) + (ff_p_pair_decomposed_product))) /\\ ((((exists ff_h_pair_decomposed_product_partial. ff_h_pair_decomposed_product_partial + S (ff_r_pair_decomposed_product) = S ((S (ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_partial. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_partial * S ((S (ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product) + (ff_r_pair_decomposed_product))) /\\ ((((exists ff_h_pair_decomposed_product_successor. ff_h_pair_decomposed_product_successor + S (ff_s_pair_decomposed_product) = S ((S (S ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product)) /\\ exists ff_q_pair_decomposed_product_successor. ff_u_pair_decomposed_product = ff_q_pair_decomposed_product_successor * S ((S (S ff_i_pair_decomposed_product)) * ff_v_pair_decomposed_product) + (ff_s_pair_decomposed_product))) /\\ ff_s_pair_decomposed_product = ff_r_pair_decomposed_product * ff_p_pair_decomposed_product)))))))) /\\ n = z * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose e", "specialize pow_successor_decompose se", "specialize pow_successor_decompose n", "apply pow_successor_decompose", "exact hse", "exact hsuccessor", "cases hstep", "cases hstep_witness", "have hz : x = r", "specialize pow_functional a", "specialize pow_functional e", "specialize pow_functional x", "specialize pow_functional r", "apply pow_functional", "exact hstep_witness_left", "exact hprevious", "trans x * a", "exact hstep_witness_right", "rewrite hz", "refl" ], "script_sha256": "7eeb7a6261fef70dcd181edff7dea72f45923859efad055d470de5689ac95bee", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a e se r n. se = S e -> (exists ff_b_pair_predecessor ff_c_pair_predecessor. ((forall ff_i_pair_predecessor_repeat. (exists ff_lt_pair_predecessor_repeat_bound. ff_lt_pair_predecessor_repeat_bound + S ff_i_pair_predecessor_repeat = e) -> (((exists ff_h_pair_predecessor_repeat_decoded. ff_h_pair_predecessor_repeat_decoded + S (a) = S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor)) /\\ exists ff_q_pair_predecessor_repeat_decoded. ff_b_pair_predecessor = ff_q_pair_predecessor_repeat_decoded * S ((S (ff_i_pair_predecessor_repeat)) * ff_c_pair_predecessor) + (a)))) /\\ (exists ff_u_pair_predecessor_product ff_v_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_start. ff_h_pair_predecessor_product_start + S (1) = S ((S (0)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_start. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_start * S ((S (0)) * ff_v_pair_predecessor_product) + (1))) /\\ ((((exists ff_h_pair_predecessor_product_terminal. ff_h_pair_predecessor_product_terminal + S (r) = S ((S (e)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_terminal. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_terminal * S ((S (e)) * ff_v_pair_predecessor_product) + (r))) /\\ forall ff_i_pair_predecessor_product. (exists ff_lt_pair_predecessor_product_bound. ff_lt_pair_predecessor_product_bound + S ff_i_pair_predecessor_product = e) -> exists ff_p_pair_predecessor_product ff_r_pair_predecessor_product ff_s_pair_predecessor_product. ((((exists ff_h_pair_predecessor_product_factor. ff_h_pair_predecessor_product_factor + S (ff_p_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor)) /\\ exists ff_q_pair_predecessor_product_factor. ff_b_pair_predecessor = ff_q_pair_predecessor_product_factor * S ((S (ff_i_pair_predecessor_product)) * ff_c_pair_predecessor) + (ff_p_pair_predecessor_product))) /\\ ((((exists ff_h_pair_predecessor_product_partial. ff_h_pair_predecessor_product_partial + S (ff_r_pair_predecessor_product) = S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_partial. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_partial * S ((S (ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_r_pair_predecessor_product))) /\\ ((((exists ff_h_pair_predecessor_product_successor. ff_h_pair_predecessor_product_successor + S (ff_s_pair_predecessor_product) = S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product)) /\\ exists ff_q_pair_predecessor_product_successor. ff_u_pair_predecessor_product = ff_q_pair_predecessor_product_successor * S ((S (S ff_i_pair_predecessor_product)) * ff_v_pair_predecessor_product) + (ff_s_pair_predecessor_product))) /\\ ff_s_pair_predecessor_product = ff_r_pair_predecessor_product * ff_p_pair_predecessor_product)))))))) -> (exists ff_b_pair_successor ff_c_pair_successor. ((forall ff_i_pair_successor_repeat. (exists ff_lt_pair_successor_repeat_bound. ff_lt_pair_successor_repeat_bound + S ff_i_pair_successor_repeat = se) -> (((exists ff_h_pair_successor_repeat_decoded. ff_h_pair_successor_repeat_decoded + S (a) = S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor)) /\\ exists ff_q_pair_successor_repeat_decoded. ff_b_pair_successor = ff_q_pair_successor_repeat_decoded * S ((S (ff_i_pair_successor_repeat)) * ff_c_pair_successor) + (a)))) /\\ (exists ff_u_pair_successor_product ff_v_pair_successor_product. ((((exists ff_h_pair_successor_product_start. ff_h_pair_successor_product_start + S (1) = S ((S (0)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_start. ff_u_pair_successor_product = ff_q_pair_successor_product_start * S ((S (0)) * ff_v_pair_successor_product) + (1))) /\\ ((((exists ff_h_pair_successor_product_terminal. ff_h_pair_successor_product_terminal + S (n) = S ((S (se)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_terminal. ff_u_pair_successor_product = ff_q_pair_successor_product_terminal * S ((S (se)) * ff_v_pair_successor_product) + (n))) /\\ forall ff_i_pair_successor_product. (exists ff_lt_pair_successor_product_bound. ff_lt_pair_successor_product_bound + S ff_i_pair_successor_product = se) -> exists ff_p_pair_successor_product ff_r_pair_successor_product ff_s_pair_successor_product. ((((exists ff_h_pair_successor_product_factor. ff_h_pair_successor_product_factor + S (ff_p_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor)) /\\ exists ff_q_pair_successor_product_factor. ff_b_pair_successor = ff_q_pair_successor_product_factor * S ((S (ff_i_pair_successor_product)) * ff_c_pair_successor) + (ff_p_pair_successor_product))) /\\ ((((exists ff_h_pair_successor_product_partial. ff_h_pair_successor_product_partial + S (ff_r_pair_successor_product) = S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_partial. ff_u_pair_successor_product = ff_q_pair_successor_product_partial * S ((S (ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_r_pair_successor_product))) /\\ ((((exists ff_h_pair_successor_product_successor. ff_h_pair_successor_product_successor + S (ff_s_pair_successor_product) = S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product)) /\\ exists ff_q_pair_successor_product_successor. ff_u_pair_successor_product = ff_q_pair_successor_product_successor * S ((S (S ff_i_pair_successor_product)) * ff_v_pair_successor_product) + (ff_s_pair_successor_product))) /\\ ff_s_pair_successor_product = ff_r_pair_successor_product * ff_p_pair_successor_product)))))))) -> n = r * a", "statement_sha256": "0057e37220405bfa53138560953bc6b0fed8aefe6c3f9babdf2fb60351e7424d" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_add", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "pow_zero", "pow_functional", "pow_successor_decompose", "mul_one", "mul_assoc" ], "dependencies_sha256": "24667f0e43acfffe965d8a46634ba23cdb0f035d921f06ab3494049ce39423cb", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "e7260f892c412b64d44eb7275736be30338be6313efc7127bd12e18effb2391e", "cut_nodes": 183, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 66, "proof_edges": 1588, "proof_nodes": 6744, "proof_objects": 1530, "reused_objects": 59, "status": "checked" }, "enrollment_index": 348, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_add]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "7b6a1f6161b6fc2e163889a0ee3c55cb89cf1463907446a8da53a8495cc94b2b", "membership": "stable", "name": "pow_add", "proof_tag": "PA005X", "provenance": [ "stable" ], "script": [ "intro a", "intro e", "induction f", "intro s", "intro x", "intro y", "intro z", "intro hs", "intro hx", "intro hy", "intro hz", "rewrite PA3 at hs", "rewrite hs at hz", "rewrite hs at hz", "rewrite hs at hz", "rewrite hs at hz", "have hzx : z = x", "specialize pow_functional a", "specialize pow_functional e", "specialize pow_functional z", "specialize pow_functional x", "apply pow_functional", "exact hz", "exact hx", "have hy1 : y = 1", "specialize pow_zero a", "specialize pow_zero 0", "specialize pow_zero y", "apply pow_zero", "refl", "exact hy", "rewrite hzx", "rewrite hy1", "specialize mul_one x", "symm", "exact mul_one", "intro s", "intro x", "intro y", "intro z", "intro hs", "intro hx", "intro hy", "intro hz", "have hy_step : exists r. (exists ff_b_add_y_prefix ff_c_add_y_prefix. ((forall ff_i_add_y_prefix_repeat. (exists ff_lt_add_y_prefix_repeat_bound. ff_lt_add_y_prefix_repeat_bound + S ff_i_add_y_prefix_repeat = f) -> (((exists ff_h_add_y_prefix_repeat_decoded. ff_h_add_y_prefix_repeat_decoded + S (a) = S ((S (ff_i_add_y_prefix_repeat)) * ff_c_add_y_prefix)) /\\ exists ff_q_add_y_prefix_repeat_decoded. ff_b_add_y_prefix = ff_q_add_y_prefix_repeat_decoded * S ((S (ff_i_add_y_prefix_repeat)) * ff_c_add_y_prefix) + (a)))) /\\ (exists ff_u_add_y_prefix_product ff_v_add_y_prefix_product. ((((exists ff_h_add_y_prefix_product_start. ff_h_add_y_prefix_product_start + S (1) = S ((S (0)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_start. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_start * S ((S (0)) * ff_v_add_y_prefix_product) + (1))) /\\ ((((exists ff_h_add_y_prefix_product_terminal. ff_h_add_y_prefix_product_terminal + S (r) = S ((S (f)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_terminal. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_terminal * S ((S (f)) * ff_v_add_y_prefix_product) + (r))) /\\ forall ff_i_add_y_prefix_product. (exists ff_lt_add_y_prefix_product_bound. ff_lt_add_y_prefix_product_bound + S ff_i_add_y_prefix_product = f) -> exists ff_p_add_y_prefix_product ff_r_add_y_prefix_product ff_s_add_y_prefix_product. ((((exists ff_h_add_y_prefix_product_factor. ff_h_add_y_prefix_product_factor + S (ff_p_add_y_prefix_product) = S ((S (ff_i_add_y_prefix_product)) * ff_c_add_y_prefix)) /\\ exists ff_q_add_y_prefix_product_factor. ff_b_add_y_prefix = ff_q_add_y_prefix_product_factor * S ((S (ff_i_add_y_prefix_product)) * ff_c_add_y_prefix) + (ff_p_add_y_prefix_product))) /\\ ((((exists ff_h_add_y_prefix_product_partial. ff_h_add_y_prefix_product_partial + S (ff_r_add_y_prefix_product) = S ((S (ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_partial. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_partial * S ((S (ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product) + (ff_r_add_y_prefix_product))) /\\ ((((exists ff_h_add_y_prefix_product_successor. ff_h_add_y_prefix_product_successor + S (ff_s_add_y_prefix_product) = S ((S (S ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_successor. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_successor * S ((S (S ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product) + (ff_s_add_y_prefix_product))) /\\ ff_s_add_y_prefix_product = ff_r_add_y_prefix_product * ff_p_add_y_prefix_product)))))))) /\\ y = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose f", "specialize pow_successor_decompose (S f)", "specialize pow_successor_decompose y", "apply pow_successor_decompose", "refl", "exact hy", "cases hy_step", "cases hy_step_witness", "have hst : s = S (e + f)", "trans e + S f", "exact hs", "apply PA4", "have hz_step : exists r. (exists pa_b_add_z_prefix pa_c_add_z_prefix. ((forall pa_i_add_z_prefix_repeat. (exists pa_lt_add_z_prefix_repeat_bound. pa_lt_add_z_prefix_repeat_bound + S pa_i_add_z_prefix_repeat = e + f) -> (((exists pa_h_add_z_prefix_repeat_decoded. pa_h_add_z_prefix_repeat_decoded + S (a) = S ((S (pa_i_add_z_prefix_repeat)) * pa_c_add_z_prefix)) /\\ exists pa_q_add_z_prefix_repeat_decoded. pa_b_add_z_prefix = pa_q_add_z_prefix_repeat_decoded * S ((S (pa_i_add_z_prefix_repeat)) * pa_c_add_z_prefix) + (a)))) /\\ (exists pa_u_add_z_prefix_product pa_v_add_z_prefix_product. ((((exists pa_h_add_z_prefix_product_start. pa_h_add_z_prefix_product_start + S (1) = S ((S (0)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_start. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_start * S ((S (0)) * pa_v_add_z_prefix_product) + (1))) /\\ ((((exists pa_h_add_z_prefix_product_terminal. pa_h_add_z_prefix_product_terminal + S (r) = S ((S (e + f)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_terminal. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_terminal * S ((S (e + f)) * pa_v_add_z_prefix_product) + (r))) /\\ forall pa_i_add_z_prefix_product. (exists pa_lt_add_z_prefix_product_bound. pa_lt_add_z_prefix_product_bound + S pa_i_add_z_prefix_product = e + f) -> exists pa_p_add_z_prefix_product pa_r_add_z_prefix_product pa_s_add_z_prefix_product. ((((exists pa_h_add_z_prefix_product_factor. pa_h_add_z_prefix_product_factor + S (pa_p_add_z_prefix_product) = S ((S (pa_i_add_z_prefix_product)) * pa_c_add_z_prefix)) /\\ exists pa_q_add_z_prefix_product_factor. pa_b_add_z_prefix = pa_q_add_z_prefix_product_factor * S ((S (pa_i_add_z_prefix_product)) * pa_c_add_z_prefix) + (pa_p_add_z_prefix_product))) /\\ ((((exists pa_h_add_z_prefix_product_partial. pa_h_add_z_prefix_product_partial + S (pa_r_add_z_prefix_product) = S ((S (pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_partial. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_partial * S ((S (pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product) + (pa_r_add_z_prefix_product))) /\\ ((((exists pa_h_add_z_prefix_product_successor. pa_h_add_z_prefix_product_successor + S (pa_s_add_z_prefix_product) = S ((S (S pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_successor. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_successor * S ((S (S pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product) + (pa_s_add_z_prefix_product))) /\\ pa_s_add_z_prefix_product = pa_r_add_z_prefix_product * pa_p_add_z_prefix_product)))))))) /\\ z = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose (e + f)", "specialize pow_successor_decompose s", "specialize pow_successor_decompose z", "apply pow_successor_decompose", "exact hst", "exact hz", "cases hz_step", "cases hz_step_witness", "have hprefix : x2 = x * x1", "specialize IH (e + f)", "specialize IH x", "specialize IH x1", "specialize IH x2", "apply IH", "refl", "exact hx", "exact hy_step_witness_left", "exact hz_step_witness_left", "trans x2 * a", "exact hz_step_witness_right", "trans (x * x1) * a", "congr", "exact hprefix", "refl", "trans x * (x1 * a)", "apply mul_assoc", "congr", "refl", "symm", "exact hy_step_witness_right" ], "script_sha256": "e82049235aa2fc11cacd0eebc5a847c4469193b55c044829d309156032de79a4", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall a e f s x y z. s = e + f -> (exists ff_b_add_left ff_c_add_left. ((forall ff_i_add_left_repeat. (exists ff_lt_add_left_repeat_bound. ff_lt_add_left_repeat_bound + S ff_i_add_left_repeat = e) -> (((exists ff_h_add_left_repeat_decoded. ff_h_add_left_repeat_decoded + S (a) = S ((S (ff_i_add_left_repeat)) * ff_c_add_left)) /\\ exists ff_q_add_left_repeat_decoded. ff_b_add_left = ff_q_add_left_repeat_decoded * S ((S (ff_i_add_left_repeat)) * ff_c_add_left) + (a)))) /\\ (exists ff_u_add_left_product ff_v_add_left_product. ((((exists ff_h_add_left_product_start. ff_h_add_left_product_start + S (1) = S ((S (0)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_start. ff_u_add_left_product = ff_q_add_left_product_start * S ((S (0)) * ff_v_add_left_product) + (1))) /\\ ((((exists ff_h_add_left_product_terminal. ff_h_add_left_product_terminal + S (x) = S ((S (e)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_terminal. ff_u_add_left_product = ff_q_add_left_product_terminal * S ((S (e)) * ff_v_add_left_product) + (x))) /\\ forall ff_i_add_left_product. (exists ff_lt_add_left_product_bound. ff_lt_add_left_product_bound + S ff_i_add_left_product = e) -> exists ff_p_add_left_product ff_r_add_left_product ff_s_add_left_product. ((((exists ff_h_add_left_product_factor. ff_h_add_left_product_factor + S (ff_p_add_left_product) = S ((S (ff_i_add_left_product)) * ff_c_add_left)) /\\ exists ff_q_add_left_product_factor. ff_b_add_left = ff_q_add_left_product_factor * S ((S (ff_i_add_left_product)) * ff_c_add_left) + (ff_p_add_left_product))) /\\ ((((exists ff_h_add_left_product_partial. ff_h_add_left_product_partial + S (ff_r_add_left_product) = S ((S (ff_i_add_left_product)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_partial. ff_u_add_left_product = ff_q_add_left_product_partial * S ((S (ff_i_add_left_product)) * ff_v_add_left_product) + (ff_r_add_left_product))) /\\ ((((exists ff_h_add_left_product_successor. ff_h_add_left_product_successor + S (ff_s_add_left_product) = S ((S (S ff_i_add_left_product)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_successor. ff_u_add_left_product = ff_q_add_left_product_successor * S ((S (S ff_i_add_left_product)) * ff_v_add_left_product) + (ff_s_add_left_product))) /\\ ff_s_add_left_product = ff_r_add_left_product * ff_p_add_left_product)))))))) -> (exists ff_b_add_right ff_c_add_right. ((forall ff_i_add_right_repeat. (exists ff_lt_add_right_repeat_bound. ff_lt_add_right_repeat_bound + S ff_i_add_right_repeat = f) -> (((exists ff_h_add_right_repeat_decoded. ff_h_add_right_repeat_decoded + S (a) = S ((S (ff_i_add_right_repeat)) * ff_c_add_right)) /\\ exists ff_q_add_right_repeat_decoded. ff_b_add_right = ff_q_add_right_repeat_decoded * S ((S (ff_i_add_right_repeat)) * ff_c_add_right) + (a)))) /\\ (exists ff_u_add_right_product ff_v_add_right_product. ((((exists ff_h_add_right_product_start. ff_h_add_right_product_start + S (1) = S ((S (0)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_start. ff_u_add_right_product = ff_q_add_right_product_start * S ((S (0)) * ff_v_add_right_product) + (1))) /\\ ((((exists ff_h_add_right_product_terminal. ff_h_add_right_product_terminal + S (y) = S ((S (f)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_terminal. ff_u_add_right_product = ff_q_add_right_product_terminal * S ((S (f)) * ff_v_add_right_product) + (y))) /\\ forall ff_i_add_right_product. (exists ff_lt_add_right_product_bound. ff_lt_add_right_product_bound + S ff_i_add_right_product = f) -> exists ff_p_add_right_product ff_r_add_right_product ff_s_add_right_product. ((((exists ff_h_add_right_product_factor. ff_h_add_right_product_factor + S (ff_p_add_right_product) = S ((S (ff_i_add_right_product)) * ff_c_add_right)) /\\ exists ff_q_add_right_product_factor. ff_b_add_right = ff_q_add_right_product_factor * S ((S (ff_i_add_right_product)) * ff_c_add_right) + (ff_p_add_right_product))) /\\ ((((exists ff_h_add_right_product_partial. ff_h_add_right_product_partial + S (ff_r_add_right_product) = S ((S (ff_i_add_right_product)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_partial. ff_u_add_right_product = ff_q_add_right_product_partial * S ((S (ff_i_add_right_product)) * ff_v_add_right_product) + (ff_r_add_right_product))) /\\ ((((exists ff_h_add_right_product_successor. ff_h_add_right_product_successor + S (ff_s_add_right_product) = S ((S (S ff_i_add_right_product)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_successor. ff_u_add_right_product = ff_q_add_right_product_successor * S ((S (S ff_i_add_right_product)) * ff_v_add_right_product) + (ff_s_add_right_product))) /\\ ff_s_add_right_product = ff_r_add_right_product * ff_p_add_right_product)))))))) -> (exists ff_b_add_total ff_c_add_total. ((forall ff_i_add_total_repeat. (exists ff_lt_add_total_repeat_bound. ff_lt_add_total_repeat_bound + S ff_i_add_total_repeat = s) -> (((exists ff_h_add_total_repeat_decoded. ff_h_add_total_repeat_decoded + S (a) = S ((S (ff_i_add_total_repeat)) * ff_c_add_total)) /\\ exists ff_q_add_total_repeat_decoded. ff_b_add_total = ff_q_add_total_repeat_decoded * S ((S (ff_i_add_total_repeat)) * ff_c_add_total) + (a)))) /\\ (exists ff_u_add_total_product ff_v_add_total_product. ((((exists ff_h_add_total_product_start. ff_h_add_total_product_start + S (1) = S ((S (0)) * ff_v_add_total_product)) /\\ exists ff_q_add_total_product_start. ff_u_add_total_product = ff_q_add_total_product_start * S ((S (0)) * ff_v_add_total_product) + (1))) /\\ ((((exists ff_h_add_total_product_terminal. ff_h_add_total_product_terminal + S (z) = S ((S (s)) * ff_v_add_total_product)) /\\ exists ff_q_add_total_product_terminal. ff_u_add_total_product = ff_q_add_total_product_terminal * S ((S (s)) * ff_v_add_total_product) + (z))) /\\ forall ff_i_add_total_product. (exists ff_lt_add_total_product_bound. ff_lt_add_total_product_bound + S ff_i_add_total_product = s) -> exists ff_p_add_total_product ff_r_add_total_product ff_s_add_total_product. ((((exists ff_h_add_total_product_factor. ff_h_add_total_product_factor + S (ff_p_add_total_product) = S ((S (ff_i_add_total_product)) * ff_c_add_total)) /\\ exists ff_q_add_total_product_factor. ff_b_add_total = ff_q_add_total_product_factor * S ((S (ff_i_add_total_product)) * ff_c_add_total) + (ff_p_add_total_product))) /\\ ((((exists ff_h_add_total_product_partial. ff_h_add_total_product_partial + S (ff_r_add_total_product) = S ((S (ff_i_add_total_product)) * ff_v_add_total_product)) /\\ exists ff_q_add_total_product_partial. ff_u_add_total_product = ff_q_add_total_product_partial * S ((S (ff_i_add_total_product)) * ff_v_add_total_product) + (ff_r_add_total_product))) /\\ ((((exists ff_h_add_total_product_successor. ff_h_add_total_product_successor + S (ff_s_add_total_product) = S ((S (S ff_i_add_total_product)) * ff_v_add_total_product)) /\\ exists ff_q_add_total_product_successor. ff_u_add_total_product = ff_q_add_total_product_successor * S ((S (S ff_i_add_total_product)) * ff_v_add_total_product) + (ff_s_add_total_product))) /\\ ff_s_add_total_product = ff_r_add_total_product * ff_p_add_total_product)))))))) -> z = x * y", "statement_sha256": "7ede92eb4825c0936cf43eb575041c886b24d07683cf2e7268d079f1a37123ab", "summary": "Relational powers turn addition of exponents into multiplication.", "summary_sha256": "03d87dbe2076024565f093b94ac2c27a5f6521e1115d43ead252ad83fbe5e613" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_zero", "pow_functional", "pow_successor_decompose", "mul_one", "mul_assoc" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=pow_add]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_add", "parent_alpha_version": "v34", "proof_bundle_node_id": 163, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_add", "script": [ "intro a", "intro e", "induction f", "intro s", "intro x", "intro y", "intro z", "intro hs", "intro hx", "intro hy", "intro hz", "rewrite PA3 at hs", "rewrite hs at hz", "rewrite hs at hz", "rewrite hs at hz", "rewrite hs at hz", "have hzx : z = x", "specialize pow_functional a", "specialize pow_functional e", "specialize pow_functional z", "specialize pow_functional x", "apply pow_functional", "exact hz", "exact hx", "have hy1 : y = 1", "specialize pow_zero a", "specialize pow_zero 0", "specialize pow_zero y", "apply pow_zero", "refl", "exact hy", "rewrite hzx", "rewrite hy1", "specialize mul_one x", "symm", "exact mul_one", "intro s", "intro x", "intro y", "intro z", "intro hs", "intro hx", "intro hy", "intro hz", "have hy_step : exists r. (exists ff_b_add_y_prefix ff_c_add_y_prefix. ((forall ff_i_add_y_prefix_repeat. (exists ff_lt_add_y_prefix_repeat_bound. ff_lt_add_y_prefix_repeat_bound + S ff_i_add_y_prefix_repeat = f) -> (((exists ff_h_add_y_prefix_repeat_decoded. ff_h_add_y_prefix_repeat_decoded + S (a) = S ((S (ff_i_add_y_prefix_repeat)) * ff_c_add_y_prefix)) /\\ exists ff_q_add_y_prefix_repeat_decoded. ff_b_add_y_prefix = ff_q_add_y_prefix_repeat_decoded * S ((S (ff_i_add_y_prefix_repeat)) * ff_c_add_y_prefix) + (a)))) /\\ (exists ff_u_add_y_prefix_product ff_v_add_y_prefix_product. ((((exists ff_h_add_y_prefix_product_start. ff_h_add_y_prefix_product_start + S (1) = S ((S (0)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_start. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_start * S ((S (0)) * ff_v_add_y_prefix_product) + (1))) /\\ ((((exists ff_h_add_y_prefix_product_terminal. ff_h_add_y_prefix_product_terminal + S (r) = S ((S (f)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_terminal. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_terminal * S ((S (f)) * ff_v_add_y_prefix_product) + (r))) /\\ forall ff_i_add_y_prefix_product. (exists ff_lt_add_y_prefix_product_bound. ff_lt_add_y_prefix_product_bound + S ff_i_add_y_prefix_product = f) -> exists ff_p_add_y_prefix_product ff_r_add_y_prefix_product ff_s_add_y_prefix_product. ((((exists ff_h_add_y_prefix_product_factor. ff_h_add_y_prefix_product_factor + S (ff_p_add_y_prefix_product) = S ((S (ff_i_add_y_prefix_product)) * ff_c_add_y_prefix)) /\\ exists ff_q_add_y_prefix_product_factor. ff_b_add_y_prefix = ff_q_add_y_prefix_product_factor * S ((S (ff_i_add_y_prefix_product)) * ff_c_add_y_prefix) + (ff_p_add_y_prefix_product))) /\\ ((((exists ff_h_add_y_prefix_product_partial. ff_h_add_y_prefix_product_partial + S (ff_r_add_y_prefix_product) = S ((S (ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_partial. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_partial * S ((S (ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product) + (ff_r_add_y_prefix_product))) /\\ ((((exists ff_h_add_y_prefix_product_successor. ff_h_add_y_prefix_product_successor + S (ff_s_add_y_prefix_product) = S ((S (S ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product)) /\\ exists ff_q_add_y_prefix_product_successor. ff_u_add_y_prefix_product = ff_q_add_y_prefix_product_successor * S ((S (S ff_i_add_y_prefix_product)) * ff_v_add_y_prefix_product) + (ff_s_add_y_prefix_product))) /\\ ff_s_add_y_prefix_product = ff_r_add_y_prefix_product * ff_p_add_y_prefix_product)))))))) /\\ y = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose f", "specialize pow_successor_decompose (S f)", "specialize pow_successor_decompose y", "apply pow_successor_decompose", "refl", "exact hy", "cases hy_step", "cases hy_step_witness", "have hst : s = S (e + f)", "trans e + S f", "exact hs", "apply PA4", "have hz_step : exists r. (exists pa_b_add_z_prefix pa_c_add_z_prefix. ((forall pa_i_add_z_prefix_repeat. (exists pa_lt_add_z_prefix_repeat_bound. pa_lt_add_z_prefix_repeat_bound + S pa_i_add_z_prefix_repeat = e + f) -> (((exists pa_h_add_z_prefix_repeat_decoded. pa_h_add_z_prefix_repeat_decoded + S (a) = S ((S (pa_i_add_z_prefix_repeat)) * pa_c_add_z_prefix)) /\\ exists pa_q_add_z_prefix_repeat_decoded. pa_b_add_z_prefix = pa_q_add_z_prefix_repeat_decoded * S ((S (pa_i_add_z_prefix_repeat)) * pa_c_add_z_prefix) + (a)))) /\\ (exists pa_u_add_z_prefix_product pa_v_add_z_prefix_product. ((((exists pa_h_add_z_prefix_product_start. pa_h_add_z_prefix_product_start + S (1) = S ((S (0)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_start. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_start * S ((S (0)) * pa_v_add_z_prefix_product) + (1))) /\\ ((((exists pa_h_add_z_prefix_product_terminal. pa_h_add_z_prefix_product_terminal + S (r) = S ((S (e + f)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_terminal. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_terminal * S ((S (e + f)) * pa_v_add_z_prefix_product) + (r))) /\\ forall pa_i_add_z_prefix_product. (exists pa_lt_add_z_prefix_product_bound. pa_lt_add_z_prefix_product_bound + S pa_i_add_z_prefix_product = e + f) -> exists pa_p_add_z_prefix_product pa_r_add_z_prefix_product pa_s_add_z_prefix_product. ((((exists pa_h_add_z_prefix_product_factor. pa_h_add_z_prefix_product_factor + S (pa_p_add_z_prefix_product) = S ((S (pa_i_add_z_prefix_product)) * pa_c_add_z_prefix)) /\\ exists pa_q_add_z_prefix_product_factor. pa_b_add_z_prefix = pa_q_add_z_prefix_product_factor * S ((S (pa_i_add_z_prefix_product)) * pa_c_add_z_prefix) + (pa_p_add_z_prefix_product))) /\\ ((((exists pa_h_add_z_prefix_product_partial. pa_h_add_z_prefix_product_partial + S (pa_r_add_z_prefix_product) = S ((S (pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_partial. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_partial * S ((S (pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product) + (pa_r_add_z_prefix_product))) /\\ ((((exists pa_h_add_z_prefix_product_successor. pa_h_add_z_prefix_product_successor + S (pa_s_add_z_prefix_product) = S ((S (S pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product)) /\\ exists pa_q_add_z_prefix_product_successor. pa_u_add_z_prefix_product = pa_q_add_z_prefix_product_successor * S ((S (S pa_i_add_z_prefix_product)) * pa_v_add_z_prefix_product) + (pa_s_add_z_prefix_product))) /\\ pa_s_add_z_prefix_product = pa_r_add_z_prefix_product * pa_p_add_z_prefix_product)))))))) /\\ z = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose (e + f)", "specialize pow_successor_decompose s", "specialize pow_successor_decompose z", "apply pow_successor_decompose", "exact hst", "exact hz", "cases hz_step", "cases hz_step_witness", "have hprefix : x2 = x * x1", "specialize IH (e + f)", "specialize IH x", "specialize IH x1", "specialize IH x2", "apply IH", "refl", "exact hx", "exact hy_step_witness_left", "exact hz_step_witness_left", "trans x2 * a", "exact hz_step_witness_right", "trans (x * x1) * a", "congr", "exact hprefix", "refl", "trans x * (x1 * a)", "apply mul_assoc", "congr", "refl", "symm", "exact hy_step_witness_right" ], "script_sha256": "e82049235aa2fc11cacd0eebc5a847c4469193b55c044829d309156032de79a4", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall a e f s x y z. s = e + f -> (exists ff_b_add_left ff_c_add_left. ((forall ff_i_add_left_repeat. (exists ff_lt_add_left_repeat_bound. ff_lt_add_left_repeat_bound + S ff_i_add_left_repeat = e) -> (((exists ff_h_add_left_repeat_decoded. ff_h_add_left_repeat_decoded + S (a) = S ((S (ff_i_add_left_repeat)) * ff_c_add_left)) /\\ exists ff_q_add_left_repeat_decoded. ff_b_add_left = ff_q_add_left_repeat_decoded * S ((S (ff_i_add_left_repeat)) * ff_c_add_left) + (a)))) /\\ (exists ff_u_add_left_product ff_v_add_left_product. ((((exists ff_h_add_left_product_start. ff_h_add_left_product_start + S (1) = S ((S (0)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_start. ff_u_add_left_product = ff_q_add_left_product_start * S ((S (0)) * ff_v_add_left_product) + (1))) /\\ ((((exists ff_h_add_left_product_terminal. ff_h_add_left_product_terminal + S (x) = S ((S (e)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_terminal. ff_u_add_left_product = ff_q_add_left_product_terminal * S ((S (e)) * ff_v_add_left_product) + (x))) /\\ forall ff_i_add_left_product. (exists ff_lt_add_left_product_bound. ff_lt_add_left_product_bound + S ff_i_add_left_product = e) -> exists ff_p_add_left_product ff_r_add_left_product ff_s_add_left_product. ((((exists ff_h_add_left_product_factor. ff_h_add_left_product_factor + S (ff_p_add_left_product) = S ((S (ff_i_add_left_product)) * ff_c_add_left)) /\\ exists ff_q_add_left_product_factor. ff_b_add_left = ff_q_add_left_product_factor * S ((S (ff_i_add_left_product)) * ff_c_add_left) + (ff_p_add_left_product))) /\\ ((((exists ff_h_add_left_product_partial. ff_h_add_left_product_partial + S (ff_r_add_left_product) = S ((S (ff_i_add_left_product)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_partial. ff_u_add_left_product = ff_q_add_left_product_partial * S ((S (ff_i_add_left_product)) * ff_v_add_left_product) + (ff_r_add_left_product))) /\\ ((((exists ff_h_add_left_product_successor. ff_h_add_left_product_successor + S (ff_s_add_left_product) = S ((S (S ff_i_add_left_product)) * ff_v_add_left_product)) /\\ exists ff_q_add_left_product_successor. ff_u_add_left_product = ff_q_add_left_product_successor * S ((S (S ff_i_add_left_product)) * ff_v_add_left_product) + (ff_s_add_left_product))) /\\ ff_s_add_left_product = ff_r_add_left_product * ff_p_add_left_product)))))))) -> (exists ff_b_add_right ff_c_add_right. ((forall ff_i_add_right_repeat. (exists ff_lt_add_right_repeat_bound. ff_lt_add_right_repeat_bound + S ff_i_add_right_repeat = f) -> (((exists ff_h_add_right_repeat_decoded. ff_h_add_right_repeat_decoded + S (a) = S ((S (ff_i_add_right_repeat)) * ff_c_add_right)) /\\ exists ff_q_add_right_repeat_decoded. ff_b_add_right = ff_q_add_right_repeat_decoded * S ((S (ff_i_add_right_repeat)) * ff_c_add_right) + (a)))) /\\ (exists ff_u_add_right_product ff_v_add_right_product. ((((exists ff_h_add_right_product_start. ff_h_add_right_product_start + S (1) = S ((S (0)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_start. ff_u_add_right_product = ff_q_add_right_product_start * S ((S (0)) * ff_v_add_right_product) + (1))) /\\ ((((exists ff_h_add_right_product_terminal. ff_h_add_right_product_terminal + S (y) = S ((S (f)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_terminal. ff_u_add_right_product = ff_q_add_right_product_terminal * S ((S (f)) * ff_v_add_right_product) + (y))) /\\ forall ff_i_add_right_product. (exists ff_lt_add_right_product_bound. ff_lt_add_right_product_bound + S ff_i_add_right_product = f) -> exists ff_p_add_right_product ff_r_add_right_product ff_s_add_right_product. ((((exists ff_h_add_right_product_factor. ff_h_add_right_product_factor + S (ff_p_add_right_product) = S ((S (ff_i_add_right_product)) * ff_c_add_right)) /\\ exists ff_q_add_right_product_factor. ff_b_add_right = ff_q_add_right_product_factor * S ((S (ff_i_add_right_product)) * ff_c_add_right) + (ff_p_add_right_product))) /\\ ((((exists ff_h_add_right_product_partial. ff_h_add_right_product_partial + S (ff_r_add_right_product) = S ((S (ff_i_add_right_product)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_partial. ff_u_add_right_product = ff_q_add_right_product_partial * S ((S (ff_i_add_right_product)) * ff_v_add_right_product) + (ff_r_add_right_product))) /\\ ((((exists ff_h_add_right_product_successor. ff_h_add_right_product_successor + S (ff_s_add_right_product) = S ((S (S ff_i_add_right_product)) * ff_v_add_right_product)) /\\ exists ff_q_add_right_product_successor. ff_u_add_right_product = ff_q_add_right_product_successor * S ((S (S ff_i_add_right_product)) * ff_v_add_right_product) + (ff_s_add_right_product))) /\\ ff_s_add_right_product = ff_r_add_right_product * ff_p_add_right_product)))))))) -> (exists ff_b_add_total ff_c_add_total. ((forall ff_i_add_total_repeat. (exists ff_lt_add_total_repeat_bound. ff_lt_add_total_repeat_bound + S ff_i_add_total_repeat = s) -> (((exists ff_h_add_total_repeat_decoded. ff_h_add_total_repeat_decoded + S (a) = S ((S (ff_i_add_total_repeat)) * ff_c_add_total)) /\\ exists ff_q_add_total_repeat_decoded. ff_b_add_total = ff_q_add_total_repeat_decoded * S ((S (ff_i_add_total_repeat)) * ff_c_add_total) + (a)))) /\\ (exists ff_u_add_total_product ff_v_add_total_product. ((((exists ff_h_add_total_product_start. ff_h_add_total_product_start + S (1) = S ((S (0)) * ff_v_add_total_product)) /\\ exists ff_q_add_total_product_start. ff_u_add_total_product = ff_q_add_total_product_start * S ((S (0)) * ff_v_add_total_product) + (1))) /\\ ((((exists ff_h_add_total_product_terminal. ff_h_add_total_product_terminal + S (z) = S ((S (s)) * ff_v_add_total_product)) /\\ exists ff_q_add_total_product_terminal. ff_u_add_total_product = ff_q_add_total_product_terminal * S ((S (s)) * ff_v_add_total_product) + (z))) /\\ forall ff_i_add_total_product. (exists ff_lt_add_total_product_bound. ff_lt_add_total_product_bound + S ff_i_add_total_product = s) -> exists ff_p_add_total_product ff_r_add_total_product ff_s_add_total_product. 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(exists h. h + S i = n) -> (((exists ff_h_swap_old_i. ff_h_swap_old_i + S (x) = S ((S (i)) * c)) /\\ exists ff_q_swap_old_i. b = ff_q_swap_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_old_n. ff_h_swap_old_n + S (y) = S ((S (n)) * c)) /\\ exists ff_q_swap_old_n. b = ff_q_swap_old_n * S ((S (n)) * c) + (y))) -> exists z d. ((((exists ff_h_swap_new_i. ff_h_swap_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_swap_new_i. z = ff_q_swap_new_i * S ((S (i)) * d) + (y))) /\\ ((((exists ff_h_swap_new_n. ff_h_swap_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_swap_new_n. z = ff_q_swap_new_n * S ((S (n)) * d) + (x))) /\\ forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_old_j. ff_h_swap_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_old_j. b = ff_q_swap_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_new_j. ff_h_swap_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_new_j. z = ff_q_swap_new_j * S ((S (j)) * d) + (a)))))", "statement_sha256": "efc7ec7a02b3e966da262daa4ea0f4cd2ea5c8c01af1515a6f7ad1d6e8ac2b24", "summary": "Swap a chosen interior beta entry with the last entry, given both decoded values.", "summary_sha256": "58dbb85c215e5972e2938d163eac30901fbfc034b09c5c54f803367932f419d2" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_prefix_replace_exists", "le_succ", "le_refl", "lt_irrefl_expanded" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=beta_prefix_swap_last_from_entries]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_prefix_swap_last_from_entries", "parent_alpha_version": "v34", "proof_bundle_node_id": 169, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_prefix_swap_last_from_entries", "script": [ "intro b", "intro c", "intro n", "intro i", "intro x", "intro y", "intro hi", "intro hxi", "intro hyn", "have hisn : exists h. h + S i = S n", "specialize le_succ (S i)", "specialize le_succ n", "apply le_succ", "exact hi", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hin : ~(i = n)", "intro hin_eq", "specialize lt_irrefl_expanded n", "apply lt_irrefl_expanded", "rewrite hin_eq at hi", "exact hi", "have hni : ~(n = i)", "intro hni_eq", "apply hin", "symm", "exact hni_eq", "have hfirst : exists z d. 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(((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. 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(forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l", "exact finite_bounded_entry_lt", "have hentry_bound_n : forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l", "exact finite_bounded_entry_lt", "have hxb : exists h. h + S x = S n", "specialize hentry_bound_i b", "specialize hentry_bound_i c", "specialize hentry_bound_i (S n)", "specialize hentry_bound_i i", "specialize hentry_bound_i x", "apply hentry_bound_i", "exact hbounded", "exact hisn", "exact hold_i", "have hyb : exists h. h + S y = S n", "specialize hentry_bound_n b", "specialize hentry_bound_n c", "specialize hentry_bound_n (S n)", "specialize hentry_bound_n n", "specialize hentry_bound_n y", "apply hentry_bound_n", "exact hbounded", "exact hnsn", "exact hold_n", "have heq_i : forall u v. u = v \\/ ~(u = v)", "exact eq_decidable", "have heq_n : forall u v. u = v \\/ ~(u = v)", "exact eq_decidable", "rewrite hsn", "rewrite hsn", "intro j", "intro hj", "specialize heq_i j", "specialize heq_i i", "cases heq_i", "exists y", "split", "rewrite heq_i_left", "rewrite heq_i_left", "exact hnew_i", "exact hyb", "specialize heq_n j", "specialize heq_n n", "cases heq_n", "exists x", "split", "rewrite heq_n_left", "rewrite heq_n_left", "exact hnew_n", "exact hxb", "specialize hbounded j", "have hold : exists a. (((exists h. h + S a = S ((S j) * c)) /\\ exists q. b = q * S ((S j) * c) + a) /\\ exists h. h + S a = S n)", "apply hbounded", "exact hj", "cases hold", "cases hold_witness", "exists x1", "split", "specialize hpreserve j", "specialize hpreserve x1", "apply hpreserve", "exact hj", "exact heq_i_right", "exact heq_n_right", "exact hold_witness_left", "exact hold_witness_right" ], "script_sha256": "954aa73b49514d2753f4d115eeb5d53eb3861755e3eca556257c0aced9a72183", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_bound_old. (exists fp_gap_swap_bound_old_index. fp_gap_swap_bound_old_index + S fp_i_swap_bound_old = sn) -> exists fp_value_swap_bound_old. ((((exists ff_h_swap_bound_old_entry. ff_h_swap_bound_old_entry + S (fp_value_swap_bound_old) = S ((S (fp_i_swap_bound_old)) * c)) /\\ exists ff_q_swap_bound_old_entry. b = ff_q_swap_bound_old_entry * S ((S (fp_i_swap_bound_old)) * c) + (fp_value_swap_bound_old))) /\\ (exists fp_gap_swap_bound_old_value. fp_gap_swap_bound_old_value + S fp_value_swap_bound_old = sn))) -> (((exists ff_h_swap_bound_old_i. ff_h_swap_bound_old_i + S (x) = S ((S (i)) * c)) /\\ exists ff_q_swap_bound_old_i. b = ff_q_swap_bound_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_bound_old_n. ff_h_swap_bound_old_n + S (y) = S ((S (n)) * c)) /\\ exists ff_q_swap_bound_old_n. b = ff_q_swap_bound_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_bound_new_i. ff_h_swap_bound_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_swap_bound_new_i. z = ff_q_swap_bound_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_bound_new_n. ff_h_swap_bound_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_swap_bound_new_n. z = ff_q_swap_bound_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_bound_old_j. ff_h_swap_bound_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_bound_old_j. b = ff_q_swap_bound_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_bound_new_j. ff_h_swap_bound_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_bound_new_j. z = ff_q_swap_bound_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_bound_new. (exists fp_gap_swap_bound_new_index. fp_gap_swap_bound_new_index + S fp_i_swap_bound_new = sn) -> exists fp_value_swap_bound_new. ((((exists ff_h_swap_bound_new_entry. ff_h_swap_bound_new_entry + S (fp_value_swap_bound_new) = S ((S (fp_i_swap_bound_new)) * d)) /\\ exists ff_q_swap_bound_new_entry. z = ff_q_swap_bound_new_entry * S ((S (fp_i_swap_bound_new)) * d) + (fp_value_swap_bound_new))) /\\ (exists fp_gap_swap_bound_new_value. fp_gap_swap_bound_new_value + S fp_value_swap_bound_new = sn)))", "statement_sha256": "cc20084340c25cc95fdb2f5fb90769164dca9a98cebd0655ad7c58939062acb9", "summary": "A swap-last recoding preserves boundedness of the full successor prefix.", "summary_sha256": "801f9b8725d3dee2eb0584c41c0bc11bd3620321fccf21543652c5a406078e93" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "finite_bounded_entry_lt", "eq_decidable", "le_succ", "le_refl" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_swap_last_bounded]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_swap_last_bounded", "parent_alpha_version": "v34", "proof_bundle_node_id": 171, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_swap_last_bounded", "script": [ "intro b", "intro c", "intro z", "intro d", "intro n", "intro sn", "intro i", "intro x", "intro y", "intro hsn", "intro hi", "intro hbounded", "intro hold_i", "intro hold_n", "intro hnew_i", "intro hnew_n", "intro hpreserve", "rewrite hsn at hbounded", "rewrite hsn at hbounded", "have hisn : exists h. h + S i = S n", "specialize le_succ (S i)", "specialize le_succ n", "apply le_succ", "exact hi", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hentry_bound_i : forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l", "exact finite_bounded_entry_lt", "have hentry_bound_n : forall b c l i x. (forall fp_i_entry_bound. (exists fp_gap_entry_bound_index. fp_gap_entry_bound_index + S fp_i_entry_bound = l) -> exists fp_value_entry_bound. ((((exists ff_h_entry_bound_entry. ff_h_entry_bound_entry + S (fp_value_entry_bound) = S ((S (fp_i_entry_bound)) * c)) /\\ exists ff_q_entry_bound_entry. b = ff_q_entry_bound_entry * S ((S (fp_i_entry_bound)) * c) + (fp_value_entry_bound))) /\\ (exists fp_gap_entry_bound_value. fp_gap_entry_bound_value + S fp_value_entry_bound = l))) -> (exists h. h + S i = l) -> (((exists ff_h_entry_bound_at. ff_h_entry_bound_at + S (x) = S ((S (i)) * c)) /\\ exists ff_q_entry_bound_at. b = ff_q_entry_bound_at * S ((S (i)) * c) + (x))) -> exists h. h + S x = l", "exact finite_bounded_entry_lt", "have hxb : exists h. h + S x = S n", "specialize hentry_bound_i b", "specialize hentry_bound_i c", "specialize hentry_bound_i (S n)", "specialize hentry_bound_i i", "specialize hentry_bound_i x", "apply hentry_bound_i", "exact hbounded", "exact hisn", "exact hold_i", "have hyb : exists h. h + S y = S n", "specialize hentry_bound_n b", "specialize hentry_bound_n c", "specialize hentry_bound_n (S n)", "specialize hentry_bound_n n", "specialize hentry_bound_n y", "apply hentry_bound_n", "exact hbounded", "exact hnsn", "exact hold_n", "have heq_i : forall u v. u = v \\/ ~(u = v)", "exact eq_decidable", "have heq_n : forall u v. u = v \\/ ~(u = v)", "exact eq_decidable", "rewrite hsn", "rewrite hsn", "intro j", "intro hj", "specialize heq_i j", "specialize heq_i i", "cases heq_i", "exists y", "split", "rewrite heq_i_left", "rewrite heq_i_left", "exact hnew_i", "exact hyb", "specialize heq_n j", "specialize heq_n n", "cases heq_n", "exists x", "split", "rewrite heq_n_left", "rewrite heq_n_left", "exact hnew_n", "exact hxb", "specialize hbounded j", "have hold : exists a. (((exists h. h + S a = S ((S j) * c)) /\\ exists q. b = q * S ((S j) * c) + a) /\\ exists h. h + S a = S n)", "apply hbounded", "exact hj", "cases hold", "cases hold_witness", "exists x1", "split", "specialize hpreserve j", "specialize hpreserve x1", "apply hpreserve", "exact hj", "exact heq_i_right", "exact heq_n_right", "exact hold_witness_left", "exact hold_witness_right" ], "script_sha256": "954aa73b49514d2753f4d115eeb5d53eb3861755e3eca556257c0aced9a72183", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_bound_old. (exists fp_gap_swap_bound_old_index. fp_gap_swap_bound_old_index + S fp_i_swap_bound_old = sn) -> exists fp_value_swap_bound_old. ((((exists ff_h_swap_bound_old_entry. ff_h_swap_bound_old_entry + S (fp_value_swap_bound_old) = S ((S (fp_i_swap_bound_old)) * c)) /\\ exists ff_q_swap_bound_old_entry. b = ff_q_swap_bound_old_entry * S ((S (fp_i_swap_bound_old)) * c) + (fp_value_swap_bound_old))) /\\ (exists fp_gap_swap_bound_old_value. fp_gap_swap_bound_old_value + S fp_value_swap_bound_old = sn))) -> (((exists ff_h_swap_bound_old_i. ff_h_swap_bound_old_i + S (x) = S ((S (i)) * c)) /\\ exists ff_q_swap_bound_old_i. b = ff_q_swap_bound_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_bound_old_n. ff_h_swap_bound_old_n + S (y) = S ((S (n)) * c)) /\\ exists ff_q_swap_bound_old_n. b = ff_q_swap_bound_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_bound_new_i. ff_h_swap_bound_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_swap_bound_new_i. z = ff_q_swap_bound_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_bound_new_n. ff_h_swap_bound_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_swap_bound_new_n. z = ff_q_swap_bound_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_bound_old_j. ff_h_swap_bound_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_bound_old_j. b = ff_q_swap_bound_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_bound_new_j. ff_h_swap_bound_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_bound_new_j. z = ff_q_swap_bound_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_bound_new. (exists fp_gap_swap_bound_new_index. fp_gap_swap_bound_new_index + S fp_i_swap_bound_new = sn) -> exists fp_value_swap_bound_new. ((((exists ff_h_swap_bound_new_entry. ff_h_swap_bound_new_entry + S (fp_value_swap_bound_new) = S ((S (fp_i_swap_bound_new)) * d)) /\\ exists ff_q_swap_bound_new_entry. z = ff_q_swap_bound_new_entry * S ((S (fp_i_swap_bound_new)) * d) + (fp_value_swap_bound_new))) /\\ (exists fp_gap_swap_bound_new_value. fp_gap_swap_bound_new_value + S fp_value_swap_bound_new = sn)))", "statement_sha256": "cc20084340c25cc95fdb2f5fb90769164dca9a98cebd0655ad7c58939062acb9" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_swap_last_injective", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_prefix_swap_last_reflect", "le_succ", "le_refl" ], "dependencies_sha256": "5a3222b4b53a921e50b3f52b9f00fd380ed6312cf1e48d175183d937dc2abb31", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "b9a49da8fb88f30576f93b62a3bac1f5dc9149dc5e8666351d1bec645c7e2212", "cut_nodes": 53, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 63, "proof_edges": 1491, "proof_nodes": 2203, "proof_objects": 1435, "reused_objects": 57, "status": "checked" }, "enrollment_index": 367, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_swap_last_injective]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "1b1dc9913d97d4a4921ad28c0e6b2fab3a9b41c360b9f2253950027395799e62", "membership": "stable", "name": "finite_swap_last_injective", "proof_tag": "PA004O", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro z", "intro d", "intro n", "intro sn", "intro i", "intro x", "intro y", "intro hsn", "intro hi", "intro hinjective", "intro hold_i", "intro hold_n", "intro hnew_i", "intro hnew_n", "intro hpreserve", "rewrite hsn at hinjective", "rewrite hsn at hinjective", "have hisn : exists h. h + S i = S n", "specialize le_succ (S i)", "specialize le_succ n", "apply le_succ", "exact hi", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hreflect_j : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "exact beta_prefix_swap_last_reflect", "have hreflect_k : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "exact beta_prefix_swap_last_reflect", "rewrite hsn", "rewrite hsn", "intro j", "intro k", "intro a", "intro hj", "intro hk", "intro hnew_j", "intro hnew_k", "specialize hreflect_j b", "specialize hreflect_j c", "specialize hreflect_j z", "specialize hreflect_j d", "specialize hreflect_j n", "specialize hreflect_j i", "specialize hreflect_j x", "specialize hreflect_j y", "have hreflect_entries_j : forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "apply hreflect_j", "exact hnew_i", "exact hnew_n", "exact hpreserve", "specialize hreflect_entries_j j", "specialize hreflect_entries_j a", "have hclass_j : ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ ((exists h. h + S a = S ((S j) * c)) /\\ exists q. b = q * S ((S j) * c) + a)))))", "apply hreflect_entries_j", "exact hj", "exact hnew_j", "specialize hreflect_k b", "specialize hreflect_k c", "specialize hreflect_k z", "specialize hreflect_k d", "specialize hreflect_k n", "specialize hreflect_k i", "specialize hreflect_k x", "specialize hreflect_k y", "have hreflect_entries_k : forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "apply hreflect_k", "exact hnew_i", "exact hnew_n", "exact hpreserve", "specialize hreflect_entries_k k", "specialize hreflect_entries_k a", "have hclass_k : ((k = i /\\ a = y) \\/ ((k = n /\\ a = x) \\/ (~(k = i) /\\ (~(k = n) /\\ ((exists h. h + S a = S ((S k) * c)) /\\ exists q. b = q * S ((S k) * c) + a)))))", "apply hreflect_entries_k", "exact hk", "exact hnew_k", "cases hclass_j", "cases hclass_j_left", "cases hclass_k", "cases hclass_k_left", "trans i", "exact hclass_j_left_left", "symm", "exact hclass_k_left_left", "cases hclass_k_right", "cases hclass_k_right_left", "have hxy : x = y", "trans a", "symm", "exact hclass_k_right_left_right", "exact hclass_j_left_right", "have hin : i = n", "specialize hinjective i", "specialize hinjective n", "specialize hinjective x", "apply hinjective", "exact hisn", "exact hnsn", "exact hold_i", "rewrite hxy", "rewrite hxy", "exact hold_n", "trans i", "exact hclass_j_left_left", "trans n", "exact hin", "symm", "exact hclass_k_right_left_left", "cases hclass_k_right_right", "cases hclass_k_right_right_right", "have hnk : n = k", "specialize hinjective n", "specialize hinjective k", "specialize hinjective y", "apply hinjective", "exact hnsn", "exact hk", "exact hold_n", "rewrite <- hclass_j_left_right", "rewrite <- hclass_j_left_right", "exact hclass_k_right_right_right_right", "exfalso", "apply hclass_k_right_right_right_left", "symm", "exact hnk", "cases hclass_j_right", "cases hclass_j_right_left", "cases hclass_k", "cases hclass_k_left", "have hxy2 : x = y", "trans a", "symm", "exact hclass_j_right_left_right", "exact hclass_k_left_right", "have hin2 : n = i", "specialize hinjective n", "specialize hinjective i", "specialize hinjective y", "apply hinjective", "exact hnsn", "exact hisn", "exact hold_n", "rewrite <- hxy2", "rewrite <- hxy2", "exact hold_i", "trans n", "exact hclass_j_right_left_left", "trans i", "exact hin2", "symm", "exact hclass_k_left_left", "cases hclass_k_right", "cases hclass_k_right_left", "trans n", "exact hclass_j_right_left_left", "symm", "exact hclass_k_right_left_left", "cases hclass_k_right_right", "cases hclass_k_right_right_right", "have hik : i = k", "specialize hinjective i", "specialize hinjective k", "specialize hinjective x", "apply hinjective", "exact hisn", "exact hk", "exact hold_i", "rewrite <- hclass_j_right_left_right", "rewrite <- hclass_j_right_left_right", "exact hclass_k_right_right_right_right", "exfalso", "apply hclass_k_right_right_left", "symm", "exact hik", "cases hclass_j_right_right", "cases hclass_j_right_right_right", "cases hclass_k", "cases hclass_k_left", "have hjn : j = n", "specialize hinjective j", "specialize hinjective n", "specialize hinjective y", "apply hinjective", "exact hj", "exact hnsn", "rewrite <- hclass_k_left_right", "rewrite <- hclass_k_left_right", "exact hclass_j_right_right_right_right", "exact hold_n", "exfalso", "apply hclass_j_right_right_right_left", "exact hjn", "cases hclass_k_right", "cases hclass_k_right_left", "have hji : j = i", "specialize hinjective j", "specialize hinjective i", "specialize hinjective x", "apply hinjective", "exact hj", "exact hisn", "rewrite <- hclass_k_right_left_right", "rewrite <- hclass_k_right_left_right", "exact hclass_j_right_right_right_right", "exact hold_i", "exfalso", "apply hclass_j_right_right_left", "exact hji", "cases hclass_k_right_right", "cases hclass_k_right_right_right", "specialize hinjective j", "specialize hinjective k", "specialize hinjective a", "apply hinjective", "exact hj", "exact hk", "exact hclass_j_right_right_right_right", "exact hclass_k_right_right_right_right" ], "script_sha256": "197325cac3b1578ad337023d095f9d62f4313d4ac0bae9daa645528ee531f5f3", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_inj_old fp_j_swap_inj_old fp_value_swap_inj_old. (exists fp_gap_swap_inj_old_i. fp_gap_swap_inj_old_i + S fp_i_swap_inj_old = sn) -> (exists fp_gap_swap_inj_old_j. fp_gap_swap_inj_old_j + S fp_j_swap_inj_old = sn) -> (((exists ff_h_swap_inj_old_left. ff_h_swap_inj_old_left + S (fp_value_swap_inj_old) = S ((S (fp_i_swap_inj_old)) * c)) /\\ exists ff_q_swap_inj_old_left. b = ff_q_swap_inj_old_left * S ((S (fp_i_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> (((exists ff_h_swap_inj_old_right. ff_h_swap_inj_old_right + S (fp_value_swap_inj_old) = S ((S (fp_j_swap_inj_old)) * c)) /\\ exists ff_q_swap_inj_old_right. b = ff_q_swap_inj_old_right * S ((S (fp_j_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> fp_i_swap_inj_old = fp_j_swap_inj_old) -> (((exists ff_h_swap_inj_old_i. ff_h_swap_inj_old_i + S (x) = S ((S (i)) * c)) /\\ exists ff_q_swap_inj_old_i. b = ff_q_swap_inj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_inj_old_n. ff_h_swap_inj_old_n + S (y) = S ((S (n)) * c)) /\\ exists ff_q_swap_inj_old_n. b = ff_q_swap_inj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_inj_new_i. ff_h_swap_inj_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_swap_inj_new_i. z = ff_q_swap_inj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_inj_new_n. ff_h_swap_inj_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_swap_inj_new_n. z = ff_q_swap_inj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_inj_old_j. ff_h_swap_inj_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_inj_old_j. b = ff_q_swap_inj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_inj_new_j. ff_h_swap_inj_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_inj_new_j. z = ff_q_swap_inj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_inj_new fp_j_swap_inj_new fp_value_swap_inj_new. (exists fp_gap_swap_inj_new_i. fp_gap_swap_inj_new_i + S fp_i_swap_inj_new = sn) -> (exists fp_gap_swap_inj_new_j. fp_gap_swap_inj_new_j + S fp_j_swap_inj_new = sn) -> (((exists ff_h_swap_inj_new_left. ff_h_swap_inj_new_left + S (fp_value_swap_inj_new) = S ((S (fp_i_swap_inj_new)) * d)) /\\ exists ff_q_swap_inj_new_left. z = ff_q_swap_inj_new_left * S ((S (fp_i_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> (((exists ff_h_swap_inj_new_right. ff_h_swap_inj_new_right + S (fp_value_swap_inj_new) = S ((S (fp_j_swap_inj_new)) * d)) /\\ exists ff_q_swap_inj_new_right. z = ff_q_swap_inj_new_right * S ((S (fp_j_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> fp_i_swap_inj_new = fp_j_swap_inj_new)", "statement_sha256": "2bcb226cbd0852d6e8ba278305e71c546c3dfacb63846432beb9ed2fa3c73a12", "summary": "A swap-last recoding preserves injectivity of the full successor prefix.", "summary_sha256": "af15839782c6eca7537a3ac9f0e41ba27b65768d15265b3a91d73674717d2abb" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_prefix_swap_last_reflect", "le_succ", "le_refl" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_swap_last_injective]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_swap_last_injective", "parent_alpha_version": "v34", "proof_bundle_node_id": 172, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_swap_last_injective", "script": [ "intro b", "intro c", "intro z", "intro d", "intro n", "intro sn", "intro i", "intro x", "intro y", "intro hsn", "intro hi", "intro hinjective", "intro hold_i", "intro hold_n", "intro hnew_i", "intro hnew_n", "intro hpreserve", "rewrite hsn at hinjective", "rewrite hsn at hinjective", "have hisn : exists h. h + S i = S n", "specialize le_succ (S i)", "specialize le_succ n", "apply le_succ", "exact hi", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hreflect_j : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "exact beta_prefix_swap_last_reflect", "have hreflect_k : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "exact beta_prefix_swap_last_reflect", "rewrite hsn", "rewrite hsn", "intro j", "intro k", "intro a", "intro hj", "intro hk", "intro hnew_j", "intro hnew_k", "specialize hreflect_j b", "specialize hreflect_j c", "specialize hreflect_j z", "specialize hreflect_j d", "specialize hreflect_j n", "specialize hreflect_j i", "specialize hreflect_j x", "specialize hreflect_j y", "have hreflect_entries_j : forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "apply hreflect_j", "exact hnew_i", "exact hnew_n", "exact hpreserve", "specialize hreflect_entries_j j", "specialize hreflect_entries_j a", "have hclass_j : ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ ((exists h. h + S a = S ((S j) * c)) /\\ exists q. b = q * S ((S j) * c) + a)))))", "apply hreflect_entries_j", "exact hj", "exact hnew_j", "specialize hreflect_k b", "specialize hreflect_k c", "specialize hreflect_k z", "specialize hreflect_k d", "specialize hreflect_k n", "specialize hreflect_k i", "specialize hreflect_k x", "specialize hreflect_k y", "have hreflect_entries_k : forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "apply hreflect_k", "exact hnew_i", "exact hnew_n", "exact hpreserve", "specialize hreflect_entries_k k", "specialize hreflect_entries_k a", "have hclass_k : ((k = i /\\ a = y) \\/ ((k = n /\\ a = x) \\/ (~(k = i) /\\ (~(k = n) /\\ ((exists h. h + S a = S ((S k) * c)) /\\ exists q. b = q * S ((S k) * c) + a)))))", "apply hreflect_entries_k", "exact hk", "exact hnew_k", "cases hclass_j", "cases hclass_j_left", "cases hclass_k", "cases hclass_k_left", "trans i", "exact hclass_j_left_left", "symm", "exact hclass_k_left_left", "cases hclass_k_right", "cases hclass_k_right_left", "have hxy : x = y", "trans a", "symm", "exact hclass_k_right_left_right", "exact hclass_j_left_right", "have hin : i = n", "specialize hinjective i", "specialize hinjective n", "specialize hinjective x", "apply hinjective", "exact hisn", "exact hnsn", "exact hold_i", "rewrite hxy", "rewrite hxy", "exact hold_n", "trans i", "exact hclass_j_left_left", "trans n", "exact hin", "symm", "exact hclass_k_right_left_left", "cases hclass_k_right_right", "cases hclass_k_right_right_right", "have hnk : n = k", "specialize hinjective n", "specialize hinjective k", "specialize hinjective y", "apply hinjective", "exact hnsn", "exact hk", "exact hold_n", "rewrite <- hclass_j_left_right", "rewrite <- hclass_j_left_right", "exact hclass_k_right_right_right_right", "exfalso", "apply hclass_k_right_right_right_left", "symm", "exact hnk", "cases hclass_j_right", "cases hclass_j_right_left", "cases hclass_k", "cases hclass_k_left", "have hxy2 : x = y", "trans a", "symm", "exact hclass_j_right_left_right", "exact hclass_k_left_right", "have hin2 : n = i", "specialize hinjective n", "specialize hinjective i", "specialize hinjective y", "apply hinjective", "exact hnsn", "exact hisn", "exact hold_n", "rewrite <- hxy2", "rewrite <- hxy2", "exact hold_i", "trans n", "exact hclass_j_right_left_left", "trans i", "exact hin2", "symm", "exact hclass_k_left_left", "cases hclass_k_right", "cases hclass_k_right_left", "trans n", "exact hclass_j_right_left_left", "symm", "exact hclass_k_right_left_left", "cases hclass_k_right_right", "cases hclass_k_right_right_right", "have hik : i = k", "specialize hinjective i", "specialize hinjective k", "specialize hinjective x", "apply hinjective", "exact hisn", "exact hk", "exact hold_i", "rewrite <- hclass_j_right_left_right", "rewrite <- hclass_j_right_left_right", "exact hclass_k_right_right_right_right", "exfalso", "apply hclass_k_right_right_left", "symm", "exact hik", "cases hclass_j_right_right", "cases hclass_j_right_right_right", "cases hclass_k", "cases hclass_k_left", "have hjn : j = n", "specialize hinjective j", "specialize hinjective n", "specialize hinjective y", "apply hinjective", "exact hj", "exact hnsn", "rewrite <- hclass_k_left_right", "rewrite <- hclass_k_left_right", "exact hclass_j_right_right_right_right", "exact hold_n", "exfalso", "apply hclass_j_right_right_right_left", "exact hjn", "cases hclass_k_right", "cases hclass_k_right_left", "have hji : j = i", "specialize hinjective j", "specialize hinjective i", "specialize hinjective x", "apply hinjective", "exact hj", "exact hisn", "rewrite <- hclass_k_right_left_right", "rewrite <- hclass_k_right_left_right", "exact hclass_j_right_right_right_right", "exact hold_i", "exfalso", "apply hclass_j_right_right_left", "exact hji", "cases hclass_k_right_right", "cases hclass_k_right_right_right", "specialize hinjective j", "specialize hinjective k", "specialize hinjective a", "apply hinjective", "exact hj", "exact hk", "exact hclass_j_right_right_right_right", "exact hclass_k_right_right_right_right" ], "script_sha256": "197325cac3b1578ad337023d095f9d62f4313d4ac0bae9daa645528ee531f5f3", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (forall fp_i_swap_inj_old fp_j_swap_inj_old fp_value_swap_inj_old. (exists fp_gap_swap_inj_old_i. fp_gap_swap_inj_old_i + S fp_i_swap_inj_old = sn) -> (exists fp_gap_swap_inj_old_j. fp_gap_swap_inj_old_j + S fp_j_swap_inj_old = sn) -> (((exists ff_h_swap_inj_old_left. ff_h_swap_inj_old_left + S (fp_value_swap_inj_old) = S ((S (fp_i_swap_inj_old)) * c)) /\\ exists ff_q_swap_inj_old_left. b = ff_q_swap_inj_old_left * S ((S (fp_i_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> (((exists ff_h_swap_inj_old_right. ff_h_swap_inj_old_right + S (fp_value_swap_inj_old) = S ((S (fp_j_swap_inj_old)) * c)) /\\ exists ff_q_swap_inj_old_right. b = ff_q_swap_inj_old_right * S ((S (fp_j_swap_inj_old)) * c) + (fp_value_swap_inj_old))) -> fp_i_swap_inj_old = fp_j_swap_inj_old) -> (((exists ff_h_swap_inj_old_i. ff_h_swap_inj_old_i + S (x) = S ((S (i)) * c)) /\\ exists ff_q_swap_inj_old_i. b = ff_q_swap_inj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_inj_old_n. ff_h_swap_inj_old_n + S (y) = S ((S (n)) * c)) /\\ exists ff_q_swap_inj_old_n. b = ff_q_swap_inj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_inj_new_i. ff_h_swap_inj_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_swap_inj_new_i. z = ff_q_swap_inj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_inj_new_n. ff_h_swap_inj_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_swap_inj_new_n. z = ff_q_swap_inj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_inj_old_j. ff_h_swap_inj_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_inj_old_j. b = ff_q_swap_inj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_inj_new_j. ff_h_swap_inj_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_inj_new_j. z = ff_q_swap_inj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_i_swap_inj_new fp_j_swap_inj_new fp_value_swap_inj_new. (exists fp_gap_swap_inj_new_i. fp_gap_swap_inj_new_i + S fp_i_swap_inj_new = sn) -> (exists fp_gap_swap_inj_new_j. fp_gap_swap_inj_new_j + S fp_j_swap_inj_new = sn) -> (((exists ff_h_swap_inj_new_left. ff_h_swap_inj_new_left + S (fp_value_swap_inj_new) = S ((S (fp_i_swap_inj_new)) * d)) /\\ exists ff_q_swap_inj_new_left. z = ff_q_swap_inj_new_left * S ((S (fp_i_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> (((exists ff_h_swap_inj_new_right. ff_h_swap_inj_new_right + S (fp_value_swap_inj_new) = S ((S (fp_j_swap_inj_new)) * d)) /\\ exists ff_q_swap_inj_new_right. z = ff_q_swap_inj_new_right * S ((S (fp_j_swap_inj_new)) * d) + (fp_value_swap_inj_new))) -> fp_i_swap_inj_new = fp_j_swap_inj_new)", "statement_sha256": "2bcb226cbd0852d6e8ba278305e71c546c3dfacb63846432beb9ed2fa3c73a12" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_swap_last_surjective_back", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "beta_prefix_swap_last_reflect", "le_succ", "le_refl" ], "dependencies_sha256": "5a3222b4b53a921e50b3f52b9f00fd380ed6312cf1e48d175183d937dc2abb31", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "689f32aef19d5615385b9a8afa366bc4c6651bcac8c27d5e27591df89330b158", "cut_nodes": 53, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 63, "proof_edges": 1217, "proof_nodes": 1929, "proof_objects": 1161, "reused_objects": 57, "status": "checked" }, "enrollment_index": 368, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_swap_last_surjective_back]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "022ad89019dcd5fcaac15766c3d41bc68cf91b127bf6b282817a160831e7d4ec", "membership": "stable", "name": "finite_swap_last_surjective_back", "proof_tag": "PA004U", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro z", "intro d", "intro n", "intro sn", "intro i", "intro x", "intro y", "intro hsn", "intro hi", "intro hold_i", "intro hold_n", "intro hnew_i", "intro hnew_n", "intro hpreserve", "intro hsurjective", "rewrite hsn at hsurjective", "rewrite hsn at hsurjective", "have hisn : exists h. h + S i = S n", "specialize le_succ (S i)", "specialize le_succ n", "apply le_succ", "exact hi", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hreflect : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "exact beta_prefix_swap_last_reflect", "specialize hreflect b", "specialize hreflect c", "specialize hreflect z", "specialize hreflect d", "specialize hreflect n", "specialize hreflect i", "specialize hreflect x", "specialize hreflect y", "have hreflect_entries : forall j a. (exists h. h + S j = S n) -> ((exists h. h + S a = S ((S j) * d)) /\\ exists q. z = q * S ((S j) * d) + a) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ ((exists h. h + S a = S ((S j) * c)) /\\ exists q. b = q * S ((S j) * c) + a)))))", "apply hreflect", "exact hnew_i", "exact hnew_n", "exact hpreserve", "rewrite hsn", "rewrite hsn", "intro a", "intro ha", "specialize hsurjective a", "have hoccurs : exists j. ((exists h. h + S j = S n) /\\ ((exists h. h + S a = S ((S j) * d)) /\\ exists q. z = q * S ((S j) * d) + a))", "apply hsurjective", "exact ha", "cases hoccurs", "cases hoccurs_witness", "specialize hreflect_entries x1", "specialize hreflect_entries a", "have hsource : ((x1 = i /\\ a = y) \\/ ((x1 = n /\\ a = x) \\/ (~(x1 = i) /\\ (~(x1 = n) /\\ ((exists h. h + S a = S ((S x1) * c)) /\\ exists q. b = q * S ((S x1) * c) + a)))))", "apply hreflect_entries", "exact hoccurs_witness_left", "exact hoccurs_witness_right", "cases hsource", "cases hsource_left", "exists n", "split", "exact hnsn", "rewrite hsource_left_right", "rewrite hsource_left_right", "exact hold_n", "cases hsource_right", "cases hsource_right_left", "exists i", "split", "exact hisn", "rewrite hsource_right_left_right", "rewrite hsource_right_left_right", "exact hold_i", "cases hsource_right_right", "cases hsource_right_right_right", "exists x1", "split", "exact hoccurs_witness_left", "exact hsource_right_right_right_right" ], "script_sha256": "269db835be895f3ae8a134a118ff505ea1c3ba3be1889359b2192fa45c98365a", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c z d n sn i x y. sn = S n -> (exists h. h + S i = n) -> (((exists ff_h_swap_surj_old_i. ff_h_swap_surj_old_i + S (x) = S ((S (i)) * c)) /\\ exists ff_q_swap_surj_old_i. b = ff_q_swap_surj_old_i * S ((S (i)) * c) + (x))) -> (((exists ff_h_swap_surj_old_n. ff_h_swap_surj_old_n + S (y) = S ((S (n)) * c)) /\\ exists ff_q_swap_surj_old_n. b = ff_q_swap_surj_old_n * S ((S (n)) * c) + (y))) -> (((exists ff_h_swap_surj_new_i. ff_h_swap_surj_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_swap_surj_new_i. z = ff_q_swap_surj_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_swap_surj_new_n. ff_h_swap_surj_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_swap_surj_new_n. z = ff_q_swap_surj_new_n * S ((S (n)) * d) + (x))) -> (forall j a. (exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_surj_old_j. ff_h_swap_surj_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_surj_old_j. b = ff_q_swap_surj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_surj_new_j. ff_h_swap_surj_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_surj_new_j. z = ff_q_swap_surj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_value_swap_surj_new. (exists fp_gap_swap_surj_new_value. fp_gap_swap_surj_new_value + S fp_value_swap_surj_new = sn) -> exists fp_i_swap_surj_new. ((exists fp_gap_swap_surj_new_index. fp_gap_swap_surj_new_index + S fp_i_swap_surj_new = sn) /\\ (((exists ff_h_swap_surj_new_entry. ff_h_swap_surj_new_entry + S (fp_value_swap_surj_new) = S ((S (fp_i_swap_surj_new)) * d)) /\\ exists ff_q_swap_surj_new_entry. z = ff_q_swap_surj_new_entry * S ((S (fp_i_swap_surj_new)) * d) + (fp_value_swap_surj_new))))) -> (forall fp_value_swap_surj_old. (exists fp_gap_swap_surj_old_value. fp_gap_swap_surj_old_value + S fp_value_swap_surj_old = sn) -> exists fp_i_swap_surj_old. ((exists fp_gap_swap_surj_old_index. fp_gap_swap_surj_old_index + S fp_i_swap_surj_old = sn) /\\ (((exists ff_h_swap_surj_old_entry. ff_h_swap_surj_old_entry + S (fp_value_swap_surj_old) = S ((S (fp_i_swap_surj_old)) * c)) /\\ exists ff_q_swap_surj_old_entry. b = ff_q_swap_surj_old_entry * S ((S (fp_i_swap_surj_old)) * c) + (fp_value_swap_surj_old)))))", "statement_sha256": "222b6436eb97efeff9d4fe3284bee64f5e58ea781d93c539805fe074b64b96ae", "summary": "Surjectivity of a swapped successor prefix transports back to the original code.", "summary_sha256": "61f453d5b6eb95e971c049c1279adced5bbb3d2c7b087facfe52e6683e555b5f" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_prefix_swap_last_reflect", "le_succ", "le_refl" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_swap_last_surjective_back]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_swap_last_surjective_back", "parent_alpha_version": "v34", "proof_bundle_node_id": 173, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_swap_last_surjective_back", "script": [ "intro b", "intro c", "intro z", "intro d", "intro n", "intro sn", "intro i", "intro x", "intro y", "intro hsn", "intro hi", "intro hold_i", "intro hold_n", "intro hnew_i", "intro hnew_n", "intro hpreserve", "intro hsurjective", "rewrite hsn at hsurjective", "rewrite hsn at hsurjective", "have hisn : exists h. h + S i = S n", "specialize le_succ (S i)", "specialize le_succ n", "apply le_succ", "exact hi", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hreflect : forall b c z d n i x y. (((exists ff_h_reflect_new_i. ff_h_reflect_new_i + S (y) = S ((S (i)) * d)) /\\ exists ff_q_reflect_new_i. z = ff_q_reflect_new_i * S ((S (i)) * d) + (y))) -> (((exists ff_h_reflect_new_n. ff_h_reflect_new_n + S (x) = S ((S (n)) * d)) /\\ exists ff_q_reflect_new_n. z = ff_q_reflect_new_n * S ((S (n)) * d) + (x))) -> (forall k v. (exists h. h + S k = S n) -> ~(k = i) -> ~(k = n) -> (((exists ff_h_reflect_old_k. ff_h_reflect_old_k + S (v) = S ((S (k)) * c)) /\\ exists ff_q_reflect_old_k. b = ff_q_reflect_old_k * S ((S (k)) * c) + (v))) -> (((exists ff_h_reflect_new_k. ff_h_reflect_new_k + S (v) = S ((S (k)) * d)) /\\ exists ff_q_reflect_new_k. z = ff_q_reflect_new_k * S ((S (k)) * d) + (v)))) -> forall j a. (exists h. h + S j = S n) -> (((exists ff_h_reflect_new_j. ff_h_reflect_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_reflect_new_j. z = ff_q_reflect_new_j * S ((S (j)) * d) + (a))) -> ((j = i /\\ a = y) \\/ ((j = n /\\ a = x) \\/ (~(j = i) /\\ (~(j = n) /\\ (((exists ff_h_reflect_old_j. ff_h_reflect_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_reflect_old_j. b = ff_q_reflect_old_j * S ((S (j)) * c) + (a)))))))", "exact beta_prefix_swap_last_reflect", "specialize hreflect b", "specialize hreflect c", "specialize hreflect z", "specialize hreflect d", "specialize hreflect n", "specialize hreflect i", "specialize hreflect x", "specialize hreflect y", "have hreflect_entries : forall j a. 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(exists h. h + S j = S n) -> ~(j = i) -> ~(j = n) -> (((exists ff_h_swap_surj_old_j. ff_h_swap_surj_old_j + S (a) = S ((S (j)) * c)) /\\ exists ff_q_swap_surj_old_j. b = ff_q_swap_surj_old_j * S ((S (j)) * c) + (a))) -> (((exists ff_h_swap_surj_new_j. ff_h_swap_surj_new_j + S (a) = S ((S (j)) * d)) /\\ exists ff_q_swap_surj_new_j. z = ff_q_swap_surj_new_j * S ((S (j)) * d) + (a)))) -> (forall fp_value_swap_surj_new. (exists fp_gap_swap_surj_new_value. fp_gap_swap_surj_new_value + S fp_value_swap_surj_new = sn) -> exists fp_i_swap_surj_new. ((exists fp_gap_swap_surj_new_index. fp_gap_swap_surj_new_index + S fp_i_swap_surj_new = sn) /\\ (((exists ff_h_swap_surj_new_entry. ff_h_swap_surj_new_entry + S (fp_value_swap_surj_new) = S ((S (fp_i_swap_surj_new)) * d)) /\\ exists ff_q_swap_surj_new_entry. z = ff_q_swap_surj_new_entry * S ((S (fp_i_swap_surj_new)) * d) + (fp_value_swap_surj_new))))) -> (forall fp_value_swap_surj_old. (exists fp_gap_swap_surj_old_value. fp_gap_swap_surj_old_value + S fp_value_swap_surj_old = sn) -> exists fp_i_swap_surj_old. ((exists fp_gap_swap_surj_old_index. fp_gap_swap_surj_old_index + S fp_i_swap_surj_old = sn) /\\ (((exists ff_h_swap_surj_old_entry. ff_h_swap_surj_old_entry + S (fp_value_swap_surj_old) = S ((S (fp_i_swap_surj_old)) * c)) /\\ exists ff_q_swap_surj_old_entry. b = ff_q_swap_surj_old_entry * S ((S (fp_i_swap_surj_old)) * c) + (fp_value_swap_surj_old)))))", "statement_sha256": "222b6436eb97efeff9d4fe3284bee64f5e58ea781d93c539805fe074b64b96ae" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_contains_decidable", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "add_eq_zero_right", "succ_ne_zero", "finite_lt_succ_eq_or_lt", "beta_at_exists", "beta_at_unique", "eq_decidable", "le_refl", "le_succ" ], "dependencies_sha256": "ac2fe062bf99dca444303d8b0c1ec9f0ee7408559e65e9ee0ad2e0565a29efd8", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "264b86a40d0a53815676d1f9066b012e2701ac578e369e7cd5b44ae72822aa0e", "cut_nodes": 60, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 64, "proof_edges": 1200, "proof_nodes": 1961, "proof_objects": 1140, "reused_objects": 61, "status": "checked" }, "enrollment_index": 369, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_contains_decidable]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "5588d19f036df9aa958fd81a00ebf53330df2fcf56a61bbc708cfc9e20aa6e56", "membership": "stable", "name": "finite_contains_decidable", "proof_tag": "PA004H", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "induction l", "intro y", "right", "intro hcontains", "cases hcontains", "cases hcontains_witness", "cases hcontains_witness_left", "have hsi : S x = 0", "specialize add_eq_zero_right x1", "specialize add_eq_zero_right (S x)", "apply add_eq_zero_right", "exact hcontains_witness_left_witness", "specialize succ_ne_zero x", "apply succ_ne_zero", "exact hsi", "intro y", "have hpresent : (exists i. 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((exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))) \\/ ~(exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))))", "statement_sha256": "b380408a4f2da9182b3e2c5a64cf9bf39b22efe5fd581a0803d9596318554204", "summary": "Occurrence of a value in a nonempty decoded prefix is constructively decidable.", "summary_sha256": "fea750d28b483e082f44fff923ea90fb065cb1cdcaefe83f78848802456ddef9" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "add_eq_zero_right", "succ_ne_zero", "finite_lt_succ_eq_or_lt", "beta_at_exists", "beta_at_unique", "eq_decidable", "le_refl", "le_succ" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_contains_decidable]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_contains_decidable", "parent_alpha_version": "v34", "proof_bundle_node_id": 174, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_contains_decidable", "script": [ "intro b", "intro c", "induction l", "intro y", "right", "intro hcontains", "cases hcontains", "cases hcontains_witness", "cases hcontains_witness_left", "have hsi : S x = 0", "specialize add_eq_zero_right x1", "specialize add_eq_zero_right (S x)", "apply add_eq_zero_right", "exact hcontains_witness_left_witness", "specialize succ_ne_zero x", "apply succ_ne_zero", "exact hsi", "intro y", "have hpresent : (exists i. 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((exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))) \\/ ~(exists fp_i_contains_l. ((exists fp_gap_contains_l_index. fp_gap_contains_l_index + S fp_i_contains_l = l) /\\ (((exists ff_h_contains_l_entry. ff_h_contains_l_entry + S (y) = S ((S (fp_i_contains_l)) * c)) /\\ exists ff_q_contains_l_entry. b = ff_q_contains_l_entry * S ((S (fp_i_contains_l)) * c) + (y))))))", "statement_sha256": "b380408a4f2da9182b3e2c5a64cf9bf39b22efe5fd581a0803d9596318554204" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_bounded_prefix_without_top", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "le_succ", "finite_lt_succ_eq_or_lt" ], "dependencies_sha256": "d528de6046ddaf2a825b4f8e3f2b57c572eabba78592817f8f0ba80acd9e27aa", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "994ca043b11bb83ec7ebc21ac19f2bd3655ac19cf7de5ed4741863937e0d7276", "cut_nodes": 8, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 23, "proof_edges": 189, "proof_nodes": 216, "proof_objects": 185, "reused_objects": 5, "status": "checked" }, "enrollment_index": 370, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_bounded_prefix_without_top]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "6a15e09ed485f9064e6fcd19f23fabbcd353f6402e1474d2a4dfac25aaa46603", "membership": "stable", "name": "finite_bounded_prefix_without_top", "proof_tag": "PA004P", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro n", "intro sn", "intro hsn", "intro hbounded", "intro hnotop", "rewrite hsn at hbounded", "rewrite hsn at hbounded", "intro i", "intro hi", "specialize hbounded i", "have hfull : exists x. 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(exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n)))", "statement_sha256": "24c86ea298426ce28f9fa7366f65fd1d2ea5f8155df4cc625450f59ec4e99f56", "summary": "A bounded injective successor sequence must place the new value last once its prefix is surjective.", "summary_sha256": "0a7b368a2ee144deaf03eefb87c698a0441db870c5e2819b33cc81eb6ff2bbfb" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "finite_bounded_last_succ", "finite_lt_succ_eq_or_lt", "le_refl", "le_succ", "lt_irrefl_expanded" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_last_is_top_from_prefix_surjective]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_last_is_top_from_prefix_surjective", "parent_alpha_version": "v34", "proof_bundle_node_id": 178, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_last_is_top_from_prefix_surjective", "script": [ "intro b", "intro c", "intro n", "intro sn", "intro hsn", "intro hbounded", "intro hinj", "intro hsurj", "rewrite hsn at hinj", "rewrite hsn at hinj", "have hlast : exists x. 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(exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n)))", "statement_sha256": "24c86ea298426ce28f9fa7366f65fd1d2ea5f8155df4cc625450f59ec4e99f56" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_surjective_succ_from_prefix", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "finite_last_is_top_from_prefix_surjective", "finite_surjective_succ_intro" ], "dependencies_sha256": "3e4a9c810d8b23f2f46f95b2c4ae2474f21234ac0ef2e5ff033752e7f2c38796", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "78bdcadaa47a156c08d300346827c1799b006a89197259dd4d58bdc4ca985e87", "cut_nodes": 28, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 31, "proof_edges": 395, "proof_nodes": 678, "proof_objects": 385, "reused_objects": 11, "status": "checked" }, "enrollment_index": 374, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_surjective_succ_from_prefix]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "f7759be9a79e93fdb31fc01c208d7c16f3d4b3da5d73aaa2f876448b4bd5cbf6", "membership": "stable", "name": "finite_surjective_succ_from_prefix", "proof_tag": "PA004T", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro n", "intro sn", "intro hsn", "intro hbounded", "intro hinj", "intro hsurj", "have hlast : ((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n))", "specialize finite_last_is_top_from_prefix_surjective b", "specialize finite_last_is_top_from_prefix_surjective c", "specialize finite_last_is_top_from_prefix_surjective n", "specialize finite_last_is_top_from_prefix_surjective sn", "apply finite_last_is_top_from_prefix_surjective", "exact hsn", "exact hbounded", "exact hinj", "exact hsurj", "specialize finite_surjective_succ_intro b", "specialize finite_surjective_succ_intro c", "specialize finite_surjective_succ_intro n", "specialize finite_surjective_succ_intro sn", "apply finite_surjective_succ_intro", "exact hsn", "exact hsurj", "exact hlast" ], "script_sha256": "de9ffe9ab5c7621efa0bd7d820e0b7de76101b65dc4ba707720e8ba959c9ea07", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))", "statement_sha256": "9a186aa36848e2e362baecbf084f1334d9f9fb6c7248c70728272fc0944e948d", "summary": "The available successor branch extends prefix surjectivity to the full prefix.", "summary_sha256": "a0684de4bfcad2794dc9283d6dbfea62c341bf132a1ad860c21b869e94709756" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "finite_last_is_top_from_prefix_surjective", "finite_surjective_succ_intro" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_surjective_succ_from_prefix]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_surjective_succ_from_prefix", "parent_alpha_version": "v34", "proof_bundle_node_id": 179, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_surjective_succ_from_prefix", "script": [ "intro b", "intro c", "intro n", "intro sn", "intro hsn", "intro hbounded", "intro hinj", "intro hsurj", "have hlast : ((exists ff_h_last_n. ff_h_last_n + S (n) = S ((S (n)) * c)) /\\ exists ff_q_last_n. b = ff_q_last_n * S ((S (n)) * c) + (n))", "specialize finite_last_is_top_from_prefix_surjective b", "specialize finite_last_is_top_from_prefix_surjective c", "specialize finite_last_is_top_from_prefix_surjective n", "specialize finite_last_is_top_from_prefix_surjective sn", "apply finite_last_is_top_from_prefix_surjective", "exact hsn", "exact hbounded", "exact hinj", "exact hsurj", "specialize finite_surjective_succ_intro b", "specialize finite_surjective_succ_intro c", "specialize finite_surjective_succ_intro n", "specialize finite_surjective_succ_intro sn", "apply finite_surjective_succ_intro", "exact hsn", "exact hsurj", "exact hlast" ], "script_sha256": "de9ffe9ab5c7621efa0bd7d820e0b7de76101b65dc4ba707720e8ba959c9ea07", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))", "statement_sha256": "9a186aa36848e2e362baecbf084f1334d9f9fb6c7248c70728272fc0944e948d" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_no_top_successor_gate", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "finite_bounded_prefix_without_top", "finite_injective_prefix_succ", "finite_surjective_succ_from_prefix" ], "dependencies_sha256": "4fe8fa5992b0f4130d314d8ae54e25fcbb1fda467d5f4db097cd32b4853a4583", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "2d6d4891922b432f50793a01195ecdab4a2ec330b30f127d903564556d2430a6", "cut_nodes": 41, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 36, "proof_edges": 566, "proof_nodes": 1054, "proof_objects": 553, "reused_objects": 14, "status": "checked" }, "enrollment_index": 375, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_no_top_successor_gate]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "240ef07360bf3cf36b500974b093dea01b1b27d82bc8bf6f0fb10fa2e9e7d189", "membership": "stable", "name": "finite_no_top_successor_gate", "proof_tag": "PA004V", "provenance": [ "stable" ], "script": [ "intro b", "intro c", "intro n", "intro sn", "intro hsn", "intro hbounded", "intro hinj", "intro hmissing", "intro hinduction", "have hnotop : forall i. (exists h. h + S i = n) -> ~((exists h. h + S n = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + n)", "intro i", "intro hi", "intro hentry", "apply hmissing", "exists i", "split", "exact hi", "exact hentry", "have hprefix_bounded : forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))", "specialize finite_bounded_prefix_without_top b", "specialize finite_bounded_prefix_without_top c", "specialize finite_bounded_prefix_without_top n", "specialize finite_bounded_prefix_without_top sn", "apply finite_bounded_prefix_without_top", "exact hsn", "exact hbounded", "exact hnotop", "have hprefix_injective : forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix", "specialize finite_injective_prefix_succ b", "specialize finite_injective_prefix_succ c", "specialize finite_injective_prefix_succ n", "specialize finite_injective_prefix_succ sn", "apply finite_injective_prefix_succ", "exact hsn", "exact hinj", "have hprefix_surjective : forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))", "apply hinduction", "exact hprefix_bounded", "exact hprefix_injective", "specialize finite_surjective_succ_from_prefix b", "specialize finite_surjective_succ_from_prefix c", "specialize finite_surjective_succ_from_prefix n", "specialize finite_surjective_succ_from_prefix sn", "apply finite_surjective_succ_from_prefix", "exact hsn", "exact hbounded", "exact hinj", "exact hprefix_surjective" ], "script_sha256": "cc1af0767a92727ff1477db30b7d3b60c78d3bcac24c7205f2294e2dd38a866b", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) -> ((forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n)))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))", "statement_sha256": "9b8d47daab6b099597b4875c374f6a5012be4f1162d33a6266bb5d934a844c88", "summary": "The no-top branch of the constructive successor induction is complete.", "summary_sha256": "2de9ccb6d0dbe70fb5144183b2406b54e531b1e6629f32a6fb688236b0467206" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "finite_bounded_prefix_without_top", "finite_injective_prefix_succ", "finite_surjective_succ_from_prefix" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_no_top_successor_gate]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_no_top_successor_gate", "parent_alpha_version": "v34", "proof_bundle_node_id": 180, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_no_top_successor_gate", "script": [ "intro b", "intro c", "intro n", "intro sn", "intro hsn", "intro hbounded", "intro hinj", "intro hmissing", "intro hinduction", "have hnotop : forall i. (exists h. h + S i = n) -> ~((exists h. h + S n = S ((S i) * c)) /\\ exists q. b = q * S ((S i) * c) + n)", "intro i", "intro hi", "intro hentry", "apply hmissing", "exists i", "split", "exact hi", "exact hentry", "have hprefix_bounded : forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))", "specialize finite_bounded_prefix_without_top b", "specialize finite_bounded_prefix_without_top c", "specialize finite_bounded_prefix_without_top n", "specialize finite_bounded_prefix_without_top sn", "apply finite_bounded_prefix_without_top", "exact hsn", "exact hbounded", "exact hnotop", "have hprefix_injective : forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix", "specialize finite_injective_prefix_succ b", "specialize finite_injective_prefix_succ c", "specialize finite_injective_prefix_succ n", "specialize finite_injective_prefix_succ sn", "apply finite_injective_prefix_succ", "exact hsn", "exact hinj", "have hprefix_surjective : forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n))))", "apply hinduction", "exact hprefix_bounded", "exact hprefix_injective", "specialize finite_surjective_succ_from_prefix b", "specialize finite_surjective_succ_from_prefix c", "specialize finite_surjective_succ_from_prefix n", "specialize finite_surjective_succ_from_prefix sn", "apply finite_surjective_succ_from_prefix", "exact hsn", "exact hbounded", "exact hinj", "exact hprefix_surjective" ], "script_sha256": "cc1af0767a92727ff1477db30b7d3b60c78d3bcac24c7205f2294e2dd38a866b", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall b c n sn. sn = S n -> (forall fp_i_bounded_succ. (exists fp_gap_bounded_succ_index. fp_gap_bounded_succ_index + S fp_i_bounded_succ = sn) -> exists fp_value_bounded_succ. ((((exists ff_h_bounded_succ_entry. ff_h_bounded_succ_entry + S (fp_value_bounded_succ) = S ((S (fp_i_bounded_succ)) * c)) /\\ exists ff_q_bounded_succ_entry. b = ff_q_bounded_succ_entry * S ((S (fp_i_bounded_succ)) * c) + (fp_value_bounded_succ))) /\\ (exists fp_gap_bounded_succ_value. fp_gap_bounded_succ_value + S fp_value_bounded_succ = sn))) -> (forall fp_i_inj_succ fp_j_inj_succ fp_value_inj_succ. (exists fp_gap_inj_succ_i. fp_gap_inj_succ_i + S fp_i_inj_succ = sn) -> (exists fp_gap_inj_succ_j. fp_gap_inj_succ_j + S fp_j_inj_succ = sn) -> (((exists ff_h_inj_succ_left. ff_h_inj_succ_left + S (fp_value_inj_succ) = S ((S (fp_i_inj_succ)) * c)) /\\ exists ff_q_inj_succ_left. b = ff_q_inj_succ_left * S ((S (fp_i_inj_succ)) * c) + (fp_value_inj_succ))) -> (((exists ff_h_inj_succ_right. ff_h_inj_succ_right + S (fp_value_inj_succ) = S ((S (fp_j_inj_succ)) * c)) /\\ exists ff_q_inj_succ_right. b = ff_q_inj_succ_right * S ((S (fp_j_inj_succ)) * c) + (fp_value_inj_succ))) -> fp_i_inj_succ = fp_j_inj_succ) -> ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) -> ((forall fp_i_bounded_prefix. (exists fp_gap_bounded_prefix_index. fp_gap_bounded_prefix_index + S fp_i_bounded_prefix = n) -> exists fp_value_bounded_prefix. ((((exists ff_h_bounded_prefix_entry. ff_h_bounded_prefix_entry + S (fp_value_bounded_prefix) = S ((S (fp_i_bounded_prefix)) * c)) /\\ exists ff_q_bounded_prefix_entry. b = ff_q_bounded_prefix_entry * S ((S (fp_i_bounded_prefix)) * c) + (fp_value_bounded_prefix))) /\\ (exists fp_gap_bounded_prefix_value. fp_gap_bounded_prefix_value + S fp_value_bounded_prefix = n))) -> (forall fp_i_inj_prefix fp_j_inj_prefix fp_value_inj_prefix. (exists fp_gap_inj_prefix_i. fp_gap_inj_prefix_i + S fp_i_inj_prefix = n) -> (exists fp_gap_inj_prefix_j. fp_gap_inj_prefix_j + S fp_j_inj_prefix = n) -> (((exists ff_h_inj_prefix_left. ff_h_inj_prefix_left + S (fp_value_inj_prefix) = S ((S (fp_i_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_left. b = ff_q_inj_prefix_left * S ((S (fp_i_inj_prefix)) * c) + (fp_value_inj_prefix))) -> (((exists ff_h_inj_prefix_right. ff_h_inj_prefix_right + S (fp_value_inj_prefix) = S ((S (fp_j_inj_prefix)) * c)) /\\ exists ff_q_inj_prefix_right. b = ff_q_inj_prefix_right * S ((S (fp_j_inj_prefix)) * c) + (fp_value_inj_prefix))) -> fp_i_inj_prefix = fp_j_inj_prefix) -> (forall fp_value_surj_n. (exists fp_gap_surj_n_value. fp_gap_surj_n_value + S fp_value_surj_n = n) -> exists fp_i_surj_n. ((exists fp_gap_surj_n_index. fp_gap_surj_n_index + S fp_i_surj_n = n) /\\ (((exists ff_h_surj_n_entry. ff_h_surj_n_entry + S (fp_value_surj_n) = S ((S (fp_i_surj_n)) * c)) /\\ exists ff_q_surj_n_entry. b = ff_q_surj_n_entry * S ((S (fp_i_surj_n)) * c) + (fp_value_surj_n)))))) -> (forall fp_value_surj_succ. (exists fp_gap_surj_succ_value. fp_gap_surj_succ_value + S fp_value_surj_succ = sn) -> exists fp_i_surj_succ. ((exists fp_gap_surj_succ_index. fp_gap_surj_succ_index + S fp_i_surj_succ = sn) /\\ (((exists ff_h_surj_succ_entry. ff_h_surj_succ_entry + S (fp_value_surj_succ) = S ((S (fp_i_surj_succ)) * c)) /\\ exists ff_q_surj_succ_entry. b = ff_q_surj_succ_entry * S ((S (fp_i_surj_succ)) * c) + (fp_value_surj_succ)))))", "statement_sha256": "9b8d47daab6b099597b4875c374f6a5012be4f1162d33a6266bb5d934a844c88" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_bounded_injective_surjective", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "finite_surjective_zero", "finite_contains_decidable", "finite_bounded_last_succ", "beta_prefix_swap_last_from_entries", "finite_swap_last_bounded", "finite_swap_last_injective", "finite_bounded_prefix_without_top", "finite_injective_prefix_succ", "finite_surjective_succ_from_prefix", "finite_swap_last_surjective_back", "finite_no_top_successor_gate", "beta_at_unique", "le_succ", "le_refl", "lt_irrefl_expanded" ], "dependencies_sha256": "49341e1b1e2301eafeb5c8f4755bb88fd43b7e5197b6e9d427909c2877e87b92", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "3e4a5f2303e7f14927ccd4b171ff6915960e99e27b1a4c1d5a08c25f5ee5d067", "cut_nodes": 1266, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 89, "proof_edges": 6672, "proof_nodes": 42463, "proof_objects": 6399, "reused_objects": 274, "status": "checked" }, "enrollment_index": 376, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_bounded_injective_surjective]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "c70f5caf840e64fef9b09fab9df97350a0dfdfdd277cc2b45532877b04c9b1f1", "membership": "stable", "name": "finite_bounded_injective_surjective", "proof_tag": "PA004W", "provenance": [ "stable" ], "script": [ "induction n", "intro b", "intro c", "intro hbounded", "intro hinjective", "specialize finite_surjective_zero b", "specialize finite_surjective_zero c", "specialize finite_surjective_zero 0", "apply finite_surjective_zero", "refl", "intro b", "intro c", "intro hbounded", "intro hinjective", "have hcontains : (exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) \\/ ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n)))))", "specialize finite_contains_decidable b", "specialize finite_contains_decidable c", "specialize finite_contains_decidable n", "specialize finite_contains_decidable n", "exact finite_contains_decidable", "cases hcontains", "cases hcontains_left", "cases hcontains_left_witness", "have hlast : exists y. (((exists h. h + S y = S ((S n) * c)) /\\ exists q. b = q * S ((S n) * c) + y) /\\ exists h. h + S y = S n)", "specialize finite_bounded_last_succ b", "specialize finite_bounded_last_succ c", "specialize finite_bounded_last_succ n", "specialize finite_bounded_last_succ (S n)", "apply finite_bounded_last_succ", "refl", "exact hbounded", "cases hlast", "cases hlast_witness", "have hswap : exists z d. ((((exists ff_h_pigeon_swap_new_i. ff_h_pigeon_swap_new_i + S (x1) = S ((S (x)) * d)) /\\ exists ff_q_pigeon_swap_new_i. z = ff_q_pigeon_swap_new_i * S ((S (x)) * d) + (x1))) /\\ ((((exists ff_h_pigeon_swap_new_n. ff_h_pigeon_swap_new_n + S (n) = S ((S (n)) * d)) /\\ exists ff_q_pigeon_swap_new_n. z = ff_q_pigeon_swap_new_n * S ((S (n)) * d) + (n))) /\\ forall j a. (exists h. h + S j = S n) -> ~(j = x) -> ~(j = n) -> (((exists ff_h_pigeon_swap_old_other. ff_h_pigeon_swap_old_other + S (a) = S ((S (j)) * c)) /\\ exists ff_q_pigeon_swap_old_other. b = ff_q_pigeon_swap_old_other * S ((S (j)) * c) + (a))) -> (((exists ff_h_pigeon_swap_new_other. ff_h_pigeon_swap_new_other + S (a) = S ((S (j)) * d)) /\\ exists ff_q_pigeon_swap_new_other. z = ff_q_pigeon_swap_new_other * S ((S (j)) * d) + (a)))))", "specialize beta_prefix_swap_last_from_entries b", "specialize beta_prefix_swap_last_from_entries c", "specialize beta_prefix_swap_last_from_entries n", "specialize beta_prefix_swap_last_from_entries x", "specialize beta_prefix_swap_last_from_entries n", "specialize beta_prefix_swap_last_from_entries x1", "apply beta_prefix_swap_last_from_entries", "exact hcontains_left_witness_left", "exact hcontains_left_witness_right", "exact hlast_witness_left", "cases hswap", "cases hswap_witness", "cases hswap_witness_witness", "cases hswap_witness_witness_right", "have hswap_bounded : forall fp_i_pigeon_swapped_bounded. (exists fp_gap_pigeon_swapped_bounded_index. fp_gap_pigeon_swapped_bounded_index + S fp_i_pigeon_swapped_bounded = S n) -> exists fp_value_pigeon_swapped_bounded. ((((exists ff_h_pigeon_swapped_bounded_entry. ff_h_pigeon_swapped_bounded_entry + S (fp_value_pigeon_swapped_bounded) = S ((S (fp_i_pigeon_swapped_bounded)) * x3)) /\\ exists ff_q_pigeon_swapped_bounded_entry. x2 = ff_q_pigeon_swapped_bounded_entry * S ((S (fp_i_pigeon_swapped_bounded)) * x3) + (fp_value_pigeon_swapped_bounded))) /\\ (exists fp_gap_pigeon_swapped_bounded_value. fp_gap_pigeon_swapped_bounded_value + S fp_value_pigeon_swapped_bounded = S n))", "specialize finite_swap_last_bounded b", "specialize finite_swap_last_bounded c", "specialize finite_swap_last_bounded x2", "specialize finite_swap_last_bounded x3", "specialize finite_swap_last_bounded n", "specialize finite_swap_last_bounded (S n)", "specialize finite_swap_last_bounded x", "specialize finite_swap_last_bounded n", "specialize finite_swap_last_bounded x1", "apply finite_swap_last_bounded", "refl", "exact hcontains_left_witness_left", "exact hbounded", "exact hcontains_left_witness_right", "exact hlast_witness_left", "exact hswap_witness_witness_left", "exact hswap_witness_witness_right_left", "exact hswap_witness_witness_right_right", "have hswap_injective : forall fp_i_pigeon_swapped_injective fp_j_pigeon_swapped_injective fp_value_pigeon_swapped_injective. (exists fp_gap_pigeon_swapped_injective_i. fp_gap_pigeon_swapped_injective_i + S fp_i_pigeon_swapped_injective = S n) -> (exists fp_gap_pigeon_swapped_injective_j. fp_gap_pigeon_swapped_injective_j + S fp_j_pigeon_swapped_injective = S n) -> (((exists ff_h_pigeon_swapped_injective_left. ff_h_pigeon_swapped_injective_left + S (fp_value_pigeon_swapped_injective) = S ((S (fp_i_pigeon_swapped_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_injective_left. x2 = ff_q_pigeon_swapped_injective_left * S ((S (fp_i_pigeon_swapped_injective)) * x3) + (fp_value_pigeon_swapped_injective))) -> (((exists ff_h_pigeon_swapped_injective_right. ff_h_pigeon_swapped_injective_right + S (fp_value_pigeon_swapped_injective) = S ((S (fp_j_pigeon_swapped_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_injective_right. x2 = ff_q_pigeon_swapped_injective_right * S ((S (fp_j_pigeon_swapped_injective)) * x3) + (fp_value_pigeon_swapped_injective))) -> fp_i_pigeon_swapped_injective = fp_j_pigeon_swapped_injective", "specialize finite_swap_last_injective b", "specialize finite_swap_last_injective c", "specialize finite_swap_last_injective x2", "specialize finite_swap_last_injective x3", "specialize finite_swap_last_injective n", "specialize finite_swap_last_injective (S n)", "specialize finite_swap_last_injective x", "specialize finite_swap_last_injective n", "specialize finite_swap_last_injective x1", "apply finite_swap_last_injective", "refl", "exact hcontains_left_witness_left", "exact hinjective", "exact hcontains_left_witness_right", "exact hlast_witness_left", "exact hswap_witness_witness_left", "exact hswap_witness_witness_right_left", "exact hswap_witness_witness_right_right", "have hnotop : forall j. (exists h. h + S j = n) -> ~(((exists ff_h_pigeon_top_j. ff_h_pigeon_top_j + S (n) = S ((S (j)) * x3)) /\\ exists ff_q_pigeon_top_j. x2 = ff_q_pigeon_top_j * S ((S (j)) * x3) + (n)))", "intro j", "intro hj", "intro htop", "have hjsn : exists h. h + S j = S n", "specialize le_succ (S j)", "specialize le_succ n", "apply le_succ", "exact hj", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hjneq : j = n", "specialize hswap_injective j", "specialize hswap_injective n", "specialize hswap_injective n", "apply hswap_injective", "exact hjsn", "exact hnsn", "exact htop", "exact hswap_witness_witness_right_left", "specialize lt_irrefl_expanded n", "apply lt_irrefl_expanded", "rewrite hjneq at hj", "exact hj", "have hprefix_bounded : forall fp_i_pigeon_swapped_prefix_bounded. (exists fp_gap_pigeon_swapped_prefix_bounded_index. fp_gap_pigeon_swapped_prefix_bounded_index + S fp_i_pigeon_swapped_prefix_bounded = n) -> exists fp_value_pigeon_swapped_prefix_bounded. ((((exists ff_h_pigeon_swapped_prefix_bounded_entry. ff_h_pigeon_swapped_prefix_bounded_entry + S (fp_value_pigeon_swapped_prefix_bounded) = S ((S (fp_i_pigeon_swapped_prefix_bounded)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_bounded_entry. x2 = ff_q_pigeon_swapped_prefix_bounded_entry * S ((S (fp_i_pigeon_swapped_prefix_bounded)) * x3) + (fp_value_pigeon_swapped_prefix_bounded))) /\\ (exists fp_gap_pigeon_swapped_prefix_bounded_value. fp_gap_pigeon_swapped_prefix_bounded_value + S fp_value_pigeon_swapped_prefix_bounded = n))", "specialize finite_bounded_prefix_without_top x2", "specialize finite_bounded_prefix_without_top x3", "specialize finite_bounded_prefix_without_top n", "specialize finite_bounded_prefix_without_top (S n)", "apply finite_bounded_prefix_without_top", "refl", "exact hswap_bounded", "exact hnotop", "have hprefix_injective : forall fp_i_pigeon_swapped_prefix_injective fp_j_pigeon_swapped_prefix_injective fp_value_pigeon_swapped_prefix_injective. (exists fp_gap_pigeon_swapped_prefix_injective_i. fp_gap_pigeon_swapped_prefix_injective_i + S fp_i_pigeon_swapped_prefix_injective = n) -> (exists fp_gap_pigeon_swapped_prefix_injective_j. fp_gap_pigeon_swapped_prefix_injective_j + S fp_j_pigeon_swapped_prefix_injective = n) -> (((exists ff_h_pigeon_swapped_prefix_injective_left. ff_h_pigeon_swapped_prefix_injective_left + S (fp_value_pigeon_swapped_prefix_injective) = S ((S (fp_i_pigeon_swapped_prefix_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_injective_left. x2 = ff_q_pigeon_swapped_prefix_injective_left * S ((S (fp_i_pigeon_swapped_prefix_injective)) * x3) + (fp_value_pigeon_swapped_prefix_injective))) -> (((exists ff_h_pigeon_swapped_prefix_injective_right. ff_h_pigeon_swapped_prefix_injective_right + S (fp_value_pigeon_swapped_prefix_injective) = S ((S (fp_j_pigeon_swapped_prefix_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_injective_right. x2 = ff_q_pigeon_swapped_prefix_injective_right * S ((S (fp_j_pigeon_swapped_prefix_injective)) * x3) + (fp_value_pigeon_swapped_prefix_injective))) -> fp_i_pigeon_swapped_prefix_injective = fp_j_pigeon_swapped_prefix_injective", "specialize finite_injective_prefix_succ x2", "specialize finite_injective_prefix_succ x3", "specialize finite_injective_prefix_succ n", "specialize finite_injective_prefix_succ (S n)", "apply finite_injective_prefix_succ", "refl", "exact hswap_injective", "have hprefix_surjective : forall fp_value_pigeon_swapped_prefix_surjective. (exists fp_gap_pigeon_swapped_prefix_surjective_value. fp_gap_pigeon_swapped_prefix_surjective_value + S fp_value_pigeon_swapped_prefix_surjective = n) -> exists fp_i_pigeon_swapped_prefix_surjective. ((exists fp_gap_pigeon_swapped_prefix_surjective_index. fp_gap_pigeon_swapped_prefix_surjective_index + S fp_i_pigeon_swapped_prefix_surjective = n) /\\ (((exists ff_h_pigeon_swapped_prefix_surjective_entry. ff_h_pigeon_swapped_prefix_surjective_entry + S (fp_value_pigeon_swapped_prefix_surjective) = S ((S (fp_i_pigeon_swapped_prefix_surjective)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_surjective_entry. x2 = ff_q_pigeon_swapped_prefix_surjective_entry * S ((S (fp_i_pigeon_swapped_prefix_surjective)) * x3) + (fp_value_pigeon_swapped_prefix_surjective))))", "specialize IH x2", "specialize IH x3", "apply IH", "exact hprefix_bounded", "exact hprefix_injective", "have hswap_surjective : forall fp_value_pigeon_swapped_surjective. (exists fp_gap_pigeon_swapped_surjective_value. fp_gap_pigeon_swapped_surjective_value + S fp_value_pigeon_swapped_surjective = S n) -> exists fp_i_pigeon_swapped_surjective. ((exists fp_gap_pigeon_swapped_surjective_index. fp_gap_pigeon_swapped_surjective_index + S fp_i_pigeon_swapped_surjective = S n) /\\ (((exists ff_h_pigeon_swapped_surjective_entry. ff_h_pigeon_swapped_surjective_entry + S (fp_value_pigeon_swapped_surjective) = S ((S (fp_i_pigeon_swapped_surjective)) * x3)) /\\ exists ff_q_pigeon_swapped_surjective_entry. x2 = ff_q_pigeon_swapped_surjective_entry * S ((S (fp_i_pigeon_swapped_surjective)) * x3) + (fp_value_pigeon_swapped_surjective))))", "specialize finite_surjective_succ_from_prefix x2", "specialize finite_surjective_succ_from_prefix x3", "specialize finite_surjective_succ_from_prefix n", "specialize finite_surjective_succ_from_prefix (S n)", "apply finite_surjective_succ_from_prefix", "refl", "exact hswap_bounded", "exact hswap_injective", "exact hprefix_surjective", "specialize finite_swap_last_surjective_back b", "specialize finite_swap_last_surjective_back c", "specialize finite_swap_last_surjective_back x2", "specialize finite_swap_last_surjective_back x3", "specialize finite_swap_last_surjective_back n", "specialize finite_swap_last_surjective_back (S n)", "specialize finite_swap_last_surjective_back x", "specialize finite_swap_last_surjective_back n", "specialize finite_swap_last_surjective_back x1", "apply finite_swap_last_surjective_back", "refl", "exact hcontains_left_witness_left", "exact hcontains_left_witness_right", "exact hlast_witness_left", "exact hswap_witness_witness_left", "exact hswap_witness_witness_right_left", "exact hswap_witness_witness_right_right", "exact hswap_surjective", "specialize finite_no_top_successor_gate b", "specialize finite_no_top_successor_gate c", "specialize finite_no_top_successor_gate n", "specialize finite_no_top_successor_gate (S n)", "apply finite_no_top_successor_gate", "refl", "exact hbounded", "exact hinjective", "exact hcontains_right", "intro hprefix_bounded", "intro hprefix_injective", "specialize IH b", "specialize IH c", "apply IH", "exact hprefix_bounded", "exact hprefix_injective" ], "script_sha256": "6f501cc65ba7d78844c5dd6f42463be97c89b32c6dc2e19d40236a7618315533", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall n b c. (forall fp_i_pigeon_bounded. (exists fp_gap_pigeon_bounded_index. fp_gap_pigeon_bounded_index + S fp_i_pigeon_bounded = n) -> exists fp_value_pigeon_bounded. ((((exists ff_h_pigeon_bounded_entry. ff_h_pigeon_bounded_entry + S (fp_value_pigeon_bounded) = S ((S (fp_i_pigeon_bounded)) * c)) /\\ exists ff_q_pigeon_bounded_entry. b = ff_q_pigeon_bounded_entry * S ((S (fp_i_pigeon_bounded)) * c) + (fp_value_pigeon_bounded))) /\\ (exists fp_gap_pigeon_bounded_value. fp_gap_pigeon_bounded_value + S fp_value_pigeon_bounded = n))) -> (forall fp_i_pigeon_injective fp_j_pigeon_injective fp_value_pigeon_injective. (exists fp_gap_pigeon_injective_i. fp_gap_pigeon_injective_i + S fp_i_pigeon_injective = n) -> (exists fp_gap_pigeon_injective_j. fp_gap_pigeon_injective_j + S fp_j_pigeon_injective = n) -> (((exists ff_h_pigeon_injective_left. ff_h_pigeon_injective_left + S (fp_value_pigeon_injective) = S ((S (fp_i_pigeon_injective)) * c)) /\\ exists ff_q_pigeon_injective_left. b = ff_q_pigeon_injective_left * S ((S (fp_i_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> (((exists ff_h_pigeon_injective_right. ff_h_pigeon_injective_right + S (fp_value_pigeon_injective) = S ((S (fp_j_pigeon_injective)) * c)) /\\ exists ff_q_pigeon_injective_right. b = ff_q_pigeon_injective_right * S ((S (fp_j_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> fp_i_pigeon_injective = fp_j_pigeon_injective) -> (forall fp_value_pigeon_surjective. (exists fp_gap_pigeon_surjective_value. fp_gap_pigeon_surjective_value + S fp_value_pigeon_surjective = n) -> exists fp_i_pigeon_surjective. ((exists fp_gap_pigeon_surjective_index. fp_gap_pigeon_surjective_index + S fp_i_pigeon_surjective = n) /\\ (((exists ff_h_pigeon_surjective_entry. ff_h_pigeon_surjective_entry + S (fp_value_pigeon_surjective) = S ((S (fp_i_pigeon_surjective)) * c)) /\\ exists ff_q_pigeon_surjective_entry. b = ff_q_pigeon_surjective_entry * S ((S (fp_i_pigeon_surjective)) * c) + (fp_value_pigeon_surjective)))))", "statement_sha256": "9e0cad653da9de17ab7bbac3cb3bf49bc6d4a1304bda508669943b25fd247257", "summary": "Every bounded injective beta-coded prefix is surjective onto its finite interval.", "summary_sha256": "a6765cf77c9942d8e885880ca41b01604f6a04b043495cf642baa696bd694456" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "finite_surjective_zero", "finite_contains_decidable", "finite_bounded_last_succ", "beta_prefix_swap_last_from_entries", "finite_swap_last_bounded", "finite_swap_last_injective", "finite_bounded_prefix_without_top", "finite_injective_prefix_succ", "finite_surjective_succ_from_prefix", "finite_swap_last_surjective_back", "finite_no_top_successor_gate", "beta_at_unique", "le_succ", "le_refl", "lt_irrefl_expanded" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=finite_bounded_injective_surjective]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_bounded_injective_surjective", "parent_alpha_version": "v34", "proof_bundle_node_id": 181, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_bounded_injective_surjective", "script": [ "induction n", "intro b", "intro c", "intro hbounded", "intro hinjective", "specialize finite_surjective_zero b", "specialize finite_surjective_zero c", "specialize finite_surjective_zero 0", "apply finite_surjective_zero", "refl", "intro b", "intro c", "intro hbounded", "intro hinjective", "have hcontains : (exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n))))) \\/ ~(exists fp_i_contains_top. ((exists fp_gap_contains_top_index. fp_gap_contains_top_index + S fp_i_contains_top = n) /\\ (((exists ff_h_contains_top_entry. ff_h_contains_top_entry + S (n) = S ((S (fp_i_contains_top)) * c)) /\\ exists ff_q_contains_top_entry. b = ff_q_contains_top_entry * S ((S (fp_i_contains_top)) * c) + (n)))))", "specialize finite_contains_decidable b", "specialize finite_contains_decidable c", "specialize finite_contains_decidable n", "specialize finite_contains_decidable n", "exact finite_contains_decidable", "cases hcontains", "cases hcontains_left", "cases hcontains_left_witness", "have hlast : exists y. (((exists h. h + S y = S ((S n) * c)) /\\ exists q. b = q * S ((S n) * c) + y) /\\ exists h. h + S y = S n)", "specialize finite_bounded_last_succ b", "specialize finite_bounded_last_succ c", "specialize finite_bounded_last_succ n", "specialize finite_bounded_last_succ (S n)", "apply finite_bounded_last_succ", "refl", "exact hbounded", "cases hlast", "cases hlast_witness", "have hswap : exists z d. ((((exists ff_h_pigeon_swap_new_i. ff_h_pigeon_swap_new_i + S (x1) = S ((S (x)) * d)) /\\ exists ff_q_pigeon_swap_new_i. z = ff_q_pigeon_swap_new_i * S ((S (x)) * d) + (x1))) /\\ ((((exists ff_h_pigeon_swap_new_n. ff_h_pigeon_swap_new_n + S (n) = S ((S (n)) * d)) /\\ exists ff_q_pigeon_swap_new_n. z = ff_q_pigeon_swap_new_n * S ((S (n)) * d) + (n))) /\\ forall j a. (exists h. h + S j = S n) -> ~(j = x) -> ~(j = n) -> (((exists ff_h_pigeon_swap_old_other. ff_h_pigeon_swap_old_other + S (a) = S ((S (j)) * c)) /\\ exists ff_q_pigeon_swap_old_other. b = ff_q_pigeon_swap_old_other * S ((S (j)) * c) + (a))) -> (((exists ff_h_pigeon_swap_new_other. ff_h_pigeon_swap_new_other + S (a) = S ((S (j)) * d)) /\\ exists ff_q_pigeon_swap_new_other. z = ff_q_pigeon_swap_new_other * S ((S (j)) * d) + (a)))))", "specialize beta_prefix_swap_last_from_entries b", "specialize beta_prefix_swap_last_from_entries c", "specialize beta_prefix_swap_last_from_entries n", "specialize beta_prefix_swap_last_from_entries x", "specialize beta_prefix_swap_last_from_entries n", "specialize beta_prefix_swap_last_from_entries x1", "apply beta_prefix_swap_last_from_entries", "exact hcontains_left_witness_left", "exact hcontains_left_witness_right", "exact hlast_witness_left", "cases hswap", "cases hswap_witness", "cases hswap_witness_witness", "cases hswap_witness_witness_right", "have hswap_bounded : forall fp_i_pigeon_swapped_bounded. (exists fp_gap_pigeon_swapped_bounded_index. fp_gap_pigeon_swapped_bounded_index + S fp_i_pigeon_swapped_bounded = S n) -> exists fp_value_pigeon_swapped_bounded. ((((exists ff_h_pigeon_swapped_bounded_entry. ff_h_pigeon_swapped_bounded_entry + S (fp_value_pigeon_swapped_bounded) = S ((S (fp_i_pigeon_swapped_bounded)) * x3)) /\\ exists ff_q_pigeon_swapped_bounded_entry. x2 = ff_q_pigeon_swapped_bounded_entry * S ((S (fp_i_pigeon_swapped_bounded)) * x3) + (fp_value_pigeon_swapped_bounded))) /\\ (exists fp_gap_pigeon_swapped_bounded_value. fp_gap_pigeon_swapped_bounded_value + S fp_value_pigeon_swapped_bounded = S n))", "specialize finite_swap_last_bounded b", "specialize finite_swap_last_bounded c", "specialize finite_swap_last_bounded x2", "specialize finite_swap_last_bounded x3", "specialize finite_swap_last_bounded n", "specialize finite_swap_last_bounded (S n)", "specialize finite_swap_last_bounded x", "specialize finite_swap_last_bounded n", "specialize finite_swap_last_bounded x1", "apply finite_swap_last_bounded", "refl", "exact hcontains_left_witness_left", "exact hbounded", "exact hcontains_left_witness_right", "exact hlast_witness_left", "exact hswap_witness_witness_left", "exact hswap_witness_witness_right_left", "exact hswap_witness_witness_right_right", "have hswap_injective : forall fp_i_pigeon_swapped_injective fp_j_pigeon_swapped_injective fp_value_pigeon_swapped_injective. (exists fp_gap_pigeon_swapped_injective_i. fp_gap_pigeon_swapped_injective_i + S fp_i_pigeon_swapped_injective = S n) -> (exists fp_gap_pigeon_swapped_injective_j. fp_gap_pigeon_swapped_injective_j + S fp_j_pigeon_swapped_injective = S n) -> (((exists ff_h_pigeon_swapped_injective_left. ff_h_pigeon_swapped_injective_left + S (fp_value_pigeon_swapped_injective) = S ((S (fp_i_pigeon_swapped_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_injective_left. x2 = ff_q_pigeon_swapped_injective_left * S ((S (fp_i_pigeon_swapped_injective)) * x3) + (fp_value_pigeon_swapped_injective))) -> (((exists ff_h_pigeon_swapped_injective_right. ff_h_pigeon_swapped_injective_right + S (fp_value_pigeon_swapped_injective) = S ((S (fp_j_pigeon_swapped_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_injective_right. x2 = ff_q_pigeon_swapped_injective_right * S ((S (fp_j_pigeon_swapped_injective)) * x3) + (fp_value_pigeon_swapped_injective))) -> fp_i_pigeon_swapped_injective = fp_j_pigeon_swapped_injective", "specialize finite_swap_last_injective b", "specialize finite_swap_last_injective c", "specialize finite_swap_last_injective x2", "specialize finite_swap_last_injective x3", "specialize finite_swap_last_injective n", "specialize finite_swap_last_injective (S n)", "specialize finite_swap_last_injective x", "specialize finite_swap_last_injective n", "specialize finite_swap_last_injective x1", "apply finite_swap_last_injective", "refl", "exact hcontains_left_witness_left", "exact hinjective", "exact hcontains_left_witness_right", "exact hlast_witness_left", "exact hswap_witness_witness_left", "exact hswap_witness_witness_right_left", "exact hswap_witness_witness_right_right", "have hnotop : forall j. (exists h. h + S j = n) -> ~(((exists ff_h_pigeon_top_j. ff_h_pigeon_top_j + S (n) = S ((S (j)) * x3)) /\\ exists ff_q_pigeon_top_j. x2 = ff_q_pigeon_top_j * S ((S (j)) * x3) + (n)))", "intro j", "intro hj", "intro htop", "have hjsn : exists h. h + S j = S n", "specialize le_succ (S j)", "specialize le_succ n", "apply le_succ", "exact hj", "have hnsn : exists h. h + S n = S n", "specialize le_refl (S n)", "exact le_refl", "have hjneq : j = n", "specialize hswap_injective j", "specialize hswap_injective n", "specialize hswap_injective n", "apply hswap_injective", "exact hjsn", "exact hnsn", "exact htop", "exact hswap_witness_witness_right_left", "specialize lt_irrefl_expanded n", "apply lt_irrefl_expanded", "rewrite hjneq at hj", "exact hj", "have hprefix_bounded : forall fp_i_pigeon_swapped_prefix_bounded. (exists fp_gap_pigeon_swapped_prefix_bounded_index. fp_gap_pigeon_swapped_prefix_bounded_index + S fp_i_pigeon_swapped_prefix_bounded = n) -> exists fp_value_pigeon_swapped_prefix_bounded. ((((exists ff_h_pigeon_swapped_prefix_bounded_entry. ff_h_pigeon_swapped_prefix_bounded_entry + S (fp_value_pigeon_swapped_prefix_bounded) = S ((S (fp_i_pigeon_swapped_prefix_bounded)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_bounded_entry. x2 = ff_q_pigeon_swapped_prefix_bounded_entry * S ((S (fp_i_pigeon_swapped_prefix_bounded)) * x3) + (fp_value_pigeon_swapped_prefix_bounded))) /\\ (exists fp_gap_pigeon_swapped_prefix_bounded_value. fp_gap_pigeon_swapped_prefix_bounded_value + S fp_value_pigeon_swapped_prefix_bounded = n))", "specialize finite_bounded_prefix_without_top x2", "specialize finite_bounded_prefix_without_top x3", "specialize finite_bounded_prefix_without_top n", "specialize finite_bounded_prefix_without_top (S n)", "apply finite_bounded_prefix_without_top", "refl", "exact hswap_bounded", "exact hnotop", "have hprefix_injective : forall fp_i_pigeon_swapped_prefix_injective fp_j_pigeon_swapped_prefix_injective fp_value_pigeon_swapped_prefix_injective. (exists fp_gap_pigeon_swapped_prefix_injective_i. fp_gap_pigeon_swapped_prefix_injective_i + S fp_i_pigeon_swapped_prefix_injective = n) -> (exists fp_gap_pigeon_swapped_prefix_injective_j. fp_gap_pigeon_swapped_prefix_injective_j + S fp_j_pigeon_swapped_prefix_injective = n) -> (((exists ff_h_pigeon_swapped_prefix_injective_left. ff_h_pigeon_swapped_prefix_injective_left + S (fp_value_pigeon_swapped_prefix_injective) = S ((S (fp_i_pigeon_swapped_prefix_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_injective_left. x2 = ff_q_pigeon_swapped_prefix_injective_left * S ((S (fp_i_pigeon_swapped_prefix_injective)) * x3) + (fp_value_pigeon_swapped_prefix_injective))) -> (((exists ff_h_pigeon_swapped_prefix_injective_right. ff_h_pigeon_swapped_prefix_injective_right + S (fp_value_pigeon_swapped_prefix_injective) = S ((S (fp_j_pigeon_swapped_prefix_injective)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_injective_right. x2 = ff_q_pigeon_swapped_prefix_injective_right * S ((S (fp_j_pigeon_swapped_prefix_injective)) * x3) + (fp_value_pigeon_swapped_prefix_injective))) -> fp_i_pigeon_swapped_prefix_injective = fp_j_pigeon_swapped_prefix_injective", "specialize finite_injective_prefix_succ x2", "specialize finite_injective_prefix_succ x3", "specialize finite_injective_prefix_succ n", "specialize finite_injective_prefix_succ (S n)", "apply finite_injective_prefix_succ", "refl", "exact hswap_injective", "have hprefix_surjective : forall fp_value_pigeon_swapped_prefix_surjective. (exists fp_gap_pigeon_swapped_prefix_surjective_value. fp_gap_pigeon_swapped_prefix_surjective_value + S fp_value_pigeon_swapped_prefix_surjective = n) -> exists fp_i_pigeon_swapped_prefix_surjective. ((exists fp_gap_pigeon_swapped_prefix_surjective_index. fp_gap_pigeon_swapped_prefix_surjective_index + S fp_i_pigeon_swapped_prefix_surjective = n) /\\ (((exists ff_h_pigeon_swapped_prefix_surjective_entry. ff_h_pigeon_swapped_prefix_surjective_entry + S (fp_value_pigeon_swapped_prefix_surjective) = S ((S (fp_i_pigeon_swapped_prefix_surjective)) * x3)) /\\ exists ff_q_pigeon_swapped_prefix_surjective_entry. x2 = ff_q_pigeon_swapped_prefix_surjective_entry * S ((S (fp_i_pigeon_swapped_prefix_surjective)) * x3) + (fp_value_pigeon_swapped_prefix_surjective))))", "specialize IH x2", "specialize IH x3", "apply IH", "exact hprefix_bounded", "exact hprefix_injective", "have hswap_surjective : forall fp_value_pigeon_swapped_surjective. (exists fp_gap_pigeon_swapped_surjective_value. fp_gap_pigeon_swapped_surjective_value + S fp_value_pigeon_swapped_surjective = S n) -> exists fp_i_pigeon_swapped_surjective. ((exists fp_gap_pigeon_swapped_surjective_index. fp_gap_pigeon_swapped_surjective_index + S fp_i_pigeon_swapped_surjective = S n) /\\ (((exists ff_h_pigeon_swapped_surjective_entry. ff_h_pigeon_swapped_surjective_entry + S (fp_value_pigeon_swapped_surjective) = S ((S (fp_i_pigeon_swapped_surjective)) * x3)) /\\ exists ff_q_pigeon_swapped_surjective_entry. x2 = ff_q_pigeon_swapped_surjective_entry * S ((S (fp_i_pigeon_swapped_surjective)) * x3) + (fp_value_pigeon_swapped_surjective))))", "specialize finite_surjective_succ_from_prefix x2", "specialize finite_surjective_succ_from_prefix x3", "specialize finite_surjective_succ_from_prefix n", "specialize finite_surjective_succ_from_prefix (S n)", "apply finite_surjective_succ_from_prefix", "refl", "exact hswap_bounded", "exact hswap_injective", "exact hprefix_surjective", "specialize finite_swap_last_surjective_back b", "specialize finite_swap_last_surjective_back c", "specialize finite_swap_last_surjective_back x2", "specialize finite_swap_last_surjective_back x3", "specialize finite_swap_last_surjective_back n", "specialize finite_swap_last_surjective_back (S n)", "specialize finite_swap_last_surjective_back x", "specialize finite_swap_last_surjective_back n", "specialize finite_swap_last_surjective_back x1", "apply finite_swap_last_surjective_back", "refl", "exact hcontains_left_witness_left", "exact hcontains_left_witness_right", "exact hlast_witness_left", "exact hswap_witness_witness_left", "exact hswap_witness_witness_right_left", "exact hswap_witness_witness_right_right", "exact hswap_surjective", "specialize finite_no_top_successor_gate b", "specialize finite_no_top_successor_gate c", "specialize finite_no_top_successor_gate n", "specialize finite_no_top_successor_gate (S n)", "apply finite_no_top_successor_gate", "refl", "exact hbounded", "exact hinjective", "exact hcontains_right", "intro hprefix_bounded", "intro hprefix_injective", "specialize IH b", "specialize IH c", "apply IH", "exact hprefix_bounded", "exact hprefix_injective" ], "script_sha256": "6f501cc65ba7d78844c5dd6f42463be97c89b32c6dc2e19d40236a7618315533", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall n b c. (forall fp_i_pigeon_bounded. (exists fp_gap_pigeon_bounded_index. fp_gap_pigeon_bounded_index + S fp_i_pigeon_bounded = n) -> exists fp_value_pigeon_bounded. ((((exists ff_h_pigeon_bounded_entry. ff_h_pigeon_bounded_entry + S (fp_value_pigeon_bounded) = S ((S (fp_i_pigeon_bounded)) * c)) /\\ exists ff_q_pigeon_bounded_entry. b = ff_q_pigeon_bounded_entry * S ((S (fp_i_pigeon_bounded)) * c) + (fp_value_pigeon_bounded))) /\\ (exists fp_gap_pigeon_bounded_value. fp_gap_pigeon_bounded_value + S fp_value_pigeon_bounded = n))) -> (forall fp_i_pigeon_injective fp_j_pigeon_injective fp_value_pigeon_injective. (exists fp_gap_pigeon_injective_i. fp_gap_pigeon_injective_i + S fp_i_pigeon_injective = n) -> (exists fp_gap_pigeon_injective_j. fp_gap_pigeon_injective_j + S fp_j_pigeon_injective = n) -> (((exists ff_h_pigeon_injective_left. ff_h_pigeon_injective_left + S (fp_value_pigeon_injective) = S ((S (fp_i_pigeon_injective)) * c)) /\\ exists ff_q_pigeon_injective_left. b = ff_q_pigeon_injective_left * S ((S (fp_i_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> (((exists ff_h_pigeon_injective_right. ff_h_pigeon_injective_right + S (fp_value_pigeon_injective) = S ((S (fp_j_pigeon_injective)) * c)) /\\ exists ff_q_pigeon_injective_right. b = ff_q_pigeon_injective_right * S ((S (fp_j_pigeon_injective)) * c) + (fp_value_pigeon_injective))) -> fp_i_pigeon_injective = fp_j_pigeon_injective) -> (forall fp_value_pigeon_surjective. (exists fp_gap_pigeon_surjective_value. fp_gap_pigeon_surjective_value + S fp_value_pigeon_surjective = n) -> exists fp_i_pigeon_surjective. ((exists fp_gap_pigeon_surjective_index. fp_gap_pigeon_surjective_index + S fp_i_pigeon_surjective = n) /\\ (((exists ff_h_pigeon_surjective_entry. ff_h_pigeon_surjective_entry + S (fp_value_pigeon_surjective) = S ((S (fp_i_pigeon_surjective)) * c)) /\\ exists ff_q_pigeon_surjective_entry. b = ff_q_pigeon_surjective_entry * S ((S (fp_i_pigeon_surjective)) * c) + (fp_value_pigeon_surjective)))))", "statement_sha256": "9e0cad653da9de17ab7bbac3cb3bf49bc6d4a1304bda508669943b25fd247257" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_is_succ_succ", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "prime_nonzero", "nonzero_is_succ" ], "dependencies_sha256": "b1541e216818b15973cde733ef2999bd49123b7ce73551dba3e47e191d014fd9", "empty_context_closure": { "certificate_representation": "python-dataclass-repr-with-cut-v2", "certificate_sha256": "e0cd8df0205c0dd8f4773a78736736929ee489871c6dc53769d1313ecb1ea1c3", "cut_nodes": 4, "digest_kind": "python-dataclass-repr-sha256", "proof_depth": 13, "proof_edges": 97, "proof_nodes": 98, "proof_objects": 98, "reused_objects": 0, "status": "checked" }, "enrollment_index": 382, "enrollment_origin": "stable", "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=prime_is_succ_succ]" } ], "evidence_status": "stable_closed", "logical_spec_sha256": "f6368cdc97040beba3dea82f55a9c9b82b0185cf1c2dcfcdfa7561e9eebc03df", "membership": "stable", "name": "prime_is_succ_succ", "proof_tag": "PA0061", "provenance": [ "stable" ], "script": [ "intro p", "intro hp", "have hp0 : ~(p = 0)", "intro hpzero", "specialize prime_nonzero p", "apply prime_nonzero", "exact hp", "exact hpzero", "have hps : exists k. p = S k", "specialize nonzero_is_succ p", "apply nonzero_is_succ", "exact hp0", "cases hps", "have hx0 : ~(x = 0)", "intro hx0", "cases hp", "apply hp_left", "rewrite hps_witness", "rewrite hx0", "refl", "have hxs : exists k. x = S k", "specialize nonzero_is_succ x", "apply nonzero_is_succ", "exact hx0", "cases hxs", "exists x1", "rewrite hps_witness", "rewrite hxs_witness", "refl" ], "script_sha256": "02fed45baab0da03ad5e149545a4241036d4203125bcda48387ec5c22143d888", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "statement": "forall p. ((~(p = 1) /\\ forall qrbu_factor_left_prime_p qrbu_factor_right_prime_p. p = qrbu_factor_left_prime_p * qrbu_factor_right_prime_p -> qrbu_factor_left_prime_p = 1 \\/ qrbu_factor_right_prime_p = 1)) -> exists k. p = S (S k)", "statement_sha256": "3ade15c63f82b8b6f96ddbc586c1b59313dee6864a8e2cc690c3359b19cebc7e", "summary": "Every prime natural is the second successor of a natural.", "summary_sha256": "6e035174598047b66442feea2743e62c3fb02eceacdc4d2b86b1b048f0e0b808" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "prime_nonzero", "nonzero_is_succ" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "87fca4ab6e66d01f728ada1d9c6442f1167b8f2a8fe51cd6ec5eda901b3daffd", "kind": "stable_closed_snapshot", "path": "artifacts/peano-library/catalog-v1.json", "role": "empty_context_closure", "selector": "theorems[name=prime_is_succ_succ]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_is_succ_succ", "parent_alpha_version": "v34", "proof_bundle_node_id": 182, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_is_succ_succ", "script": [ "intro p", "intro hp", "have hp0 : ~(p = 0)", "intro hpzero", "specialize prime_nonzero p", "apply prime_nonzero", "exact hp", "exact hpzero", "have hps : exists k. p = S k", "specialize nonzero_is_succ p", "apply nonzero_is_succ", "exact hp0", "cases hps", "have hx0 : ~(x = 0)", "intro hx0", "cases hp", "apply hp_left", "rewrite hps_witness", "rewrite hx0", "refl", "have hxs : exists k. x = S k", "specialize nonzero_is_succ x", "apply nonzero_is_succ", "exact hx0", "cases hxs", "exists x1", "rewrite hps_witness", "rewrite hxs_witness", "refl" ], "script_sha256": "02fed45baab0da03ad5e149545a4241036d4203125bcda48387ec5c22143d888", "source": { "kind": "stable_registry", "path": "peano-lab/py/peano_lab/library/theorems.py", "sha256": "05a17b1f33a1c415582785885ca428ce2acb0f3da72700b2b25ad17e890b8919" }, "stable_member": true, "statement": "forall p. 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((((exists hlcm_left_factor_least. l = a * hlcm_left_factor_least) /\\ (exists hlcm_right_factor_least. l = b * hlcm_right_factor_least)) /\\ forall hlcm_common_least. 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(((exists ff_h_fdp_quotient_extension_last. ff_h_fdp_quotient_extension_last + S (x1) = S ((S (l)) * d)) /\\ exists ff_q_fdp_quotient_extension_last. z = ff_q_fdp_quotient_extension_last * S ((S (l)) * d) + (x1))) /\\ forall i q0. (exists gsp_lt_gap_fdp_quotient_extension_old_bound. gsp_lt_gap_fdp_quotient_extension_old_bound + S i = l) -> (((exists ff_h_fdp_quotient_extension_old_source. ff_h_fdp_quotient_extension_old_source + S (q0) = S ((S (i)) * qc)) /\\ exists ff_q_fdp_quotient_extension_old_source. qb = ff_q_fdp_quotient_extension_old_source * S ((S (i)) * qc) + (q0))) -> (((exists ff_h_fdp_quotient_extension_old_target_symbolic. ff_h_fdp_quotient_extension_old_target_symbolic + S (q0) = S ((S (i)) * d)) /\\ exists ff_q_fdp_quotient_extension_old_target_symbolic. z = ff_q_fdp_quotient_extension_old_target_symbolic * S ((S (i)) * d) + (q0)))", "specialize beta_prefix_extend l", "specialize beta_prefix_extend qb", "specialize beta_prefix_extend qc", "specialize beta_prefix_extend x1", "exact beta_prefix_extend", "cases hqextend", "cases hqextend_witness", "cases hqextend_witness_witness", "have hrextend : exists u v. (((exists ff_h_fdp_remainder_extension_last. ff_h_fdp_remainder_extension_last + S (x2) = S ((S (l)) * v)) /\\ exists ff_q_fdp_remainder_extension_last. u = ff_q_fdp_remainder_extension_last * S ((S (l)) * v) + (x2))) /\\ forall i r0. (exists gsp_lt_gap_fdp_remainder_extension_old_bound. gsp_lt_gap_fdp_remainder_extension_old_bound + S i = l) -> (((exists ff_h_fdp_remainder_extension_old_source. ff_h_fdp_remainder_extension_old_source + S (r0) = S ((S (i)) * rc)) /\\ exists ff_q_fdp_remainder_extension_old_source. rb = ff_q_fdp_remainder_extension_old_source * S ((S (i)) * rc) + (r0))) -> (((exists ff_h_fdp_remainder_extension_old_target. ff_h_fdp_remainder_extension_old_target + S (r0) = S ((S (i)) * v)) /\\ exists ff_q_fdp_remainder_extension_old_target. u = ff_q_fdp_remainder_extension_old_target * S ((S (i)) * v) + (r0)))", "specialize beta_prefix_extend l", "specialize beta_prefix_extend rb", "specialize beta_prefix_extend rc", "specialize beta_prefix_extend x2", "exact beta_prefix_extend", "cases hrextend", "cases hrextend_witness", "cases hrextend_witness_witness", "exists x3", "exists x4", "exists x5", "exists x6", "intro i", "intro hi", "have hsplit : i = l \\/ exists gap. gap + S i = l", "specialize finite_lt_succ_eq_or_lt l", "specialize finite_lt_succ_eq_or_lt i", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hsplit", "exists x", "exists x1", "exists x2", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hchoice_witness_witness_witness_left", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hqextend_witness_witness_left", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hrextend_witness_witness_left", "split", "exact hchoice_witness_witness_witness_right_left", "exact hchoice_witness_witness_witness_right_right", "have hold : exists x q r. (((exists ff_h_fdp_previous_source. ff_h_fdp_previous_source + S (x) = S ((S (i)) * c)) /\\ exists ff_q_fdp_previous_source. b = ff_q_fdp_previous_source * S ((S (i)) * c) + (x))) /\\ ((((exists ff_h_fdp_previous_quotient. ff_h_fdp_previous_quotient + S (q) = S ((S (i)) * qc)) /\\ exists ff_q_fdp_previous_quotient. qb = ff_q_fdp_previous_quotient * S ((S (i)) * qc) + (q))) /\\ ((((exists ff_h_fdp_previous_remainder. ff_h_fdp_previous_remainder + S (r) = S ((S (i)) * rc)) /\\ exists ff_q_fdp_previous_remainder. rb = ff_q_fdp_previous_remainder * S ((S (i)) * rc) + (r))) /\\ (x = p * q + r /\\ (exists gsp_lt_gap_fdp_previous_remainder_bound. gsp_lt_gap_fdp_previous_remainder_bound + S r = p))))", "specialize hprefix i", "apply hprefix", "exact hsplit_right", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "cases hold_witness_witness_witness_right", "cases hold_witness_witness_witness_right_right", "cases hold_witness_witness_witness_right_right_right", "exists x7", "exists x8", "exists x9", "split", "exact hold_witness_witness_witness_left", "split", "specialize hqextend_witness_witness_right i", "specialize hqextend_witness_witness_right x8", "apply hqextend_witness_witness_right", "exact hsplit_right", "exact hold_witness_witness_witness_right_left", "split", "specialize hrextend_witness_witness_right i", "specialize hrextend_witness_witness_right x9", "apply hrextend_witness_witness_right", "exact hsplit_right", "exact hold_witness_witness_witness_right_right_left", "split", "exact hold_witness_witness_witness_right_right_right_left", "exact hold_witness_witness_witness_right_right_right_right" ], "script_sha256": "fb8634e5490f5ba6520eaecb42cc596b1822c657b839657a5127acbd23b712ad", "source": { "href": "https://github.com/nasqret/vietnam2026/blob/peano-lab/peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py#L168", "kind": "declaration", "line": 168, "owner_module": "finite_division_prefix_candidate", "path": "peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py", "sha256": "a6af47a7d918d46cdd4b83f60524d3c7afad42886ebb8e560bda5a1318f0b606" }, "statement": "forall p b c qb qc rb rc l. (forall fdp_index_before. (exists gsp_lt_gap_before_index_bound. gsp_lt_gap_before_index_bound + S fdp_index_before = l) -> exists fdp_value_before fdp_quotient_before fdp_remainder_before. (((exists ff_h_fdp_before_source. ff_h_fdp_before_source + S (fdp_value_before) = S ((S (fdp_index_before)) * c)) /\\ exists ff_q_fdp_before_source. b = ff_q_fdp_before_source * S ((S (fdp_index_before)) * c) + (fdp_value_before))) /\\ ((((exists ff_h_fdp_before_quotient_entry. ff_h_fdp_before_quotient_entry + S (fdp_quotient_before) = S ((S (fdp_index_before)) * qc)) /\\ exists ff_q_fdp_before_quotient_entry. qb = ff_q_fdp_before_quotient_entry * S ((S (fdp_index_before)) * qc) + (fdp_quotient_before))) /\\ ((((exists ff_h_fdp_before_remainder_entry. ff_h_fdp_before_remainder_entry + S (fdp_remainder_before) = S ((S (fdp_index_before)) * rc)) /\\ exists ff_q_fdp_before_remainder_entry. rb = ff_q_fdp_before_remainder_entry * S ((S (fdp_index_before)) * rc) + (fdp_remainder_before))) /\\ (fdp_value_before = p * fdp_quotient_before + fdp_remainder_before /\\ (exists gsp_lt_gap_before_remainder_bound. gsp_lt_gap_before_remainder_bound + S fdp_remainder_before = p))))) -> (exists x q r. (((exists ff_h_fdp_choice_source. ff_h_fdp_choice_source + S (x) = S ((S (l)) * c)) /\\ exists ff_q_fdp_choice_source. b = ff_q_fdp_choice_source * S ((S (l)) * c) + (x))) /\\ (x = p * q + r /\\ (exists gsp_lt_gap_fdp_choice_remainder_bound. gsp_lt_gap_fdp_choice_remainder_bound + S r = p))) -> exists z d u v. (forall fdp_index_after. (exists gsp_lt_gap_after_index_bound. gsp_lt_gap_after_index_bound + S fdp_index_after = S l) -> exists fdp_value_after fdp_quotient_after fdp_remainder_after. (((exists ff_h_fdp_after_source. ff_h_fdp_after_source + S (fdp_value_after) = S ((S (fdp_index_after)) * c)) /\\ exists ff_q_fdp_after_source. b = ff_q_fdp_after_source * S ((S (fdp_index_after)) * c) + (fdp_value_after))) /\\ ((((exists ff_h_fdp_after_quotient_entry. ff_h_fdp_after_quotient_entry + S (fdp_quotient_after) = S ((S (fdp_index_after)) * d)) /\\ exists ff_q_fdp_after_quotient_entry. z = ff_q_fdp_after_quotient_entry * S ((S (fdp_index_after)) * d) + (fdp_quotient_after))) /\\ ((((exists ff_h_fdp_after_remainder_entry. ff_h_fdp_after_remainder_entry + S (fdp_remainder_after) = S ((S (fdp_index_after)) * v)) /\\ exists ff_q_fdp_after_remainder_entry. u = ff_q_fdp_after_remainder_entry * S ((S (fdp_index_after)) * v) + (fdp_remainder_after))) /\\ (fdp_value_after = p * fdp_quotient_after + fdp_remainder_after /\\ (exists gsp_lt_gap_after_remainder_bound. gsp_lt_gap_after_remainder_bound + S fdp_remainder_after = p)))))", "statement_sha256": "cec6006a1941b572a48c95a09f08c4d8bf3322a3560c43f2a949767dd791ff87", "summary": "Append one quotient/remainder pair while preserving the decoded prefix.", "summary_sha256": "9be24e68a4aa11b1d72aea8d3e5944bd043cde6c99f361f2b8030ffb8de881d3" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "beta_prefix_extend", "finite_lt_succ_eq_or_lt" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "ebc78a0c16fe6e9123a52363a69929590d8ca875380431776ef0de28b9b1193a", "kind": "qr_body_inventory", "path": "book/_static/pa-proof-explorer/api/corpus.json", "role": "dependency_curried_body", "selector": "theorems[name=beta_division_prefix_extend]" }, { "document_sha256": "b7774571ff25d0ab1c35707e4aa8b074b584179307bc25c2d9bcb5dc7a17f960", "kind": "qr_full_audit_status", "path": "research/arithmetic-library/wmi-qr-replay.md", "role": "promotion_blocker", "selector": "document" }, { "document_sha256": "3cd040d145f1004d07d277c66a3ffbcb355cd9c4b21938d79a6ec51b4258709c", "kind": "qr_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/quadratic-reciprocity-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=417]" }, { "document_sha256": "c69cfd6b9db2e8e60f77e593e744d1830c985d316fe9f82b2955b314435021c4", "kind": "qr_ordinary_empty_context_closure_receipt", "path": "research/arithmetic-library/quadratic-reciprocity-closure-receipt.md", "role": "original_kernel_full_root_and_independent_lean_verification", "selector": "document" }, { "document_sha256": "0123e5938f43cf67833751e2a6102d6598ac24c9be6db9a0d353ec3f55e5f32c", "kind": "sealed_alpha_v15_parent", "path": "artifacts/peano-library/alpha/catalog-v15.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=beta_division_prefix_extend]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_division_prefix_extend", "parent_alpha_version": "v34", "proof_bundle_node_id": 189, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_division_prefix_extend", "script": [ "intro p", "intro b", "intro c", "intro qb", "intro qc", "intro rb", "intro rc", "intro l", "intro hprefix", "intro hchoice", "cases hchoice", "cases hchoice_witness", "cases hchoice_witness_witness", "cases hchoice_witness_witness_witness", "cases hchoice_witness_witness_witness_right", "have hqextend : exists z d. (((exists ff_h_fdp_quotient_extension_last. ff_h_fdp_quotient_extension_last + S (x1) = S ((S (l)) * d)) /\\ exists ff_q_fdp_quotient_extension_last. z = ff_q_fdp_quotient_extension_last * S ((S (l)) * d) + (x1))) /\\ forall i q0. (exists gsp_lt_gap_fdp_quotient_extension_old_bound. gsp_lt_gap_fdp_quotient_extension_old_bound + S i = l) -> (((exists ff_h_fdp_quotient_extension_old_source. ff_h_fdp_quotient_extension_old_source + S (q0) = S ((S (i)) * qc)) /\\ exists ff_q_fdp_quotient_extension_old_source. qb = ff_q_fdp_quotient_extension_old_source * S ((S (i)) * qc) + (q0))) -> (((exists ff_h_fdp_quotient_extension_old_target_symbolic. ff_h_fdp_quotient_extension_old_target_symbolic + S (q0) = S ((S (i)) * d)) /\\ exists ff_q_fdp_quotient_extension_old_target_symbolic. z = ff_q_fdp_quotient_extension_old_target_symbolic * S ((S (i)) * d) + (q0)))", "specialize beta_prefix_extend l", "specialize beta_prefix_extend qb", "specialize beta_prefix_extend qc", "specialize beta_prefix_extend x1", "exact beta_prefix_extend", "cases hqextend", "cases hqextend_witness", "cases hqextend_witness_witness", "have hrextend : exists u v. (((exists ff_h_fdp_remainder_extension_last. ff_h_fdp_remainder_extension_last + S (x2) = S ((S (l)) * v)) /\\ exists ff_q_fdp_remainder_extension_last. u = ff_q_fdp_remainder_extension_last * S ((S (l)) * v) + (x2))) /\\ forall i r0. (exists gsp_lt_gap_fdp_remainder_extension_old_bound. gsp_lt_gap_fdp_remainder_extension_old_bound + S i = l) -> (((exists ff_h_fdp_remainder_extension_old_source. ff_h_fdp_remainder_extension_old_source + S (r0) = S ((S (i)) * rc)) /\\ exists ff_q_fdp_remainder_extension_old_source. rb = ff_q_fdp_remainder_extension_old_source * S ((S (i)) * rc) + (r0))) -> (((exists ff_h_fdp_remainder_extension_old_target. ff_h_fdp_remainder_extension_old_target + S (r0) = S ((S (i)) * v)) /\\ exists ff_q_fdp_remainder_extension_old_target. u = ff_q_fdp_remainder_extension_old_target * S ((S (i)) * v) + (r0)))", "specialize beta_prefix_extend l", "specialize beta_prefix_extend rb", "specialize beta_prefix_extend rc", "specialize beta_prefix_extend x2", "exact beta_prefix_extend", "cases hrextend", "cases hrextend_witness", "cases hrextend_witness_witness", "exists x3", "exists x4", "exists x5", "exists x6", "intro i", "intro hi", "have hsplit : i = l \\/ exists gap. gap + S i = l", "specialize finite_lt_succ_eq_or_lt l", "specialize finite_lt_succ_eq_or_lt i", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hsplit", "exists x", "exists x1", "exists x2", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hchoice_witness_witness_witness_left", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hqextend_witness_witness_left", "split", "rewrite hsplit_left", "rewrite hsplit_left", "exact hrextend_witness_witness_left", "split", "exact hchoice_witness_witness_witness_right_left", "exact hchoice_witness_witness_witness_right_right", "have hold : exists x q r. (((exists ff_h_fdp_previous_source. ff_h_fdp_previous_source + S (x) = S ((S (i)) * c)) /\\ exists ff_q_fdp_previous_source. b = ff_q_fdp_previous_source * S ((S (i)) * c) + (x))) /\\ ((((exists ff_h_fdp_previous_quotient. ff_h_fdp_previous_quotient + S (q) = S ((S (i)) * qc)) /\\ exists ff_q_fdp_previous_quotient. qb = ff_q_fdp_previous_quotient * S ((S (i)) * qc) + (q))) /\\ ((((exists ff_h_fdp_previous_remainder. ff_h_fdp_previous_remainder + S (r) = S ((S (i)) * rc)) /\\ exists ff_q_fdp_previous_remainder. rb = ff_q_fdp_previous_remainder * S ((S (i)) * rc) + (r))) /\\ (x = p * q + r /\\ (exists gsp_lt_gap_fdp_previous_remainder_bound. gsp_lt_gap_fdp_previous_remainder_bound + S r = p))))", "specialize hprefix i", "apply hprefix", "exact hsplit_right", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "cases hold_witness_witness_witness_right", "cases hold_witness_witness_witness_right_right", "cases hold_witness_witness_witness_right_right_right", "exists x7", "exists x8", "exists x9", "split", "exact hold_witness_witness_witness_left", "split", "specialize hqextend_witness_witness_right i", "specialize hqextend_witness_witness_right x8", "apply hqextend_witness_witness_right", "exact hsplit_right", "exact hold_witness_witness_witness_right_left", "split", "specialize hrextend_witness_witness_right i", "specialize hrextend_witness_witness_right x9", "apply hrextend_witness_witness_right", "exact hsplit_right", "exact hold_witness_witness_witness_right_right_left", "split", "exact hold_witness_witness_witness_right_right_right_left", "exact hold_witness_witness_witness_right_right_right_right" ], "script_sha256": "fb8634e5490f5ba6520eaecb42cc596b1822c657b839657a5127acbd23b712ad", "source": { "href": "https://github.com/nasqret/vietnam2026/blob/peano-lab/peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py#L168", "kind": "declaration", "line": 168, "owner_module": "finite_division_prefix_candidate", "path": "peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py", "sha256": "a6af47a7d918d46cdd4b83f60524d3c7afad42886ebb8e560bda5a1318f0b606" }, "stable_member": false, "statement": "forall p b c qb qc rb rc l. (forall fdp_index_before. (exists gsp_lt_gap_before_index_bound. gsp_lt_gap_before_index_bound + S fdp_index_before = l) -> exists fdp_value_before fdp_quotient_before fdp_remainder_before. (((exists ff_h_fdp_before_source. ff_h_fdp_before_source + S (fdp_value_before) = S ((S (fdp_index_before)) * c)) /\\ exists ff_q_fdp_before_source. b = ff_q_fdp_before_source * S ((S (fdp_index_before)) * c) + (fdp_value_before))) /\\ ((((exists ff_h_fdp_before_quotient_entry. ff_h_fdp_before_quotient_entry + S (fdp_quotient_before) = S ((S (fdp_index_before)) * qc)) /\\ exists ff_q_fdp_before_quotient_entry. qb = ff_q_fdp_before_quotient_entry * S ((S (fdp_index_before)) * qc) + (fdp_quotient_before))) /\\ ((((exists ff_h_fdp_before_remainder_entry. ff_h_fdp_before_remainder_entry + S (fdp_remainder_before) = S ((S (fdp_index_before)) * rc)) /\\ exists ff_q_fdp_before_remainder_entry. rb = ff_q_fdp_before_remainder_entry * S ((S (fdp_index_before)) * rc) + (fdp_remainder_before))) /\\ (fdp_value_before = p * fdp_quotient_before + fdp_remainder_before /\\ (exists gsp_lt_gap_before_remainder_bound. gsp_lt_gap_before_remainder_bound + S fdp_remainder_before = p))))) -> (exists x q r. (((exists ff_h_fdp_choice_source. ff_h_fdp_choice_source + S (x) = S ((S (l)) * c)) /\\ exists ff_q_fdp_choice_source. b = ff_q_fdp_choice_source * S ((S (l)) * c) + (x))) /\\ (x = p * q + r /\\ (exists gsp_lt_gap_fdp_choice_remainder_bound. gsp_lt_gap_fdp_choice_remainder_bound + S r = p))) -> exists z d u v. (forall fdp_index_after. (exists gsp_lt_gap_after_index_bound. gsp_lt_gap_after_index_bound + S fdp_index_after = S l) -> exists fdp_value_after fdp_quotient_after fdp_remainder_after. (((exists ff_h_fdp_after_source. ff_h_fdp_after_source + S (fdp_value_after) = S ((S (fdp_index_after)) * c)) /\\ exists ff_q_fdp_after_source. b = ff_q_fdp_after_source * S ((S (fdp_index_after)) * c) + (fdp_value_after))) /\\ ((((exists ff_h_fdp_after_quotient_entry. ff_h_fdp_after_quotient_entry + S (fdp_quotient_after) = S ((S (fdp_index_after)) * d)) /\\ exists ff_q_fdp_after_quotient_entry. z = ff_q_fdp_after_quotient_entry * S ((S (fdp_index_after)) * d) + (fdp_quotient_after))) /\\ ((((exists ff_h_fdp_after_remainder_entry. ff_h_fdp_after_remainder_entry + S (fdp_remainder_after) = S ((S (fdp_index_after)) * v)) /\\ exists ff_q_fdp_after_remainder_entry. u = ff_q_fdp_after_remainder_entry * S ((S (fdp_index_after)) * v) + (fdp_remainder_after))) /\\ (fdp_value_after = p * fdp_quotient_after + fdp_remainder_after /\\ (exists gsp_lt_gap_after_remainder_bound. gsp_lt_gap_after_remainder_bound + S fdp_remainder_after = p)))))", "statement_sha256": "cec6006a1941b572a48c95a09f08c4d8bf3322a3560c43f2a949767dd791ff87" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "beta_division_prefix_exists", "canonical_catalog_record": { "alpha_v16_promotion": { "bundle_node_id": 418, "bundle_sha256": "3cd040d145f1004d07d277c66a3ffbcb355cd9c4b21938d79a6ec51b4258709c", "parent_catalog_sha256": "0123e5938f43cf67833751e2a6102d6598ac24c9be6db9a0d353ec3f55e5f32c", "parent_evidence_status": "body_checked", "parent_row_sha256": "2d698b1762f14fd77ad654dfc3702d70e708104ca1393ca8ca0246507f4f9672" }, "body_checked": true, "checked_use": true, "dependencies": [ "add_eq_zero_right", "succ_ne_zero", "beta_at_exists", "division_remainder_exists", "beta_division_prefix_extend" ], "dependencies_sha256": "3a0e844cd913f8050cb2a432f5dab00e1bb9b280755e3199df56fd56b3be4326", "empty_context_closure": { "body_proof_depth": 30, "body_proof_nodes": 71, "bundle_dependency_edge_count": 1787, "bundle_node_count": 557, "bundle_node_id": 418, "bundle_path": "research/arithmetic-library/artifacts/quadratic-reciprocity-proof-bundle-v1.json", "bundle_root_id": 556, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "3cd040d145f1004d07d277c66a3ffbcb355cd9c4b21938d79a6ec51b4258709c", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "d82de890a8fdd2afd3f31bb1621391d7f8385f5ca7d8f78ca5556f6c5f40ec89", "status": "checked" }, "enrollment_index": 609, "enrollment_origin": "qr", "evidence_links": [ { "document_sha256": "ebc78a0c16fe6e9123a52363a69929590d8ca875380431776ef0de28b9b1193a", "kind": "qr_body_inventory", "path": "book/_static/pa-proof-explorer/api/corpus.json", "role": "dependency_curried_body", "selector": "theorems[name=beta_division_prefix_exists]" }, { "document_sha256": "b7774571ff25d0ab1c35707e4aa8b074b584179307bc25c2d9bcb5dc7a17f960", "kind": "qr_full_audit_status", "path": "research/arithmetic-library/wmi-qr-replay.md", "role": "promotion_blocker", "selector": "document" }, { "document_sha256": "3cd040d145f1004d07d277c66a3ffbcb355cd9c4b21938d79a6ec51b4258709c", "kind": "qr_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/quadratic-reciprocity-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=418]" }, { "document_sha256": "c69cfd6b9db2e8e60f77e593e744d1830c985d316fe9f82b2955b314435021c4", "kind": "qr_ordinary_empty_context_closure_receipt", "path": "research/arithmetic-library/quadratic-reciprocity-closure-receipt.md", "role": "original_kernel_full_root_and_independent_lean_verification", "selector": "document" }, { "document_sha256": "0123e5938f43cf67833751e2a6102d6598ac24c9be6db9a0d353ec3f55e5f32c", "kind": "sealed_alpha_v15_parent", "path": "artifacts/peano-library/alpha/catalog-v15.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=beta_division_prefix_exists]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "adb049a87c171cb750b5caa45e16301e9d461d329867a7d3c37043149cac625a", "membership": "alpha_only", "name": "beta_division_prefix_exists", "proof_tag": "PA00BY", "provenance": [ "qr" ], "script": [ "intro p", "intro b", "intro c", "induction l", "intro hp0", "exists 0", "exists 0", "exists 0", "exists 0", "intro i", "intro hi", "exfalso", "cases hi", "have hsi : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi", "intro hp0", "have hprevious : exists qb qc rb rc. (forall fdp_index_exists_previous. (exists gsp_lt_gap_exists_previous_index_bound. gsp_lt_gap_exists_previous_index_bound + S fdp_index_exists_previous = l) -> exists fdp_value_exists_previous fdp_quotient_exists_previous fdp_remainder_exists_previous. (((exists ff_h_fdp_exists_previous_source. ff_h_fdp_exists_previous_source + S (fdp_value_exists_previous) = S ((S (fdp_index_exists_previous)) * c)) /\\ exists ff_q_fdp_exists_previous_source. b = ff_q_fdp_exists_previous_source * S ((S (fdp_index_exists_previous)) * c) + (fdp_value_exists_previous))) /\\ ((((exists ff_h_fdp_exists_previous_quotient_entry. ff_h_fdp_exists_previous_quotient_entry + S (fdp_quotient_exists_previous) = S ((S (fdp_index_exists_previous)) * qc)) /\\ exists ff_q_fdp_exists_previous_quotient_entry. qb = ff_q_fdp_exists_previous_quotient_entry * S ((S (fdp_index_exists_previous)) * qc) + (fdp_quotient_exists_previous))) /\\ ((((exists ff_h_fdp_exists_previous_remainder_entry. ff_h_fdp_exists_previous_remainder_entry + S (fdp_remainder_exists_previous) = S ((S (fdp_index_exists_previous)) * rc)) /\\ exists ff_q_fdp_exists_previous_remainder_entry. rb = ff_q_fdp_exists_previous_remainder_entry * S ((S (fdp_index_exists_previous)) * rc) + (fdp_remainder_exists_previous))) /\\ (fdp_value_exists_previous = p * fdp_quotient_exists_previous + fdp_remainder_exists_previous /\\ (exists gsp_lt_gap_exists_previous_remainder_bound. gsp_lt_gap_exists_previous_remainder_bound + S fdp_remainder_exists_previous = p)))))", "apply IH", "exact hp0", "cases hprevious", "cases hprevious_witness", "cases hprevious_witness_witness", "cases hprevious_witness_witness_witness", "have hdecoded : exists x. (((exists ff_h_fdp_exists_last_source. ff_h_fdp_exists_last_source + S (x) = S ((S (l)) * c)) /\\ exists ff_q_fdp_exists_last_source. b = ff_q_fdp_exists_last_source * S ((S (l)) * c) + (x)))", "specialize beta_at_exists b", "specialize beta_at_exists c", "specialize beta_at_exists l", "exact beta_at_exists", "cases hdecoded", "have hdivision : exists q r. x4 = p * q + r /\\ (exists gsp_lt_gap_fdp_exists_last_remainder_bound. gsp_lt_gap_fdp_exists_last_remainder_bound + S r = p)", "specialize division_remainder_exists p", "specialize division_remainder_exists x4", "apply division_remainder_exists", "exact hp0", "cases hdivision", "cases hdivision_witness", "have hchoice : exists x q r. (((exists ff_h_fdp_choice_source. ff_h_fdp_choice_source + S (x) = S ((S (l)) * c)) /\\ exists ff_q_fdp_choice_source. b = ff_q_fdp_choice_source * S ((S (l)) * c) + (x))) /\\ (x = p * q + r /\\ (exists gsp_lt_gap_fdp_choice_remainder_bound. gsp_lt_gap_fdp_choice_remainder_bound + S r = p))", "exists x4", "exists x5", "exists x6", "split", "exact hdecoded_witness", "exact hdivision_witness_witness", "have hnext : exists qb qc rb rc. (forall fdp_index_exists_next. (exists gsp_lt_gap_exists_next_index_bound. gsp_lt_gap_exists_next_index_bound + S fdp_index_exists_next = S l) -> exists fdp_value_exists_next fdp_quotient_exists_next fdp_remainder_exists_next. (((exists ff_h_fdp_exists_next_source. ff_h_fdp_exists_next_source + S (fdp_value_exists_next) = S ((S (fdp_index_exists_next)) * c)) /\\ exists ff_q_fdp_exists_next_source. b = ff_q_fdp_exists_next_source * S ((S (fdp_index_exists_next)) * c) + (fdp_value_exists_next))) /\\ ((((exists ff_h_fdp_exists_next_quotient_entry. ff_h_fdp_exists_next_quotient_entry + S (fdp_quotient_exists_next) = S ((S (fdp_index_exists_next)) * qc)) /\\ exists ff_q_fdp_exists_next_quotient_entry. qb = ff_q_fdp_exists_next_quotient_entry * S ((S (fdp_index_exists_next)) * qc) + (fdp_quotient_exists_next))) /\\ ((((exists ff_h_fdp_exists_next_remainder_entry. ff_h_fdp_exists_next_remainder_entry + S (fdp_remainder_exists_next) = S ((S (fdp_index_exists_next)) * rc)) /\\ exists ff_q_fdp_exists_next_remainder_entry. rb = ff_q_fdp_exists_next_remainder_entry * S ((S (fdp_index_exists_next)) * rc) + (fdp_remainder_exists_next))) /\\ (fdp_value_exists_next = p * fdp_quotient_exists_next + fdp_remainder_exists_next /\\ (exists gsp_lt_gap_exists_next_remainder_bound. gsp_lt_gap_exists_next_remainder_bound + S fdp_remainder_exists_next = p)))))", "specialize beta_division_prefix_extend p", "specialize beta_division_prefix_extend b", "specialize beta_division_prefix_extend c", "specialize beta_division_prefix_extend x", "specialize beta_division_prefix_extend x1", "specialize beta_division_prefix_extend x2", "specialize beta_division_prefix_extend x3", "specialize beta_division_prefix_extend l", "apply beta_division_prefix_extend", "exact hprevious_witness_witness_witness_witness", "exact hchoice", "exact hnext" ], "script_sha256": "a037ac26e343b3cee9d81278d7ad2e2f02486d782cf80bf084a8adeb03dc46b0", "source": { "href": "https://github.com/nasqret/vietnam2026/blob/peano-lab/peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py#L271", "kind": "declaration", "line": 271, "owner_module": "finite_division_prefix_candidate", "path": "peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py", "sha256": "a6af47a7d918d46cdd4b83f60524d3c7afad42886ebb8e560bda5a1318f0b606" }, "statement": "forall p b c l. ~(p = 0) -> exists qb qc rb rc. (forall fdp_index_exists_result. (exists gsp_lt_gap_exists_result_index_bound. gsp_lt_gap_exists_result_index_bound + S fdp_index_exists_result = l) -> exists fdp_value_exists_result fdp_quotient_exists_result fdp_remainder_exists_result. (((exists ff_h_fdp_exists_result_source. ff_h_fdp_exists_result_source + S (fdp_value_exists_result) = S ((S (fdp_index_exists_result)) * c)) /\\ exists ff_q_fdp_exists_result_source. b = ff_q_fdp_exists_result_source * S ((S (fdp_index_exists_result)) * c) + (fdp_value_exists_result))) /\\ ((((exists ff_h_fdp_exists_result_quotient_entry. ff_h_fdp_exists_result_quotient_entry + S (fdp_quotient_exists_result) = S ((S (fdp_index_exists_result)) * qc)) /\\ exists ff_q_fdp_exists_result_quotient_entry. qb = ff_q_fdp_exists_result_quotient_entry * S ((S (fdp_index_exists_result)) * qc) + (fdp_quotient_exists_result))) /\\ ((((exists ff_h_fdp_exists_result_remainder_entry. ff_h_fdp_exists_result_remainder_entry + S (fdp_remainder_exists_result) = S ((S (fdp_index_exists_result)) * rc)) /\\ exists ff_q_fdp_exists_result_remainder_entry. rb = ff_q_fdp_exists_result_remainder_entry * S ((S (fdp_index_exists_result)) * rc) + (fdp_remainder_exists_result))) /\\ (fdp_value_exists_result = p * fdp_quotient_exists_result + fdp_remainder_exists_result /\\ (exists gsp_lt_gap_exists_result_remainder_bound. gsp_lt_gap_exists_result_remainder_bound + S fdp_remainder_exists_result = p)))))", "statement_sha256": "d82de890a8fdd2afd3f31bb1621391d7f8385f5ca7d8f78ca5556f6c5f40ec89", "summary": "Every finite beta source prefix has beta-coded quotients and bounded remainders for a nonzero modulus.", "summary_sha256": "eddfd7b3b3cb9eb793de76d04077a39129774774fbf3e9427b3b312efdfd6422" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "add_eq_zero_right", "succ_ne_zero", "beta_at_exists", "division_remainder_exists", "beta_division_prefix_extend" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "ebc78a0c16fe6e9123a52363a69929590d8ca875380431776ef0de28b9b1193a", "kind": "qr_body_inventory", "path": "book/_static/pa-proof-explorer/api/corpus.json", "role": "dependency_curried_body", "selector": "theorems[name=beta_division_prefix_exists]" }, { "document_sha256": "b7774571ff25d0ab1c35707e4aa8b074b584179307bc25c2d9bcb5dc7a17f960", "kind": "qr_full_audit_status", "path": "research/arithmetic-library/wmi-qr-replay.md", "role": "promotion_blocker", "selector": "document" }, { "document_sha256": "3cd040d145f1004d07d277c66a3ffbcb355cd9c4b21938d79a6ec51b4258709c", "kind": "qr_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/quadratic-reciprocity-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=418]" }, { "document_sha256": "c69cfd6b9db2e8e60f77e593e744d1830c985d316fe9f82b2955b314435021c4", "kind": "qr_ordinary_empty_context_closure_receipt", "path": "research/arithmetic-library/quadratic-reciprocity-closure-receipt.md", "role": "original_kernel_full_root_and_independent_lean_verification", "selector": "document" }, { "document_sha256": "0123e5938f43cf67833751e2a6102d6598ac24c9be6db9a0d353ec3f55e5f32c", "kind": "sealed_alpha_v15_parent", "path": "artifacts/peano-library/alpha/catalog-v15.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=beta_division_prefix_exists]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "beta_division_prefix_exists", "parent_alpha_version": "v34", "proof_bundle_node_id": 190, "reference_route": "jordan-totient/checkpoint.html#theorem-beta_division_prefix_exists", "script": [ "intro p", "intro b", "intro c", "induction l", "intro hp0", "exists 0", "exists 0", "exists 0", "exists 0", "intro i", "intro hi", "exfalso", "cases hi", "have hsi : S i = 0", "specialize add_eq_zero_right x", "specialize add_eq_zero_right (S i)", "apply add_eq_zero_right", "exact hi_witness", "specialize succ_ne_zero i", "apply succ_ne_zero", "exact hsi", "intro hp0", "have hprevious : exists qb qc rb rc. (forall fdp_index_exists_previous. (exists gsp_lt_gap_exists_previous_index_bound. gsp_lt_gap_exists_previous_index_bound + S fdp_index_exists_previous = l) -> exists fdp_value_exists_previous fdp_quotient_exists_previous fdp_remainder_exists_previous. (((exists ff_h_fdp_exists_previous_source. ff_h_fdp_exists_previous_source + S (fdp_value_exists_previous) = S ((S (fdp_index_exists_previous)) * c)) /\\ exists ff_q_fdp_exists_previous_source. b = ff_q_fdp_exists_previous_source * S ((S (fdp_index_exists_previous)) * c) + (fdp_value_exists_previous))) /\\ ((((exists ff_h_fdp_exists_previous_quotient_entry. ff_h_fdp_exists_previous_quotient_entry + S (fdp_quotient_exists_previous) = S ((S (fdp_index_exists_previous)) * qc)) /\\ exists ff_q_fdp_exists_previous_quotient_entry. qb = ff_q_fdp_exists_previous_quotient_entry * S ((S (fdp_index_exists_previous)) * qc) + (fdp_quotient_exists_previous))) /\\ ((((exists ff_h_fdp_exists_previous_remainder_entry. ff_h_fdp_exists_previous_remainder_entry + S (fdp_remainder_exists_previous) = S ((S (fdp_index_exists_previous)) * rc)) /\\ exists ff_q_fdp_exists_previous_remainder_entry. rb = ff_q_fdp_exists_previous_remainder_entry * S ((S (fdp_index_exists_previous)) * rc) + (fdp_remainder_exists_previous))) /\\ (fdp_value_exists_previous = p * fdp_quotient_exists_previous + fdp_remainder_exists_previous /\\ (exists gsp_lt_gap_exists_previous_remainder_bound. gsp_lt_gap_exists_previous_remainder_bound + S fdp_remainder_exists_previous = p)))))", "apply IH", "exact hp0", "cases hprevious", "cases hprevious_witness", "cases hprevious_witness_witness", "cases hprevious_witness_witness_witness", "have hdecoded : exists x. 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(((exists ff_h_fdp_choice_source. ff_h_fdp_choice_source + S (x) = S ((S (l)) * c)) /\\ exists ff_q_fdp_choice_source. b = ff_q_fdp_choice_source * S ((S (l)) * c) + (x))) /\\ (x = p * q + r /\\ (exists gsp_lt_gap_fdp_choice_remainder_bound. gsp_lt_gap_fdp_choice_remainder_bound + S r = p))", "exists x4", "exists x5", "exists x6", "split", "exact hdecoded_witness", "exact hdivision_witness_witness", "have hnext : exists qb qc rb rc. (forall fdp_index_exists_next. (exists gsp_lt_gap_exists_next_index_bound. gsp_lt_gap_exists_next_index_bound + S fdp_index_exists_next = S l) -> exists fdp_value_exists_next fdp_quotient_exists_next fdp_remainder_exists_next. (((exists ff_h_fdp_exists_next_source. ff_h_fdp_exists_next_source + S (fdp_value_exists_next) = S ((S (fdp_index_exists_next)) * c)) /\\ exists ff_q_fdp_exists_next_source. b = ff_q_fdp_exists_next_source * S ((S (fdp_index_exists_next)) * c) + (fdp_value_exists_next))) /\\ ((((exists ff_h_fdp_exists_next_quotient_entry. ff_h_fdp_exists_next_quotient_entry + S (fdp_quotient_exists_next) = S ((S (fdp_index_exists_next)) * qc)) /\\ exists ff_q_fdp_exists_next_quotient_entry. qb = ff_q_fdp_exists_next_quotient_entry * S ((S (fdp_index_exists_next)) * qc) + (fdp_quotient_exists_next))) /\\ ((((exists ff_h_fdp_exists_next_remainder_entry. ff_h_fdp_exists_next_remainder_entry + S (fdp_remainder_exists_next) = S ((S (fdp_index_exists_next)) * rc)) /\\ exists ff_q_fdp_exists_next_remainder_entry. rb = ff_q_fdp_exists_next_remainder_entry * S ((S (fdp_index_exists_next)) * rc) + (fdp_remainder_exists_next))) /\\ (fdp_value_exists_next = p * fdp_quotient_exists_next + fdp_remainder_exists_next /\\ (exists gsp_lt_gap_exists_next_remainder_bound. gsp_lt_gap_exists_next_remainder_bound + S fdp_remainder_exists_next = p)))))", "specialize beta_division_prefix_extend p", "specialize beta_division_prefix_extend b", "specialize beta_division_prefix_extend c", "specialize beta_division_prefix_extend x", "specialize beta_division_prefix_extend x1", "specialize beta_division_prefix_extend x2", "specialize beta_division_prefix_extend x3", "specialize beta_division_prefix_extend l", "apply beta_division_prefix_extend", "exact hprevious_witness_witness_witness_witness", "exact hchoice", "exact hnext" ], "script_sha256": "a037ac26e343b3cee9d81278d7ad2e2f02486d782cf80bf084a8adeb03dc46b0", "source": { "href": "https://github.com/nasqret/vietnam2026/blob/peano-lab/peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py#L271", "kind": "declaration", "line": 271, "owner_module": "finite_division_prefix_candidate", "path": "peano-lab/py/peano_lab/library/finite_division_prefix_candidate.py", "sha256": "a6af47a7d918d46cdd4b83f60524d3c7afad42886ebb8e560bda5a1318f0b606" }, "stable_member": false, "statement": "forall p b c l. ~(p = 0) -> exists qb qc rb rc. (forall fdp_index_exists_result. (exists gsp_lt_gap_exists_result_index_bound. gsp_lt_gap_exists_result_index_bound + S fdp_index_exists_result = l) -> exists fdp_value_exists_result fdp_quotient_exists_result fdp_remainder_exists_result. (((exists ff_h_fdp_exists_result_source. ff_h_fdp_exists_result_source + S (fdp_value_exists_result) = S ((S (fdp_index_exists_result)) * c)) /\\ exists ff_q_fdp_exists_result_source. b = ff_q_fdp_exists_result_source * S ((S (fdp_index_exists_result)) * c) + (fdp_value_exists_result))) /\\ ((((exists ff_h_fdp_exists_result_quotient_entry. ff_h_fdp_exists_result_quotient_entry + S (fdp_quotient_exists_result) = S ((S (fdp_index_exists_result)) * qc)) /\\ exists ff_q_fdp_exists_result_quotient_entry. qb = ff_q_fdp_exists_result_quotient_entry * S ((S (fdp_index_exists_result)) * qc) + (fdp_quotient_exists_result))) /\\ ((((exists ff_h_fdp_exists_result_remainder_entry. ff_h_fdp_exists_result_remainder_entry + S (fdp_remainder_exists_result) = S ((S (fdp_index_exists_result)) * rc)) /\\ exists ff_q_fdp_exists_result_remainder_entry. rb = ff_q_fdp_exists_result_remainder_entry * S ((S (fdp_index_exists_result)) * rc) + (fdp_remainder_exists_result))) /\\ (fdp_value_exists_result = p * fdp_quotient_exists_result + fdp_remainder_exists_result /\\ (exists gsp_lt_gap_exists_result_remainder_bound. gsp_lt_gap_exists_result_remainder_bound + S fdp_remainder_exists_result = p)))))", "statement_sha256": "d82de890a8fdd2afd3f31bb1621391d7f8385f5ca7d8f78ca5556f6c5f40ec89" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "canonical_gcd_exists", "canonical_catalog_record": { "body_checked": true, "checked_use": true, "dependencies": [ "gcd_exists_relational" ], "dependencies_sha256": "be753cc231ed3480213b2e4d7cdccca545212d31d62bdd2ff52b2aba6c90a236", "empty_context_closure": { "certificate_sha256": "8e3b24c937725618f58d1fddaf36bac33d4ce5f16ae1f3f5da4e254763468350", "cut_nodes": 36, "digest_kind": "content-proof-dag-sha256", "proof_depth": 47, "proof_edges": 892, "proof_nodes": 1280, "proof_objects": 845, "reused_objects": 48, "status": "checked" }, "enrollment_index": 748, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "6adfed3ba5ef4186589a212b60cf46b10e3344eea43e0a39dab948886ee0035a", "kind": "ha_campaign_closure", "path": "research/arithmetic-library/ha-number-theory-campaign.json", "role": "empty_context_closure", "selector": "theorem_evidence.theorems[name=canonical_gcd_exists]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "317788f0db25f5b26960bc60209989a7a0ae4496c6a7b997e6cf50d17ca9681b", "membership": "alpha_only", "name": "canonical_gcd_exists", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro a", "intro b", "specialize gcd_exists_relational a", "specialize gcd_exists_relational b", "exact gcd_exists_relational" ], "script_sha256": "197b11356f59c4c6bb32f7a3d54bd90c78578d3ba4475017d47fa6c39e3cb9b7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/ha_canonical_gcd_candidate.py", "sha256": "8d6b7675a520726ee945ad165b23cd75d820561a67596d40906f5c61859edc2b" }, "statement": "forall a b. exists g. 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((((exists ff_h_bpg_prefix_product_start. ff_h_bpg_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpg_prefix_product)) /\\ exists ff_q_bpg_prefix_product_start. ff_u_bpg_prefix_product = ff_q_bpg_prefix_product_start * S ((S (0)) * ff_v_bpg_prefix_product) + (1))) /\\ ((((exists ff_h_bpg_prefix_product_terminal. ff_h_bpg_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpg_prefix_product)) /\\ exists ff_q_bpg_prefix_product_terminal. ff_u_bpg_prefix_product = ff_q_bpg_prefix_product_terminal * S ((S (e)) * ff_v_bpg_prefix_product) + (r))) /\\ forall ff_i_bpg_prefix_product. (exists ff_lt_bpg_prefix_product_bound. ff_lt_bpg_prefix_product_bound + S ff_i_bpg_prefix_product = e) -> exists ff_p_bpg_prefix_product ff_r_bpg_prefix_product ff_s_bpg_prefix_product. 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(exists bpg_gap_base. bpg_gap_base + (1) = (a)) -> (exists ff_b_bpg_value ff_c_bpg_value. ((forall ff_i_bpg_value_repeat. (exists ff_lt_bpg_value_repeat_bound. ff_lt_bpg_value_repeat_bound + S ff_i_bpg_value_repeat = e) -> (((exists ff_h_bpg_value_repeat_decoded. ff_h_bpg_value_repeat_decoded + S (a) = S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_repeat_decoded. ff_b_bpg_value = ff_q_bpg_value_repeat_decoded * S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value) + (a)))) /\\ (exists ff_u_bpg_value_product ff_v_bpg_value_product. ((((exists ff_h_bpg_value_product_start. ff_h_bpg_value_product_start + S (1) = S ((S (0)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_start. ff_u_bpg_value_product = ff_q_bpg_value_product_start * S ((S (0)) * ff_v_bpg_value_product) + (1))) /\\ ((((exists ff_h_bpg_value_product_terminal. ff_h_bpg_value_product_terminal + S (x) = S ((S (e)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_terminal. ff_u_bpg_value_product = ff_q_bpg_value_product_terminal * S ((S (e)) * ff_v_bpg_value_product) + (x))) /\\ forall ff_i_bpg_value_product. (exists ff_lt_bpg_value_product_bound. ff_lt_bpg_value_product_bound + S ff_i_bpg_value_product = e) -> exists ff_p_bpg_value_product ff_r_bpg_value_product ff_s_bpg_value_product. ((((exists ff_h_bpg_value_product_factor. ff_h_bpg_value_product_factor + S (ff_p_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_product_factor. ff_b_bpg_value = ff_q_bpg_value_product_factor * S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value) + (ff_p_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_partial. ff_h_bpg_value_product_partial + S (ff_r_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_partial. ff_u_bpg_value_product = ff_q_bpg_value_product_partial * S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_r_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_successor. ff_h_bpg_value_product_successor + S (ff_s_bpg_value_product) = S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_successor. ff_u_bpg_value_product = ff_q_bpg_value_product_successor * S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_s_bpg_value_product))) /\\ ff_s_bpg_value_product = ff_r_bpg_value_product * ff_p_bpg_value_product)))))))) -> (exists bpg_gap_value. bpg_gap_value + (1) = (x))", "statement_sha256": "52c15a620b3303df1dd639722843445b568cd9587e53556acde3522cc0166a05", "summary": "Every relational power of a base at least one is at least one.", "summary_sha256": "0198e705ab70d0a2997c105a298925a4e09039ff49cf101801c324a93bb8c15f" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_zero", "pow_successor_decompose", "le_refl", "le_mul_of_one_le_right", "le_trans" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "41584397a149b7af19891bdd7b0f6b6366f6412c4c636508921af85d7220bfab", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_growth_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b0efd50a155c8f24eefdf23fe1d90b97708887b9fdb574a02d99e5d902162802", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_growth_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=180]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=one_le_pow]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "one_le_pow", "parent_alpha_version": "v34", "proof_bundle_node_id": 194, "reference_route": "jordan-totient/checkpoint.html#theorem-one_le_pow", "script": [ "intro a", "intro e", "induction e", "intro x", "intro ha", "intro hx", "have hx1 : x = 1", "specialize pow_zero a", "specialize pow_zero 0", "specialize pow_zero x", "apply pow_zero", "refl", "exact hx", "rewrite hx1", "specialize le_refl 1", "exact le_refl", "intro x", "intro ha", "intro hx", "have hstep : exists r. (exists ff_b_bpg_prefix ff_c_bpg_prefix. ((forall ff_i_bpg_prefix_repeat. (exists ff_lt_bpg_prefix_repeat_bound. ff_lt_bpg_prefix_repeat_bound + S ff_i_bpg_prefix_repeat = e) -> (((exists ff_h_bpg_prefix_repeat_decoded. ff_h_bpg_prefix_repeat_decoded + S (a) = S ((S (ff_i_bpg_prefix_repeat)) * ff_c_bpg_prefix)) /\\ exists ff_q_bpg_prefix_repeat_decoded. ff_b_bpg_prefix = ff_q_bpg_prefix_repeat_decoded * S ((S (ff_i_bpg_prefix_repeat)) * ff_c_bpg_prefix) + (a)))) /\\ (exists ff_u_bpg_prefix_product ff_v_bpg_prefix_product. ((((exists ff_h_bpg_prefix_product_start. ff_h_bpg_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpg_prefix_product)) /\\ exists ff_q_bpg_prefix_product_start. ff_u_bpg_prefix_product = ff_q_bpg_prefix_product_start * S ((S (0)) * ff_v_bpg_prefix_product) + (1))) /\\ ((((exists ff_h_bpg_prefix_product_terminal. ff_h_bpg_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpg_prefix_product)) /\\ exists ff_q_bpg_prefix_product_terminal. ff_u_bpg_prefix_product = ff_q_bpg_prefix_product_terminal * S ((S (e)) * ff_v_bpg_prefix_product) + (r))) /\\ forall ff_i_bpg_prefix_product. (exists ff_lt_bpg_prefix_product_bound. ff_lt_bpg_prefix_product_bound + S ff_i_bpg_prefix_product = e) -> exists ff_p_bpg_prefix_product ff_r_bpg_prefix_product ff_s_bpg_prefix_product. ((((exists ff_h_bpg_prefix_product_factor. ff_h_bpg_prefix_product_factor + S (ff_p_bpg_prefix_product) = S ((S (ff_i_bpg_prefix_product)) * ff_c_bpg_prefix)) /\\ exists ff_q_bpg_prefix_product_factor. ff_b_bpg_prefix = ff_q_bpg_prefix_product_factor * S ((S (ff_i_bpg_prefix_product)) * ff_c_bpg_prefix) + (ff_p_bpg_prefix_product))) /\\ ((((exists ff_h_bpg_prefix_product_partial. ff_h_bpg_prefix_product_partial + S (ff_r_bpg_prefix_product) = S ((S (ff_i_bpg_prefix_product)) * ff_v_bpg_prefix_product)) /\\ exists ff_q_bpg_prefix_product_partial. ff_u_bpg_prefix_product = ff_q_bpg_prefix_product_partial * S ((S (ff_i_bpg_prefix_product)) * ff_v_bpg_prefix_product) + (ff_r_bpg_prefix_product))) /\\ ((((exists ff_h_bpg_prefix_product_successor. ff_h_bpg_prefix_product_successor + S (ff_s_bpg_prefix_product) = S ((S (S ff_i_bpg_prefix_product)) * ff_v_bpg_prefix_product)) /\\ exists ff_q_bpg_prefix_product_successor. ff_u_bpg_prefix_product = ff_q_bpg_prefix_product_successor * S ((S (S ff_i_bpg_prefix_product)) * ff_v_bpg_prefix_product) + (ff_s_bpg_prefix_product))) /\\ ff_s_bpg_prefix_product = ff_r_bpg_prefix_product * ff_p_bpg_prefix_product)))))))) /\\ x = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose e", "specialize pow_successor_decompose (S e)", "specialize pow_successor_decompose x", "apply pow_successor_decompose", "refl", "exact hx", "cases hstep", "cases hstep_witness", "have hr : exists k. k + 1 = x1", "specialize IH x1", "apply IH", "exact ha", "exact hstep_witness_left", "have hrproduct : exists k. k + x1 = x1 * a", "specialize le_mul_of_one_le_right x1", "specialize le_mul_of_one_le_right a", "apply le_mul_of_one_le_right", "exact ha", "rewrite hstep_witness_right", "specialize le_trans 1", "specialize le_trans x1", "specialize le_trans (x1 * a)", "apply le_trans", "exact hr", "exact hrproduct" ], "script_sha256": "7c0df347297084d5a7645a264c9198d16518d57d29fea54e49e6f0f6c6122000", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_growth_candidate.py", "sha256": "41584397a149b7af19891bdd7b0f6b6366f6412c4c636508921af85d7220bfab" }, "stable_member": false, "statement": "forall a e x. (exists bpg_gap_base. bpg_gap_base + (1) = (a)) -> (exists ff_b_bpg_value ff_c_bpg_value. ((forall ff_i_bpg_value_repeat. (exists ff_lt_bpg_value_repeat_bound. ff_lt_bpg_value_repeat_bound + S ff_i_bpg_value_repeat = e) -> (((exists ff_h_bpg_value_repeat_decoded. ff_h_bpg_value_repeat_decoded + S (a) = S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_repeat_decoded. ff_b_bpg_value = ff_q_bpg_value_repeat_decoded * S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value) + (a)))) /\\ (exists ff_u_bpg_value_product ff_v_bpg_value_product. ((((exists ff_h_bpg_value_product_start. ff_h_bpg_value_product_start + S (1) = S ((S (0)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_start. ff_u_bpg_value_product = ff_q_bpg_value_product_start * S ((S (0)) * ff_v_bpg_value_product) + (1))) /\\ ((((exists ff_h_bpg_value_product_terminal. ff_h_bpg_value_product_terminal + S (x) = S ((S (e)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_terminal. ff_u_bpg_value_product = ff_q_bpg_value_product_terminal * S ((S (e)) * ff_v_bpg_value_product) + (x))) /\\ forall ff_i_bpg_value_product. (exists ff_lt_bpg_value_product_bound. ff_lt_bpg_value_product_bound + S ff_i_bpg_value_product = e) -> exists ff_p_bpg_value_product ff_r_bpg_value_product ff_s_bpg_value_product. ((((exists ff_h_bpg_value_product_factor. ff_h_bpg_value_product_factor + S (ff_p_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_product_factor. ff_b_bpg_value = ff_q_bpg_value_product_factor * S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value) + (ff_p_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_partial. ff_h_bpg_value_product_partial + S (ff_r_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_partial. ff_u_bpg_value_product = ff_q_bpg_value_product_partial * S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_r_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_successor. ff_h_bpg_value_product_successor + S (ff_s_bpg_value_product) = S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_successor. ff_u_bpg_value_product = ff_q_bpg_value_product_successor * S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_s_bpg_value_product))) /\\ ff_s_bpg_value_product = ff_r_bpg_value_product * ff_p_bpg_value_product)))))))) -> (exists bpg_gap_value. bpg_gap_value + (1) = (x))", "statement_sha256": "52c15a620b3303df1dd639722843445b568cd9587e53556acde3522cc0166a05" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_nonzero_of_one_le", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 181, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "e6849c32e866a988f7bec83f1e81cf6ecfc07f92d87fcc9fa9f194e5d5a6c9fc" }, "bertrand_evidence_bundle_sha256": "0d09b378b307dada7784e2f71550d8a3c7f1f8cf998501cd536da08b5e958b3f", "body_checked": true, "body_receipt": { "command_count": 17, "dependency_count": 2, "dne_command_count": 0, "name": "pow_nonzero_of_one_le", "proof_depth": 16, "proof_edges": 20, "proof_nodes": 21, "proof_objects": 21, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "one_le_pow", "ne_zero_of_one_le" ], "dependencies_sha256": "11d26cb2c797bf1cfc7ffb50fbc4e86e54dba1b3fbea52891be58ae7cf726014", "empty_context_closure": { "body_proof_depth": 16, "body_proof_nodes": 21, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 181, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", 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"reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=181]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=pow_nonzero_of_one_le]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "8c331fb80e87c797ceedf420f49e7c2ddb777d874b1f787a0d6da2930226b874", "membership": "alpha_only", "name": "pow_nonzero_of_one_le", "proof_tag": null, "provenance": [ "bertrand_b1_power_growth" ], "script": [ "intro a", "intro e", "intro x", "intro ha", "intro hx", "have hx1 : exists bpg_gap_value. bpg_gap_value + (1) = (x)", "specialize one_le_pow a", "specialize one_le_pow e", "specialize one_le_pow x", "apply one_le_pow", "exact ha", "exact hx", "intro hx0", "specialize ne_zero_of_one_le x", "apply ne_zero_of_one_le", "exact hx1", "exact hx0" ], "script_sha256": "05d5f96772bd92e9c78310fb6d407949fc20523d92b985ae60cab73772d31c66", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_growth_candidate.py", "sha256": "41584397a149b7af19891bdd7b0f6b6366f6412c4c636508921af85d7220bfab" }, "statement": "forall a e x. (exists bpg_gap_base. bpg_gap_base + (1) = (a)) -> (exists ff_b_bpg_value ff_c_bpg_value. ((forall ff_i_bpg_value_repeat. (exists ff_lt_bpg_value_repeat_bound. ff_lt_bpg_value_repeat_bound + S ff_i_bpg_value_repeat = e) -> (((exists ff_h_bpg_value_repeat_decoded. ff_h_bpg_value_repeat_decoded + S (a) = S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_repeat_decoded. ff_b_bpg_value = ff_q_bpg_value_repeat_decoded * S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value) + (a)))) /\\ (exists ff_u_bpg_value_product ff_v_bpg_value_product. ((((exists ff_h_bpg_value_product_start. ff_h_bpg_value_product_start + S (1) = S ((S (0)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_start. ff_u_bpg_value_product = ff_q_bpg_value_product_start * S ((S (0)) * ff_v_bpg_value_product) + (1))) /\\ ((((exists ff_h_bpg_value_product_terminal. ff_h_bpg_value_product_terminal + S (x) = S ((S (e)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_terminal. ff_u_bpg_value_product = ff_q_bpg_value_product_terminal * S ((S (e)) * ff_v_bpg_value_product) + (x))) /\\ forall ff_i_bpg_value_product. (exists ff_lt_bpg_value_product_bound. ff_lt_bpg_value_product_bound + S ff_i_bpg_value_product = e) -> exists ff_p_bpg_value_product ff_r_bpg_value_product ff_s_bpg_value_product. ((((exists ff_h_bpg_value_product_factor. ff_h_bpg_value_product_factor + S (ff_p_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_product_factor. ff_b_bpg_value = ff_q_bpg_value_product_factor * S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value) + (ff_p_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_partial. ff_h_bpg_value_product_partial + S (ff_r_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_partial. ff_u_bpg_value_product = ff_q_bpg_value_product_partial * S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_r_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_successor. ff_h_bpg_value_product_successor + S (ff_s_bpg_value_product) = S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_successor. ff_u_bpg_value_product = ff_q_bpg_value_product_successor * S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_s_bpg_value_product))) /\\ ff_s_bpg_value_product = ff_r_bpg_value_product * ff_p_bpg_value_product)))))))) -> ~(x = 0)", "statement_sha256": "0cfed99b44eeebba9e89ae38dac56cf5769114b989d4b5f96512c4c65d4f2899", "summary": "A relational power of a base at least one cannot be zero.", "summary_sha256": "dbc458eb2b1e09bba66a01b9b2b51fed3accc305d24e839dc9f2d008ef44f2cc" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "one_le_pow", "ne_zero_of_one_le" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "41584397a149b7af19891bdd7b0f6b6366f6412c4c636508921af85d7220bfab", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_growth_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b0efd50a155c8f24eefdf23fe1d90b97708887b9fdb574a02d99e5d902162802", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_growth_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=181]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=pow_nonzero_of_one_le]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_nonzero_of_one_le", "parent_alpha_version": "v34", "proof_bundle_node_id": 195, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_nonzero_of_one_le", "script": [ "intro a", "intro e", "intro x", "intro ha", "intro hx", "have hx1 : exists bpg_gap_value. bpg_gap_value + (1) = (x)", "specialize one_le_pow a", "specialize one_le_pow e", "specialize one_le_pow x", "apply one_le_pow", "exact ha", "exact hx", "intro hx0", "specialize ne_zero_of_one_le x", "apply ne_zero_of_one_le", "exact hx1", "exact hx0" ], "script_sha256": "05d5f96772bd92e9c78310fb6d407949fc20523d92b985ae60cab73772d31c66", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_growth_candidate.py", "sha256": "41584397a149b7af19891bdd7b0f6b6366f6412c4c636508921af85d7220bfab" }, "stable_member": false, "statement": "forall a e x. (exists bpg_gap_base. bpg_gap_base + (1) = (a)) -> (exists ff_b_bpg_value ff_c_bpg_value. ((forall ff_i_bpg_value_repeat. (exists ff_lt_bpg_value_repeat_bound. ff_lt_bpg_value_repeat_bound + S ff_i_bpg_value_repeat = e) -> (((exists ff_h_bpg_value_repeat_decoded. ff_h_bpg_value_repeat_decoded + S (a) = S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_repeat_decoded. ff_b_bpg_value = ff_q_bpg_value_repeat_decoded * S ((S (ff_i_bpg_value_repeat)) * ff_c_bpg_value) + (a)))) /\\ (exists ff_u_bpg_value_product ff_v_bpg_value_product. ((((exists ff_h_bpg_value_product_start. ff_h_bpg_value_product_start + S (1) = S ((S (0)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_start. ff_u_bpg_value_product = ff_q_bpg_value_product_start * S ((S (0)) * ff_v_bpg_value_product) + (1))) /\\ ((((exists ff_h_bpg_value_product_terminal. ff_h_bpg_value_product_terminal + S (x) = S ((S (e)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_terminal. ff_u_bpg_value_product = ff_q_bpg_value_product_terminal * S ((S (e)) * ff_v_bpg_value_product) + (x))) /\\ forall ff_i_bpg_value_product. (exists ff_lt_bpg_value_product_bound. ff_lt_bpg_value_product_bound + S ff_i_bpg_value_product = e) -> exists ff_p_bpg_value_product ff_r_bpg_value_product ff_s_bpg_value_product. ((((exists ff_h_bpg_value_product_factor. ff_h_bpg_value_product_factor + S (ff_p_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value)) /\\ exists ff_q_bpg_value_product_factor. ff_b_bpg_value = ff_q_bpg_value_product_factor * S ((S (ff_i_bpg_value_product)) * ff_c_bpg_value) + (ff_p_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_partial. ff_h_bpg_value_product_partial + S (ff_r_bpg_value_product) = S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_partial. ff_u_bpg_value_product = ff_q_bpg_value_product_partial * S ((S (ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_r_bpg_value_product))) /\\ ((((exists ff_h_bpg_value_product_successor. ff_h_bpg_value_product_successor + S (ff_s_bpg_value_product) = S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product)) /\\ exists ff_q_bpg_value_product_successor. ff_u_bpg_value_product = ff_q_bpg_value_product_successor * S ((S (S ff_i_bpg_value_product)) * ff_v_bpg_value_product) + (ff_s_bpg_value_product))) /\\ ff_s_bpg_value_product = ff_r_bpg_value_product * ff_p_bpg_value_product)))))))) -> ~(x = 0)", "statement_sha256": "0cfed99b44eeebba9e89ae38dac56cf5769114b989d4b5f96512c4c65d4f2899" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_divides_decidable", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 182, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "30c06759f2e146fd13a73d7892e3c0c4156e02d683efea08d24fa8d9e82ac4b0" }, "bertrand_evidence_bundle_sha256": "1e135c5bd13cf97ee13724706756cd13ec5a04f7a1b9911fd0dd528e596fc459", "body_checked": true, "body_receipt": { "command_count": 33, "dependency_count": 3, "dne_command_count": 0, "name": "power_divides_decidable", "proof_depth": 20, "proof_edges": 37, "proof_nodes": 38, "proof_objects": 38, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "pow_exists", "multiple_decidable", "pow_functional" ], "dependencies_sha256": "1e97244837bb96be367178355f2c97372758377e0d127a5985924ae3d2542fa7", "empty_context_closure": { "body_proof_depth": 20, "body_proof_nodes": 38, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 182, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "056bf86316331243e9372e8b59cd02ca2f4a88cee0f43b5900e299ef979e227e", "status": "checked" }, "enrollment_index": 913, "enrollment_origin": "bertrand_b2_bounded_valuation", "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=182]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_decidable]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "aceaef5b2aa71a1cb01117d20da471823625832278bcf998b97b76d2199a37e9", "membership": "alpha_only", "name": "power_divides_decidable", "proof_tag": null, "provenance": [ "bertrand_b2_bounded_valuation" ], "script": [ "intro p", "intro e", "intro a", "have hpower : exists r. (exists ff_b_decision_witness ff_c_decision_witness. ((forall ff_i_decision_witness_repeat. (exists ff_lt_decision_witness_repeat_bound. ff_lt_decision_witness_repeat_bound + S ff_i_decision_witness_repeat = e) -> (((exists ff_h_decision_witness_repeat_decoded. ff_h_decision_witness_repeat_decoded + S (p) = S ((S (ff_i_decision_witness_repeat)) * ff_c_decision_witness)) /\\ exists ff_q_decision_witness_repeat_decoded. ff_b_decision_witness = ff_q_decision_witness_repeat_decoded * S ((S (ff_i_decision_witness_repeat)) * ff_c_decision_witness) + (p)))) /\\ (exists ff_u_decision_witness_product ff_v_decision_witness_product. ((((exists ff_h_decision_witness_product_start. ff_h_decision_witness_product_start + S (1) = S ((S (0)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_start. ff_u_decision_witness_product = ff_q_decision_witness_product_start * S ((S (0)) * ff_v_decision_witness_product) + (1))) /\\ ((((exists ff_h_decision_witness_product_terminal. ff_h_decision_witness_product_terminal + S (r) = S ((S (e)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_terminal. ff_u_decision_witness_product = ff_q_decision_witness_product_terminal * S ((S (e)) * ff_v_decision_witness_product) + (r))) /\\ forall ff_i_decision_witness_product. (exists ff_lt_decision_witness_product_bound. ff_lt_decision_witness_product_bound + S ff_i_decision_witness_product = e) -> exists ff_p_decision_witness_product ff_r_decision_witness_product ff_s_decision_witness_product. ((((exists ff_h_decision_witness_product_factor. ff_h_decision_witness_product_factor + S (ff_p_decision_witness_product) = S ((S (ff_i_decision_witness_product)) * ff_c_decision_witness)) /\\ exists ff_q_decision_witness_product_factor. ff_b_decision_witness = ff_q_decision_witness_product_factor * S ((S (ff_i_decision_witness_product)) * ff_c_decision_witness) + (ff_p_decision_witness_product))) /\\ ((((exists ff_h_decision_witness_product_partial. ff_h_decision_witness_product_partial + S (ff_r_decision_witness_product) = S ((S (ff_i_decision_witness_product)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_partial. ff_u_decision_witness_product = ff_q_decision_witness_product_partial * S ((S (ff_i_decision_witness_product)) * ff_v_decision_witness_product) + (ff_r_decision_witness_product))) /\\ ((((exists ff_h_decision_witness_product_successor. ff_h_decision_witness_product_successor + S (ff_s_decision_witness_product) = S ((S (S ff_i_decision_witness_product)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_successor. ff_u_decision_witness_product = ff_q_decision_witness_product_successor * S ((S (S ff_i_decision_witness_product)) * ff_v_decision_witness_product) + (ff_s_decision_witness_product))) /\\ ff_s_decision_witness_product = ff_r_decision_witness_product * ff_p_decision_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists e", "exact pow_exists", "cases hpower", "have hdiv : (exists q. a = x * q) \\/ ~(exists q. a = x * q)", "specialize multiple_decidable x", "specialize multiple_decidable a", "exact multiple_decidable", "cases hdiv", "left", "exists x", "split", "exact hpower_witness", "exact hdiv_left", "right", "intro hother", "cases hother", "cases hother_witness", "have heq : x1 = x", "specialize pow_functional p", "specialize pow_functional e", "specialize pow_functional x1", "specialize pow_functional x", "apply pow_functional", "exact hother_witness_left", "exact hpower_witness", "apply hdiv_right", "rewrite heq at hother_witness_right", "exact hother_witness_right" ], "script_sha256": "c95214e16d0f4ed6de95f4d38bbf754f9ba75e031083e4f49b255fd8f7e9de6c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "statement": "forall p e a. (exists bpv_result_decision. ((exists ff_b_decision_power ff_c_decision_power. ((forall ff_i_decision_power_repeat. (exists ff_lt_decision_power_repeat_bound. ff_lt_decision_power_repeat_bound + S ff_i_decision_power_repeat = e) -> (((exists ff_h_decision_power_repeat_decoded. ff_h_decision_power_repeat_decoded + S (p) = S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_repeat_decoded. ff_b_decision_power = ff_q_decision_power_repeat_decoded * S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power) + (p)))) /\\ (exists ff_u_decision_power_product ff_v_decision_power_product. ((((exists ff_h_decision_power_product_start. ff_h_decision_power_product_start + S (1) = S ((S (0)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_start. ff_u_decision_power_product = ff_q_decision_power_product_start * S ((S (0)) * ff_v_decision_power_product) + (1))) /\\ ((((exists ff_h_decision_power_product_terminal. ff_h_decision_power_product_terminal + S (bpv_result_decision) = S ((S (e)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_terminal. ff_u_decision_power_product = ff_q_decision_power_product_terminal * S ((S (e)) * ff_v_decision_power_product) + (bpv_result_decision))) /\\ forall ff_i_decision_power_product. (exists ff_lt_decision_power_product_bound. ff_lt_decision_power_product_bound + S ff_i_decision_power_product = e) -> exists ff_p_decision_power_product ff_r_decision_power_product ff_s_decision_power_product. ((((exists ff_h_decision_power_product_factor. ff_h_decision_power_product_factor + S (ff_p_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_product_factor. ff_b_decision_power = ff_q_decision_power_product_factor * S ((S (ff_i_decision_power_product)) * ff_c_decision_power) + (ff_p_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_partial. ff_h_decision_power_product_partial + S (ff_r_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_partial. ff_u_decision_power_product = ff_q_decision_power_product_partial * S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_r_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_successor. ff_h_decision_power_product_successor + S (ff_s_decision_power_product) = S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_successor. ff_u_decision_power_product = ff_q_decision_power_product_successor * S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_s_decision_power_product))) /\\ ff_s_decision_power_product = ff_r_decision_power_product * ff_p_decision_power_product)))))))) /\\ (exists bpv_factor_decision_divides. a = bpv_result_decision * bpv_factor_decision_divides))) \\/ ~(exists bpv_result_decision. ((exists ff_b_decision_power ff_c_decision_power. ((forall ff_i_decision_power_repeat. (exists ff_lt_decision_power_repeat_bound. ff_lt_decision_power_repeat_bound + S ff_i_decision_power_repeat = e) -> (((exists ff_h_decision_power_repeat_decoded. ff_h_decision_power_repeat_decoded + S (p) = S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_repeat_decoded. ff_b_decision_power = ff_q_decision_power_repeat_decoded * S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power) + (p)))) /\\ (exists ff_u_decision_power_product ff_v_decision_power_product. ((((exists ff_h_decision_power_product_start. ff_h_decision_power_product_start + S (1) = S ((S (0)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_start. ff_u_decision_power_product = ff_q_decision_power_product_start * S ((S (0)) * ff_v_decision_power_product) + (1))) /\\ ((((exists ff_h_decision_power_product_terminal. ff_h_decision_power_product_terminal + S (bpv_result_decision) = S ((S (e)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_terminal. ff_u_decision_power_product = ff_q_decision_power_product_terminal * S ((S (e)) * ff_v_decision_power_product) + (bpv_result_decision))) /\\ forall ff_i_decision_power_product. (exists ff_lt_decision_power_product_bound. ff_lt_decision_power_product_bound + S ff_i_decision_power_product = e) -> exists ff_p_decision_power_product ff_r_decision_power_product ff_s_decision_power_product. ((((exists ff_h_decision_power_product_factor. ff_h_decision_power_product_factor + S (ff_p_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_product_factor. ff_b_decision_power = ff_q_decision_power_product_factor * S ((S (ff_i_decision_power_product)) * ff_c_decision_power) + (ff_p_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_partial. ff_h_decision_power_product_partial + S (ff_r_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_partial. ff_u_decision_power_product = ff_q_decision_power_product_partial * S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_r_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_successor. ff_h_decision_power_product_successor + S (ff_s_decision_power_product) = S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_successor. ff_u_decision_power_product = ff_q_decision_power_product_successor * S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_s_decision_power_product))) /\\ ff_s_decision_power_product = ff_r_decision_power_product * ff_p_decision_power_product)))))))) /\\ (exists bpv_factor_decision_divides. a = bpv_result_decision * bpv_factor_decision_divides)))", "statement_sha256": "056bf86316331243e9372e8b59cd02ca2f4a88cee0f43b5900e299ef979e227e", "summary": "Divisibility by a relational power is constructively decidable.", "summary_sha256": "e4158649b28956f5bf4f8fabe7177fbad8c99866f3ace5014428b22228d8500b" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_exists", "multiple_decidable", "pow_functional" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=182]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_decidable]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_divides_decidable", "parent_alpha_version": "v34", "proof_bundle_node_id": 196, "reference_route": "jordan-totient/checkpoint.html#theorem-power_divides_decidable", "script": [ "intro p", "intro e", "intro a", "have hpower : exists r. (exists ff_b_decision_witness ff_c_decision_witness. ((forall ff_i_decision_witness_repeat. (exists ff_lt_decision_witness_repeat_bound. ff_lt_decision_witness_repeat_bound + S ff_i_decision_witness_repeat = e) -> (((exists ff_h_decision_witness_repeat_decoded. ff_h_decision_witness_repeat_decoded + S (p) = S ((S (ff_i_decision_witness_repeat)) * ff_c_decision_witness)) /\\ exists ff_q_decision_witness_repeat_decoded. ff_b_decision_witness = ff_q_decision_witness_repeat_decoded * S ((S (ff_i_decision_witness_repeat)) * ff_c_decision_witness) + (p)))) /\\ (exists ff_u_decision_witness_product ff_v_decision_witness_product. ((((exists ff_h_decision_witness_product_start. ff_h_decision_witness_product_start + S (1) = S ((S (0)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_start. ff_u_decision_witness_product = ff_q_decision_witness_product_start * S ((S (0)) * ff_v_decision_witness_product) + (1))) /\\ ((((exists ff_h_decision_witness_product_terminal. ff_h_decision_witness_product_terminal + S (r) = S ((S (e)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_terminal. ff_u_decision_witness_product = ff_q_decision_witness_product_terminal * S ((S (e)) * ff_v_decision_witness_product) + (r))) /\\ forall ff_i_decision_witness_product. (exists ff_lt_decision_witness_product_bound. ff_lt_decision_witness_product_bound + S ff_i_decision_witness_product = e) -> exists ff_p_decision_witness_product ff_r_decision_witness_product ff_s_decision_witness_product. ((((exists ff_h_decision_witness_product_factor. ff_h_decision_witness_product_factor + S (ff_p_decision_witness_product) = S ((S (ff_i_decision_witness_product)) * ff_c_decision_witness)) /\\ exists ff_q_decision_witness_product_factor. ff_b_decision_witness = ff_q_decision_witness_product_factor * S ((S (ff_i_decision_witness_product)) * ff_c_decision_witness) + (ff_p_decision_witness_product))) /\\ ((((exists ff_h_decision_witness_product_partial. ff_h_decision_witness_product_partial + S (ff_r_decision_witness_product) = S ((S (ff_i_decision_witness_product)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_partial. ff_u_decision_witness_product = ff_q_decision_witness_product_partial * S ((S (ff_i_decision_witness_product)) * ff_v_decision_witness_product) + (ff_r_decision_witness_product))) /\\ ((((exists ff_h_decision_witness_product_successor. ff_h_decision_witness_product_successor + S (ff_s_decision_witness_product) = S ((S (S ff_i_decision_witness_product)) * ff_v_decision_witness_product)) /\\ exists ff_q_decision_witness_product_successor. ff_u_decision_witness_product = ff_q_decision_witness_product_successor * S ((S (S ff_i_decision_witness_product)) * ff_v_decision_witness_product) + (ff_s_decision_witness_product))) /\\ ff_s_decision_witness_product = ff_r_decision_witness_product * ff_p_decision_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists e", "exact pow_exists", "cases hpower", "have hdiv : (exists q. a = x * q) \\/ ~(exists q. a = x * q)", "specialize multiple_decidable x", "specialize multiple_decidable a", "exact multiple_decidable", "cases hdiv", "left", "exists x", "split", "exact hpower_witness", "exact hdiv_left", "right", "intro hother", "cases hother", "cases hother_witness", "have heq : x1 = x", "specialize pow_functional p", "specialize pow_functional e", "specialize pow_functional x1", "specialize pow_functional x", "apply pow_functional", "exact hother_witness_left", "exact hpower_witness", "apply hdiv_right", "rewrite heq at hother_witness_right", "exact hother_witness_right" ], "script_sha256": "c95214e16d0f4ed6de95f4d38bbf754f9ba75e031083e4f49b255fd8f7e9de6c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall p e a. (exists bpv_result_decision. ((exists ff_b_decision_power ff_c_decision_power. ((forall ff_i_decision_power_repeat. (exists ff_lt_decision_power_repeat_bound. ff_lt_decision_power_repeat_bound + S ff_i_decision_power_repeat = e) -> (((exists ff_h_decision_power_repeat_decoded. ff_h_decision_power_repeat_decoded + S (p) = S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_repeat_decoded. ff_b_decision_power = ff_q_decision_power_repeat_decoded * S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power) + (p)))) /\\ (exists ff_u_decision_power_product ff_v_decision_power_product. ((((exists ff_h_decision_power_product_start. ff_h_decision_power_product_start + S (1) = S ((S (0)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_start. ff_u_decision_power_product = ff_q_decision_power_product_start * S ((S (0)) * ff_v_decision_power_product) + (1))) /\\ ((((exists ff_h_decision_power_product_terminal. ff_h_decision_power_product_terminal + S (bpv_result_decision) = S ((S (e)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_terminal. ff_u_decision_power_product = ff_q_decision_power_product_terminal * S ((S (e)) * ff_v_decision_power_product) + (bpv_result_decision))) /\\ forall ff_i_decision_power_product. (exists ff_lt_decision_power_product_bound. ff_lt_decision_power_product_bound + S ff_i_decision_power_product = e) -> exists ff_p_decision_power_product ff_r_decision_power_product ff_s_decision_power_product. ((((exists ff_h_decision_power_product_factor. ff_h_decision_power_product_factor + S (ff_p_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_product_factor. ff_b_decision_power = ff_q_decision_power_product_factor * S ((S (ff_i_decision_power_product)) * ff_c_decision_power) + (ff_p_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_partial. ff_h_decision_power_product_partial + S (ff_r_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_partial. ff_u_decision_power_product = ff_q_decision_power_product_partial * S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_r_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_successor. ff_h_decision_power_product_successor + S (ff_s_decision_power_product) = S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_successor. ff_u_decision_power_product = ff_q_decision_power_product_successor * S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_s_decision_power_product))) /\\ ff_s_decision_power_product = ff_r_decision_power_product * ff_p_decision_power_product)))))))) /\\ (exists bpv_factor_decision_divides. a = bpv_result_decision * bpv_factor_decision_divides))) \\/ ~(exists bpv_result_decision. ((exists ff_b_decision_power ff_c_decision_power. ((forall ff_i_decision_power_repeat. (exists ff_lt_decision_power_repeat_bound. ff_lt_decision_power_repeat_bound + S ff_i_decision_power_repeat = e) -> (((exists ff_h_decision_power_repeat_decoded. ff_h_decision_power_repeat_decoded + S (p) = S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_repeat_decoded. ff_b_decision_power = ff_q_decision_power_repeat_decoded * S ((S (ff_i_decision_power_repeat)) * ff_c_decision_power) + (p)))) /\\ (exists ff_u_decision_power_product ff_v_decision_power_product. ((((exists ff_h_decision_power_product_start. ff_h_decision_power_product_start + S (1) = S ((S (0)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_start. ff_u_decision_power_product = ff_q_decision_power_product_start * S ((S (0)) * ff_v_decision_power_product) + (1))) /\\ ((((exists ff_h_decision_power_product_terminal. ff_h_decision_power_product_terminal + S (bpv_result_decision) = S ((S (e)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_terminal. ff_u_decision_power_product = ff_q_decision_power_product_terminal * S ((S (e)) * ff_v_decision_power_product) + (bpv_result_decision))) /\\ forall ff_i_decision_power_product. (exists ff_lt_decision_power_product_bound. ff_lt_decision_power_product_bound + S ff_i_decision_power_product = e) -> exists ff_p_decision_power_product ff_r_decision_power_product ff_s_decision_power_product. ((((exists ff_h_decision_power_product_factor. ff_h_decision_power_product_factor + S (ff_p_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_c_decision_power)) /\\ exists ff_q_decision_power_product_factor. ff_b_decision_power = ff_q_decision_power_product_factor * S ((S (ff_i_decision_power_product)) * ff_c_decision_power) + (ff_p_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_partial. ff_h_decision_power_product_partial + S (ff_r_decision_power_product) = S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_partial. ff_u_decision_power_product = ff_q_decision_power_product_partial * S ((S (ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_r_decision_power_product))) /\\ ((((exists ff_h_decision_power_product_successor. ff_h_decision_power_product_successor + S (ff_s_decision_power_product) = S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product)) /\\ exists ff_q_decision_power_product_successor. ff_u_decision_power_product = ff_q_decision_power_product_successor * S ((S (S ff_i_decision_power_product)) * ff_v_decision_power_product) + (ff_s_decision_power_product))) /\\ ff_s_decision_power_product = ff_r_decision_power_product * ff_p_decision_power_product)))))))) /\\ (exists bpv_factor_decision_divides. a = bpv_result_decision * bpv_factor_decision_divides)))", "statement_sha256": "056bf86316331243e9372e8b59cd02ca2f4a88cee0f43b5900e299ef979e227e" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_divides_zero", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 183, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "af803b22daa2befbbaffe997bc45159dfc6894cca89b8d542a1aa2bd5b068d10" }, "bertrand_evidence_bundle_sha256": "f872b094a598ac14f7a15ce663fe9d8cef65e18eb7e87d5863ec21c5c7e04e37", "body_checked": true, "body_receipt": { "command_count": 22, "dependency_count": 3, "dne_command_count": 0, "name": "power_divides_zero", "proof_depth": 18, "proof_edges": 25, "proof_nodes": 26, "proof_objects": 26, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "pow_exists", "pow_zero", "one_multiple" ], "dependencies_sha256": "2333eadc1eb8b7d367c4eb89a7e122f50d3b3e504045f25999085a942cbccca5", "empty_context_closure": { "body_proof_depth": 18, "body_proof_nodes": 26, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 183, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "c1da6eefcb1b59ca8732c537ab4f565427595dfbd4c38d9b0ff5352bf799b8ef", "status": "checked" }, "enrollment_index": 914, "enrollment_origin": "bertrand_b2_bounded_valuation", "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=183]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_zero]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "7db1cf70425d654aeeba4350f5eaa7ed5648927919e2ad53bf50057352e15995", "membership": "alpha_only", "name": "power_divides_zero", "proof_tag": null, "provenance": [ "bertrand_b2_bounded_valuation" ], "script": [ "intro p", "intro a", "intro z", "intro hz", "have hpower : exists r. (exists ff_b_zero_witness ff_c_zero_witness. ((forall ff_i_zero_witness_repeat. (exists ff_lt_zero_witness_repeat_bound. ff_lt_zero_witness_repeat_bound + S ff_i_zero_witness_repeat = z) -> (((exists ff_h_zero_witness_repeat_decoded. ff_h_zero_witness_repeat_decoded + S (p) = S ((S (ff_i_zero_witness_repeat)) * ff_c_zero_witness)) /\\ exists ff_q_zero_witness_repeat_decoded. ff_b_zero_witness = ff_q_zero_witness_repeat_decoded * S ((S (ff_i_zero_witness_repeat)) * ff_c_zero_witness) + (p)))) /\\ (exists ff_u_zero_witness_product ff_v_zero_witness_product. ((((exists ff_h_zero_witness_product_start. ff_h_zero_witness_product_start + S (1) = S ((S (0)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_start. ff_u_zero_witness_product = ff_q_zero_witness_product_start * S ((S (0)) * ff_v_zero_witness_product) + (1))) /\\ ((((exists ff_h_zero_witness_product_terminal. ff_h_zero_witness_product_terminal + S (r) = S ((S (z)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_terminal. ff_u_zero_witness_product = ff_q_zero_witness_product_terminal * S ((S (z)) * ff_v_zero_witness_product) + (r))) /\\ forall ff_i_zero_witness_product. (exists ff_lt_zero_witness_product_bound. ff_lt_zero_witness_product_bound + S ff_i_zero_witness_product = z) -> exists ff_p_zero_witness_product ff_r_zero_witness_product ff_s_zero_witness_product. ((((exists ff_h_zero_witness_product_factor. ff_h_zero_witness_product_factor + S (ff_p_zero_witness_product) = S ((S (ff_i_zero_witness_product)) * ff_c_zero_witness)) /\\ exists ff_q_zero_witness_product_factor. ff_b_zero_witness = ff_q_zero_witness_product_factor * S ((S (ff_i_zero_witness_product)) * ff_c_zero_witness) + (ff_p_zero_witness_product))) /\\ ((((exists ff_h_zero_witness_product_partial. ff_h_zero_witness_product_partial + S (ff_r_zero_witness_product) = S ((S (ff_i_zero_witness_product)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_partial. ff_u_zero_witness_product = ff_q_zero_witness_product_partial * S ((S (ff_i_zero_witness_product)) * ff_v_zero_witness_product) + (ff_r_zero_witness_product))) /\\ ((((exists ff_h_zero_witness_product_successor. ff_h_zero_witness_product_successor + S (ff_s_zero_witness_product) = S ((S (S ff_i_zero_witness_product)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_successor. ff_u_zero_witness_product = ff_q_zero_witness_product_successor * S ((S (S ff_i_zero_witness_product)) * ff_v_zero_witness_product) + (ff_s_zero_witness_product))) /\\ ff_s_zero_witness_product = ff_r_zero_witness_product * ff_p_zero_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists z", "exact pow_exists", "cases hpower", "have hr : x = 1", "specialize pow_zero p", "specialize pow_zero z", "specialize pow_zero x", "apply pow_zero", "exact hz", "exact hpower_witness", "exists x", "split", "exact hpower_witness", "rewrite hr", "specialize one_multiple a", "exact one_multiple" ], "script_sha256": "386ccf7b1071c44c1b4db75be28a4b78a98a5f9127f75cc3522ddc0c024868b2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "statement": "forall p a z. z = 0 -> (exists bpv_result_zero. ((exists ff_b_zero_power ff_c_zero_power. ((forall ff_i_zero_power_repeat. (exists ff_lt_zero_power_repeat_bound. ff_lt_zero_power_repeat_bound + S ff_i_zero_power_repeat = z) -> (((exists ff_h_zero_power_repeat_decoded. ff_h_zero_power_repeat_decoded + S (p) = S ((S (ff_i_zero_power_repeat)) * ff_c_zero_power)) /\\ exists ff_q_zero_power_repeat_decoded. ff_b_zero_power = ff_q_zero_power_repeat_decoded * S ((S (ff_i_zero_power_repeat)) * ff_c_zero_power) + (p)))) /\\ (exists ff_u_zero_power_product ff_v_zero_power_product. ((((exists ff_h_zero_power_product_start. ff_h_zero_power_product_start + S (1) = S ((S (0)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_start. ff_u_zero_power_product = ff_q_zero_power_product_start * S ((S (0)) * ff_v_zero_power_product) + (1))) /\\ ((((exists ff_h_zero_power_product_terminal. ff_h_zero_power_product_terminal + S (bpv_result_zero) = S ((S (z)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_terminal. ff_u_zero_power_product = ff_q_zero_power_product_terminal * S ((S (z)) * ff_v_zero_power_product) + (bpv_result_zero))) /\\ forall ff_i_zero_power_product. (exists ff_lt_zero_power_product_bound. ff_lt_zero_power_product_bound + S ff_i_zero_power_product = z) -> exists ff_p_zero_power_product ff_r_zero_power_product ff_s_zero_power_product. ((((exists ff_h_zero_power_product_factor. ff_h_zero_power_product_factor + S (ff_p_zero_power_product) = S ((S (ff_i_zero_power_product)) * ff_c_zero_power)) /\\ exists ff_q_zero_power_product_factor. ff_b_zero_power = ff_q_zero_power_product_factor * S ((S (ff_i_zero_power_product)) * ff_c_zero_power) + (ff_p_zero_power_product))) /\\ ((((exists ff_h_zero_power_product_partial. ff_h_zero_power_product_partial + S (ff_r_zero_power_product) = S ((S (ff_i_zero_power_product)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_partial. ff_u_zero_power_product = ff_q_zero_power_product_partial * S ((S (ff_i_zero_power_product)) * ff_v_zero_power_product) + (ff_r_zero_power_product))) /\\ ((((exists ff_h_zero_power_product_successor. ff_h_zero_power_product_successor + S (ff_s_zero_power_product) = S ((S (S ff_i_zero_power_product)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_successor. ff_u_zero_power_product = ff_q_zero_power_product_successor * S ((S (S ff_i_zero_power_product)) * ff_v_zero_power_product) + (ff_s_zero_power_product))) /\\ ff_s_zero_power_product = ff_r_zero_power_product * ff_p_zero_power_product)))))))) /\\ (exists bpv_factor_zero_divides. a = bpv_result_zero * bpv_factor_zero_divides)))", "statement_sha256": "c1da6eefcb1b59ca8732c537ab4f565427595dfbd4c38d9b0ff5352bf799b8ef", "summary": "The zeroth relational power divides every natural.", "summary_sha256": "e4a4699e7e273bc12ec356c97e49d88ac7cccafdd294b908681b0124d9b5bf74" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_exists", "pow_zero", "one_multiple" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=183]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_zero]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_divides_zero", "parent_alpha_version": "v34", "proof_bundle_node_id": 197, "reference_route": "jordan-totient/checkpoint.html#theorem-power_divides_zero", "script": [ "intro p", "intro a", "intro z", "intro hz", "have hpower : exists r. (exists ff_b_zero_witness ff_c_zero_witness. ((forall ff_i_zero_witness_repeat. (exists ff_lt_zero_witness_repeat_bound. ff_lt_zero_witness_repeat_bound + S ff_i_zero_witness_repeat = z) -> (((exists ff_h_zero_witness_repeat_decoded. ff_h_zero_witness_repeat_decoded + S (p) = S ((S (ff_i_zero_witness_repeat)) * ff_c_zero_witness)) /\\ exists ff_q_zero_witness_repeat_decoded. ff_b_zero_witness = ff_q_zero_witness_repeat_decoded * S ((S (ff_i_zero_witness_repeat)) * ff_c_zero_witness) + (p)))) /\\ (exists ff_u_zero_witness_product ff_v_zero_witness_product. ((((exists ff_h_zero_witness_product_start. ff_h_zero_witness_product_start + S (1) = S ((S (0)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_start. ff_u_zero_witness_product = ff_q_zero_witness_product_start * S ((S (0)) * ff_v_zero_witness_product) + (1))) /\\ ((((exists ff_h_zero_witness_product_terminal. ff_h_zero_witness_product_terminal + S (r) = S ((S (z)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_terminal. ff_u_zero_witness_product = ff_q_zero_witness_product_terminal * S ((S (z)) * ff_v_zero_witness_product) + (r))) /\\ forall ff_i_zero_witness_product. (exists ff_lt_zero_witness_product_bound. ff_lt_zero_witness_product_bound + S ff_i_zero_witness_product = z) -> exists ff_p_zero_witness_product ff_r_zero_witness_product ff_s_zero_witness_product. ((((exists ff_h_zero_witness_product_factor. ff_h_zero_witness_product_factor + S (ff_p_zero_witness_product) = S ((S (ff_i_zero_witness_product)) * ff_c_zero_witness)) /\\ exists ff_q_zero_witness_product_factor. ff_b_zero_witness = ff_q_zero_witness_product_factor * S ((S (ff_i_zero_witness_product)) * ff_c_zero_witness) + (ff_p_zero_witness_product))) /\\ ((((exists ff_h_zero_witness_product_partial. ff_h_zero_witness_product_partial + S (ff_r_zero_witness_product) = S ((S (ff_i_zero_witness_product)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_partial. ff_u_zero_witness_product = ff_q_zero_witness_product_partial * S ((S (ff_i_zero_witness_product)) * ff_v_zero_witness_product) + (ff_r_zero_witness_product))) /\\ ((((exists ff_h_zero_witness_product_successor. ff_h_zero_witness_product_successor + S (ff_s_zero_witness_product) = S ((S (S ff_i_zero_witness_product)) * ff_v_zero_witness_product)) /\\ exists ff_q_zero_witness_product_successor. ff_u_zero_witness_product = ff_q_zero_witness_product_successor * S ((S (S ff_i_zero_witness_product)) * ff_v_zero_witness_product) + (ff_s_zero_witness_product))) /\\ ff_s_zero_witness_product = ff_r_zero_witness_product * ff_p_zero_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists z", "exact pow_exists", "cases hpower", "have hr : x = 1", "specialize pow_zero p", "specialize pow_zero z", "specialize pow_zero x", "apply pow_zero", "exact hz", "exact hpower_witness", "exists x", "split", "exact hpower_witness", "rewrite hr", "specialize one_multiple a", "exact one_multiple" ], "script_sha256": "386ccf7b1071c44c1b4db75be28a4b78a98a5f9127f75cc3522ddc0c024868b2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall p a z. z = 0 -> (exists bpv_result_zero. ((exists ff_b_zero_power ff_c_zero_power. ((forall ff_i_zero_power_repeat. (exists ff_lt_zero_power_repeat_bound. ff_lt_zero_power_repeat_bound + S ff_i_zero_power_repeat = z) -> (((exists ff_h_zero_power_repeat_decoded. ff_h_zero_power_repeat_decoded + S (p) = S ((S (ff_i_zero_power_repeat)) * ff_c_zero_power)) /\\ exists ff_q_zero_power_repeat_decoded. ff_b_zero_power = ff_q_zero_power_repeat_decoded * S ((S (ff_i_zero_power_repeat)) * ff_c_zero_power) + (p)))) /\\ (exists ff_u_zero_power_product ff_v_zero_power_product. ((((exists ff_h_zero_power_product_start. ff_h_zero_power_product_start + S (1) = S ((S (0)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_start. ff_u_zero_power_product = ff_q_zero_power_product_start * S ((S (0)) * ff_v_zero_power_product) + (1))) /\\ ((((exists ff_h_zero_power_product_terminal. ff_h_zero_power_product_terminal + S (bpv_result_zero) = S ((S (z)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_terminal. ff_u_zero_power_product = ff_q_zero_power_product_terminal * S ((S (z)) * ff_v_zero_power_product) + (bpv_result_zero))) /\\ forall ff_i_zero_power_product. (exists ff_lt_zero_power_product_bound. ff_lt_zero_power_product_bound + S ff_i_zero_power_product = z) -> exists ff_p_zero_power_product ff_r_zero_power_product ff_s_zero_power_product. ((((exists ff_h_zero_power_product_factor. ff_h_zero_power_product_factor + S (ff_p_zero_power_product) = S ((S (ff_i_zero_power_product)) * ff_c_zero_power)) /\\ exists ff_q_zero_power_product_factor. ff_b_zero_power = ff_q_zero_power_product_factor * S ((S (ff_i_zero_power_product)) * ff_c_zero_power) + (ff_p_zero_power_product))) /\\ ((((exists ff_h_zero_power_product_partial. ff_h_zero_power_product_partial + S (ff_r_zero_power_product) = S ((S (ff_i_zero_power_product)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_partial. ff_u_zero_power_product = ff_q_zero_power_product_partial * S ((S (ff_i_zero_power_product)) * ff_v_zero_power_product) + (ff_r_zero_power_product))) /\\ ((((exists ff_h_zero_power_product_successor. ff_h_zero_power_product_successor + S (ff_s_zero_power_product) = S ((S (S ff_i_zero_power_product)) * ff_v_zero_power_product)) /\\ exists ff_q_zero_power_product_successor. ff_u_zero_power_product = ff_q_zero_power_product_successor * S ((S (S ff_i_zero_power_product)) * ff_v_zero_power_product) + (ff_s_zero_power_product))) /\\ ff_s_zero_power_product = ff_r_zero_power_product * ff_p_zero_power_product)))))))) /\\ (exists bpv_factor_zero_divides. a = bpv_result_zero * bpv_factor_zero_divides)))", "statement_sha256": "c1da6eefcb1b59ca8732c537ab4f565427595dfbd4c38d9b0ff5352bf799b8ef" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "bounded_power_valuation_search", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 184, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "7cd947e0041d7f0f2e25899d357c1d921aedf30bf3bb9280dcea684d93e92b45" }, "bertrand_evidence_bundle_sha256": "def5e5bbee8a4cc618c581517201ed719454c507a074e5272a2acdd0e6d1b7d6", "body_checked": true, "body_receipt": { "command_count": 122, "dependency_count": 6, "dne_command_count": 0, "name": "bounded_power_valuation_search", "proof_depth": 28, "proof_edges": 161, "proof_nodes": 162, "proof_objects": 162, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "power_divides_decidable", "le_zero", "le_refl", "le_eq_or_lt", "le_of_succ_le_succ", "le_succ" ], "dependencies_sha256": "8ebd3d248f3c2dbcecce5d7c60e6146a426e2c4b7c1f70f518ab4f8c6ff2808b", "empty_context_closure": { "body_proof_depth": 28, "body_proof_nodes": 162, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 184, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "1b8ba19947c6b40b4e3d2bbde7f110eda02a7fcc4e23bd247d13fac51e6878f5", "status": "checked" }, "enrollment_index": 915, "enrollment_origin": "bertrand_b2_bounded_valuation", "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=184]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=bounded_power_valuation_search]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "4561d6949900b5ab046b901b26687d116e4bf5d99df2f6d216d3d1bad5357605", "membership": "alpha_only", "name": "bounded_power_valuation_search", "proof_tag": null, "provenance": [ "bertrand_b2_bounded_valuation" ], "script": [ "induction B", "intro p", "intro a", "have hboundary : (exists bpvi_result_search_base_boundary. ((exists bpvi_b_search_base_boundary_power bpvi_c_search_base_boundary_power. ((forall bpvi_i_search_base_boundary_power. (exists bpvi_repeat_gap_search_base_boundary_power. bpvi_repeat_gap_search_base_boundary_power + S bpvi_i_search_base_boundary_power = 0) -> (((exists bpvi_h_search_base_boundary_power_repeat. bpvi_h_search_base_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_repeat. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_repeat * S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (p)))) /\\ (exists bpvi_u_search_base_boundary_power bpvi_v_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_start. bpvi_h_search_base_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_start. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_start * S ((S (0)) * bpvi_v_search_base_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_base_boundary_power_terminal. bpvi_h_search_base_boundary_power_terminal + S (bpvi_result_search_base_boundary) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_terminal. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_terminal * S ((S (0)) * bpvi_v_search_base_boundary_power) + (bpvi_result_search_base_boundary))) /\\ forall bpvi_j_search_base_boundary_power. (exists bpvi_product_gap_search_base_boundary_power. bpvi_product_gap_search_base_boundary_power + S bpvi_j_search_base_boundary_power = 0) -> exists bpvi_factor_search_base_boundary_power bpvi_partial_search_base_boundary_power bpvi_successor_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_factor. bpvi_h_search_base_boundary_power_factor + S (bpvi_factor_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_factor. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_factor * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (bpvi_factor_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_partial. bpvi_h_search_base_boundary_power_partial + S (bpvi_partial_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_partial. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_partial * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_partial_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_successor. bpvi_h_search_base_boundary_power_successor + S (bpvi_successor_search_base_boundary_power) = S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_successor. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_successor * S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_successor_search_base_boundary_power))) /\\ bpvi_successor_search_base_boundary_power = bpvi_partial_search_base_boundary_power * bpvi_factor_search_base_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_base_boundary. a = bpvi_result_search_base_boundary * bpvi_divisor_factor_search_base_boundary)) \\/ ~(exists bpvi_result_search_base_boundary. ((exists bpvi_b_search_base_boundary_power bpvi_c_search_base_boundary_power. ((forall bpvi_i_search_base_boundary_power. (exists bpvi_repeat_gap_search_base_boundary_power. bpvi_repeat_gap_search_base_boundary_power + S bpvi_i_search_base_boundary_power = 0) -> (((exists bpvi_h_search_base_boundary_power_repeat. bpvi_h_search_base_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_repeat. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_repeat * S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (p)))) /\\ (exists bpvi_u_search_base_boundary_power bpvi_v_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_start. bpvi_h_search_base_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_start. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_start * S ((S (0)) * bpvi_v_search_base_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_base_boundary_power_terminal. bpvi_h_search_base_boundary_power_terminal + S (bpvi_result_search_base_boundary) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_terminal. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_terminal * S ((S (0)) * bpvi_v_search_base_boundary_power) + (bpvi_result_search_base_boundary))) /\\ forall bpvi_j_search_base_boundary_power. (exists bpvi_product_gap_search_base_boundary_power. bpvi_product_gap_search_base_boundary_power + S bpvi_j_search_base_boundary_power = 0) -> exists bpvi_factor_search_base_boundary_power bpvi_partial_search_base_boundary_power bpvi_successor_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_factor. bpvi_h_search_base_boundary_power_factor + S (bpvi_factor_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_factor. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_factor * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (bpvi_factor_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_partial. bpvi_h_search_base_boundary_power_partial + S (bpvi_partial_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_partial. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_partial * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_partial_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_successor. bpvi_h_search_base_boundary_power_successor + S (bpvi_successor_search_base_boundary_power) = S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_successor. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_successor * S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_successor_search_base_boundary_power))) /\\ bpvi_successor_search_base_boundary_power = bpvi_partial_search_base_boundary_power * bpvi_factor_search_base_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_base_boundary. a = bpvi_result_search_base_boundary * bpvi_divisor_factor_search_base_boundary))", "specialize power_divides_decidable p", "specialize power_divides_decidable 0", "specialize power_divides_decidable a", "exact power_divides_decidable", "cases hboundary", "right", "exists 0", "split", "split", "specialize le_refl 0", "exact le_refl", "exact hboundary_left", "intro f", "intro hf", "intro hproperty", "have hf0 : f = 0", "specialize le_zero f", "apply le_zero", "exact hf", "rewrite hf0", "specialize le_refl 0", "exact le_refl", "left", "intro f", "intro hf", "intro hproperty", "have hf0 : f = 0", "specialize le_zero f", "apply le_zero", "exact hf", "apply hboundary_right", "rewrite hf0 at hproperty", "rewrite hf0 at hproperty", "rewrite hf0 at hproperty", "rewrite hf0 at hproperty", "exact hproperty", "intro p", "intro a", "have hboundary : (exists bpvi_result_search_succ_boundary. ((exists bpvi_b_search_succ_boundary_power bpvi_c_search_succ_boundary_power. ((forall bpvi_i_search_succ_boundary_power. (exists bpvi_repeat_gap_search_succ_boundary_power. bpvi_repeat_gap_search_succ_boundary_power + S bpvi_i_search_succ_boundary_power = S B) -> (((exists bpvi_h_search_succ_boundary_power_repeat. bpvi_h_search_succ_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_repeat. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_repeat * S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (p)))) /\\ (exists bpvi_u_search_succ_boundary_power bpvi_v_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_start. bpvi_h_search_succ_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_start. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_start * S ((S (0)) * bpvi_v_search_succ_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_succ_boundary_power_terminal. bpvi_h_search_succ_boundary_power_terminal + S (bpvi_result_search_succ_boundary) = S ((S (S B)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_terminal. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_terminal * S ((S (S B)) * bpvi_v_search_succ_boundary_power) + (bpvi_result_search_succ_boundary))) /\\ forall bpvi_j_search_succ_boundary_power. (exists bpvi_product_gap_search_succ_boundary_power. bpvi_product_gap_search_succ_boundary_power + S bpvi_j_search_succ_boundary_power = S B) -> exists bpvi_factor_search_succ_boundary_power bpvi_partial_search_succ_boundary_power bpvi_successor_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_factor. bpvi_h_search_succ_boundary_power_factor + S (bpvi_factor_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_factor. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_factor * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (bpvi_factor_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_partial. bpvi_h_search_succ_boundary_power_partial + S (bpvi_partial_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_partial. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_partial * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_partial_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_successor. bpvi_h_search_succ_boundary_power_successor + S (bpvi_successor_search_succ_boundary_power) = S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_successor. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_successor * S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_successor_search_succ_boundary_power))) /\\ bpvi_successor_search_succ_boundary_power = bpvi_partial_search_succ_boundary_power * bpvi_factor_search_succ_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_succ_boundary. a = bpvi_result_search_succ_boundary * bpvi_divisor_factor_search_succ_boundary)) \\/ ~(exists bpvi_result_search_succ_boundary. ((exists bpvi_b_search_succ_boundary_power bpvi_c_search_succ_boundary_power. ((forall bpvi_i_search_succ_boundary_power. (exists bpvi_repeat_gap_search_succ_boundary_power. bpvi_repeat_gap_search_succ_boundary_power + S bpvi_i_search_succ_boundary_power = S B) -> (((exists bpvi_h_search_succ_boundary_power_repeat. bpvi_h_search_succ_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_repeat. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_repeat * S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (p)))) /\\ (exists bpvi_u_search_succ_boundary_power bpvi_v_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_start. bpvi_h_search_succ_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_start. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_start * S ((S (0)) * bpvi_v_search_succ_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_succ_boundary_power_terminal. bpvi_h_search_succ_boundary_power_terminal + S (bpvi_result_search_succ_boundary) = S ((S (S B)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_terminal. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_terminal * S ((S (S B)) * bpvi_v_search_succ_boundary_power) + (bpvi_result_search_succ_boundary))) /\\ forall bpvi_j_search_succ_boundary_power. (exists bpvi_product_gap_search_succ_boundary_power. bpvi_product_gap_search_succ_boundary_power + S bpvi_j_search_succ_boundary_power = S B) -> exists bpvi_factor_search_succ_boundary_power bpvi_partial_search_succ_boundary_power bpvi_successor_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_factor. bpvi_h_search_succ_boundary_power_factor + S (bpvi_factor_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_factor. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_factor * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (bpvi_factor_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_partial. bpvi_h_search_succ_boundary_power_partial + S (bpvi_partial_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_partial. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_partial * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_partial_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_successor. bpvi_h_search_succ_boundary_power_successor + S (bpvi_successor_search_succ_boundary_power) = S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_successor. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_successor * S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_successor_search_succ_boundary_power))) /\\ bpvi_successor_search_succ_boundary_power = bpvi_partial_search_succ_boundary_power * bpvi_factor_search_succ_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_succ_boundary. a = bpvi_result_search_succ_boundary * bpvi_divisor_factor_search_succ_boundary))", "specialize power_divides_decidable p", "specialize power_divides_decidable (S B)", "specialize power_divides_decidable a", "exact power_divides_decidable", "cases hboundary", "right", "exists S B", "split", "split", "specialize le_refl (S B)", "exact le_refl", "exact hboundary_left", "intro f", "intro hf", "intro hproperty", "exact hf", "have hprevious : (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \\/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))", "specialize IH p", "specialize IH a", "exact IH", "cases hprevious", "left", "intro f", "intro hf", "intro hproperty", "have hsplit : f = S B \\/ exists h. h + S f = S B", "specialize le_eq_or_lt f", "specialize le_eq_or_lt (S B)", "apply le_eq_or_lt", "exact hf", "cases hsplit", "apply hboundary_right", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "exact hproperty", "specialize hprevious_left f", "apply hprevious_left", "specialize le_of_succ_le_succ f", "specialize le_of_succ_le_succ B", "apply le_of_succ_le_succ", "exact hsplit_right", "exact hproperty", "right", "cases hprevious_right", "cases hprevious_right_witness", "cases hprevious_right_witness_left", "exists x", "split", "split", "specialize le_succ x", "specialize le_succ B", "apply le_succ", "exact hprevious_right_witness_left_left", "exact hprevious_right_witness_left_right", "intro f", "intro hf", "intro hproperty", "have hsplit : f = S B \\/ exists h. h + S f = S B", "specialize le_eq_or_lt f", "specialize le_eq_or_lt (S B)", "apply le_eq_or_lt", "exact hf", "cases hsplit", "exfalso", "apply hboundary_right", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "exact hproperty", "specialize hprevious_right_witness_right f", "apply hprevious_right_witness_right", "specialize le_of_succ_le_succ f", "specialize le_of_succ_le_succ B", "apply le_of_succ_le_succ", "exact hsplit_right", "exact hproperty" ], "script_sha256": "b5eb26ae07e9ee65904122d6bad51bb5d65da067a62359f708691eb3c0383246", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "statement": "forall B p a. (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \\/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))", "statement_sha256": "1b8ba19947c6b40b4e3d2bbde7f110eda02a7fcc4e23bd247d13fac51e6878f5", "summary": "Finite search either excludes every power divisor or returns a greatest exponent.", "summary_sha256": "128be1d048a72b36ff38207df1e7011806f75aad8237c053a35e8b020b9be7ad" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_divides_decidable", "le_zero", "le_refl", "le_eq_or_lt", "le_of_succ_le_succ", "le_succ" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=184]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=bounded_power_valuation_search]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "bounded_power_valuation_search", "parent_alpha_version": "v34", "proof_bundle_node_id": 198, "reference_route": "jordan-totient/checkpoint.html#theorem-bounded_power_valuation_search", "script": [ "induction B", "intro p", "intro a", "have hboundary : (exists bpvi_result_search_base_boundary. ((exists bpvi_b_search_base_boundary_power bpvi_c_search_base_boundary_power. ((forall bpvi_i_search_base_boundary_power. (exists bpvi_repeat_gap_search_base_boundary_power. bpvi_repeat_gap_search_base_boundary_power + S bpvi_i_search_base_boundary_power = 0) -> (((exists bpvi_h_search_base_boundary_power_repeat. bpvi_h_search_base_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_repeat. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_repeat * S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (p)))) /\\ (exists bpvi_u_search_base_boundary_power bpvi_v_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_start. bpvi_h_search_base_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_start. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_start * S ((S (0)) * bpvi_v_search_base_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_base_boundary_power_terminal. bpvi_h_search_base_boundary_power_terminal + S (bpvi_result_search_base_boundary) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_terminal. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_terminal * S ((S (0)) * bpvi_v_search_base_boundary_power) + (bpvi_result_search_base_boundary))) /\\ forall bpvi_j_search_base_boundary_power. (exists bpvi_product_gap_search_base_boundary_power. bpvi_product_gap_search_base_boundary_power + S bpvi_j_search_base_boundary_power = 0) -> exists bpvi_factor_search_base_boundary_power bpvi_partial_search_base_boundary_power bpvi_successor_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_factor. bpvi_h_search_base_boundary_power_factor + S (bpvi_factor_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_factor. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_factor * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (bpvi_factor_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_partial. bpvi_h_search_base_boundary_power_partial + S (bpvi_partial_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_partial. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_partial * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_partial_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_successor. bpvi_h_search_base_boundary_power_successor + S (bpvi_successor_search_base_boundary_power) = S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_successor. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_successor * S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_successor_search_base_boundary_power))) /\\ bpvi_successor_search_base_boundary_power = bpvi_partial_search_base_boundary_power * bpvi_factor_search_base_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_base_boundary. a = bpvi_result_search_base_boundary * bpvi_divisor_factor_search_base_boundary)) \\/ ~(exists bpvi_result_search_base_boundary. ((exists bpvi_b_search_base_boundary_power bpvi_c_search_base_boundary_power. ((forall bpvi_i_search_base_boundary_power. (exists bpvi_repeat_gap_search_base_boundary_power. bpvi_repeat_gap_search_base_boundary_power + S bpvi_i_search_base_boundary_power = 0) -> (((exists bpvi_h_search_base_boundary_power_repeat. bpvi_h_search_base_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_repeat. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_repeat * S ((S (bpvi_i_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (p)))) /\\ (exists bpvi_u_search_base_boundary_power bpvi_v_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_start. bpvi_h_search_base_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_start. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_start * S ((S (0)) * bpvi_v_search_base_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_base_boundary_power_terminal. bpvi_h_search_base_boundary_power_terminal + S (bpvi_result_search_base_boundary) = S ((S (0)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_terminal. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_terminal * S ((S (0)) * bpvi_v_search_base_boundary_power) + (bpvi_result_search_base_boundary))) /\\ forall bpvi_j_search_base_boundary_power. (exists bpvi_product_gap_search_base_boundary_power. bpvi_product_gap_search_base_boundary_power + S bpvi_j_search_base_boundary_power = 0) -> exists bpvi_factor_search_base_boundary_power bpvi_partial_search_base_boundary_power bpvi_successor_search_base_boundary_power. ((((exists bpvi_h_search_base_boundary_power_factor. bpvi_h_search_base_boundary_power_factor + S (bpvi_factor_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_factor. bpvi_b_search_base_boundary_power = bpvi_q_search_base_boundary_power_factor * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_c_search_base_boundary_power) + (bpvi_factor_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_partial. bpvi_h_search_base_boundary_power_partial + S (bpvi_partial_search_base_boundary_power) = S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_partial. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_partial * S ((S (bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_partial_search_base_boundary_power))) /\\ ((((exists bpvi_h_search_base_boundary_power_successor. bpvi_h_search_base_boundary_power_successor + S (bpvi_successor_search_base_boundary_power) = S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power)) /\\ exists bpvi_q_search_base_boundary_power_successor. bpvi_u_search_base_boundary_power = bpvi_q_search_base_boundary_power_successor * S ((S (S bpvi_j_search_base_boundary_power)) * bpvi_v_search_base_boundary_power) + (bpvi_successor_search_base_boundary_power))) /\\ bpvi_successor_search_base_boundary_power = bpvi_partial_search_base_boundary_power * bpvi_factor_search_base_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_base_boundary. a = bpvi_result_search_base_boundary * bpvi_divisor_factor_search_base_boundary))", "specialize power_divides_decidable p", "specialize power_divides_decidable 0", "specialize power_divides_decidable a", "exact power_divides_decidable", "cases hboundary", "right", "exists 0", "split", "split", "specialize le_refl 0", "exact le_refl", "exact hboundary_left", "intro f", "intro hf", "intro hproperty", "have hf0 : f = 0", "specialize le_zero f", "apply le_zero", "exact hf", "rewrite hf0", "specialize le_refl 0", "exact le_refl", "left", "intro f", "intro hf", "intro hproperty", "have hf0 : f = 0", "specialize le_zero f", "apply le_zero", "exact hf", "apply hboundary_right", "rewrite hf0 at hproperty", "rewrite hf0 at hproperty", "rewrite hf0 at hproperty", "rewrite hf0 at hproperty", "exact hproperty", "intro p", "intro a", "have hboundary : (exists bpvi_result_search_succ_boundary. ((exists bpvi_b_search_succ_boundary_power bpvi_c_search_succ_boundary_power. ((forall bpvi_i_search_succ_boundary_power. (exists bpvi_repeat_gap_search_succ_boundary_power. bpvi_repeat_gap_search_succ_boundary_power + S bpvi_i_search_succ_boundary_power = S B) -> (((exists bpvi_h_search_succ_boundary_power_repeat. bpvi_h_search_succ_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_repeat. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_repeat * S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (p)))) /\\ (exists bpvi_u_search_succ_boundary_power bpvi_v_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_start. bpvi_h_search_succ_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_start. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_start * S ((S (0)) * bpvi_v_search_succ_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_succ_boundary_power_terminal. bpvi_h_search_succ_boundary_power_terminal + S (bpvi_result_search_succ_boundary) = S ((S (S B)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_terminal. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_terminal * S ((S (S B)) * bpvi_v_search_succ_boundary_power) + (bpvi_result_search_succ_boundary))) /\\ forall bpvi_j_search_succ_boundary_power. (exists bpvi_product_gap_search_succ_boundary_power. bpvi_product_gap_search_succ_boundary_power + S bpvi_j_search_succ_boundary_power = S B) -> exists bpvi_factor_search_succ_boundary_power bpvi_partial_search_succ_boundary_power bpvi_successor_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_factor. bpvi_h_search_succ_boundary_power_factor + S (bpvi_factor_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_factor. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_factor * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (bpvi_factor_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_partial. bpvi_h_search_succ_boundary_power_partial + S (bpvi_partial_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_partial. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_partial * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_partial_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_successor. bpvi_h_search_succ_boundary_power_successor + S (bpvi_successor_search_succ_boundary_power) = S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_successor. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_successor * S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_successor_search_succ_boundary_power))) /\\ bpvi_successor_search_succ_boundary_power = bpvi_partial_search_succ_boundary_power * bpvi_factor_search_succ_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_succ_boundary. a = bpvi_result_search_succ_boundary * bpvi_divisor_factor_search_succ_boundary)) \\/ ~(exists bpvi_result_search_succ_boundary. ((exists bpvi_b_search_succ_boundary_power bpvi_c_search_succ_boundary_power. ((forall bpvi_i_search_succ_boundary_power. (exists bpvi_repeat_gap_search_succ_boundary_power. bpvi_repeat_gap_search_succ_boundary_power + S bpvi_i_search_succ_boundary_power = S B) -> (((exists bpvi_h_search_succ_boundary_power_repeat. bpvi_h_search_succ_boundary_power_repeat + S (p) = S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_repeat. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_repeat * S ((S (bpvi_i_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (p)))) /\\ (exists bpvi_u_search_succ_boundary_power bpvi_v_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_start. bpvi_h_search_succ_boundary_power_start + S (1) = S ((S (0)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_start. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_start * S ((S (0)) * bpvi_v_search_succ_boundary_power) + (1))) /\\ ((((exists bpvi_h_search_succ_boundary_power_terminal. bpvi_h_search_succ_boundary_power_terminal + S (bpvi_result_search_succ_boundary) = S ((S (S B)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_terminal. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_terminal * S ((S (S B)) * bpvi_v_search_succ_boundary_power) + (bpvi_result_search_succ_boundary))) /\\ forall bpvi_j_search_succ_boundary_power. (exists bpvi_product_gap_search_succ_boundary_power. bpvi_product_gap_search_succ_boundary_power + S bpvi_j_search_succ_boundary_power = S B) -> exists bpvi_factor_search_succ_boundary_power bpvi_partial_search_succ_boundary_power bpvi_successor_search_succ_boundary_power. ((((exists bpvi_h_search_succ_boundary_power_factor. bpvi_h_search_succ_boundary_power_factor + S (bpvi_factor_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_factor. bpvi_b_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_factor * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_c_search_succ_boundary_power) + (bpvi_factor_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_partial. bpvi_h_search_succ_boundary_power_partial + S (bpvi_partial_search_succ_boundary_power) = S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_partial. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_partial * S ((S (bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_partial_search_succ_boundary_power))) /\\ ((((exists bpvi_h_search_succ_boundary_power_successor. bpvi_h_search_succ_boundary_power_successor + S (bpvi_successor_search_succ_boundary_power) = S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power)) /\\ exists bpvi_q_search_succ_boundary_power_successor. bpvi_u_search_succ_boundary_power = bpvi_q_search_succ_boundary_power_successor * S ((S (S bpvi_j_search_succ_boundary_power)) * bpvi_v_search_succ_boundary_power) + (bpvi_successor_search_succ_boundary_power))) /\\ bpvi_successor_search_succ_boundary_power = bpvi_partial_search_succ_boundary_power * bpvi_factor_search_succ_boundary_power)))))))) /\\ exists bpvi_divisor_factor_search_succ_boundary. a = bpvi_result_search_succ_boundary * bpvi_divisor_factor_search_succ_boundary))", "specialize power_divides_decidable p", "specialize power_divides_decidable (S B)", "specialize power_divides_decidable a", "exact power_divides_decidable", "cases hboundary", "right", "exists S B", "split", "split", "specialize le_refl (S B)", "exact le_refl", "exact hboundary_left", "intro f", "intro hf", "intro hproperty", "exact hf", "have hprevious : (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \\/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))", "specialize IH p", "specialize IH a", "exact IH", "cases hprevious", "left", "intro f", "intro hf", "intro hproperty", "have hsplit : f = S B \\/ exists h. h + S f = S B", "specialize le_eq_or_lt f", "specialize le_eq_or_lt (S B)", "apply le_eq_or_lt", "exact hf", "cases hsplit", "apply hboundary_right", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "exact hproperty", "specialize hprevious_left f", "apply hprevious_left", "specialize le_of_succ_le_succ f", "specialize le_of_succ_le_succ B", "apply le_of_succ_le_succ", "exact hsplit_right", "exact hproperty", "right", "cases hprevious_right", "cases hprevious_right_witness", "cases hprevious_right_witness_left", "exists x", "split", "split", "specialize le_succ x", "specialize le_succ B", "apply le_succ", "exact hprevious_right_witness_left_left", "exact hprevious_right_witness_left_right", "intro f", "intro hf", "intro hproperty", "have hsplit : f = S B \\/ exists h. h + S f = S B", "specialize le_eq_or_lt f", "specialize le_eq_or_lt (S B)", "apply le_eq_or_lt", "exact hf", "cases hsplit", "exfalso", "apply hboundary_right", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "rewrite hsplit_left at hproperty", "exact hproperty", "specialize hprevious_right_witness_right f", "apply hprevious_right_witness_right", "specialize le_of_succ_le_succ f", "specialize le_of_succ_le_succ B", "apply le_of_succ_le_succ", "exact hsplit_right", "exact hproperty" ], "script_sha256": "b5eb26ae07e9ee65904122d6bad51bb5d65da067a62359f708691eb3c0383246", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall B p a. (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \\/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))", "statement_sha256": "1b8ba19947c6b40b4e3d2bbde7f110eda02a7fcc4e23bd247d13fac51e6878f5" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "bounded_power_valuation_exists", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 185, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "e1045e88198500378d4a66fd0cdb1ee82c02bfe76fc708fa6fa0a5175b3a40c9" }, "bertrand_evidence_bundle_sha256": "bc452b045afd6edecb951c87bb1576b1283154a6539c38671d26ebccb592a7c9", "body_checked": true, "body_receipt": { "command_count": 24, "dependency_count": 3, "dne_command_count": 0, "name": "bounded_power_valuation_exists", "proof_depth": 14, "proof_edges": 27, "proof_nodes": 28, "proof_objects": 28, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "bounded_power_valuation_search", "power_divides_zero", "zero_le" ], "dependencies_sha256": "1fd7735ad32aaad416a0a7f62ae116a7c46b06d2ecb453f1631369c983e6c708", "empty_context_closure": { "body_proof_depth": 14, "body_proof_nodes": 28, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 185, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "e3360e9cfb622363275168e8f789dc68fdbe6996c5e646ea21a1afff5e993829", "status": "checked" }, "enrollment_index": 916, "enrollment_origin": "bertrand_b2_bounded_valuation", "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=185]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=bounded_power_valuation_exists]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "507afa0a0e72d44edc7fb8458c75cd0e70f866e271187d8b6c88ee7daa12c4d9", "membership": "alpha_only", "name": "bounded_power_valuation_exists", "proof_tag": null, "provenance": [ "bertrand_b2_bounded_valuation" ], "script": [ "intro p", "intro a", "intro B", "have hsearch : (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \\/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))", "specialize bounded_power_valuation_search B", "specialize bounded_power_valuation_search p", "specialize bounded_power_valuation_search a", "exact bounded_power_valuation_search", "cases hsearch", "have hzero : (exists bpvi_result_exists_zero. ((exists bpvi_b_exists_zero_power bpvi_c_exists_zero_power. ((forall bpvi_i_exists_zero_power. (exists bpvi_repeat_gap_exists_zero_power. bpvi_repeat_gap_exists_zero_power + S bpvi_i_exists_zero_power = 0) -> (((exists bpvi_h_exists_zero_power_repeat. bpvi_h_exists_zero_power_repeat + S (p) = S ((S (bpvi_i_exists_zero_power)) * bpvi_c_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_repeat. bpvi_b_exists_zero_power = bpvi_q_exists_zero_power_repeat * S ((S (bpvi_i_exists_zero_power)) * bpvi_c_exists_zero_power) + (p)))) /\\ (exists bpvi_u_exists_zero_power bpvi_v_exists_zero_power. ((((exists bpvi_h_exists_zero_power_start. bpvi_h_exists_zero_power_start + S (1) = S ((S (0)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_start. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_start * S ((S (0)) * bpvi_v_exists_zero_power) + (1))) /\\ ((((exists bpvi_h_exists_zero_power_terminal. bpvi_h_exists_zero_power_terminal + S (bpvi_result_exists_zero) = S ((S (0)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_terminal. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_terminal * S ((S (0)) * bpvi_v_exists_zero_power) + (bpvi_result_exists_zero))) /\\ forall bpvi_j_exists_zero_power. (exists bpvi_product_gap_exists_zero_power. bpvi_product_gap_exists_zero_power + S bpvi_j_exists_zero_power = 0) -> exists bpvi_factor_exists_zero_power bpvi_partial_exists_zero_power bpvi_successor_exists_zero_power. ((((exists bpvi_h_exists_zero_power_factor. bpvi_h_exists_zero_power_factor + S (bpvi_factor_exists_zero_power) = S ((S (bpvi_j_exists_zero_power)) * bpvi_c_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_factor. bpvi_b_exists_zero_power = bpvi_q_exists_zero_power_factor * S ((S (bpvi_j_exists_zero_power)) * bpvi_c_exists_zero_power) + (bpvi_factor_exists_zero_power))) /\\ ((((exists bpvi_h_exists_zero_power_partial. bpvi_h_exists_zero_power_partial + S (bpvi_partial_exists_zero_power) = S ((S (bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_partial. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_partial * S ((S (bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power) + (bpvi_partial_exists_zero_power))) /\\ ((((exists bpvi_h_exists_zero_power_successor. bpvi_h_exists_zero_power_successor + S (bpvi_successor_exists_zero_power) = S ((S (S bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_successor. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_successor * S ((S (S bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power) + (bpvi_successor_exists_zero_power))) /\\ bpvi_successor_exists_zero_power = bpvi_partial_exists_zero_power * bpvi_factor_exists_zero_power)))))))) /\\ exists bpvi_divisor_factor_exists_zero. a = bpvi_result_exists_zero * bpvi_divisor_factor_exists_zero))", "specialize power_divides_zero p", "specialize power_divides_zero a", "specialize power_divides_zero 0", "apply power_divides_zero", "refl", "specialize hsearch_left 0", "exfalso", "apply hsearch_left", "specialize zero_le B", "exact zero_le", "exact hzero", "cases hsearch_right", "exists x", "exact hsearch_right_witness" ], "script_sha256": "0cf79cce89992fe5d0aaffa214d15af9d0c57c51e69677d57a9f5b18d3077600", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "statement": "forall p a B. exists e. (((exists bpv_gap_bounded_exponent_bound. bpv_gap_bounded_exponent_bound + e = B) /\\ (exists bpv_result_bounded_selected. ((exists ff_b_bounded_selected_power ff_c_bounded_selected_power. ((forall ff_i_bounded_selected_power_repeat. (exists ff_lt_bounded_selected_power_repeat_bound. ff_lt_bounded_selected_power_repeat_bound + S ff_i_bounded_selected_power_repeat = e) -> (((exists ff_h_bounded_selected_power_repeat_decoded. ff_h_bounded_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bounded_selected_power_repeat)) * ff_c_bounded_selected_power)) /\\ exists ff_q_bounded_selected_power_repeat_decoded. ff_b_bounded_selected_power = ff_q_bounded_selected_power_repeat_decoded * S ((S (ff_i_bounded_selected_power_repeat)) * ff_c_bounded_selected_power) + (p)))) /\\ (exists ff_u_bounded_selected_power_product ff_v_bounded_selected_power_product. ((((exists ff_h_bounded_selected_power_product_start. ff_h_bounded_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_start. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_start * S ((S (0)) * ff_v_bounded_selected_power_product) + (1))) /\\ ((((exists ff_h_bounded_selected_power_product_terminal. ff_h_bounded_selected_power_product_terminal + S (bpv_result_bounded_selected) = S ((S (e)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_terminal. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_terminal * S ((S (e)) * ff_v_bounded_selected_power_product) + (bpv_result_bounded_selected))) /\\ forall ff_i_bounded_selected_power_product. (exists ff_lt_bounded_selected_power_product_bound. ff_lt_bounded_selected_power_product_bound + S ff_i_bounded_selected_power_product = e) -> exists ff_p_bounded_selected_power_product ff_r_bounded_selected_power_product ff_s_bounded_selected_power_product. ((((exists ff_h_bounded_selected_power_product_factor. ff_h_bounded_selected_power_product_factor + S (ff_p_bounded_selected_power_product) = S ((S (ff_i_bounded_selected_power_product)) * ff_c_bounded_selected_power)) /\\ exists ff_q_bounded_selected_power_product_factor. ff_b_bounded_selected_power = ff_q_bounded_selected_power_product_factor * S ((S (ff_i_bounded_selected_power_product)) * ff_c_bounded_selected_power) + (ff_p_bounded_selected_power_product))) /\\ ((((exists ff_h_bounded_selected_power_product_partial. ff_h_bounded_selected_power_product_partial + S (ff_r_bounded_selected_power_product) = S ((S (ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_partial. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_partial * S ((S (ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product) + (ff_r_bounded_selected_power_product))) /\\ ((((exists ff_h_bounded_selected_power_product_successor. ff_h_bounded_selected_power_product_successor + S (ff_s_bounded_selected_power_product) = S ((S (S ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_successor. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_successor * S ((S (S ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product) + (ff_s_bounded_selected_power_product))) /\\ ff_s_bounded_selected_power_product = ff_r_bounded_selected_power_product * ff_p_bounded_selected_power_product)))))))) /\\ (exists bpv_factor_bounded_selected_divides. a = bpv_result_bounded_selected * bpv_factor_bounded_selected_divides)))) /\\ forall bpv_candidate_bounded. (exists bpv_gap_bounded_candidate_bound. bpv_gap_bounded_candidate_bound + bpv_candidate_bounded = B) -> (exists bpv_result_bounded_candidate. ((exists ff_b_bounded_candidate_power ff_c_bounded_candidate_power. ((forall ff_i_bounded_candidate_power_repeat. (exists ff_lt_bounded_candidate_power_repeat_bound. ff_lt_bounded_candidate_power_repeat_bound + S ff_i_bounded_candidate_power_repeat = bpv_candidate_bounded) -> (((exists ff_h_bounded_candidate_power_repeat_decoded. ff_h_bounded_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bounded_candidate_power_repeat)) * ff_c_bounded_candidate_power)) /\\ exists ff_q_bounded_candidate_power_repeat_decoded. ff_b_bounded_candidate_power = ff_q_bounded_candidate_power_repeat_decoded * S ((S (ff_i_bounded_candidate_power_repeat)) * ff_c_bounded_candidate_power) + (p)))) /\\ (exists ff_u_bounded_candidate_power_product ff_v_bounded_candidate_power_product. ((((exists ff_h_bounded_candidate_power_product_start. ff_h_bounded_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_start. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_start * S ((S (0)) * ff_v_bounded_candidate_power_product) + (1))) /\\ ((((exists ff_h_bounded_candidate_power_product_terminal. ff_h_bounded_candidate_power_product_terminal + S (bpv_result_bounded_candidate) = S ((S (bpv_candidate_bounded)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_terminal. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_terminal * S ((S (bpv_candidate_bounded)) * ff_v_bounded_candidate_power_product) + (bpv_result_bounded_candidate))) /\\ forall ff_i_bounded_candidate_power_product. (exists ff_lt_bounded_candidate_power_product_bound. ff_lt_bounded_candidate_power_product_bound + S ff_i_bounded_candidate_power_product = bpv_candidate_bounded) -> exists ff_p_bounded_candidate_power_product ff_r_bounded_candidate_power_product ff_s_bounded_candidate_power_product. ((((exists ff_h_bounded_candidate_power_product_factor. ff_h_bounded_candidate_power_product_factor + S (ff_p_bounded_candidate_power_product) = S ((S (ff_i_bounded_candidate_power_product)) * ff_c_bounded_candidate_power)) /\\ exists ff_q_bounded_candidate_power_product_factor. ff_b_bounded_candidate_power = ff_q_bounded_candidate_power_product_factor * S ((S (ff_i_bounded_candidate_power_product)) * ff_c_bounded_candidate_power) + (ff_p_bounded_candidate_power_product))) /\\ ((((exists ff_h_bounded_candidate_power_product_partial. ff_h_bounded_candidate_power_product_partial + S (ff_r_bounded_candidate_power_product) = S ((S (ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_partial. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_partial * S ((S (ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product) + (ff_r_bounded_candidate_power_product))) /\\ ((((exists ff_h_bounded_candidate_power_product_successor. ff_h_bounded_candidate_power_product_successor + S (ff_s_bounded_candidate_power_product) = S ((S (S ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_successor. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_successor * S ((S (S ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product) + (ff_s_bounded_candidate_power_product))) /\\ ff_s_bounded_candidate_power_product = ff_r_bounded_candidate_power_product * ff_p_bounded_candidate_power_product)))))))) /\\ (exists bpv_factor_bounded_candidate_divides. a = bpv_result_bounded_candidate * bpv_factor_bounded_candidate_divides))) -> (exists bpv_gap_bounded_maximal. bpv_gap_bounded_maximal + bpv_candidate_bounded = e))", "statement_sha256": "e3360e9cfb622363275168e8f789dc68fdbe6996c5e646ea21a1afff5e993829", "summary": "Every explicit exponent bound has a greatest power-divisor exponent.", "summary_sha256": "01932dfd7fab3150d34c6c478e65eacf812c58ba9a142867b7829a37765a72f8" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "bounded_power_valuation_search", "power_divides_zero", "zero_le" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=185]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=bounded_power_valuation_exists]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "bounded_power_valuation_exists", "parent_alpha_version": "v34", "proof_bundle_node_id": 199, "reference_route": "jordan-totient/checkpoint.html#theorem-bounded_power_valuation_exists", "script": [ "intro p", "intro a", "intro B", "have hsearch : (forall f. (exists bpv_gap_search_none_bound. bpv_gap_search_none_bound + f = B) -> ~(exists bpv_result_search_property. ((exists ff_b_search_property_power ff_c_search_property_power. ((forall ff_i_search_property_power_repeat. (exists ff_lt_search_property_power_repeat_bound. ff_lt_search_property_power_repeat_bound + S ff_i_search_property_power_repeat = f) -> (((exists ff_h_search_property_power_repeat_decoded. ff_h_search_property_power_repeat_decoded + S (p) = S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_repeat_decoded. ff_b_search_property_power = ff_q_search_property_power_repeat_decoded * S ((S (ff_i_search_property_power_repeat)) * ff_c_search_property_power) + (p)))) /\\ (exists ff_u_search_property_power_product ff_v_search_property_power_product. ((((exists ff_h_search_property_power_product_start. ff_h_search_property_power_product_start + S (1) = S ((S (0)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_start. ff_u_search_property_power_product = ff_q_search_property_power_product_start * S ((S (0)) * ff_v_search_property_power_product) + (1))) /\\ ((((exists ff_h_search_property_power_product_terminal. ff_h_search_property_power_product_terminal + S (bpv_result_search_property) = S ((S (f)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_terminal. ff_u_search_property_power_product = ff_q_search_property_power_product_terminal * S ((S (f)) * ff_v_search_property_power_product) + (bpv_result_search_property))) /\\ forall ff_i_search_property_power_product. (exists ff_lt_search_property_power_product_bound. ff_lt_search_property_power_product_bound + S ff_i_search_property_power_product = f) -> exists ff_p_search_property_power_product ff_r_search_property_power_product ff_s_search_property_power_product. ((((exists ff_h_search_property_power_product_factor. ff_h_search_property_power_product_factor + S (ff_p_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power)) /\\ exists ff_q_search_property_power_product_factor. ff_b_search_property_power = ff_q_search_property_power_product_factor * S ((S (ff_i_search_property_power_product)) * ff_c_search_property_power) + (ff_p_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_partial. ff_h_search_property_power_product_partial + S (ff_r_search_property_power_product) = S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_partial. ff_u_search_property_power_product = ff_q_search_property_power_product_partial * S ((S (ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_r_search_property_power_product))) /\\ ((((exists ff_h_search_property_power_product_successor. ff_h_search_property_power_product_successor + S (ff_s_search_property_power_product) = S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product)) /\\ exists ff_q_search_property_power_product_successor. ff_u_search_property_power_product = ff_q_search_property_power_product_successor * S ((S (S ff_i_search_property_power_product)) * ff_v_search_property_power_product) + (ff_s_search_property_power_product))) /\\ ff_s_search_property_power_product = ff_r_search_property_power_product * ff_p_search_property_power_product)))))))) /\\ (exists bpv_factor_search_property_divides. a = bpv_result_search_property * bpv_factor_search_property_divides)))) \\/ (exists e. ((exists bpv_gap_search_selected_bound. bpv_gap_search_selected_bound + e = B) /\\ (exists bpv_result_search_selected. ((exists ff_b_search_selected_power ff_c_search_selected_power. ((forall ff_i_search_selected_power_repeat. (exists ff_lt_search_selected_power_repeat_bound. ff_lt_search_selected_power_repeat_bound + S ff_i_search_selected_power_repeat = e) -> (((exists ff_h_search_selected_power_repeat_decoded. ff_h_search_selected_power_repeat_decoded + S (p) = S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_repeat_decoded. ff_b_search_selected_power = ff_q_search_selected_power_repeat_decoded * S ((S (ff_i_search_selected_power_repeat)) * ff_c_search_selected_power) + (p)))) /\\ (exists ff_u_search_selected_power_product ff_v_search_selected_power_product. ((((exists ff_h_search_selected_power_product_start. ff_h_search_selected_power_product_start + S (1) = S ((S (0)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_start. ff_u_search_selected_power_product = ff_q_search_selected_power_product_start * S ((S (0)) * ff_v_search_selected_power_product) + (1))) /\\ ((((exists ff_h_search_selected_power_product_terminal. ff_h_search_selected_power_product_terminal + S (bpv_result_search_selected) = S ((S (e)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_terminal. ff_u_search_selected_power_product = ff_q_search_selected_power_product_terminal * S ((S (e)) * ff_v_search_selected_power_product) + (bpv_result_search_selected))) /\\ forall ff_i_search_selected_power_product. (exists ff_lt_search_selected_power_product_bound. ff_lt_search_selected_power_product_bound + S ff_i_search_selected_power_product = e) -> exists ff_p_search_selected_power_product ff_r_search_selected_power_product ff_s_search_selected_power_product. ((((exists ff_h_search_selected_power_product_factor. ff_h_search_selected_power_product_factor + S (ff_p_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power)) /\\ exists ff_q_search_selected_power_product_factor. ff_b_search_selected_power = ff_q_search_selected_power_product_factor * S ((S (ff_i_search_selected_power_product)) * ff_c_search_selected_power) + (ff_p_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_partial. ff_h_search_selected_power_product_partial + S (ff_r_search_selected_power_product) = S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_partial. ff_u_search_selected_power_product = ff_q_search_selected_power_product_partial * S ((S (ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_r_search_selected_power_product))) /\\ ((((exists ff_h_search_selected_power_product_successor. ff_h_search_selected_power_product_successor + S (ff_s_search_selected_power_product) = S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product)) /\\ exists ff_q_search_selected_power_product_successor. ff_u_search_selected_power_product = ff_q_search_selected_power_product_successor * S ((S (S ff_i_search_selected_power_product)) * ff_v_search_selected_power_product) + (ff_s_search_selected_power_product))) /\\ ff_s_search_selected_power_product = ff_r_search_selected_power_product * ff_p_search_selected_power_product)))))))) /\\ (exists bpv_factor_search_selected_divides. a = bpv_result_search_selected * bpv_factor_search_selected_divides)))) /\\ forall f. (exists bpv_gap_search_candidate_bound. bpv_gap_search_candidate_bound + f = B) -> (exists bpv_result_search_candidate. ((exists ff_b_search_candidate_power ff_c_search_candidate_power. ((forall ff_i_search_candidate_power_repeat. (exists ff_lt_search_candidate_power_repeat_bound. ff_lt_search_candidate_power_repeat_bound + S ff_i_search_candidate_power_repeat = f) -> (((exists ff_h_search_candidate_power_repeat_decoded. ff_h_search_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_repeat_decoded. ff_b_search_candidate_power = ff_q_search_candidate_power_repeat_decoded * S ((S (ff_i_search_candidate_power_repeat)) * ff_c_search_candidate_power) + (p)))) /\\ (exists ff_u_search_candidate_power_product ff_v_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_start. ff_h_search_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_start. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_start * S ((S (0)) * ff_v_search_candidate_power_product) + (1))) /\\ ((((exists ff_h_search_candidate_power_product_terminal. ff_h_search_candidate_power_product_terminal + S (bpv_result_search_candidate) = S ((S (f)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_terminal. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_terminal * S ((S (f)) * ff_v_search_candidate_power_product) + (bpv_result_search_candidate))) /\\ forall ff_i_search_candidate_power_product. (exists ff_lt_search_candidate_power_product_bound. ff_lt_search_candidate_power_product_bound + S ff_i_search_candidate_power_product = f) -> exists ff_p_search_candidate_power_product ff_r_search_candidate_power_product ff_s_search_candidate_power_product. ((((exists ff_h_search_candidate_power_product_factor. ff_h_search_candidate_power_product_factor + S (ff_p_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power)) /\\ exists ff_q_search_candidate_power_product_factor. ff_b_search_candidate_power = ff_q_search_candidate_power_product_factor * S ((S (ff_i_search_candidate_power_product)) * ff_c_search_candidate_power) + (ff_p_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_partial. ff_h_search_candidate_power_product_partial + S (ff_r_search_candidate_power_product) = S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_partial. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_partial * S ((S (ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_r_search_candidate_power_product))) /\\ ((((exists ff_h_search_candidate_power_product_successor. ff_h_search_candidate_power_product_successor + S (ff_s_search_candidate_power_product) = S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product)) /\\ exists ff_q_search_candidate_power_product_successor. ff_u_search_candidate_power_product = ff_q_search_candidate_power_product_successor * S ((S (S ff_i_search_candidate_power_product)) * ff_v_search_candidate_power_product) + (ff_s_search_candidate_power_product))) /\\ ff_s_search_candidate_power_product = ff_r_search_candidate_power_product * ff_p_search_candidate_power_product)))))))) /\\ (exists bpv_factor_search_candidate_divides. a = bpv_result_search_candidate * bpv_factor_search_candidate_divides))) -> (exists bpv_gap_search_maximal. bpv_gap_search_maximal + f = e))", "specialize bounded_power_valuation_search B", "specialize bounded_power_valuation_search p", "specialize bounded_power_valuation_search a", "exact bounded_power_valuation_search", "cases hsearch", "have hzero : (exists bpvi_result_exists_zero. ((exists bpvi_b_exists_zero_power bpvi_c_exists_zero_power. ((forall bpvi_i_exists_zero_power. (exists bpvi_repeat_gap_exists_zero_power. bpvi_repeat_gap_exists_zero_power + S bpvi_i_exists_zero_power = 0) -> (((exists bpvi_h_exists_zero_power_repeat. bpvi_h_exists_zero_power_repeat + S (p) = S ((S (bpvi_i_exists_zero_power)) * bpvi_c_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_repeat. bpvi_b_exists_zero_power = bpvi_q_exists_zero_power_repeat * S ((S (bpvi_i_exists_zero_power)) * bpvi_c_exists_zero_power) + (p)))) /\\ (exists bpvi_u_exists_zero_power bpvi_v_exists_zero_power. ((((exists bpvi_h_exists_zero_power_start. bpvi_h_exists_zero_power_start + S (1) = S ((S (0)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_start. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_start * S ((S (0)) * bpvi_v_exists_zero_power) + (1))) /\\ ((((exists bpvi_h_exists_zero_power_terminal. bpvi_h_exists_zero_power_terminal + S (bpvi_result_exists_zero) = S ((S (0)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_terminal. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_terminal * S ((S (0)) * bpvi_v_exists_zero_power) + (bpvi_result_exists_zero))) /\\ forall bpvi_j_exists_zero_power. (exists bpvi_product_gap_exists_zero_power. bpvi_product_gap_exists_zero_power + S bpvi_j_exists_zero_power = 0) -> exists bpvi_factor_exists_zero_power bpvi_partial_exists_zero_power bpvi_successor_exists_zero_power. ((((exists bpvi_h_exists_zero_power_factor. bpvi_h_exists_zero_power_factor + S (bpvi_factor_exists_zero_power) = S ((S (bpvi_j_exists_zero_power)) * bpvi_c_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_factor. bpvi_b_exists_zero_power = bpvi_q_exists_zero_power_factor * S ((S (bpvi_j_exists_zero_power)) * bpvi_c_exists_zero_power) + (bpvi_factor_exists_zero_power))) /\\ ((((exists bpvi_h_exists_zero_power_partial. bpvi_h_exists_zero_power_partial + S (bpvi_partial_exists_zero_power) = S ((S (bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_partial. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_partial * S ((S (bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power) + (bpvi_partial_exists_zero_power))) /\\ ((((exists bpvi_h_exists_zero_power_successor. bpvi_h_exists_zero_power_successor + S (bpvi_successor_exists_zero_power) = S ((S (S bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power)) /\\ exists bpvi_q_exists_zero_power_successor. bpvi_u_exists_zero_power = bpvi_q_exists_zero_power_successor * S ((S (S bpvi_j_exists_zero_power)) * bpvi_v_exists_zero_power) + (bpvi_successor_exists_zero_power))) /\\ bpvi_successor_exists_zero_power = bpvi_partial_exists_zero_power * bpvi_factor_exists_zero_power)))))))) /\\ exists bpvi_divisor_factor_exists_zero. a = bpvi_result_exists_zero * bpvi_divisor_factor_exists_zero))", "specialize power_divides_zero p", "specialize power_divides_zero a", "specialize power_divides_zero 0", "apply power_divides_zero", "refl", "specialize hsearch_left 0", "exfalso", "apply hsearch_left", "specialize zero_le B", "exact zero_le", "exact hzero", "cases hsearch_right", "exists x", "exact hsearch_right_witness" ], "script_sha256": "0cf79cce89992fe5d0aaffa214d15af9d0c57c51e69677d57a9f5b18d3077600", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall p a B. exists e. (((exists bpv_gap_bounded_exponent_bound. bpv_gap_bounded_exponent_bound + e = B) /\\ (exists bpv_result_bounded_selected. ((exists ff_b_bounded_selected_power ff_c_bounded_selected_power. ((forall ff_i_bounded_selected_power_repeat. (exists ff_lt_bounded_selected_power_repeat_bound. ff_lt_bounded_selected_power_repeat_bound + S ff_i_bounded_selected_power_repeat = e) -> (((exists ff_h_bounded_selected_power_repeat_decoded. ff_h_bounded_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bounded_selected_power_repeat)) * ff_c_bounded_selected_power)) /\\ exists ff_q_bounded_selected_power_repeat_decoded. ff_b_bounded_selected_power = ff_q_bounded_selected_power_repeat_decoded * S ((S (ff_i_bounded_selected_power_repeat)) * ff_c_bounded_selected_power) + (p)))) /\\ (exists ff_u_bounded_selected_power_product ff_v_bounded_selected_power_product. ((((exists ff_h_bounded_selected_power_product_start. ff_h_bounded_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_start. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_start * S ((S (0)) * ff_v_bounded_selected_power_product) + (1))) /\\ ((((exists ff_h_bounded_selected_power_product_terminal. ff_h_bounded_selected_power_product_terminal + S (bpv_result_bounded_selected) = S ((S (e)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_terminal. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_terminal * S ((S (e)) * ff_v_bounded_selected_power_product) + (bpv_result_bounded_selected))) /\\ forall ff_i_bounded_selected_power_product. (exists ff_lt_bounded_selected_power_product_bound. ff_lt_bounded_selected_power_product_bound + S ff_i_bounded_selected_power_product = e) -> exists ff_p_bounded_selected_power_product ff_r_bounded_selected_power_product ff_s_bounded_selected_power_product. ((((exists ff_h_bounded_selected_power_product_factor. ff_h_bounded_selected_power_product_factor + S (ff_p_bounded_selected_power_product) = S ((S (ff_i_bounded_selected_power_product)) * ff_c_bounded_selected_power)) /\\ exists ff_q_bounded_selected_power_product_factor. ff_b_bounded_selected_power = ff_q_bounded_selected_power_product_factor * S ((S (ff_i_bounded_selected_power_product)) * ff_c_bounded_selected_power) + (ff_p_bounded_selected_power_product))) /\\ ((((exists ff_h_bounded_selected_power_product_partial. ff_h_bounded_selected_power_product_partial + S (ff_r_bounded_selected_power_product) = S ((S (ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_partial. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_partial * S ((S (ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product) + (ff_r_bounded_selected_power_product))) /\\ ((((exists ff_h_bounded_selected_power_product_successor. ff_h_bounded_selected_power_product_successor + S (ff_s_bounded_selected_power_product) = S ((S (S ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product)) /\\ exists ff_q_bounded_selected_power_product_successor. ff_u_bounded_selected_power_product = ff_q_bounded_selected_power_product_successor * S ((S (S ff_i_bounded_selected_power_product)) * ff_v_bounded_selected_power_product) + (ff_s_bounded_selected_power_product))) /\\ ff_s_bounded_selected_power_product = ff_r_bounded_selected_power_product * ff_p_bounded_selected_power_product)))))))) /\\ (exists bpv_factor_bounded_selected_divides. a = bpv_result_bounded_selected * bpv_factor_bounded_selected_divides)))) /\\ forall bpv_candidate_bounded. (exists bpv_gap_bounded_candidate_bound. bpv_gap_bounded_candidate_bound + bpv_candidate_bounded = B) -> (exists bpv_result_bounded_candidate. ((exists ff_b_bounded_candidate_power ff_c_bounded_candidate_power. ((forall ff_i_bounded_candidate_power_repeat. (exists ff_lt_bounded_candidate_power_repeat_bound. ff_lt_bounded_candidate_power_repeat_bound + S ff_i_bounded_candidate_power_repeat = bpv_candidate_bounded) -> (((exists ff_h_bounded_candidate_power_repeat_decoded. ff_h_bounded_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bounded_candidate_power_repeat)) * ff_c_bounded_candidate_power)) /\\ exists ff_q_bounded_candidate_power_repeat_decoded. ff_b_bounded_candidate_power = ff_q_bounded_candidate_power_repeat_decoded * S ((S (ff_i_bounded_candidate_power_repeat)) * ff_c_bounded_candidate_power) + (p)))) /\\ (exists ff_u_bounded_candidate_power_product ff_v_bounded_candidate_power_product. ((((exists ff_h_bounded_candidate_power_product_start. ff_h_bounded_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_start. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_start * S ((S (0)) * ff_v_bounded_candidate_power_product) + (1))) /\\ ((((exists ff_h_bounded_candidate_power_product_terminal. ff_h_bounded_candidate_power_product_terminal + S (bpv_result_bounded_candidate) = S ((S (bpv_candidate_bounded)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_terminal. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_terminal * S ((S (bpv_candidate_bounded)) * ff_v_bounded_candidate_power_product) + (bpv_result_bounded_candidate))) /\\ forall ff_i_bounded_candidate_power_product. (exists ff_lt_bounded_candidate_power_product_bound. ff_lt_bounded_candidate_power_product_bound + S ff_i_bounded_candidate_power_product = bpv_candidate_bounded) -> exists ff_p_bounded_candidate_power_product ff_r_bounded_candidate_power_product ff_s_bounded_candidate_power_product. ((((exists ff_h_bounded_candidate_power_product_factor. ff_h_bounded_candidate_power_product_factor + S (ff_p_bounded_candidate_power_product) = S ((S (ff_i_bounded_candidate_power_product)) * ff_c_bounded_candidate_power)) /\\ exists ff_q_bounded_candidate_power_product_factor. ff_b_bounded_candidate_power = ff_q_bounded_candidate_power_product_factor * S ((S (ff_i_bounded_candidate_power_product)) * ff_c_bounded_candidate_power) + (ff_p_bounded_candidate_power_product))) /\\ ((((exists ff_h_bounded_candidate_power_product_partial. ff_h_bounded_candidate_power_product_partial + S (ff_r_bounded_candidate_power_product) = S ((S (ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_partial. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_partial * S ((S (ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product) + (ff_r_bounded_candidate_power_product))) /\\ ((((exists ff_h_bounded_candidate_power_product_successor. ff_h_bounded_candidate_power_product_successor + S (ff_s_bounded_candidate_power_product) = S ((S (S ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product)) /\\ exists ff_q_bounded_candidate_power_product_successor. ff_u_bounded_candidate_power_product = ff_q_bounded_candidate_power_product_successor * S ((S (S ff_i_bounded_candidate_power_product)) * ff_v_bounded_candidate_power_product) + (ff_s_bounded_candidate_power_product))) /\\ ff_s_bounded_candidate_power_product = ff_r_bounded_candidate_power_product * ff_p_bounded_candidate_power_product)))))))) /\\ (exists bpv_factor_bounded_candidate_divides. a = bpv_result_bounded_candidate * bpv_factor_bounded_candidate_divides))) -> (exists bpv_gap_bounded_maximal. bpv_gap_bounded_maximal + bpv_candidate_bounded = e))", "statement_sha256": "e3360e9cfb622363275168e8f789dc68fdbe6996c5e646ea21a1afff5e993829" }, { "admitted_to_alpha": true, 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(((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e))", "statement_sha256": 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"e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall p a. exists e. (((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e))", "statement_sha256": "e1b175e27a5a13c926aa2215122a3a89d7a9d85fca190895f73279a3b993c306" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_valuation_power_divides", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 187, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "4f233833a03fcf117d2749751216e6327ae1a06d04f34bef65d1a593bdd16bf7" }, "bertrand_evidence_bundle_sha256": "b760593fa3d69f6421922d7930869ae2418cbfaa600ca858c536a43312e3eecc", "body_checked": true, "body_receipt": { "command_count": 7, "dependency_count": 0, "dne_command_count": 0, "name": "power_valuation_power_divides", "proof_depth": 13, "proof_edges": 20, "proof_nodes": 21, "proof_objects": 21, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [], "dependencies_sha256": "01ba4719c80b6fe911b091a7c05124b64eeece964e09c058ef8f9805daca546b", "empty_context_closure": { "body_proof_depth": 13, "body_proof_nodes": 21, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 187, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "e6209bee9e63324cdf55b24e1fe19e6e140c7f658ede37c16ed8aa46e2de12d5", "status": "checked" }, "enrollment_index": 919, "enrollment_origin": "bertrand_b2_bounded_valuation", "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=187]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_power_divides]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "691b6daa3b5429cd9f3979e27c15fac3718db7901bf148d53465b0344ebebc32", "membership": "alpha_only", "name": "power_valuation_power_divides", "proof_tag": null, "provenance": [ "bertrand_b2_bounded_valuation" ], "script": [ "intro p", "intro a", "intro e", "intro hvaluation", "cases hvaluation", "cases hvaluation_left", "exact hvaluation_left_right" ], "script_sha256": "7dec934c55b786f6a1bbae7a25710590a18cd1f68e3c7318e01c657d52c83abc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "statement": "forall p a e. (((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e)) -> (exists bpv_result_projection. ((exists ff_b_projection_power ff_c_projection_power. ((forall ff_i_projection_power_repeat. (exists ff_lt_projection_power_repeat_bound. ff_lt_projection_power_repeat_bound + S ff_i_projection_power_repeat = e) -> (((exists ff_h_projection_power_repeat_decoded. ff_h_projection_power_repeat_decoded + S (p) = S ((S (ff_i_projection_power_repeat)) * ff_c_projection_power)) /\\ exists ff_q_projection_power_repeat_decoded. ff_b_projection_power = ff_q_projection_power_repeat_decoded * S ((S (ff_i_projection_power_repeat)) * ff_c_projection_power) + (p)))) /\\ (exists ff_u_projection_power_product ff_v_projection_power_product. ((((exists ff_h_projection_power_product_start. ff_h_projection_power_product_start + S (1) = S ((S (0)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_start. ff_u_projection_power_product = ff_q_projection_power_product_start * S ((S (0)) * ff_v_projection_power_product) + (1))) /\\ ((((exists ff_h_projection_power_product_terminal. ff_h_projection_power_product_terminal + S (bpv_result_projection) = S ((S (e)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_terminal. ff_u_projection_power_product = ff_q_projection_power_product_terminal * S ((S (e)) * ff_v_projection_power_product) + (bpv_result_projection))) /\\ forall ff_i_projection_power_product. (exists ff_lt_projection_power_product_bound. ff_lt_projection_power_product_bound + S ff_i_projection_power_product = e) -> exists ff_p_projection_power_product ff_r_projection_power_product ff_s_projection_power_product. ((((exists ff_h_projection_power_product_factor. ff_h_projection_power_product_factor + S (ff_p_projection_power_product) = S ((S (ff_i_projection_power_product)) * ff_c_projection_power)) /\\ exists ff_q_projection_power_product_factor. ff_b_projection_power = ff_q_projection_power_product_factor * S ((S (ff_i_projection_power_product)) * ff_c_projection_power) + (ff_p_projection_power_product))) /\\ ((((exists ff_h_projection_power_product_partial. ff_h_projection_power_product_partial + S (ff_r_projection_power_product) = S ((S (ff_i_projection_power_product)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_partial. ff_u_projection_power_product = ff_q_projection_power_product_partial * S ((S (ff_i_projection_power_product)) * ff_v_projection_power_product) + (ff_r_projection_power_product))) /\\ ((((exists ff_h_projection_power_product_successor. ff_h_projection_power_product_successor + S (ff_s_projection_power_product) = S ((S (S ff_i_projection_power_product)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_successor. ff_u_projection_power_product = ff_q_projection_power_product_successor * S ((S (S ff_i_projection_power_product)) * ff_v_projection_power_product) + (ff_s_projection_power_product))) /\\ ff_s_projection_power_product = ff_r_projection_power_product * ff_p_projection_power_product)))))))) /\\ (exists bpv_factor_projection_divides. a = bpv_result_projection * bpv_factor_projection_divides)))", "statement_sha256": "e6209bee9e63324cdf55b24e1fe19e6e140c7f658ede37c16ed8aa46e2de12d5", "summary": "A valuation exponent has a relational power dividing the value.", "summary_sha256": "cd8cd4682b10a789e78d5f49b2bc1051d9ef0a117f8fbc64a8fb2c3dca94a2f3" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=187]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_power_divides]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_power_divides", "parent_alpha_version": "v34", "proof_bundle_node_id": 201, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_power_divides", "script": [ "intro p", "intro a", "intro e", "intro hvaluation", "cases hvaluation", "cases hvaluation_left", "exact hvaluation_left_right" ], "script_sha256": "7dec934c55b786f6a1bbae7a25710590a18cd1f68e3c7318e01c657d52c83abc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall p a e. (((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e)) -> (exists bpv_result_projection. ((exists ff_b_projection_power ff_c_projection_power. ((forall ff_i_projection_power_repeat. (exists ff_lt_projection_power_repeat_bound. ff_lt_projection_power_repeat_bound + S ff_i_projection_power_repeat = e) -> (((exists ff_h_projection_power_repeat_decoded. ff_h_projection_power_repeat_decoded + S (p) = S ((S (ff_i_projection_power_repeat)) * ff_c_projection_power)) /\\ exists ff_q_projection_power_repeat_decoded. ff_b_projection_power = ff_q_projection_power_repeat_decoded * S ((S (ff_i_projection_power_repeat)) * ff_c_projection_power) + (p)))) /\\ (exists ff_u_projection_power_product ff_v_projection_power_product. ((((exists ff_h_projection_power_product_start. ff_h_projection_power_product_start + S (1) = S ((S (0)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_start. ff_u_projection_power_product = ff_q_projection_power_product_start * S ((S (0)) * ff_v_projection_power_product) + (1))) /\\ ((((exists ff_h_projection_power_product_terminal. ff_h_projection_power_product_terminal + S (bpv_result_projection) = S ((S (e)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_terminal. ff_u_projection_power_product = ff_q_projection_power_product_terminal * S ((S (e)) * ff_v_projection_power_product) + (bpv_result_projection))) /\\ forall ff_i_projection_power_product. (exists ff_lt_projection_power_product_bound. ff_lt_projection_power_product_bound + S ff_i_projection_power_product = e) -> exists ff_p_projection_power_product ff_r_projection_power_product ff_s_projection_power_product. ((((exists ff_h_projection_power_product_factor. ff_h_projection_power_product_factor + S (ff_p_projection_power_product) = S ((S (ff_i_projection_power_product)) * ff_c_projection_power)) /\\ exists ff_q_projection_power_product_factor. ff_b_projection_power = ff_q_projection_power_product_factor * S ((S (ff_i_projection_power_product)) * ff_c_projection_power) + (ff_p_projection_power_product))) /\\ ((((exists ff_h_projection_power_product_partial. ff_h_projection_power_product_partial + S (ff_r_projection_power_product) = S ((S (ff_i_projection_power_product)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_partial. ff_u_projection_power_product = ff_q_projection_power_product_partial * S ((S (ff_i_projection_power_product)) * ff_v_projection_power_product) + (ff_r_projection_power_product))) /\\ ((((exists ff_h_projection_power_product_successor. ff_h_projection_power_product_successor + S (ff_s_projection_power_product) = S ((S (S ff_i_projection_power_product)) * ff_v_projection_power_product)) /\\ exists ff_q_projection_power_product_successor. ff_u_projection_power_product = ff_q_projection_power_product_successor * S ((S (S ff_i_projection_power_product)) * ff_v_projection_power_product) + (ff_s_projection_power_product))) /\\ ff_s_projection_power_product = ff_r_projection_power_product * ff_p_projection_power_product)))))))) /\\ (exists bpv_factor_projection_divides. a = bpv_result_projection * bpv_factor_projection_divides)))", "statement_sha256": "e6209bee9e63324cdf55b24e1fe19e6e140c7f658ede37c16ed8aa46e2de12d5" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_valuation_dominates", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 188, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "0b3e2461ace629274ddcdc96fca55f1fc7e02fe0107fb555ccf7144a3dd93590" }, "bertrand_evidence_bundle_sha256": "29bd870577a8752994c98b531bee07956e53b8bfc53231bedd82c6fd94904d88", "body_checked": true, "body_receipt": { "command_count": 12, "dependency_count": 0, "dne_command_count": 0, "name": "power_valuation_dominates", "proof_depth": 16, "proof_edges": 23, "proof_nodes": 24, "proof_objects": 24, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [], "dependencies_sha256": "01ba4719c80b6fe911b091a7c05124b64eeece964e09c058ef8f9805daca546b", "empty_context_closure": { "body_proof_depth": 16, "body_proof_nodes": 24, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 188, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "11940140cf2fb3b274b4bc444e9c547d88c09813bdbae6245dda5ef1e4f77487", "status": "checked" }, "enrollment_index": 920, "enrollment_origin": "bertrand_b2_bounded_valuation", "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=188]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_dominates]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "d9f03dbc9bb62de00de5516c28ffa233bf80f8cd571ab868169aa12c4e1d2a1e", "membership": "alpha_only", "name": "power_valuation_dominates", "proof_tag": null, "provenance": [ "bertrand_b2_bounded_valuation" ], "script": [ "intro p", "intro a", "intro e", "intro f", "intro hvaluation", "intro hbound", "intro hdivides", "cases hvaluation", "specialize hvaluation_right f", "apply hvaluation_right", "exact hbound", "exact hdivides" ], "script_sha256": "b3896861f6be282952c88d6d7dd463d5f1bc64a91c882d080abefb29e69aaabc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "statement": "forall p a e f. (((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e)) -> (exists bpv_gap_dominates_bound. bpv_gap_dominates_bound + f = a) -> (exists bpv_result_dominates_candidate. ((exists ff_b_dominates_candidate_power ff_c_dominates_candidate_power. ((forall ff_i_dominates_candidate_power_repeat. (exists ff_lt_dominates_candidate_power_repeat_bound. ff_lt_dominates_candidate_power_repeat_bound + S ff_i_dominates_candidate_power_repeat = f) -> (((exists ff_h_dominates_candidate_power_repeat_decoded. ff_h_dominates_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_dominates_candidate_power_repeat)) * ff_c_dominates_candidate_power)) /\\ exists ff_q_dominates_candidate_power_repeat_decoded. ff_b_dominates_candidate_power = ff_q_dominates_candidate_power_repeat_decoded * S ((S (ff_i_dominates_candidate_power_repeat)) * ff_c_dominates_candidate_power) + (p)))) /\\ (exists ff_u_dominates_candidate_power_product ff_v_dominates_candidate_power_product. ((((exists ff_h_dominates_candidate_power_product_start. ff_h_dominates_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_start. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_start * S ((S (0)) * ff_v_dominates_candidate_power_product) + (1))) /\\ ((((exists ff_h_dominates_candidate_power_product_terminal. ff_h_dominates_candidate_power_product_terminal + S (bpv_result_dominates_candidate) = S ((S (f)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_terminal. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_terminal * S ((S (f)) * ff_v_dominates_candidate_power_product) + (bpv_result_dominates_candidate))) /\\ forall ff_i_dominates_candidate_power_product. (exists ff_lt_dominates_candidate_power_product_bound. ff_lt_dominates_candidate_power_product_bound + S ff_i_dominates_candidate_power_product = f) -> exists ff_p_dominates_candidate_power_product ff_r_dominates_candidate_power_product ff_s_dominates_candidate_power_product. ((((exists ff_h_dominates_candidate_power_product_factor. ff_h_dominates_candidate_power_product_factor + S (ff_p_dominates_candidate_power_product) = S ((S (ff_i_dominates_candidate_power_product)) * ff_c_dominates_candidate_power)) /\\ exists ff_q_dominates_candidate_power_product_factor. ff_b_dominates_candidate_power = ff_q_dominates_candidate_power_product_factor * S ((S (ff_i_dominates_candidate_power_product)) * ff_c_dominates_candidate_power) + (ff_p_dominates_candidate_power_product))) /\\ ((((exists ff_h_dominates_candidate_power_product_partial. ff_h_dominates_candidate_power_product_partial + S (ff_r_dominates_candidate_power_product) = S ((S (ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_partial. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_partial * S ((S (ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product) + (ff_r_dominates_candidate_power_product))) /\\ ((((exists ff_h_dominates_candidate_power_product_successor. ff_h_dominates_candidate_power_product_successor + S (ff_s_dominates_candidate_power_product) = S ((S (S ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_successor. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_successor * S ((S (S ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product) + (ff_s_dominates_candidate_power_product))) /\\ ff_s_dominates_candidate_power_product = ff_r_dominates_candidate_power_product * ff_p_dominates_candidate_power_product)))))))) /\\ (exists bpv_factor_dominates_candidate_divides. a = bpv_result_dominates_candidate * bpv_factor_dominates_candidate_divides))) -> (exists bpv_gap_dominates_result. bpv_gap_dominates_result + f = e)", "statement_sha256": "11940140cf2fb3b274b4bc444e9c547d88c09813bdbae6245dda5ef1e4f77487", "summary": "Every bounded power-divisor exponent lies below the valuation exponent.", "summary_sha256": "9b96431d78eca874c3a2eac72380c0b629979b801997a04fe1b2d2b2f50d0209" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d857c1cd33f38b277f1e4c593b0024e1a65b76a435d397dabf43a6f4ad416679", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=188]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_dominates]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_dominates", "parent_alpha_version": "v34", "proof_bundle_node_id": 202, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_dominates", "script": [ "intro p", "intro a", "intro e", "intro f", "intro hvaluation", "intro hbound", "intro hdivides", "cases hvaluation", "specialize hvaluation_right f", "apply hvaluation_right", "exact hbound", "exact hdivides" ], "script_sha256": "b3896861f6be282952c88d6d7dd463d5f1bc64a91c882d080abefb29e69aaabc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_candidate.py", "sha256": "e1d7177ba713425dd3545fa7de2d78dae73ce155e09fabcfe6cd46fcf562fd57" }, "stable_member": false, "statement": "forall p a e f. (((exists bpv_gap_canonical_exponent_bound. bpv_gap_canonical_exponent_bound + e = a) /\\ (exists bpv_result_canonical_selected. ((exists ff_b_canonical_selected_power ff_c_canonical_selected_power. ((forall ff_i_canonical_selected_power_repeat. (exists ff_lt_canonical_selected_power_repeat_bound. ff_lt_canonical_selected_power_repeat_bound + S ff_i_canonical_selected_power_repeat = e) -> (((exists ff_h_canonical_selected_power_repeat_decoded. ff_h_canonical_selected_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_repeat_decoded. ff_b_canonical_selected_power = ff_q_canonical_selected_power_repeat_decoded * S ((S (ff_i_canonical_selected_power_repeat)) * ff_c_canonical_selected_power) + (p)))) /\\ (exists ff_u_canonical_selected_power_product ff_v_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_start. ff_h_canonical_selected_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_start. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_start * S ((S (0)) * ff_v_canonical_selected_power_product) + (1))) /\\ ((((exists ff_h_canonical_selected_power_product_terminal. ff_h_canonical_selected_power_product_terminal + S (bpv_result_canonical_selected) = S ((S (e)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_terminal. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_terminal * S ((S (e)) * ff_v_canonical_selected_power_product) + (bpv_result_canonical_selected))) /\\ forall ff_i_canonical_selected_power_product. (exists ff_lt_canonical_selected_power_product_bound. ff_lt_canonical_selected_power_product_bound + S ff_i_canonical_selected_power_product = e) -> exists ff_p_canonical_selected_power_product ff_r_canonical_selected_power_product ff_s_canonical_selected_power_product. ((((exists ff_h_canonical_selected_power_product_factor. ff_h_canonical_selected_power_product_factor + S (ff_p_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power)) /\\ exists ff_q_canonical_selected_power_product_factor. ff_b_canonical_selected_power = ff_q_canonical_selected_power_product_factor * S ((S (ff_i_canonical_selected_power_product)) * ff_c_canonical_selected_power) + (ff_p_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_partial. ff_h_canonical_selected_power_product_partial + S (ff_r_canonical_selected_power_product) = S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_partial. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_partial * S ((S (ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_r_canonical_selected_power_product))) /\\ ((((exists ff_h_canonical_selected_power_product_successor. ff_h_canonical_selected_power_product_successor + S (ff_s_canonical_selected_power_product) = S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product)) /\\ exists ff_q_canonical_selected_power_product_successor. ff_u_canonical_selected_power_product = ff_q_canonical_selected_power_product_successor * S ((S (S ff_i_canonical_selected_power_product)) * ff_v_canonical_selected_power_product) + (ff_s_canonical_selected_power_product))) /\\ ff_s_canonical_selected_power_product = ff_r_canonical_selected_power_product * ff_p_canonical_selected_power_product)))))))) /\\ (exists bpv_factor_canonical_selected_divides. a = bpv_result_canonical_selected * bpv_factor_canonical_selected_divides)))) /\\ forall bpv_candidate_canonical. (exists bpv_gap_canonical_candidate_bound. bpv_gap_canonical_candidate_bound + bpv_candidate_canonical = a) -> (exists bpv_result_canonical_candidate. ((exists ff_b_canonical_candidate_power ff_c_canonical_candidate_power. ((forall ff_i_canonical_candidate_power_repeat. (exists ff_lt_canonical_candidate_power_repeat_bound. ff_lt_canonical_candidate_power_repeat_bound + S ff_i_canonical_candidate_power_repeat = bpv_candidate_canonical) -> (((exists ff_h_canonical_candidate_power_repeat_decoded. ff_h_canonical_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_repeat_decoded. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_repeat_decoded * S ((S (ff_i_canonical_candidate_power_repeat)) * ff_c_canonical_candidate_power) + (p)))) /\\ (exists ff_u_canonical_candidate_power_product ff_v_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_start. ff_h_canonical_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_start. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_start * S ((S (0)) * ff_v_canonical_candidate_power_product) + (1))) /\\ ((((exists ff_h_canonical_candidate_power_product_terminal. ff_h_canonical_candidate_power_product_terminal + S (bpv_result_canonical_candidate) = S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_terminal. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_terminal * S ((S (bpv_candidate_canonical)) * ff_v_canonical_candidate_power_product) + (bpv_result_canonical_candidate))) /\\ forall ff_i_canonical_candidate_power_product. (exists ff_lt_canonical_candidate_power_product_bound. ff_lt_canonical_candidate_power_product_bound + S ff_i_canonical_candidate_power_product = bpv_candidate_canonical) -> exists ff_p_canonical_candidate_power_product ff_r_canonical_candidate_power_product ff_s_canonical_candidate_power_product. ((((exists ff_h_canonical_candidate_power_product_factor. ff_h_canonical_candidate_power_product_factor + S (ff_p_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power)) /\\ exists ff_q_canonical_candidate_power_product_factor. ff_b_canonical_candidate_power = ff_q_canonical_candidate_power_product_factor * S ((S (ff_i_canonical_candidate_power_product)) * ff_c_canonical_candidate_power) + (ff_p_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_partial. ff_h_canonical_candidate_power_product_partial + S (ff_r_canonical_candidate_power_product) = S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_partial. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_partial * S ((S (ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_r_canonical_candidate_power_product))) /\\ ((((exists ff_h_canonical_candidate_power_product_successor. ff_h_canonical_candidate_power_product_successor + S (ff_s_canonical_candidate_power_product) = S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product)) /\\ exists ff_q_canonical_candidate_power_product_successor. ff_u_canonical_candidate_power_product = ff_q_canonical_candidate_power_product_successor * S ((S (S ff_i_canonical_candidate_power_product)) * ff_v_canonical_candidate_power_product) + (ff_s_canonical_candidate_power_product))) /\\ ff_s_canonical_candidate_power_product = ff_r_canonical_candidate_power_product * ff_p_canonical_candidate_power_product)))))))) /\\ (exists bpv_factor_canonical_candidate_divides. a = bpv_result_canonical_candidate * bpv_factor_canonical_candidate_divides))) -> (exists bpv_gap_canonical_maximal. bpv_gap_canonical_maximal + bpv_candidate_canonical = e)) -> (exists bpv_gap_dominates_bound. bpv_gap_dominates_bound + f = a) -> (exists bpv_result_dominates_candidate. ((exists ff_b_dominates_candidate_power ff_c_dominates_candidate_power. ((forall ff_i_dominates_candidate_power_repeat. (exists ff_lt_dominates_candidate_power_repeat_bound. ff_lt_dominates_candidate_power_repeat_bound + S ff_i_dominates_candidate_power_repeat = f) -> (((exists ff_h_dominates_candidate_power_repeat_decoded. ff_h_dominates_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_dominates_candidate_power_repeat)) * ff_c_dominates_candidate_power)) /\\ exists ff_q_dominates_candidate_power_repeat_decoded. ff_b_dominates_candidate_power = ff_q_dominates_candidate_power_repeat_decoded * S ((S (ff_i_dominates_candidate_power_repeat)) * ff_c_dominates_candidate_power) + (p)))) /\\ (exists ff_u_dominates_candidate_power_product ff_v_dominates_candidate_power_product. ((((exists ff_h_dominates_candidate_power_product_start. ff_h_dominates_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_start. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_start * S ((S (0)) * ff_v_dominates_candidate_power_product) + (1))) /\\ ((((exists ff_h_dominates_candidate_power_product_terminal. ff_h_dominates_candidate_power_product_terminal + S (bpv_result_dominates_candidate) = S ((S (f)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_terminal. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_terminal * S ((S (f)) * ff_v_dominates_candidate_power_product) + (bpv_result_dominates_candidate))) /\\ forall ff_i_dominates_candidate_power_product. (exists ff_lt_dominates_candidate_power_product_bound. ff_lt_dominates_candidate_power_product_bound + S ff_i_dominates_candidate_power_product = f) -> exists ff_p_dominates_candidate_power_product ff_r_dominates_candidate_power_product ff_s_dominates_candidate_power_product. ((((exists ff_h_dominates_candidate_power_product_factor. ff_h_dominates_candidate_power_product_factor + S (ff_p_dominates_candidate_power_product) = S ((S (ff_i_dominates_candidate_power_product)) * ff_c_dominates_candidate_power)) /\\ exists ff_q_dominates_candidate_power_product_factor. ff_b_dominates_candidate_power = ff_q_dominates_candidate_power_product_factor * S ((S (ff_i_dominates_candidate_power_product)) * ff_c_dominates_candidate_power) + (ff_p_dominates_candidate_power_product))) /\\ ((((exists ff_h_dominates_candidate_power_product_partial. ff_h_dominates_candidate_power_product_partial + S (ff_r_dominates_candidate_power_product) = S ((S (ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_partial. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_partial * S ((S (ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product) + (ff_r_dominates_candidate_power_product))) /\\ ((((exists ff_h_dominates_candidate_power_product_successor. ff_h_dominates_candidate_power_product_successor + S (ff_s_dominates_candidate_power_product) = S ((S (S ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product)) /\\ exists ff_q_dominates_candidate_power_product_successor. ff_u_dominates_candidate_power_product = ff_q_dominates_candidate_power_product_successor * S ((S (S ff_i_dominates_candidate_power_product)) * ff_v_dominates_candidate_power_product) + (ff_s_dominates_candidate_power_product))) /\\ ff_s_dominates_candidate_power_product = ff_r_dominates_candidate_power_product * ff_p_dominates_candidate_power_product)))))))) /\\ (exists bpv_factor_dominates_candidate_divides. a = bpv_result_dominates_candidate * bpv_factor_dominates_candidate_divides))) -> (exists bpv_gap_dominates_result. bpv_gap_dominates_result + f = e)", "statement_sha256": 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"theorems[name=prime_two_le]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "416d1f2b7897799d6491aea5221e0f8eed6c0206fc58024b970225a7d6076ba7", "membership": "alpha_only", "name": "prime_two_le", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_laws" ], "script": [ "intro p", "intro hp", "have hshape : exists k. p = S (S k)", "specialize prime_is_succ_succ p", "apply prime_is_succ_succ", "exact hp", "cases hshape", "exists x", "trans S (S x)", "rewrite PA4", "rewrite PA4", "rewrite PA3", "refl", "symm", "exact hshape_witness" ], "script_sha256": "3c9e57f0bcfa156e36681907a8991edbf10eaecab4bf4242ad171182d591b1a8", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "statement": "forall p. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> (exists bpvl_gap_prime_two. bpvl_gap_prime_two + (2) = (p))", "statement_sha256": "1b3c7c140e6ce0e53d771c6580a3b8cf081835013d994878a22e5de1d1a04e7c", "summary": "Every prime is at least two in witness-defined order.", "summary_sha256": "333134960edece0f95da9531799ebc1b407b2e103da528ee65a1f5df9bf65326" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "prime_is_succ_succ" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "3686a414ab5aeb8dcb4c064e030dcab57129744845737b708354814823898ef1", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_laws_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "02b1eef360dce55f0156bda2029e64567b8b83b5d58833d6c4f8695ab8d41832", "kind": "lucas_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/lucas-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=140]" }, { "document_sha256": "f0e8b0b74dde3f940036c3c38b1307b01f7a42501182392a90a635214440686e", "kind": "lucas_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/lucas-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_two_le]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_two_le", "parent_alpha_version": "v34", "proof_bundle_node_id": 203, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_two_le", "script": [ "intro p", "intro hp", "have hshape : exists k. p = S (S k)", "specialize prime_is_succ_succ p", "apply prime_is_succ_succ", "exact hp", "cases hshape", "exists x", "trans S (S x)", "rewrite PA4", "rewrite PA4", "rewrite PA3", "refl", "symm", "exact hshape_witness" ], "script_sha256": "3c9e57f0bcfa156e36681907a8991edbf10eaecab4bf4242ad171182d591b1a8", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "stable_member": false, "statement": "forall p. 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mul_one at hstep", "have hscaled : exists k. k + r * 2 = r * p", "specialize mul_le_mul_left 2", "specialize mul_le_mul_left p", "specialize mul_le_mul_left r", "apply mul_le_mul_left", "exact hp", "specialize le_trans (S r)", "specialize le_trans (r * 2)", "specialize le_trans (r * p)", "apply le_trans", "exact hstep", "exact hscaled" ], "script_sha256": "0416cd8d1f1597146b95209de77e081ee4fe88c3853eeda7385802c448c2f626", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "statement": "forall r p. ~(r = 0) -> (exists bpvl_gap_factor_two. bpvl_gap_factor_two + (2) = (p)) -> (exists bpvl_gap_factor_result. bpvl_gap_factor_result + (S r) = (r * p))", "statement_sha256": "201972639fe0cde658f6249560fe41d3a8904c77500aa72787162006a556b8c5", "summary": "Multiplying a nonzero natural by a factor at least two exceeds it.", "summary_sha256": 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(exists ff_b_bpvl_prefix ff_c_bpvl_prefix. ((forall ff_i_bpvl_prefix_repeat. (exists ff_lt_bpvl_prefix_repeat_bound. ff_lt_bpvl_prefix_repeat_bound + S ff_i_bpvl_prefix_repeat = e) -> (((exists ff_h_bpvl_prefix_repeat_decoded. ff_h_bpvl_prefix_repeat_decoded + S (p) = S ((S (ff_i_bpvl_prefix_repeat)) * ff_c_bpvl_prefix)) /\\ exists ff_q_bpvl_prefix_repeat_decoded. ff_b_bpvl_prefix = ff_q_bpvl_prefix_repeat_decoded * S ((S (ff_i_bpvl_prefix_repeat)) * ff_c_bpvl_prefix) + (p)))) /\\ (exists ff_u_bpvl_prefix_product ff_v_bpvl_prefix_product. ((((exists ff_h_bpvl_prefix_product_start. ff_h_bpvl_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_start. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_start * S ((S (0)) * ff_v_bpvl_prefix_product) + (1))) /\\ ((((exists ff_h_bpvl_prefix_product_terminal. ff_h_bpvl_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_terminal. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_terminal * S ((S (e)) * ff_v_bpvl_prefix_product) + (r))) /\\ forall ff_i_bpvl_prefix_product. (exists ff_lt_bpvl_prefix_product_bound. ff_lt_bpvl_prefix_product_bound + S ff_i_bpvl_prefix_product = e) -> exists ff_p_bpvl_prefix_product ff_r_bpvl_prefix_product ff_s_bpvl_prefix_product. ((((exists ff_h_bpvl_prefix_product_factor. ff_h_bpvl_prefix_product_factor + S (ff_p_bpvl_prefix_product) = S ((S (ff_i_bpvl_prefix_product)) * ff_c_bpvl_prefix)) /\\ exists ff_q_bpvl_prefix_product_factor. ff_b_bpvl_prefix = ff_q_bpvl_prefix_product_factor * S ((S (ff_i_bpvl_prefix_product)) * ff_c_bpvl_prefix) + (ff_p_bpvl_prefix_product))) /\\ ((((exists ff_h_bpvl_prefix_product_partial. ff_h_bpvl_prefix_product_partial + S (ff_r_bpvl_prefix_product) = S ((S (ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_partial. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_partial * S ((S (ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product) + (ff_r_bpvl_prefix_product))) /\\ ((((exists ff_h_bpvl_prefix_product_successor. ff_h_bpvl_prefix_product_successor + S (ff_s_bpvl_prefix_product) = S ((S (S ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_successor. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_successor * S ((S (S ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product) + (ff_s_bpvl_prefix_product))) /\\ ff_s_bpvl_prefix_product = ff_r_bpvl_prefix_product * ff_p_bpvl_prefix_product)))))))) /\\ x = r * p", "specialize pow_successor_decompose p", "specialize pow_successor_decompose e", "specialize pow_successor_decompose (S e)", "specialize pow_successor_decompose x", "apply pow_successor_decompose", "refl", "exact hx", "cases hstep", "cases hstep_witness", "have he_prefix : exists k. k + e = x1", "specialize IH x1", "apply IH", "exact hp", "exact hstep_witness_left", "have hp0 : ~(p = 0)", "intro hpzero", "specialize prime_nonzero p", "apply prime_nonzero", "exact hp", "exact hpzero", "have hp1 : exists k. k + 1 = p", "specialize one_le_of_ne_zero p", "apply one_le_of_ne_zero", "exact hp0", "have hprefix0 : ~(x1 = 0)", "intro hprefixzero", "specialize pow_nonzero_of_one_le p", "specialize pow_nonzero_of_one_le e", "specialize pow_nonzero_of_one_le x1", "apply pow_nonzero_of_one_le", "exact hp1", "exact hstep_witness_left", "exact hprefixzero", "have hp2 : exists k. k + 2 = p", "specialize prime_two_le p", "apply prime_two_le", "exact hp", "have hprefix_step : exists k. k + S x1 = x1 * p", "specialize succ_le_mul_of_two_le_right x1", "specialize succ_le_mul_of_two_le_right p", "apply succ_le_mul_of_two_le_right", "exact hprefix0", "exact hp2", "have he_step : exists k. k + S e = S x1", "specialize succ_le_succ e", "specialize succ_le_succ x1", "apply succ_le_succ", "exact he_prefix", "rewrite hstep_witness_right", "specialize le_trans (S e)", "specialize le_trans (S x1)", "specialize le_trans (x1 * p)", "apply le_trans", "exact he_step", "exact hprefix_step" ], "script_sha256": "5174ee8a1a5fb369d2e5ac441cf99a27735f9bd5aa86a7e61fbbab86ffbf2390", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "statement": "forall p e x. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> (exists ff_b_bpvl_exponent_bound ff_c_bpvl_exponent_bound. ((forall ff_i_bpvl_exponent_bound_repeat. (exists ff_lt_bpvl_exponent_bound_repeat_bound. ff_lt_bpvl_exponent_bound_repeat_bound + S ff_i_bpvl_exponent_bound_repeat = e) -> (((exists ff_h_bpvl_exponent_bound_repeat_decoded. ff_h_bpvl_exponent_bound_repeat_decoded + S (p) = S ((S (ff_i_bpvl_exponent_bound_repeat)) * ff_c_bpvl_exponent_bound)) /\\ exists ff_q_bpvl_exponent_bound_repeat_decoded. ff_b_bpvl_exponent_bound = ff_q_bpvl_exponent_bound_repeat_decoded * S ((S (ff_i_bpvl_exponent_bound_repeat)) * ff_c_bpvl_exponent_bound) + (p)))) /\\ (exists ff_u_bpvl_exponent_bound_product ff_v_bpvl_exponent_bound_product. ((((exists ff_h_bpvl_exponent_bound_product_start. ff_h_bpvl_exponent_bound_product_start + S (1) = S ((S (0)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_start. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_start * S ((S (0)) * ff_v_bpvl_exponent_bound_product) + (1))) /\\ ((((exists ff_h_bpvl_exponent_bound_product_terminal. ff_h_bpvl_exponent_bound_product_terminal + S (x) = S ((S (e)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_terminal. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_terminal * S ((S (e)) * ff_v_bpvl_exponent_bound_product) + (x))) /\\ forall ff_i_bpvl_exponent_bound_product. (exists ff_lt_bpvl_exponent_bound_product_bound. ff_lt_bpvl_exponent_bound_product_bound + S ff_i_bpvl_exponent_bound_product = e) -> exists ff_p_bpvl_exponent_bound_product ff_r_bpvl_exponent_bound_product ff_s_bpvl_exponent_bound_product. ((((exists ff_h_bpvl_exponent_bound_product_factor. ff_h_bpvl_exponent_bound_product_factor + S (ff_p_bpvl_exponent_bound_product) = S ((S (ff_i_bpvl_exponent_bound_product)) * ff_c_bpvl_exponent_bound)) /\\ exists ff_q_bpvl_exponent_bound_product_factor. ff_b_bpvl_exponent_bound = ff_q_bpvl_exponent_bound_product_factor * S ((S (ff_i_bpvl_exponent_bound_product)) * ff_c_bpvl_exponent_bound) + (ff_p_bpvl_exponent_bound_product))) /\\ ((((exists ff_h_bpvl_exponent_bound_product_partial. ff_h_bpvl_exponent_bound_product_partial + S (ff_r_bpvl_exponent_bound_product) = S ((S (ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_partial. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_partial * S ((S (ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product) + (ff_r_bpvl_exponent_bound_product))) /\\ ((((exists ff_h_bpvl_exponent_bound_product_successor. ff_h_bpvl_exponent_bound_product_successor + S (ff_s_bpvl_exponent_bound_product) = S ((S (S ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_successor. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_successor * S ((S (S ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product) + (ff_s_bpvl_exponent_bound_product))) /\\ ff_s_bpvl_exponent_bound_product = ff_r_bpvl_exponent_bound_product * ff_p_bpvl_exponent_bound_product)))))))) -> (exists bpv_gap_power_exponent. bpv_gap_power_exponent + e = x)", "statement_sha256": "8867c628e09cfa8890c2644b7b3df87d9951b10041f0e91da0d7d870ff54f53b", "summary": "The exponent of a relational power at a prime base is bounded by its value.", "summary_sha256": "92cdab29d4c99ad13777786bcf5a710d74cbe28ab55f04e935a925dfbf3ccc60" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_successor_decompose", "zero_le", "prime_nonzero", "one_le_of_ne_zero", "pow_nonzero_of_one_le", "prime_two_le", "succ_le_succ", "succ_le_mul_of_two_le_right", "le_trans" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "3686a414ab5aeb8dcb4c064e030dcab57129744845737b708354814823898ef1", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_laws_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=191]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_power_exponent_le]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_power_exponent_le", "parent_alpha_version": "v34", "proof_bundle_node_id": 205, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_power_exponent_le", "script": [ "intro p", "intro e", "induction e", "intro x", "intro hp", "intro hx", "specialize zero_le x", "exact zero_le", "intro x", "intro hp", "intro hx", "have hstep : exists r. (exists ff_b_bpvl_prefix ff_c_bpvl_prefix. ((forall ff_i_bpvl_prefix_repeat. (exists ff_lt_bpvl_prefix_repeat_bound. ff_lt_bpvl_prefix_repeat_bound + S ff_i_bpvl_prefix_repeat = e) -> (((exists ff_h_bpvl_prefix_repeat_decoded. ff_h_bpvl_prefix_repeat_decoded + S (p) = S ((S (ff_i_bpvl_prefix_repeat)) * ff_c_bpvl_prefix)) /\\ exists ff_q_bpvl_prefix_repeat_decoded. ff_b_bpvl_prefix = ff_q_bpvl_prefix_repeat_decoded * S ((S (ff_i_bpvl_prefix_repeat)) * ff_c_bpvl_prefix) + (p)))) /\\ (exists ff_u_bpvl_prefix_product ff_v_bpvl_prefix_product. ((((exists ff_h_bpvl_prefix_product_start. ff_h_bpvl_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_start. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_start * S ((S (0)) * ff_v_bpvl_prefix_product) + (1))) /\\ ((((exists ff_h_bpvl_prefix_product_terminal. ff_h_bpvl_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_terminal. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_terminal * S ((S (e)) * ff_v_bpvl_prefix_product) + (r))) /\\ forall ff_i_bpvl_prefix_product. (exists ff_lt_bpvl_prefix_product_bound. ff_lt_bpvl_prefix_product_bound + S ff_i_bpvl_prefix_product = e) -> exists ff_p_bpvl_prefix_product ff_r_bpvl_prefix_product ff_s_bpvl_prefix_product. ((((exists ff_h_bpvl_prefix_product_factor. ff_h_bpvl_prefix_product_factor + S (ff_p_bpvl_prefix_product) = S ((S (ff_i_bpvl_prefix_product)) * ff_c_bpvl_prefix)) /\\ exists ff_q_bpvl_prefix_product_factor. ff_b_bpvl_prefix = ff_q_bpvl_prefix_product_factor * S ((S (ff_i_bpvl_prefix_product)) * ff_c_bpvl_prefix) + (ff_p_bpvl_prefix_product))) /\\ ((((exists ff_h_bpvl_prefix_product_partial. ff_h_bpvl_prefix_product_partial + S (ff_r_bpvl_prefix_product) = S ((S (ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_partial. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_partial * S ((S (ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product) + (ff_r_bpvl_prefix_product))) /\\ ((((exists ff_h_bpvl_prefix_product_successor. ff_h_bpvl_prefix_product_successor + S (ff_s_bpvl_prefix_product) = S ((S (S ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product)) /\\ exists ff_q_bpvl_prefix_product_successor. ff_u_bpvl_prefix_product = ff_q_bpvl_prefix_product_successor * S ((S (S ff_i_bpvl_prefix_product)) * ff_v_bpvl_prefix_product) + (ff_s_bpvl_prefix_product))) /\\ ff_s_bpvl_prefix_product = ff_r_bpvl_prefix_product * ff_p_bpvl_prefix_product)))))))) /\\ x = r * p", "specialize pow_successor_decompose p", "specialize pow_successor_decompose e", "specialize pow_successor_decompose (S e)", "specialize pow_successor_decompose x", "apply pow_successor_decompose", "refl", "exact hx", "cases hstep", "cases hstep_witness", "have he_prefix : exists k. k + e = x1", "specialize IH x1", "apply IH", "exact hp", "exact hstep_witness_left", "have hp0 : ~(p = 0)", "intro hpzero", "specialize prime_nonzero p", "apply prime_nonzero", "exact hp", "exact hpzero", "have hp1 : exists k. k + 1 = p", "specialize one_le_of_ne_zero p", "apply one_le_of_ne_zero", "exact hp0", "have hprefix0 : ~(x1 = 0)", "intro hprefixzero", "specialize pow_nonzero_of_one_le p", "specialize pow_nonzero_of_one_le e", "specialize pow_nonzero_of_one_le x1", "apply pow_nonzero_of_one_le", "exact hp1", "exact hstep_witness_left", "exact hprefixzero", "have hp2 : exists k. k + 2 = p", "specialize prime_two_le p", "apply prime_two_le", "exact hp", "have hprefix_step : exists k. k + S x1 = x1 * p", "specialize succ_le_mul_of_two_le_right x1", "specialize succ_le_mul_of_two_le_right p", "apply succ_le_mul_of_two_le_right", "exact hprefix0", "exact hp2", "have he_step : exists k. k + S e = S x1", "specialize succ_le_succ e", "specialize succ_le_succ x1", "apply succ_le_succ", "exact he_prefix", "rewrite hstep_witness_right", "specialize le_trans (S e)", "specialize le_trans (S x1)", "specialize le_trans (x1 * p)", "apply le_trans", "exact he_step", "exact hprefix_step" ], "script_sha256": "5174ee8a1a5fb369d2e5ac441cf99a27735f9bd5aa86a7e61fbbab86ffbf2390", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "stable_member": false, "statement": "forall p e x. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> (exists ff_b_bpvl_exponent_bound ff_c_bpvl_exponent_bound. ((forall ff_i_bpvl_exponent_bound_repeat. (exists ff_lt_bpvl_exponent_bound_repeat_bound. ff_lt_bpvl_exponent_bound_repeat_bound + S ff_i_bpvl_exponent_bound_repeat = e) -> (((exists ff_h_bpvl_exponent_bound_repeat_decoded. ff_h_bpvl_exponent_bound_repeat_decoded + S (p) = S ((S (ff_i_bpvl_exponent_bound_repeat)) * ff_c_bpvl_exponent_bound)) /\\ exists ff_q_bpvl_exponent_bound_repeat_decoded. ff_b_bpvl_exponent_bound = ff_q_bpvl_exponent_bound_repeat_decoded * S ((S (ff_i_bpvl_exponent_bound_repeat)) * ff_c_bpvl_exponent_bound) + (p)))) /\\ (exists ff_u_bpvl_exponent_bound_product ff_v_bpvl_exponent_bound_product. ((((exists ff_h_bpvl_exponent_bound_product_start. ff_h_bpvl_exponent_bound_product_start + S (1) = S ((S (0)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_start. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_start * S ((S (0)) * ff_v_bpvl_exponent_bound_product) + (1))) /\\ ((((exists ff_h_bpvl_exponent_bound_product_terminal. ff_h_bpvl_exponent_bound_product_terminal + S (x) = S ((S (e)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_terminal. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_terminal * S ((S (e)) * ff_v_bpvl_exponent_bound_product) + (x))) /\\ forall ff_i_bpvl_exponent_bound_product. (exists ff_lt_bpvl_exponent_bound_product_bound. ff_lt_bpvl_exponent_bound_product_bound + S ff_i_bpvl_exponent_bound_product = e) -> exists ff_p_bpvl_exponent_bound_product ff_r_bpvl_exponent_bound_product ff_s_bpvl_exponent_bound_product. ((((exists ff_h_bpvl_exponent_bound_product_factor. ff_h_bpvl_exponent_bound_product_factor + S (ff_p_bpvl_exponent_bound_product) = S ((S (ff_i_bpvl_exponent_bound_product)) * ff_c_bpvl_exponent_bound)) /\\ exists ff_q_bpvl_exponent_bound_product_factor. ff_b_bpvl_exponent_bound = ff_q_bpvl_exponent_bound_product_factor * S ((S (ff_i_bpvl_exponent_bound_product)) * ff_c_bpvl_exponent_bound) + (ff_p_bpvl_exponent_bound_product))) /\\ ((((exists ff_h_bpvl_exponent_bound_product_partial. ff_h_bpvl_exponent_bound_product_partial + S (ff_r_bpvl_exponent_bound_product) = S ((S (ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_partial. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_partial * S ((S (ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product) + (ff_r_bpvl_exponent_bound_product))) /\\ ((((exists ff_h_bpvl_exponent_bound_product_successor. ff_h_bpvl_exponent_bound_product_successor + S (ff_s_bpvl_exponent_bound_product) = S ((S (S ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product)) /\\ exists ff_q_bpvl_exponent_bound_product_successor. ff_u_bpvl_exponent_bound_product = ff_q_bpvl_exponent_bound_product_successor * S ((S (S ff_i_bpvl_exponent_bound_product)) * ff_v_bpvl_exponent_bound_product) + (ff_s_bpvl_exponent_bound_product))) /\\ ff_s_bpvl_exponent_bound_product = ff_r_bpvl_exponent_bound_product * ff_p_bpvl_exponent_bound_product)))))))) -> (exists bpv_gap_power_exponent. bpv_gap_power_exponent + e = x)", "statement_sha256": "8867c628e09cfa8890c2644b7b3df87d9951b10041f0e91da0d7d870ff54f53b" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_power_divides_exponent_le_value", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 192, 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"7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "statement": "forall p e a. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (exists bpv_result_bpvl_bound_divides. ((exists ff_b_bpvl_bound_divides_power ff_c_bpvl_bound_divides_power. ((forall ff_i_bpvl_bound_divides_power_repeat. (exists ff_lt_bpvl_bound_divides_power_repeat_bound. ff_lt_bpvl_bound_divides_power_repeat_bound + S ff_i_bpvl_bound_divides_power_repeat = e) -> (((exists ff_h_bpvl_bound_divides_power_repeat_decoded. ff_h_bpvl_bound_divides_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_bound_divides_power_repeat)) * ff_c_bpvl_bound_divides_power)) /\\ exists ff_q_bpvl_bound_divides_power_repeat_decoded. ff_b_bpvl_bound_divides_power = ff_q_bpvl_bound_divides_power_repeat_decoded * S ((S (ff_i_bpvl_bound_divides_power_repeat)) * ff_c_bpvl_bound_divides_power) + (p)))) /\\ (exists ff_u_bpvl_bound_divides_power_product ff_v_bpvl_bound_divides_power_product. ((((exists ff_h_bpvl_bound_divides_power_product_start. ff_h_bpvl_bound_divides_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_start. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_start * S ((S (0)) * ff_v_bpvl_bound_divides_power_product) + (1))) /\\ ((((exists ff_h_bpvl_bound_divides_power_product_terminal. ff_h_bpvl_bound_divides_power_product_terminal + S (bpv_result_bpvl_bound_divides) = S ((S (e)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_terminal. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_terminal * S ((S (e)) * ff_v_bpvl_bound_divides_power_product) + (bpv_result_bpvl_bound_divides))) /\\ forall ff_i_bpvl_bound_divides_power_product. (exists ff_lt_bpvl_bound_divides_power_product_bound. ff_lt_bpvl_bound_divides_power_product_bound + S ff_i_bpvl_bound_divides_power_product = e) -> exists ff_p_bpvl_bound_divides_power_product ff_r_bpvl_bound_divides_power_product ff_s_bpvl_bound_divides_power_product. ((((exists ff_h_bpvl_bound_divides_power_product_factor. ff_h_bpvl_bound_divides_power_product_factor + S (ff_p_bpvl_bound_divides_power_product) = S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_c_bpvl_bound_divides_power)) /\\ exists ff_q_bpvl_bound_divides_power_product_factor. ff_b_bpvl_bound_divides_power = ff_q_bpvl_bound_divides_power_product_factor * S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_c_bpvl_bound_divides_power) + (ff_p_bpvl_bound_divides_power_product))) /\\ ((((exists ff_h_bpvl_bound_divides_power_product_partial. ff_h_bpvl_bound_divides_power_product_partial + S (ff_r_bpvl_bound_divides_power_product) = S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_partial. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_partial * S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product) + (ff_r_bpvl_bound_divides_power_product))) /\\ ((((exists ff_h_bpvl_bound_divides_power_product_successor. ff_h_bpvl_bound_divides_power_product_successor + S (ff_s_bpvl_bound_divides_power_product) = S ((S (S ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_successor. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_successor * S ((S (S ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product) + (ff_s_bpvl_bound_divides_power_product))) /\\ ff_s_bpvl_bound_divides_power_product = ff_r_bpvl_bound_divides_power_product * ff_p_bpvl_bound_divides_power_product)))))))) /\\ (exists bpv_factor_bpvl_bound_divides_divides. a = bpv_result_bpvl_bound_divides * bpv_factor_bpvl_bound_divides_divides))) -> (exists bpv_gap_divides_exponent_value. bpv_gap_divides_exponent_value + e = a)", "statement_sha256": 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((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (exists bpv_result_bpvl_bound_divides. ((exists ff_b_bpvl_bound_divides_power ff_c_bpvl_bound_divides_power. ((forall ff_i_bpvl_bound_divides_power_repeat. (exists ff_lt_bpvl_bound_divides_power_repeat_bound. ff_lt_bpvl_bound_divides_power_repeat_bound + S ff_i_bpvl_bound_divides_power_repeat = e) -> (((exists ff_h_bpvl_bound_divides_power_repeat_decoded. ff_h_bpvl_bound_divides_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_bound_divides_power_repeat)) * ff_c_bpvl_bound_divides_power)) /\\ exists ff_q_bpvl_bound_divides_power_repeat_decoded. ff_b_bpvl_bound_divides_power = ff_q_bpvl_bound_divides_power_repeat_decoded * S ((S (ff_i_bpvl_bound_divides_power_repeat)) * ff_c_bpvl_bound_divides_power) + (p)))) /\\ (exists ff_u_bpvl_bound_divides_power_product ff_v_bpvl_bound_divides_power_product. ((((exists ff_h_bpvl_bound_divides_power_product_start. ff_h_bpvl_bound_divides_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_start. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_start * S ((S (0)) * ff_v_bpvl_bound_divides_power_product) + (1))) /\\ ((((exists ff_h_bpvl_bound_divides_power_product_terminal. ff_h_bpvl_bound_divides_power_product_terminal + S (bpv_result_bpvl_bound_divides) = S ((S (e)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_terminal. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_terminal * S ((S (e)) * ff_v_bpvl_bound_divides_power_product) + (bpv_result_bpvl_bound_divides))) /\\ forall ff_i_bpvl_bound_divides_power_product. (exists ff_lt_bpvl_bound_divides_power_product_bound. ff_lt_bpvl_bound_divides_power_product_bound + S ff_i_bpvl_bound_divides_power_product = e) -> exists ff_p_bpvl_bound_divides_power_product ff_r_bpvl_bound_divides_power_product ff_s_bpvl_bound_divides_power_product. ((((exists ff_h_bpvl_bound_divides_power_product_factor. ff_h_bpvl_bound_divides_power_product_factor + S (ff_p_bpvl_bound_divides_power_product) = S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_c_bpvl_bound_divides_power)) /\\ exists ff_q_bpvl_bound_divides_power_product_factor. ff_b_bpvl_bound_divides_power = ff_q_bpvl_bound_divides_power_product_factor * S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_c_bpvl_bound_divides_power) + (ff_p_bpvl_bound_divides_power_product))) /\\ ((((exists ff_h_bpvl_bound_divides_power_product_partial. ff_h_bpvl_bound_divides_power_product_partial + S (ff_r_bpvl_bound_divides_power_product) = S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_partial. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_partial * S ((S (ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product) + (ff_r_bpvl_bound_divides_power_product))) /\\ ((((exists ff_h_bpvl_bound_divides_power_product_successor. ff_h_bpvl_bound_divides_power_product_successor + S (ff_s_bpvl_bound_divides_power_product) = S ((S (S ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product)) /\\ exists ff_q_bpvl_bound_divides_power_product_successor. ff_u_bpvl_bound_divides_power_product = ff_q_bpvl_bound_divides_power_product_successor * S ((S (S ff_i_bpvl_bound_divides_power_product)) * ff_v_bpvl_bound_divides_power_product) + (ff_s_bpvl_bound_divides_power_product))) /\\ ff_s_bpvl_bound_divides_power_product = ff_r_bpvl_bound_divides_power_product * ff_p_bpvl_bound_divides_power_product)))))))) /\\ (exists bpv_factor_bpvl_bound_divides_divides. a = bpv_result_bpvl_bound_divides * bpv_factor_bpvl_bound_divides_divides))) -> (exists bpv_gap_divides_exponent_value. bpv_gap_divides_exponent_value + e = a)", "statement_sha256": 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34, "proof_objects": 34, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_power_divides_exponent_le_value", "zero_add", "lt_not_le" ], "dependencies_sha256": "caa774ae231033ac697db16f3a417b0d0d9760a0c2d97defddc3a276e295c4c0", "empty_context_closure": { "body_proof_depth": 20, "body_proof_nodes": 34, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 193, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "2db989e4c49867e39aede51d867d65c496cd47d56f69fcd139f6db30a66603f6", "status": "checked" }, 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"artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_successor_not_divides]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "e7fc1b611ac9f439aabd22290fa9cc78f5a9f5db91cd11fe0d3f38af3706644a", "membership": "alpha_only", "name": "power_valuation_successor_not_divides", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_laws" ], "script": [ "intro p", "intro a", "intro e", "intro hp", "intro ha", "intro hvaluation", "intro hsuccessor", "have hbound : exists k. k + S e = a", "specialize prime_power_divides_exponent_le_value p", "specialize prime_power_divides_exponent_le_value (S e)", "specialize prime_power_divides_exponent_le_value a", "apply prime_power_divides_exponent_le_value", "exact hp", "exact ha", "exact hsuccessor", "cases hvaluation", "have himpossible : exists k. k + S e = e", "specialize hvaluation_right (S e)", "apply hvaluation_right", "exact hbound", "exact hsuccessor", "have hstrict : exists k. k + S e = S e", "exists 0", "specialize zero_add (S e)", "exact zero_add", "specialize lt_not_le e", "specialize lt_not_le (S e)", "apply lt_not_le", "exact hstrict", "exact himpossible" ], "script_sha256": "da98c70a360e73da8c43363d6cf7fa613a56e218212abe9d4148fd2ed494ad55", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "statement": "forall p a e. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpvl_valuation_exponent_bound. bpv_gap_bpvl_valuation_exponent_bound + e = a) /\\ (exists bpv_result_bpvl_valuation_selected. ((exists ff_b_bpvl_valuation_selected_power ff_c_bpvl_valuation_selected_power. ((forall ff_i_bpvl_valuation_selected_power_repeat. (exists ff_lt_bpvl_valuation_selected_power_repeat_bound. ff_lt_bpvl_valuation_selected_power_repeat_bound + S ff_i_bpvl_valuation_selected_power_repeat = e) -> (((exists ff_h_bpvl_valuation_selected_power_repeat_decoded. ff_h_bpvl_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_repeat_decoded. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_selected_power_product ff_v_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_start. ff_h_bpvl_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_start. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_terminal. ff_h_bpvl_valuation_selected_power_product_terminal + S (bpv_result_bpvl_valuation_selected) = S ((S (e)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_terminal. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bpvl_valuation_selected_power_product) + (bpv_result_bpvl_valuation_selected))) /\\ forall ff_i_bpvl_valuation_selected_power_product. (exists ff_lt_bpvl_valuation_selected_power_product_bound. ff_lt_bpvl_valuation_selected_power_product_bound + S ff_i_bpvl_valuation_selected_power_product = e) -> exists ff_p_bpvl_valuation_selected_power_product ff_r_bpvl_valuation_selected_power_product ff_s_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_factor. ff_h_bpvl_valuation_selected_power_product_factor + S (ff_p_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_product_factor. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_product_factor * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power) + (ff_p_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_partial. ff_h_bpvl_valuation_selected_power_product_partial + S (ff_r_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_partial. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_partial * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_r_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_successor. ff_h_bpvl_valuation_selected_power_product_successor + S (ff_s_bpvl_valuation_selected_power_product) = S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_successor. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_successor * S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_s_bpvl_valuation_selected_power_product))) /\\ ff_s_bpvl_valuation_selected_power_product = ff_r_bpvl_valuation_selected_power_product * ff_p_bpvl_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_selected_divides. a = bpv_result_bpvl_valuation_selected * bpv_factor_bpvl_valuation_selected_divides)))) /\\ forall bpv_candidate_bpvl_valuation. (exists bpv_gap_bpvl_valuation_candidate_bound. bpv_gap_bpvl_valuation_candidate_bound + bpv_candidate_bpvl_valuation = a) -> (exists bpv_result_bpvl_valuation_candidate. ((exists ff_b_bpvl_valuation_candidate_power ff_c_bpvl_valuation_candidate_power. ((forall ff_i_bpvl_valuation_candidate_power_repeat. (exists ff_lt_bpvl_valuation_candidate_power_repeat_bound. ff_lt_bpvl_valuation_candidate_power_repeat_bound + S ff_i_bpvl_valuation_candidate_power_repeat = bpv_candidate_bpvl_valuation) -> (((exists ff_h_bpvl_valuation_candidate_power_repeat_decoded. ff_h_bpvl_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_repeat_decoded. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_candidate_power_product ff_v_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_start. ff_h_bpvl_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_start. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_terminal. ff_h_bpvl_valuation_candidate_power_product_terminal + S (bpv_result_bpvl_valuation_candidate) = S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_terminal. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product) + (bpv_result_bpvl_valuation_candidate))) /\\ forall ff_i_bpvl_valuation_candidate_power_product. (exists ff_lt_bpvl_valuation_candidate_power_product_bound. ff_lt_bpvl_valuation_candidate_power_product_bound + S ff_i_bpvl_valuation_candidate_power_product = bpv_candidate_bpvl_valuation) -> exists ff_p_bpvl_valuation_candidate_power_product ff_r_bpvl_valuation_candidate_power_product ff_s_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_factor. ff_h_bpvl_valuation_candidate_power_product_factor + S (ff_p_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_factor. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_product_factor * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power) + (ff_p_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_partial. ff_h_bpvl_valuation_candidate_power_product_partial + S (ff_r_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_partial. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_partial * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_r_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_successor. ff_h_bpvl_valuation_candidate_power_product_successor + S (ff_s_bpvl_valuation_candidate_power_product) = S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_successor. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_successor * S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_s_bpvl_valuation_candidate_power_product))) /\\ ff_s_bpvl_valuation_candidate_power_product = ff_r_bpvl_valuation_candidate_power_product * ff_p_bpvl_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_candidate_divides. a = bpv_result_bpvl_valuation_candidate * bpv_factor_bpvl_valuation_candidate_divides))) -> (exists bpv_gap_bpvl_valuation_maximal. bpv_gap_bpvl_valuation_maximal + bpv_candidate_bpvl_valuation = e)) -> ~(exists bpvi_result_bpvl_successor_divides. ((exists bpvi_b_bpvl_successor_divides_power bpvi_c_bpvl_successor_divides_power. ((forall bpvi_i_bpvl_successor_divides_power. (exists bpvi_repeat_gap_bpvl_successor_divides_power. bpvi_repeat_gap_bpvl_successor_divides_power + S bpvi_i_bpvl_successor_divides_power = S e) -> (((exists bpvi_h_bpvl_successor_divides_power_repeat. bpvi_h_bpvl_successor_divides_power_repeat + S (p) = S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_repeat. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_repeat * S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (p)))) /\\ (exists bpvi_u_bpvl_successor_divides_power bpvi_v_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_start. bpvi_h_bpvl_successor_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_start. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_start * S ((S (0)) * bpvi_v_bpvl_successor_divides_power) + (1))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_terminal. bpvi_h_bpvl_successor_divides_power_terminal + S (bpvi_result_bpvl_successor_divides) = S ((S (S e)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_terminal. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_terminal * S ((S (S e)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_result_bpvl_successor_divides))) /\\ forall bpvi_j_bpvl_successor_divides_power. (exists bpvi_product_gap_bpvl_successor_divides_power. bpvi_product_gap_bpvl_successor_divides_power + S bpvi_j_bpvl_successor_divides_power = S e) -> exists bpvi_factor_bpvl_successor_divides_power bpvi_partial_bpvl_successor_divides_power bpvi_successor_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_factor. bpvi_h_bpvl_successor_divides_power_factor + S (bpvi_factor_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_factor. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_factor * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (bpvi_factor_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_partial. bpvi_h_bpvl_successor_divides_power_partial + S (bpvi_partial_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_partial. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_partial * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_partial_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_successor. bpvi_h_bpvl_successor_divides_power_successor + S (bpvi_successor_bpvl_successor_divides_power) = S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_successor. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_successor * S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_successor_bpvl_successor_divides_power))) /\\ bpvi_successor_bpvl_successor_divides_power = bpvi_partial_bpvl_successor_divides_power * bpvi_factor_bpvl_successor_divides_power)))))))) /\\ exists bpvi_divisor_factor_bpvl_successor_divides. a = bpvi_result_bpvl_successor_divides * bpvi_divisor_factor_bpvl_successor_divides))", "statement_sha256": "2db989e4c49867e39aede51d867d65c496cd47d56f69fcd139f6db30a66603f6", "summary": "A canonical valuation at a prime cannot admit the next power divisor.", "summary_sha256": "5968a22202bc3da48ab042f108cc52dc6d0445cdfbefffacd30581255144dfbd" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "prime_power_divides_exponent_le_value", "zero_add", "lt_not_le" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "3686a414ab5aeb8dcb4c064e030dcab57129744845737b708354814823898ef1", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_laws_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": 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"specialize prime_power_divides_exponent_le_value (S e)", "specialize prime_power_divides_exponent_le_value a", "apply prime_power_divides_exponent_le_value", "exact hp", "exact ha", "exact hsuccessor", "cases hvaluation", "have himpossible : exists k. k + S e = e", "specialize hvaluation_right (S e)", "apply hvaluation_right", "exact hbound", "exact hsuccessor", "have hstrict : exists k. k + S e = S e", "exists 0", "specialize zero_add (S e)", "exact zero_add", "specialize lt_not_le e", "specialize lt_not_le (S e)", "apply lt_not_le", "exact hstrict", "exact himpossible" ], "script_sha256": "da98c70a360e73da8c43363d6cf7fa613a56e218212abe9d4148fd2ed494ad55", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "stable_member": false, "statement": "forall p a e. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpvl_valuation_exponent_bound. bpv_gap_bpvl_valuation_exponent_bound + e = a) /\\ (exists bpv_result_bpvl_valuation_selected. ((exists ff_b_bpvl_valuation_selected_power ff_c_bpvl_valuation_selected_power. ((forall ff_i_bpvl_valuation_selected_power_repeat. (exists ff_lt_bpvl_valuation_selected_power_repeat_bound. ff_lt_bpvl_valuation_selected_power_repeat_bound + S ff_i_bpvl_valuation_selected_power_repeat = e) -> (((exists ff_h_bpvl_valuation_selected_power_repeat_decoded. ff_h_bpvl_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_repeat_decoded. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_selected_power_product ff_v_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_start. ff_h_bpvl_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_start. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_terminal. ff_h_bpvl_valuation_selected_power_product_terminal + S (bpv_result_bpvl_valuation_selected) = S ((S (e)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_terminal. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bpvl_valuation_selected_power_product) + (bpv_result_bpvl_valuation_selected))) /\\ forall ff_i_bpvl_valuation_selected_power_product. (exists ff_lt_bpvl_valuation_selected_power_product_bound. ff_lt_bpvl_valuation_selected_power_product_bound + S ff_i_bpvl_valuation_selected_power_product = e) -> exists ff_p_bpvl_valuation_selected_power_product ff_r_bpvl_valuation_selected_power_product ff_s_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_factor. ff_h_bpvl_valuation_selected_power_product_factor + S (ff_p_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_product_factor. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_product_factor * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power) + (ff_p_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_partial. ff_h_bpvl_valuation_selected_power_product_partial + S (ff_r_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_partial. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_partial * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_r_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_successor. ff_h_bpvl_valuation_selected_power_product_successor + S (ff_s_bpvl_valuation_selected_power_product) = S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_successor. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_successor * S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_s_bpvl_valuation_selected_power_product))) /\\ ff_s_bpvl_valuation_selected_power_product = ff_r_bpvl_valuation_selected_power_product * ff_p_bpvl_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_selected_divides. a = bpv_result_bpvl_valuation_selected * bpv_factor_bpvl_valuation_selected_divides)))) /\\ forall bpv_candidate_bpvl_valuation. (exists bpv_gap_bpvl_valuation_candidate_bound. bpv_gap_bpvl_valuation_candidate_bound + bpv_candidate_bpvl_valuation = a) -> (exists bpv_result_bpvl_valuation_candidate. ((exists ff_b_bpvl_valuation_candidate_power ff_c_bpvl_valuation_candidate_power. ((forall ff_i_bpvl_valuation_candidate_power_repeat. (exists ff_lt_bpvl_valuation_candidate_power_repeat_bound. ff_lt_bpvl_valuation_candidate_power_repeat_bound + S ff_i_bpvl_valuation_candidate_power_repeat = bpv_candidate_bpvl_valuation) -> (((exists ff_h_bpvl_valuation_candidate_power_repeat_decoded. ff_h_bpvl_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_repeat_decoded. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_candidate_power_product ff_v_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_start. ff_h_bpvl_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_start. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_terminal. ff_h_bpvl_valuation_candidate_power_product_terminal + S (bpv_result_bpvl_valuation_candidate) = S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_terminal. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product) + (bpv_result_bpvl_valuation_candidate))) /\\ forall ff_i_bpvl_valuation_candidate_power_product. (exists ff_lt_bpvl_valuation_candidate_power_product_bound. ff_lt_bpvl_valuation_candidate_power_product_bound + S ff_i_bpvl_valuation_candidate_power_product = bpv_candidate_bpvl_valuation) -> exists ff_p_bpvl_valuation_candidate_power_product ff_r_bpvl_valuation_candidate_power_product ff_s_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_factor. ff_h_bpvl_valuation_candidate_power_product_factor + S (ff_p_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_factor. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_product_factor * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power) + (ff_p_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_partial. ff_h_bpvl_valuation_candidate_power_product_partial + S (ff_r_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_partial. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_partial * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_r_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_successor. ff_h_bpvl_valuation_candidate_power_product_successor + S (ff_s_bpvl_valuation_candidate_power_product) = S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_successor. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_successor * S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_s_bpvl_valuation_candidate_power_product))) /\\ ff_s_bpvl_valuation_candidate_power_product = ff_r_bpvl_valuation_candidate_power_product * ff_p_bpvl_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_candidate_divides. a = bpv_result_bpvl_valuation_candidate * bpv_factor_bpvl_valuation_candidate_divides))) -> (exists bpv_gap_bpvl_valuation_maximal. bpv_gap_bpvl_valuation_maximal + bpv_candidate_bpvl_valuation = e)) -> ~(exists bpvi_result_bpvl_successor_divides. ((exists bpvi_b_bpvl_successor_divides_power bpvi_c_bpvl_successor_divides_power. ((forall bpvi_i_bpvl_successor_divides_power. (exists bpvi_repeat_gap_bpvl_successor_divides_power. bpvi_repeat_gap_bpvl_successor_divides_power + S bpvi_i_bpvl_successor_divides_power = S e) -> (((exists bpvi_h_bpvl_successor_divides_power_repeat. bpvi_h_bpvl_successor_divides_power_repeat + S (p) = S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_repeat. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_repeat * S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (p)))) /\\ (exists bpvi_u_bpvl_successor_divides_power bpvi_v_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_start. bpvi_h_bpvl_successor_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_start. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_start * S ((S (0)) * bpvi_v_bpvl_successor_divides_power) + (1))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_terminal. bpvi_h_bpvl_successor_divides_power_terminal + S (bpvi_result_bpvl_successor_divides) = S ((S (S e)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_terminal. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_terminal * S ((S (S e)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_result_bpvl_successor_divides))) /\\ forall bpvi_j_bpvl_successor_divides_power. (exists bpvi_product_gap_bpvl_successor_divides_power. bpvi_product_gap_bpvl_successor_divides_power + S bpvi_j_bpvl_successor_divides_power = S e) -> exists bpvi_factor_bpvl_successor_divides_power bpvi_partial_bpvl_successor_divides_power bpvi_successor_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_factor. bpvi_h_bpvl_successor_divides_power_factor + S (bpvi_factor_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_factor. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_factor * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (bpvi_factor_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_partial. bpvi_h_bpvl_successor_divides_power_partial + S (bpvi_partial_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_partial. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_partial * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_partial_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_successor. bpvi_h_bpvl_successor_divides_power_successor + S (bpvi_successor_bpvl_successor_divides_power) = S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_successor. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_successor * S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_successor_bpvl_successor_divides_power))) /\\ bpvi_successor_bpvl_successor_divides_power = bpvi_partial_bpvl_successor_divides_power * bpvi_factor_bpvl_successor_divides_power)))))))) /\\ exists bpvi_divisor_factor_bpvl_successor_divides. a = bpvi_result_bpvl_successor_divides * bpvi_divisor_factor_bpvl_successor_divides))", "statement_sha256": "2db989e4c49867e39aede51d867d65c496cd47d56f69fcd139f6db30a66603f6" }, { "admitted_to_alpha": true, "alpha_checked_use": 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"kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "power_valuation_power_divides", "power_valuation_successor_not_divides" ], "dependencies_sha256": "c1a607b252622046baeb34ef171278dc342ec79c67be1b1448d59cfec8981c33", "empty_context_closure": { "body_proof_depth": 21, "body_proof_nodes": 46, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 194, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "c2f7d22ed9ae46992ff28f8511511554c5876f0309a13e0f476b14ed741f4ff5", "status": "checked" }, "enrollment_index": 928, "enrollment_origin": 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"exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_selected_and_successor_not_divides]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "3b8e2042b55e4fb0c384d936b8ece00a7aee8b4c5d6bbc7ce647b5e5c42d7342", "membership": "alpha_only", "name": "power_valuation_selected_and_successor_not_divides", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_laws" ], "script": [ "intro p", "intro a", "intro e", "intro hp", "intro ha", "intro hvaluation", "split", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides a", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation", "intro hsuccessor", "specialize power_valuation_successor_not_divides p", "specialize power_valuation_successor_not_divides a", "specialize power_valuation_successor_not_divides e", "apply power_valuation_successor_not_divides", "exact hp", "exact ha", "exact hvaluation", "exact hsuccessor" ], "script_sha256": "0e4263fd02839d3719cb85163ff96c2486cfe0987c0a5a32c558b5b38a297cfb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "statement": "forall p a e. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpvl_valuation_exponent_bound. bpv_gap_bpvl_valuation_exponent_bound + e = a) /\\ (exists bpv_result_bpvl_valuation_selected. ((exists ff_b_bpvl_valuation_selected_power ff_c_bpvl_valuation_selected_power. ((forall ff_i_bpvl_valuation_selected_power_repeat. (exists ff_lt_bpvl_valuation_selected_power_repeat_bound. ff_lt_bpvl_valuation_selected_power_repeat_bound + S ff_i_bpvl_valuation_selected_power_repeat = e) -> (((exists ff_h_bpvl_valuation_selected_power_repeat_decoded. ff_h_bpvl_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_repeat_decoded. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_selected_power_product ff_v_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_start. ff_h_bpvl_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_start. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_terminal. ff_h_bpvl_valuation_selected_power_product_terminal + S (bpv_result_bpvl_valuation_selected) = S ((S (e)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_terminal. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bpvl_valuation_selected_power_product) + (bpv_result_bpvl_valuation_selected))) /\\ forall ff_i_bpvl_valuation_selected_power_product. (exists ff_lt_bpvl_valuation_selected_power_product_bound. ff_lt_bpvl_valuation_selected_power_product_bound + S ff_i_bpvl_valuation_selected_power_product = e) -> exists ff_p_bpvl_valuation_selected_power_product ff_r_bpvl_valuation_selected_power_product ff_s_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_factor. ff_h_bpvl_valuation_selected_power_product_factor + S (ff_p_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_product_factor. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_product_factor * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power) + (ff_p_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_partial. ff_h_bpvl_valuation_selected_power_product_partial + S (ff_r_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_partial. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_partial * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_r_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_successor. ff_h_bpvl_valuation_selected_power_product_successor + S (ff_s_bpvl_valuation_selected_power_product) = S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_successor. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_successor * S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_s_bpvl_valuation_selected_power_product))) /\\ ff_s_bpvl_valuation_selected_power_product = ff_r_bpvl_valuation_selected_power_product * ff_p_bpvl_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_selected_divides. a = bpv_result_bpvl_valuation_selected * bpv_factor_bpvl_valuation_selected_divides)))) /\\ forall bpv_candidate_bpvl_valuation. (exists bpv_gap_bpvl_valuation_candidate_bound. bpv_gap_bpvl_valuation_candidate_bound + bpv_candidate_bpvl_valuation = a) -> (exists bpv_result_bpvl_valuation_candidate. ((exists ff_b_bpvl_valuation_candidate_power ff_c_bpvl_valuation_candidate_power. ((forall ff_i_bpvl_valuation_candidate_power_repeat. (exists ff_lt_bpvl_valuation_candidate_power_repeat_bound. ff_lt_bpvl_valuation_candidate_power_repeat_bound + S ff_i_bpvl_valuation_candidate_power_repeat = bpv_candidate_bpvl_valuation) -> (((exists ff_h_bpvl_valuation_candidate_power_repeat_decoded. ff_h_bpvl_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_repeat_decoded. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_candidate_power_product ff_v_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_start. ff_h_bpvl_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_start. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_terminal. ff_h_bpvl_valuation_candidate_power_product_terminal + S (bpv_result_bpvl_valuation_candidate) = S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_terminal. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product) + (bpv_result_bpvl_valuation_candidate))) /\\ forall ff_i_bpvl_valuation_candidate_power_product. (exists ff_lt_bpvl_valuation_candidate_power_product_bound. ff_lt_bpvl_valuation_candidate_power_product_bound + S ff_i_bpvl_valuation_candidate_power_product = bpv_candidate_bpvl_valuation) -> exists ff_p_bpvl_valuation_candidate_power_product ff_r_bpvl_valuation_candidate_power_product ff_s_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_factor. ff_h_bpvl_valuation_candidate_power_product_factor + S (ff_p_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_factor. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_product_factor * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power) + (ff_p_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_partial. ff_h_bpvl_valuation_candidate_power_product_partial + S (ff_r_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_partial. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_partial * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_r_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_successor. ff_h_bpvl_valuation_candidate_power_product_successor + S (ff_s_bpvl_valuation_candidate_power_product) = S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_successor. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_successor * S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_s_bpvl_valuation_candidate_power_product))) /\\ ff_s_bpvl_valuation_candidate_power_product = ff_r_bpvl_valuation_candidate_power_product * ff_p_bpvl_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_candidate_divides. a = bpv_result_bpvl_valuation_candidate * bpv_factor_bpvl_valuation_candidate_divides))) -> (exists bpv_gap_bpvl_valuation_maximal. bpv_gap_bpvl_valuation_maximal + bpv_candidate_bpvl_valuation = e)) -> ((exists bpv_result_bpvl_selected_divides. ((exists ff_b_bpvl_selected_divides_power ff_c_bpvl_selected_divides_power. ((forall ff_i_bpvl_selected_divides_power_repeat. (exists ff_lt_bpvl_selected_divides_power_repeat_bound. ff_lt_bpvl_selected_divides_power_repeat_bound + S ff_i_bpvl_selected_divides_power_repeat = e) -> (((exists ff_h_bpvl_selected_divides_power_repeat_decoded. ff_h_bpvl_selected_divides_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_selected_divides_power_repeat)) * ff_c_bpvl_selected_divides_power)) /\\ exists ff_q_bpvl_selected_divides_power_repeat_decoded. ff_b_bpvl_selected_divides_power = ff_q_bpvl_selected_divides_power_repeat_decoded * S ((S (ff_i_bpvl_selected_divides_power_repeat)) * ff_c_bpvl_selected_divides_power) + (p)))) /\\ (exists ff_u_bpvl_selected_divides_power_product ff_v_bpvl_selected_divides_power_product. ((((exists ff_h_bpvl_selected_divides_power_product_start. ff_h_bpvl_selected_divides_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_start. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_start * S ((S (0)) * ff_v_bpvl_selected_divides_power_product) + (1))) /\\ ((((exists ff_h_bpvl_selected_divides_power_product_terminal. ff_h_bpvl_selected_divides_power_product_terminal + S (bpv_result_bpvl_selected_divides) = S ((S (e)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_terminal. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_terminal * S ((S (e)) * ff_v_bpvl_selected_divides_power_product) + (bpv_result_bpvl_selected_divides))) /\\ forall ff_i_bpvl_selected_divides_power_product. (exists ff_lt_bpvl_selected_divides_power_product_bound. ff_lt_bpvl_selected_divides_power_product_bound + S ff_i_bpvl_selected_divides_power_product = e) -> exists ff_p_bpvl_selected_divides_power_product ff_r_bpvl_selected_divides_power_product ff_s_bpvl_selected_divides_power_product. ((((exists ff_h_bpvl_selected_divides_power_product_factor. ff_h_bpvl_selected_divides_power_product_factor + S (ff_p_bpvl_selected_divides_power_product) = S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_c_bpvl_selected_divides_power)) /\\ exists ff_q_bpvl_selected_divides_power_product_factor. ff_b_bpvl_selected_divides_power = ff_q_bpvl_selected_divides_power_product_factor * S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_c_bpvl_selected_divides_power) + (ff_p_bpvl_selected_divides_power_product))) /\\ ((((exists ff_h_bpvl_selected_divides_power_product_partial. ff_h_bpvl_selected_divides_power_product_partial + S (ff_r_bpvl_selected_divides_power_product) = S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_partial. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_partial * S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product) + (ff_r_bpvl_selected_divides_power_product))) /\\ ((((exists ff_h_bpvl_selected_divides_power_product_successor. ff_h_bpvl_selected_divides_power_product_successor + S (ff_s_bpvl_selected_divides_power_product) = S ((S (S ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_successor. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_successor * S ((S (S ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product) + (ff_s_bpvl_selected_divides_power_product))) /\\ ff_s_bpvl_selected_divides_power_product = ff_r_bpvl_selected_divides_power_product * ff_p_bpvl_selected_divides_power_product)))))))) /\\ (exists bpv_factor_bpvl_selected_divides_divides. a = bpv_result_bpvl_selected_divides * bpv_factor_bpvl_selected_divides_divides))) /\\ ~(exists bpvi_result_bpvl_successor_divides. ((exists bpvi_b_bpvl_successor_divides_power bpvi_c_bpvl_successor_divides_power. ((forall bpvi_i_bpvl_successor_divides_power. (exists bpvi_repeat_gap_bpvl_successor_divides_power. bpvi_repeat_gap_bpvl_successor_divides_power + S bpvi_i_bpvl_successor_divides_power = S e) -> (((exists bpvi_h_bpvl_successor_divides_power_repeat. bpvi_h_bpvl_successor_divides_power_repeat + S (p) = S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_repeat. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_repeat * S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (p)))) /\\ (exists bpvi_u_bpvl_successor_divides_power bpvi_v_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_start. bpvi_h_bpvl_successor_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_start. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_start * S ((S (0)) * bpvi_v_bpvl_successor_divides_power) + (1))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_terminal. bpvi_h_bpvl_successor_divides_power_terminal + S (bpvi_result_bpvl_successor_divides) = S ((S (S e)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_terminal. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_terminal * S ((S (S e)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_result_bpvl_successor_divides))) /\\ forall bpvi_j_bpvl_successor_divides_power. (exists bpvi_product_gap_bpvl_successor_divides_power. bpvi_product_gap_bpvl_successor_divides_power + S bpvi_j_bpvl_successor_divides_power = S e) -> exists bpvi_factor_bpvl_successor_divides_power bpvi_partial_bpvl_successor_divides_power bpvi_successor_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_factor. bpvi_h_bpvl_successor_divides_power_factor + S (bpvi_factor_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_factor. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_factor * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (bpvi_factor_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_partial. bpvi_h_bpvl_successor_divides_power_partial + S (bpvi_partial_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_partial. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_partial * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_partial_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_successor. bpvi_h_bpvl_successor_divides_power_successor + S (bpvi_successor_bpvl_successor_divides_power) = S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_successor. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_successor * S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_successor_bpvl_successor_divides_power))) /\\ bpvi_successor_bpvl_successor_divides_power = bpvi_partial_bpvl_successor_divides_power * bpvi_factor_bpvl_successor_divides_power)))))))) /\\ exists bpvi_divisor_factor_bpvl_successor_divides. a = bpvi_result_bpvl_successor_divides * bpvi_divisor_factor_bpvl_successor_divides)))", "statement_sha256": "c2f7d22ed9ae46992ff28f8511511554c5876f0309a13e0f476b14ed741f4ff5", "summary": "Canonical prime valuations have the usual maximal-power characterization.", "summary_sha256": "ae2b5ab2f96c9018db874ea3fc26d302bb5e38ed91ca150ee9a5a8c71e79d652" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_power_divides", "power_valuation_successor_not_divides" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "3686a414ab5aeb8dcb4c064e030dcab57129744845737b708354814823898ef1", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_valuation_laws_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": 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"research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_selected_and_successor_not_divides]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_selected_and_successor_not_divides", "parent_alpha_version": "v34", "proof_bundle_node_id": 208, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_selected_and_successor_not_divides", "script": [ "intro p", "intro a", "intro e", "intro hp", "intro ha", "intro hvaluation", "split", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides a", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation", "intro hsuccessor", "specialize power_valuation_successor_not_divides p", "specialize power_valuation_successor_not_divides a", "specialize power_valuation_successor_not_divides e", "apply power_valuation_successor_not_divides", "exact hp", "exact ha", "exact hvaluation", "exact hsuccessor" ], "script_sha256": "0e4263fd02839d3719cb85163ff96c2486cfe0987c0a5a32c558b5b38a297cfb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_valuation_laws_candidate.py", "sha256": "7b95e4f2a16df3866cb3e01f17d1b455000706454a1a241948957c4548a0a17f" }, "stable_member": false, "statement": "forall p a e. ((~(p = 1) /\\ forall frm_prime_left_bpvl_prime frm_prime_right_bpvl_prime. p = frm_prime_left_bpvl_prime * frm_prime_right_bpvl_prime -> frm_prime_left_bpvl_prime = 1 \\/ frm_prime_right_bpvl_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpvl_valuation_exponent_bound. bpv_gap_bpvl_valuation_exponent_bound + e = a) /\\ (exists bpv_result_bpvl_valuation_selected. ((exists ff_b_bpvl_valuation_selected_power ff_c_bpvl_valuation_selected_power. ((forall ff_i_bpvl_valuation_selected_power_repeat. (exists ff_lt_bpvl_valuation_selected_power_repeat_bound. ff_lt_bpvl_valuation_selected_power_repeat_bound + S ff_i_bpvl_valuation_selected_power_repeat = e) -> (((exists ff_h_bpvl_valuation_selected_power_repeat_decoded. ff_h_bpvl_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_repeat_decoded. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_selected_power_repeat)) * ff_c_bpvl_valuation_selected_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_selected_power_product ff_v_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_start. ff_h_bpvl_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_start. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_terminal. ff_h_bpvl_valuation_selected_power_product_terminal + S (bpv_result_bpvl_valuation_selected) = S ((S (e)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_terminal. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_bpvl_valuation_selected_power_product) + (bpv_result_bpvl_valuation_selected))) /\\ forall ff_i_bpvl_valuation_selected_power_product. (exists ff_lt_bpvl_valuation_selected_power_product_bound. ff_lt_bpvl_valuation_selected_power_product_bound + S ff_i_bpvl_valuation_selected_power_product = e) -> exists ff_p_bpvl_valuation_selected_power_product ff_r_bpvl_valuation_selected_power_product ff_s_bpvl_valuation_selected_power_product. ((((exists ff_h_bpvl_valuation_selected_power_product_factor. ff_h_bpvl_valuation_selected_power_product_factor + S (ff_p_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power)) /\\ exists ff_q_bpvl_valuation_selected_power_product_factor. ff_b_bpvl_valuation_selected_power = ff_q_bpvl_valuation_selected_power_product_factor * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_c_bpvl_valuation_selected_power) + (ff_p_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_partial. ff_h_bpvl_valuation_selected_power_product_partial + S (ff_r_bpvl_valuation_selected_power_product) = S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_partial. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_partial * S ((S (ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_r_bpvl_valuation_selected_power_product))) /\\ ((((exists ff_h_bpvl_valuation_selected_power_product_successor. ff_h_bpvl_valuation_selected_power_product_successor + S (ff_s_bpvl_valuation_selected_power_product) = S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product)) /\\ exists ff_q_bpvl_valuation_selected_power_product_successor. ff_u_bpvl_valuation_selected_power_product = ff_q_bpvl_valuation_selected_power_product_successor * S ((S (S ff_i_bpvl_valuation_selected_power_product)) * ff_v_bpvl_valuation_selected_power_product) + (ff_s_bpvl_valuation_selected_power_product))) /\\ ff_s_bpvl_valuation_selected_power_product = ff_r_bpvl_valuation_selected_power_product * ff_p_bpvl_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_selected_divides. a = bpv_result_bpvl_valuation_selected * bpv_factor_bpvl_valuation_selected_divides)))) /\\ forall bpv_candidate_bpvl_valuation. (exists bpv_gap_bpvl_valuation_candidate_bound. bpv_gap_bpvl_valuation_candidate_bound + bpv_candidate_bpvl_valuation = a) -> (exists bpv_result_bpvl_valuation_candidate. ((exists ff_b_bpvl_valuation_candidate_power ff_c_bpvl_valuation_candidate_power. ((forall ff_i_bpvl_valuation_candidate_power_repeat. (exists ff_lt_bpvl_valuation_candidate_power_repeat_bound. ff_lt_bpvl_valuation_candidate_power_repeat_bound + S ff_i_bpvl_valuation_candidate_power_repeat = bpv_candidate_bpvl_valuation) -> (((exists ff_h_bpvl_valuation_candidate_power_repeat_decoded. ff_h_bpvl_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_repeat_decoded. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_repeat_decoded * S ((S (ff_i_bpvl_valuation_candidate_power_repeat)) * ff_c_bpvl_valuation_candidate_power) + (p)))) /\\ (exists ff_u_bpvl_valuation_candidate_power_product ff_v_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_start. ff_h_bpvl_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_start. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_start * S ((S (0)) * ff_v_bpvl_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_terminal. ff_h_bpvl_valuation_candidate_power_product_terminal + S (bpv_result_bpvl_valuation_candidate) = S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_terminal. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_bpvl_valuation)) * ff_v_bpvl_valuation_candidate_power_product) + (bpv_result_bpvl_valuation_candidate))) /\\ forall ff_i_bpvl_valuation_candidate_power_product. (exists ff_lt_bpvl_valuation_candidate_power_product_bound. ff_lt_bpvl_valuation_candidate_power_product_bound + S ff_i_bpvl_valuation_candidate_power_product = bpv_candidate_bpvl_valuation) -> exists ff_p_bpvl_valuation_candidate_power_product ff_r_bpvl_valuation_candidate_power_product ff_s_bpvl_valuation_candidate_power_product. ((((exists ff_h_bpvl_valuation_candidate_power_product_factor. ff_h_bpvl_valuation_candidate_power_product_factor + S (ff_p_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_factor. ff_b_bpvl_valuation_candidate_power = ff_q_bpvl_valuation_candidate_power_product_factor * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_c_bpvl_valuation_candidate_power) + (ff_p_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_partial. ff_h_bpvl_valuation_candidate_power_product_partial + S (ff_r_bpvl_valuation_candidate_power_product) = S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_partial. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_partial * S ((S (ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_r_bpvl_valuation_candidate_power_product))) /\\ ((((exists ff_h_bpvl_valuation_candidate_power_product_successor. ff_h_bpvl_valuation_candidate_power_product_successor + S (ff_s_bpvl_valuation_candidate_power_product) = S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product)) /\\ exists ff_q_bpvl_valuation_candidate_power_product_successor. ff_u_bpvl_valuation_candidate_power_product = ff_q_bpvl_valuation_candidate_power_product_successor * S ((S (S ff_i_bpvl_valuation_candidate_power_product)) * ff_v_bpvl_valuation_candidate_power_product) + (ff_s_bpvl_valuation_candidate_power_product))) /\\ ff_s_bpvl_valuation_candidate_power_product = ff_r_bpvl_valuation_candidate_power_product * ff_p_bpvl_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_bpvl_valuation_candidate_divides. a = bpv_result_bpvl_valuation_candidate * bpv_factor_bpvl_valuation_candidate_divides))) -> (exists bpv_gap_bpvl_valuation_maximal. bpv_gap_bpvl_valuation_maximal + bpv_candidate_bpvl_valuation = e)) -> ((exists bpv_result_bpvl_selected_divides. ((exists ff_b_bpvl_selected_divides_power ff_c_bpvl_selected_divides_power. ((forall ff_i_bpvl_selected_divides_power_repeat. (exists ff_lt_bpvl_selected_divides_power_repeat_bound. ff_lt_bpvl_selected_divides_power_repeat_bound + S ff_i_bpvl_selected_divides_power_repeat = e) -> (((exists ff_h_bpvl_selected_divides_power_repeat_decoded. ff_h_bpvl_selected_divides_power_repeat_decoded + S (p) = S ((S (ff_i_bpvl_selected_divides_power_repeat)) * ff_c_bpvl_selected_divides_power)) /\\ exists ff_q_bpvl_selected_divides_power_repeat_decoded. ff_b_bpvl_selected_divides_power = ff_q_bpvl_selected_divides_power_repeat_decoded * S ((S (ff_i_bpvl_selected_divides_power_repeat)) * ff_c_bpvl_selected_divides_power) + (p)))) /\\ (exists ff_u_bpvl_selected_divides_power_product ff_v_bpvl_selected_divides_power_product. ((((exists ff_h_bpvl_selected_divides_power_product_start. ff_h_bpvl_selected_divides_power_product_start + S (1) = S ((S (0)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_start. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_start * S ((S (0)) * ff_v_bpvl_selected_divides_power_product) + (1))) /\\ ((((exists ff_h_bpvl_selected_divides_power_product_terminal. ff_h_bpvl_selected_divides_power_product_terminal + S (bpv_result_bpvl_selected_divides) = S ((S (e)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_terminal. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_terminal * S ((S (e)) * ff_v_bpvl_selected_divides_power_product) + (bpv_result_bpvl_selected_divides))) /\\ forall ff_i_bpvl_selected_divides_power_product. (exists ff_lt_bpvl_selected_divides_power_product_bound. ff_lt_bpvl_selected_divides_power_product_bound + S ff_i_bpvl_selected_divides_power_product = e) -> exists ff_p_bpvl_selected_divides_power_product ff_r_bpvl_selected_divides_power_product ff_s_bpvl_selected_divides_power_product. ((((exists ff_h_bpvl_selected_divides_power_product_factor. ff_h_bpvl_selected_divides_power_product_factor + S (ff_p_bpvl_selected_divides_power_product) = S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_c_bpvl_selected_divides_power)) /\\ exists ff_q_bpvl_selected_divides_power_product_factor. ff_b_bpvl_selected_divides_power = ff_q_bpvl_selected_divides_power_product_factor * S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_c_bpvl_selected_divides_power) + (ff_p_bpvl_selected_divides_power_product))) /\\ ((((exists ff_h_bpvl_selected_divides_power_product_partial. ff_h_bpvl_selected_divides_power_product_partial + S (ff_r_bpvl_selected_divides_power_product) = S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_partial. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_partial * S ((S (ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product) + (ff_r_bpvl_selected_divides_power_product))) /\\ ((((exists ff_h_bpvl_selected_divides_power_product_successor. ff_h_bpvl_selected_divides_power_product_successor + S (ff_s_bpvl_selected_divides_power_product) = S ((S (S ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product)) /\\ exists ff_q_bpvl_selected_divides_power_product_successor. ff_u_bpvl_selected_divides_power_product = ff_q_bpvl_selected_divides_power_product_successor * S ((S (S ff_i_bpvl_selected_divides_power_product)) * ff_v_bpvl_selected_divides_power_product) + (ff_s_bpvl_selected_divides_power_product))) /\\ ff_s_bpvl_selected_divides_power_product = ff_r_bpvl_selected_divides_power_product * ff_p_bpvl_selected_divides_power_product)))))))) /\\ (exists bpv_factor_bpvl_selected_divides_divides. a = bpv_result_bpvl_selected_divides * bpv_factor_bpvl_selected_divides_divides))) /\\ ~(exists bpvi_result_bpvl_successor_divides. ((exists bpvi_b_bpvl_successor_divides_power bpvi_c_bpvl_successor_divides_power. ((forall bpvi_i_bpvl_successor_divides_power. (exists bpvi_repeat_gap_bpvl_successor_divides_power. bpvi_repeat_gap_bpvl_successor_divides_power + S bpvi_i_bpvl_successor_divides_power = S e) -> (((exists bpvi_h_bpvl_successor_divides_power_repeat. bpvi_h_bpvl_successor_divides_power_repeat + S (p) = S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_repeat. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_repeat * S ((S (bpvi_i_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (p)))) /\\ (exists bpvi_u_bpvl_successor_divides_power bpvi_v_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_start. bpvi_h_bpvl_successor_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_start. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_start * S ((S (0)) * bpvi_v_bpvl_successor_divides_power) + (1))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_terminal. bpvi_h_bpvl_successor_divides_power_terminal + S (bpvi_result_bpvl_successor_divides) = S ((S (S e)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_terminal. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_terminal * S ((S (S e)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_result_bpvl_successor_divides))) /\\ forall bpvi_j_bpvl_successor_divides_power. (exists bpvi_product_gap_bpvl_successor_divides_power. bpvi_product_gap_bpvl_successor_divides_power + S bpvi_j_bpvl_successor_divides_power = S e) -> exists bpvi_factor_bpvl_successor_divides_power bpvi_partial_bpvl_successor_divides_power bpvi_successor_bpvl_successor_divides_power. ((((exists bpvi_h_bpvl_successor_divides_power_factor. bpvi_h_bpvl_successor_divides_power_factor + S (bpvi_factor_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_factor. bpvi_b_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_factor * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_c_bpvl_successor_divides_power) + (bpvi_factor_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_partial. bpvi_h_bpvl_successor_divides_power_partial + S (bpvi_partial_bpvl_successor_divides_power) = S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_partial. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_partial * S ((S (bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_partial_bpvl_successor_divides_power))) /\\ ((((exists bpvi_h_bpvl_successor_divides_power_successor. bpvi_h_bpvl_successor_divides_power_successor + S (bpvi_successor_bpvl_successor_divides_power) = S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power)) /\\ exists bpvi_q_bpvl_successor_divides_power_successor. bpvi_u_bpvl_successor_divides_power = bpvi_q_bpvl_successor_divides_power_successor * S ((S (S bpvi_j_bpvl_successor_divides_power)) * bpvi_v_bpvl_successor_divides_power) + (bpvi_successor_bpvl_successor_divides_power))) /\\ bpvi_successor_bpvl_successor_divides_power = bpvi_partial_bpvl_successor_divides_power * bpvi_factor_bpvl_successor_divides_power)))))))) /\\ exists bpvi_divisor_factor_bpvl_successor_divides. a = bpvi_result_bpvl_successor_divides * bpvi_divisor_factor_bpvl_successor_divides)))", "statement_sha256": "c2f7d22ed9ae46992ff28f8511511554c5876f0309a13e0f476b14ed741f4ff5" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "mul_shuffle_four", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 195, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "aa7545f9ef2ad1f493fbe05df6493c0572d386d1559bfa2e9c959bc7d122c036" }, "bertrand_v4_evidence_bundle_sha256": "60d0b0f7674b3d36223e8418ed2e19d2cc06678f71fda132464d51d7a3c2f9e7", "body_checked": true, "body_receipt": { "command_count": 23, "dependency_count": 2, "dne_command_count": 0, "name": "mul_shuffle_four", "proof_depth": 15, "proof_edges": 38, "proof_nodes": 39, "proof_objects": 39, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "mul_assoc", "mul_comm" ], "dependencies_sha256": "aaf14086bacde0f1ae6df1d1349eef5b9864caf0c80c5e41a01342cdff689126", "empty_context_closure": { "body_proof_depth": 15, "body_proof_nodes": 39, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 195, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "19b66d6067bb43e0b83b85fb2608c48f5a9ffd6b14f34b217c852b9e0820cb25", "status": "checked" }, "enrollment_index": 929, "enrollment_origin": "bertrand_b2_valuation_multiplication", "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=195]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=mul_shuffle_four]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "b22e425be8fd79dae3d622c8a2162792ed59ea8230afd5289e2427ed1a60c472", "membership": "alpha_only", "name": "mul_shuffle_four", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro a", "intro b", "intro c", "intro d", "trans a * (b * (c * d))", "apply mul_assoc", "trans a * ((b * c) * d)", "congr", "refl", "symm", "apply mul_assoc", "trans a * ((c * b) * d)", "congr", "refl", "congr", "apply mul_comm", "refl", "trans a * (c * (b * d))", "congr", "refl", "apply mul_assoc", "symm", "apply mul_assoc" ], "script_sha256": "a17254672d78bdbd01fb52bba6cdcbb55f0c5b8e8a883107c18d38bcc8701f33", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall a b c d. 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(a * b) * (c * d) = (a * c) * (b * d)", "statement_sha256": "19b66d6067bb43e0b83b85fb2608c48f5a9ffd6b14f34b217c852b9e0820cb25" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_divides_exponent_antitone", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 196, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "acda496b86e11b401152ff3f85f51acf8518d11898c5ffcb607e7cab3e1b40ec" }, "bertrand_v4_evidence_bundle_sha256": "bc7b5d498b38c6c395a2d5a9049054f5a575ac103de3f32204fb602099310a3c", "body_checked": true, "body_receipt": { "command_count": 51, "dependency_count": 4, "dne_command_count": 0, "name": "power_divides_exponent_antitone", "proof_depth": 33, "proof_edges": 59, "proof_nodes": 60, "proof_objects": 60, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "pow_exists", "pow_add", "add_comm", "mul_assoc" ], "dependencies_sha256": "61ca6b15d35e225e7e5bcc3f5b399cc0a36bbba0ac33fcb17b15a6b4235531ba", "empty_context_closure": { "body_proof_depth": 33, "body_proof_nodes": 60, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 196, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "a55f7b6504d1c5aa7d841ff0127e2ddae4899ef611623bc0994e54f7bc2f878a", "status": "checked" }, "enrollment_index": 930, "enrollment_origin": "bertrand_b2_valuation_multiplication", "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=196]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_exponent_antitone]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "7c7192d729807b610c7a94b93806e27c52cb12142c01c903deffc7ce13159192", "membership": "alpha_only", "name": "power_divides_exponent_antitone", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro e", "intro f", "intro a", "intro hef", "intro hhigh", "cases hef", "cases hhigh", "cases hhigh_witness", "cases hhigh_witness_right", "have hsum : f = e + x", "trans x + e", "symm", "exact hef_witness", "specialize add_comm x", "specialize add_comm e", "exact add_comm", "have hlow_power : exists r. (exists ff_b_bpd_antitone_low_witness ff_c_bpd_antitone_low_witness. ((forall ff_i_bpd_antitone_low_witness_repeat. (exists ff_lt_bpd_antitone_low_witness_repeat_bound. ff_lt_bpd_antitone_low_witness_repeat_bound + S ff_i_bpd_antitone_low_witness_repeat = e) -> (((exists ff_h_bpd_antitone_low_witness_repeat_decoded. ff_h_bpd_antitone_low_witness_repeat_decoded + S (p) = S ((S (ff_i_bpd_antitone_low_witness_repeat)) * ff_c_bpd_antitone_low_witness)) /\\ exists ff_q_bpd_antitone_low_witness_repeat_decoded. ff_b_bpd_antitone_low_witness = ff_q_bpd_antitone_low_witness_repeat_decoded * S ((S (ff_i_bpd_antitone_low_witness_repeat)) * ff_c_bpd_antitone_low_witness) + (p)))) /\\ (exists ff_u_bpd_antitone_low_witness_product ff_v_bpd_antitone_low_witness_product. ((((exists ff_h_bpd_antitone_low_witness_product_start. ff_h_bpd_antitone_low_witness_product_start + S (1) = S ((S (0)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_start. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_start * S ((S (0)) * ff_v_bpd_antitone_low_witness_product) + (1))) /\\ ((((exists ff_h_bpd_antitone_low_witness_product_terminal. ff_h_bpd_antitone_low_witness_product_terminal + S (r) = S ((S (e)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_terminal. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_terminal * S ((S (e)) * ff_v_bpd_antitone_low_witness_product) + (r))) /\\ forall ff_i_bpd_antitone_low_witness_product. (exists ff_lt_bpd_antitone_low_witness_product_bound. ff_lt_bpd_antitone_low_witness_product_bound + S ff_i_bpd_antitone_low_witness_product = e) -> exists ff_p_bpd_antitone_low_witness_product ff_r_bpd_antitone_low_witness_product ff_s_bpd_antitone_low_witness_product. ((((exists ff_h_bpd_antitone_low_witness_product_factor. ff_h_bpd_antitone_low_witness_product_factor + S (ff_p_bpd_antitone_low_witness_product) = S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_c_bpd_antitone_low_witness)) /\\ exists ff_q_bpd_antitone_low_witness_product_factor. ff_b_bpd_antitone_low_witness = ff_q_bpd_antitone_low_witness_product_factor * S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_c_bpd_antitone_low_witness) + (ff_p_bpd_antitone_low_witness_product))) /\\ ((((exists ff_h_bpd_antitone_low_witness_product_partial. ff_h_bpd_antitone_low_witness_product_partial + S (ff_r_bpd_antitone_low_witness_product) = S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_partial. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_partial * S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product) + (ff_r_bpd_antitone_low_witness_product))) /\\ ((((exists ff_h_bpd_antitone_low_witness_product_successor. ff_h_bpd_antitone_low_witness_product_successor + S (ff_s_bpd_antitone_low_witness_product) = S ((S (S ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_successor. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_successor * S ((S (S ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product) + (ff_s_bpd_antitone_low_witness_product))) /\\ ff_s_bpd_antitone_low_witness_product = ff_r_bpd_antitone_low_witness_product * ff_p_bpd_antitone_low_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists e", "exact pow_exists", "cases hlow_power", "have hgap_power : exists r. (exists bpvi_b_bpd_antitone_gap_witness bpvi_c_bpd_antitone_gap_witness. ((forall bpvi_i_bpd_antitone_gap_witness. (exists bpvi_repeat_gap_bpd_antitone_gap_witness. bpvi_repeat_gap_bpd_antitone_gap_witness + S bpvi_i_bpd_antitone_gap_witness = x) -> (((exists bpvi_h_bpd_antitone_gap_witness_repeat. bpvi_h_bpd_antitone_gap_witness_repeat + S (p) = S ((S (bpvi_i_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_repeat. bpvi_b_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_repeat * S ((S (bpvi_i_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness) + (p)))) /\\ (exists bpvi_u_bpd_antitone_gap_witness bpvi_v_bpd_antitone_gap_witness. ((((exists bpvi_h_bpd_antitone_gap_witness_start. bpvi_h_bpd_antitone_gap_witness_start + S (1) = S ((S (0)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_start. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_start * S ((S (0)) * bpvi_v_bpd_antitone_gap_witness) + (1))) /\\ ((((exists bpvi_h_bpd_antitone_gap_witness_terminal. bpvi_h_bpd_antitone_gap_witness_terminal + S (r) = S ((S (x)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_terminal. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_terminal * S ((S (x)) * bpvi_v_bpd_antitone_gap_witness) + (r))) /\\ forall bpvi_j_bpd_antitone_gap_witness. (exists bpvi_product_gap_bpd_antitone_gap_witness. bpvi_product_gap_bpd_antitone_gap_witness + S bpvi_j_bpd_antitone_gap_witness = x) -> exists bpvi_factor_bpd_antitone_gap_witness bpvi_partial_bpd_antitone_gap_witness bpvi_successor_bpd_antitone_gap_witness. ((((exists bpvi_h_bpd_antitone_gap_witness_factor. bpvi_h_bpd_antitone_gap_witness_factor + S (bpvi_factor_bpd_antitone_gap_witness) = S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_factor. bpvi_b_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_factor * S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness) + (bpvi_factor_bpd_antitone_gap_witness))) /\\ ((((exists bpvi_h_bpd_antitone_gap_witness_partial. bpvi_h_bpd_antitone_gap_witness_partial + S (bpvi_partial_bpd_antitone_gap_witness) = S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_partial. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_partial * S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness) + (bpvi_partial_bpd_antitone_gap_witness))) /\\ ((((exists bpvi_h_bpd_antitone_gap_witness_successor. bpvi_h_bpd_antitone_gap_witness_successor + S (bpvi_successor_bpd_antitone_gap_witness) = S ((S (S bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_successor. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_successor * S ((S (S bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness) + (bpvi_successor_bpd_antitone_gap_witness))) /\\ bpvi_successor_bpd_antitone_gap_witness = bpvi_partial_bpd_antitone_gap_witness * bpvi_factor_bpd_antitone_gap_witness))))))))", "specialize pow_exists p", "specialize pow_exists x", "exact pow_exists", "cases hgap_power", "have hfactor : x1 = x3 * x4", "specialize pow_add p", "specialize pow_add e", "specialize pow_add x", "specialize pow_add f", "specialize pow_add x3", "specialize pow_add x4", "specialize pow_add x1", "apply pow_add", "exact hsum", "exact hlow_power_witness", "exact hgap_power_witness", "exact hhigh_witness_left", "exists x3", "split", "exact hlow_power_witness", "exists x4 * x2", "trans x1 * x2", "exact hhigh_witness_right_witness", "trans (x3 * x4) * x2", "congr", "exact hfactor", "refl", "apply mul_assoc" ], "script_sha256": "ce4ed1ef332dd1d0ba25c26b09066a177f0c1888f90785a836d72331068a1423", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p e f a. (exists bpd_gap_antitone_exponents. bpd_gap_antitone_exponents + (e) = (f)) -> (exists bpv_result_antitone_high. ((exists ff_b_antitone_high_power ff_c_antitone_high_power. ((forall ff_i_antitone_high_power_repeat. (exists ff_lt_antitone_high_power_repeat_bound. ff_lt_antitone_high_power_repeat_bound + S ff_i_antitone_high_power_repeat = f) -> (((exists ff_h_antitone_high_power_repeat_decoded. ff_h_antitone_high_power_repeat_decoded + S (p) = S ((S (ff_i_antitone_high_power_repeat)) * ff_c_antitone_high_power)) /\\ exists ff_q_antitone_high_power_repeat_decoded. ff_b_antitone_high_power = ff_q_antitone_high_power_repeat_decoded * S ((S (ff_i_antitone_high_power_repeat)) * ff_c_antitone_high_power) + (p)))) /\\ (exists ff_u_antitone_high_power_product ff_v_antitone_high_power_product. ((((exists ff_h_antitone_high_power_product_start. ff_h_antitone_high_power_product_start + S (1) = S ((S (0)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_start. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_start * S ((S (0)) * ff_v_antitone_high_power_product) + (1))) /\\ ((((exists ff_h_antitone_high_power_product_terminal. ff_h_antitone_high_power_product_terminal + S (bpv_result_antitone_high) = S ((S (f)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_terminal. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_terminal * S ((S (f)) * ff_v_antitone_high_power_product) + (bpv_result_antitone_high))) /\\ forall ff_i_antitone_high_power_product. (exists ff_lt_antitone_high_power_product_bound. ff_lt_antitone_high_power_product_bound + S ff_i_antitone_high_power_product = f) -> exists ff_p_antitone_high_power_product ff_r_antitone_high_power_product ff_s_antitone_high_power_product. ((((exists ff_h_antitone_high_power_product_factor. ff_h_antitone_high_power_product_factor + S (ff_p_antitone_high_power_product) = S ((S (ff_i_antitone_high_power_product)) * ff_c_antitone_high_power)) /\\ exists ff_q_antitone_high_power_product_factor. ff_b_antitone_high_power = ff_q_antitone_high_power_product_factor * S ((S (ff_i_antitone_high_power_product)) * ff_c_antitone_high_power) + (ff_p_antitone_high_power_product))) /\\ ((((exists ff_h_antitone_high_power_product_partial. ff_h_antitone_high_power_product_partial + S (ff_r_antitone_high_power_product) = S ((S (ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_partial. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_partial * S ((S (ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product) + (ff_r_antitone_high_power_product))) /\\ ((((exists ff_h_antitone_high_power_product_successor. ff_h_antitone_high_power_product_successor + S (ff_s_antitone_high_power_product) = S ((S (S ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_successor. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_successor * S ((S (S ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product) + (ff_s_antitone_high_power_product))) /\\ ff_s_antitone_high_power_product = ff_r_antitone_high_power_product * ff_p_antitone_high_power_product)))))))) /\\ (exists bpv_factor_antitone_high_divides. a = bpv_result_antitone_high * bpv_factor_antitone_high_divides))) -> (exists bpv_result_antitone_low. ((exists ff_b_antitone_low_power ff_c_antitone_low_power. ((forall ff_i_antitone_low_power_repeat. (exists ff_lt_antitone_low_power_repeat_bound. ff_lt_antitone_low_power_repeat_bound + S ff_i_antitone_low_power_repeat = e) -> (((exists ff_h_antitone_low_power_repeat_decoded. ff_h_antitone_low_power_repeat_decoded + S (p) = S ((S (ff_i_antitone_low_power_repeat)) * ff_c_antitone_low_power)) /\\ exists ff_q_antitone_low_power_repeat_decoded. ff_b_antitone_low_power = ff_q_antitone_low_power_repeat_decoded * S ((S (ff_i_antitone_low_power_repeat)) * ff_c_antitone_low_power) + (p)))) /\\ (exists ff_u_antitone_low_power_product ff_v_antitone_low_power_product. ((((exists ff_h_antitone_low_power_product_start. ff_h_antitone_low_power_product_start + S (1) = S ((S (0)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_start. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_start * S ((S (0)) * ff_v_antitone_low_power_product) + (1))) /\\ ((((exists ff_h_antitone_low_power_product_terminal. ff_h_antitone_low_power_product_terminal + S (bpv_result_antitone_low) = S ((S (e)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_terminal. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_terminal * S ((S (e)) * ff_v_antitone_low_power_product) + (bpv_result_antitone_low))) /\\ forall ff_i_antitone_low_power_product. (exists ff_lt_antitone_low_power_product_bound. ff_lt_antitone_low_power_product_bound + S ff_i_antitone_low_power_product = e) -> exists ff_p_antitone_low_power_product ff_r_antitone_low_power_product ff_s_antitone_low_power_product. ((((exists ff_h_antitone_low_power_product_factor. ff_h_antitone_low_power_product_factor + S (ff_p_antitone_low_power_product) = S ((S (ff_i_antitone_low_power_product)) * ff_c_antitone_low_power)) /\\ exists ff_q_antitone_low_power_product_factor. ff_b_antitone_low_power = ff_q_antitone_low_power_product_factor * S ((S (ff_i_antitone_low_power_product)) * ff_c_antitone_low_power) + (ff_p_antitone_low_power_product))) /\\ ((((exists ff_h_antitone_low_power_product_partial. ff_h_antitone_low_power_product_partial + S (ff_r_antitone_low_power_product) = S ((S (ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_partial. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_partial * S ((S (ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product) + (ff_r_antitone_low_power_product))) /\\ ((((exists ff_h_antitone_low_power_product_successor. ff_h_antitone_low_power_product_successor + S (ff_s_antitone_low_power_product) = S ((S (S ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_successor. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_successor * S ((S (S ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product) + (ff_s_antitone_low_power_product))) /\\ ff_s_antitone_low_power_product = ff_r_antitone_low_power_product * ff_p_antitone_low_power_product)))))))) /\\ (exists bpv_factor_antitone_low_divides. a = bpv_result_antitone_low * bpv_factor_antitone_low_divides)))", "statement_sha256": "a55f7b6504d1c5aa7d841ff0127e2ddae4899ef611623bc0994e54f7bc2f878a", "summary": "Divisibility by a higher relational power entails every lower exponent.", "summary_sha256": "4631c9623fb5d287aad36be859de350f9950a3afe0de73c8e48ee4ddefaa9a48" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_exists", "pow_add", "add_comm", "mul_assoc" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=196]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_exponent_antitone]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_divides_exponent_antitone", "parent_alpha_version": "v34", "proof_bundle_node_id": 210, "reference_route": "jordan-totient/checkpoint.html#theorem-power_divides_exponent_antitone", "script": [ "intro p", "intro e", "intro f", "intro a", "intro hef", "intro hhigh", "cases hef", "cases hhigh", "cases hhigh_witness", "cases hhigh_witness_right", "have hsum : f = e + x", "trans x + e", "symm", "exact hef_witness", "specialize add_comm x", "specialize add_comm e", "exact add_comm", "have hlow_power : exists r. (exists ff_b_bpd_antitone_low_witness ff_c_bpd_antitone_low_witness. ((forall ff_i_bpd_antitone_low_witness_repeat. (exists ff_lt_bpd_antitone_low_witness_repeat_bound. ff_lt_bpd_antitone_low_witness_repeat_bound + S ff_i_bpd_antitone_low_witness_repeat = e) -> (((exists ff_h_bpd_antitone_low_witness_repeat_decoded. ff_h_bpd_antitone_low_witness_repeat_decoded + S (p) = S ((S (ff_i_bpd_antitone_low_witness_repeat)) * ff_c_bpd_antitone_low_witness)) /\\ exists ff_q_bpd_antitone_low_witness_repeat_decoded. ff_b_bpd_antitone_low_witness = ff_q_bpd_antitone_low_witness_repeat_decoded * S ((S (ff_i_bpd_antitone_low_witness_repeat)) * ff_c_bpd_antitone_low_witness) + (p)))) /\\ (exists ff_u_bpd_antitone_low_witness_product ff_v_bpd_antitone_low_witness_product. ((((exists ff_h_bpd_antitone_low_witness_product_start. ff_h_bpd_antitone_low_witness_product_start + S (1) = S ((S (0)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_start. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_start * S ((S (0)) * ff_v_bpd_antitone_low_witness_product) + (1))) /\\ ((((exists ff_h_bpd_antitone_low_witness_product_terminal. ff_h_bpd_antitone_low_witness_product_terminal + S (r) = S ((S (e)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_terminal. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_terminal * S ((S (e)) * ff_v_bpd_antitone_low_witness_product) + (r))) /\\ forall ff_i_bpd_antitone_low_witness_product. (exists ff_lt_bpd_antitone_low_witness_product_bound. ff_lt_bpd_antitone_low_witness_product_bound + S ff_i_bpd_antitone_low_witness_product = e) -> exists ff_p_bpd_antitone_low_witness_product ff_r_bpd_antitone_low_witness_product ff_s_bpd_antitone_low_witness_product. ((((exists ff_h_bpd_antitone_low_witness_product_factor. ff_h_bpd_antitone_low_witness_product_factor + S (ff_p_bpd_antitone_low_witness_product) = S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_c_bpd_antitone_low_witness)) /\\ exists ff_q_bpd_antitone_low_witness_product_factor. ff_b_bpd_antitone_low_witness = ff_q_bpd_antitone_low_witness_product_factor * S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_c_bpd_antitone_low_witness) + (ff_p_bpd_antitone_low_witness_product))) /\\ ((((exists ff_h_bpd_antitone_low_witness_product_partial. ff_h_bpd_antitone_low_witness_product_partial + S (ff_r_bpd_antitone_low_witness_product) = S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_partial. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_partial * S ((S (ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product) + (ff_r_bpd_antitone_low_witness_product))) /\\ ((((exists ff_h_bpd_antitone_low_witness_product_successor. ff_h_bpd_antitone_low_witness_product_successor + S (ff_s_bpd_antitone_low_witness_product) = S ((S (S ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product)) /\\ exists ff_q_bpd_antitone_low_witness_product_successor. ff_u_bpd_antitone_low_witness_product = ff_q_bpd_antitone_low_witness_product_successor * S ((S (S ff_i_bpd_antitone_low_witness_product)) * ff_v_bpd_antitone_low_witness_product) + (ff_s_bpd_antitone_low_witness_product))) /\\ ff_s_bpd_antitone_low_witness_product = ff_r_bpd_antitone_low_witness_product * ff_p_bpd_antitone_low_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists e", "exact pow_exists", "cases hlow_power", "have hgap_power : exists r. (exists bpvi_b_bpd_antitone_gap_witness bpvi_c_bpd_antitone_gap_witness. ((forall bpvi_i_bpd_antitone_gap_witness. (exists bpvi_repeat_gap_bpd_antitone_gap_witness. bpvi_repeat_gap_bpd_antitone_gap_witness + S bpvi_i_bpd_antitone_gap_witness = x) -> (((exists bpvi_h_bpd_antitone_gap_witness_repeat. bpvi_h_bpd_antitone_gap_witness_repeat + S (p) = S ((S (bpvi_i_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_repeat. bpvi_b_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_repeat * S ((S (bpvi_i_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness) + (p)))) /\\ (exists bpvi_u_bpd_antitone_gap_witness bpvi_v_bpd_antitone_gap_witness. ((((exists bpvi_h_bpd_antitone_gap_witness_start. bpvi_h_bpd_antitone_gap_witness_start + S (1) = S ((S (0)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_start. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_start * S ((S (0)) * bpvi_v_bpd_antitone_gap_witness) + (1))) /\\ ((((exists bpvi_h_bpd_antitone_gap_witness_terminal. bpvi_h_bpd_antitone_gap_witness_terminal + S (r) = S ((S (x)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_terminal. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_terminal * S ((S (x)) * bpvi_v_bpd_antitone_gap_witness) + (r))) /\\ forall bpvi_j_bpd_antitone_gap_witness. (exists bpvi_product_gap_bpd_antitone_gap_witness. bpvi_product_gap_bpd_antitone_gap_witness + S bpvi_j_bpd_antitone_gap_witness = x) -> exists bpvi_factor_bpd_antitone_gap_witness bpvi_partial_bpd_antitone_gap_witness bpvi_successor_bpd_antitone_gap_witness. ((((exists bpvi_h_bpd_antitone_gap_witness_factor. bpvi_h_bpd_antitone_gap_witness_factor + S (bpvi_factor_bpd_antitone_gap_witness) = S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_factor. bpvi_b_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_factor * S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_c_bpd_antitone_gap_witness) + (bpvi_factor_bpd_antitone_gap_witness))) /\\ ((((exists bpvi_h_bpd_antitone_gap_witness_partial. bpvi_h_bpd_antitone_gap_witness_partial + S (bpvi_partial_bpd_antitone_gap_witness) = S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_partial. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_partial * S ((S (bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness) + (bpvi_partial_bpd_antitone_gap_witness))) /\\ ((((exists bpvi_h_bpd_antitone_gap_witness_successor. bpvi_h_bpd_antitone_gap_witness_successor + S (bpvi_successor_bpd_antitone_gap_witness) = S ((S (S bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness)) /\\ exists bpvi_q_bpd_antitone_gap_witness_successor. bpvi_u_bpd_antitone_gap_witness = bpvi_q_bpd_antitone_gap_witness_successor * S ((S (S bpvi_j_bpd_antitone_gap_witness)) * bpvi_v_bpd_antitone_gap_witness) + (bpvi_successor_bpd_antitone_gap_witness))) /\\ bpvi_successor_bpd_antitone_gap_witness = bpvi_partial_bpd_antitone_gap_witness * bpvi_factor_bpd_antitone_gap_witness))))))))", "specialize pow_exists p", "specialize pow_exists x", "exact pow_exists", "cases hgap_power", "have hfactor : x1 = x3 * x4", "specialize pow_add p", "specialize pow_add e", "specialize pow_add x", "specialize pow_add f", "specialize pow_add x3", "specialize pow_add x4", "specialize pow_add x1", "apply pow_add", "exact hsum", "exact hlow_power_witness", "exact hgap_power_witness", "exact hhigh_witness_left", "exists x3", "split", "exact hlow_power_witness", "exists x4 * x2", "trans x1 * x2", "exact hhigh_witness_right_witness", "trans (x3 * x4) * x2", "congr", "exact hfactor", "refl", "apply mul_assoc" ], "script_sha256": "ce4ed1ef332dd1d0ba25c26b09066a177f0c1888f90785a836d72331068a1423", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p e f a. (exists bpd_gap_antitone_exponents. bpd_gap_antitone_exponents + (e) = (f)) -> (exists bpv_result_antitone_high. ((exists ff_b_antitone_high_power ff_c_antitone_high_power. ((forall ff_i_antitone_high_power_repeat. (exists ff_lt_antitone_high_power_repeat_bound. ff_lt_antitone_high_power_repeat_bound + S ff_i_antitone_high_power_repeat = f) -> (((exists ff_h_antitone_high_power_repeat_decoded. ff_h_antitone_high_power_repeat_decoded + S (p) = S ((S (ff_i_antitone_high_power_repeat)) * ff_c_antitone_high_power)) /\\ exists ff_q_antitone_high_power_repeat_decoded. ff_b_antitone_high_power = ff_q_antitone_high_power_repeat_decoded * S ((S (ff_i_antitone_high_power_repeat)) * ff_c_antitone_high_power) + (p)))) /\\ (exists ff_u_antitone_high_power_product ff_v_antitone_high_power_product. ((((exists ff_h_antitone_high_power_product_start. ff_h_antitone_high_power_product_start + S (1) = S ((S (0)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_start. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_start * S ((S (0)) * ff_v_antitone_high_power_product) + (1))) /\\ ((((exists ff_h_antitone_high_power_product_terminal. ff_h_antitone_high_power_product_terminal + S (bpv_result_antitone_high) = S ((S (f)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_terminal. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_terminal * S ((S (f)) * ff_v_antitone_high_power_product) + (bpv_result_antitone_high))) /\\ forall ff_i_antitone_high_power_product. (exists ff_lt_antitone_high_power_product_bound. ff_lt_antitone_high_power_product_bound + S ff_i_antitone_high_power_product = f) -> exists ff_p_antitone_high_power_product ff_r_antitone_high_power_product ff_s_antitone_high_power_product. ((((exists ff_h_antitone_high_power_product_factor. ff_h_antitone_high_power_product_factor + S (ff_p_antitone_high_power_product) = S ((S (ff_i_antitone_high_power_product)) * ff_c_antitone_high_power)) /\\ exists ff_q_antitone_high_power_product_factor. ff_b_antitone_high_power = ff_q_antitone_high_power_product_factor * S ((S (ff_i_antitone_high_power_product)) * ff_c_antitone_high_power) + (ff_p_antitone_high_power_product))) /\\ ((((exists ff_h_antitone_high_power_product_partial. ff_h_antitone_high_power_product_partial + S (ff_r_antitone_high_power_product) = S ((S (ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_partial. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_partial * S ((S (ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product) + (ff_r_antitone_high_power_product))) /\\ ((((exists ff_h_antitone_high_power_product_successor. ff_h_antitone_high_power_product_successor + S (ff_s_antitone_high_power_product) = S ((S (S ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product)) /\\ exists ff_q_antitone_high_power_product_successor. ff_u_antitone_high_power_product = ff_q_antitone_high_power_product_successor * S ((S (S ff_i_antitone_high_power_product)) * ff_v_antitone_high_power_product) + (ff_s_antitone_high_power_product))) /\\ ff_s_antitone_high_power_product = ff_r_antitone_high_power_product * ff_p_antitone_high_power_product)))))))) /\\ (exists bpv_factor_antitone_high_divides. a = bpv_result_antitone_high * bpv_factor_antitone_high_divides))) -> (exists bpv_result_antitone_low. ((exists ff_b_antitone_low_power ff_c_antitone_low_power. ((forall ff_i_antitone_low_power_repeat. (exists ff_lt_antitone_low_power_repeat_bound. ff_lt_antitone_low_power_repeat_bound + S ff_i_antitone_low_power_repeat = e) -> (((exists ff_h_antitone_low_power_repeat_decoded. ff_h_antitone_low_power_repeat_decoded + S (p) = S ((S (ff_i_antitone_low_power_repeat)) * ff_c_antitone_low_power)) /\\ exists ff_q_antitone_low_power_repeat_decoded. ff_b_antitone_low_power = ff_q_antitone_low_power_repeat_decoded * S ((S (ff_i_antitone_low_power_repeat)) * ff_c_antitone_low_power) + (p)))) /\\ (exists ff_u_antitone_low_power_product ff_v_antitone_low_power_product. ((((exists ff_h_antitone_low_power_product_start. ff_h_antitone_low_power_product_start + S (1) = S ((S (0)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_start. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_start * S ((S (0)) * ff_v_antitone_low_power_product) + (1))) /\\ ((((exists ff_h_antitone_low_power_product_terminal. ff_h_antitone_low_power_product_terminal + S (bpv_result_antitone_low) = S ((S (e)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_terminal. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_terminal * S ((S (e)) * ff_v_antitone_low_power_product) + (bpv_result_antitone_low))) /\\ forall ff_i_antitone_low_power_product. (exists ff_lt_antitone_low_power_product_bound. ff_lt_antitone_low_power_product_bound + S ff_i_antitone_low_power_product = e) -> exists ff_p_antitone_low_power_product ff_r_antitone_low_power_product ff_s_antitone_low_power_product. ((((exists ff_h_antitone_low_power_product_factor. ff_h_antitone_low_power_product_factor + S (ff_p_antitone_low_power_product) = S ((S (ff_i_antitone_low_power_product)) * ff_c_antitone_low_power)) /\\ exists ff_q_antitone_low_power_product_factor. ff_b_antitone_low_power = ff_q_antitone_low_power_product_factor * S ((S (ff_i_antitone_low_power_product)) * ff_c_antitone_low_power) + (ff_p_antitone_low_power_product))) /\\ ((((exists ff_h_antitone_low_power_product_partial. ff_h_antitone_low_power_product_partial + S (ff_r_antitone_low_power_product) = S ((S (ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_partial. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_partial * S ((S (ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product) + (ff_r_antitone_low_power_product))) /\\ ((((exists ff_h_antitone_low_power_product_successor. ff_h_antitone_low_power_product_successor + S (ff_s_antitone_low_power_product) = S ((S (S ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product)) /\\ exists ff_q_antitone_low_power_product_successor. ff_u_antitone_low_power_product = ff_q_antitone_low_power_product_successor * S ((S (S ff_i_antitone_low_power_product)) * ff_v_antitone_low_power_product) + (ff_s_antitone_low_power_product))) /\\ ff_s_antitone_low_power_product = ff_r_antitone_low_power_product * ff_p_antitone_low_power_product)))))))) /\\ (exists bpv_factor_antitone_low_divides. a = bpv_result_antitone_low * bpv_factor_antitone_low_divides)))", "statement_sha256": "a55f7b6504d1c5aa7d841ff0127e2ddae4899ef611623bc0994e54f7bc2f878a" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_divides_add_mul", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 197, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "a5b74d0733760ca1ae4a737a71a47f12971146637ba126ec15e832a137cd9898" }, "bertrand_v4_evidence_bundle_sha256": "a66a0ffb7823893af9730283c6b8c69617253e653743159d2cee66834561a586", "body_checked": true, "body_receipt": { "command_count": 47, "dependency_count": 3, "dne_command_count": 0, "name": "power_divides_add_mul", "proof_depth": 36, "proof_edges": 60, "proof_nodes": 61, "proof_objects": 61, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "pow_exists", "pow_add", "mul_shuffle_four" ], "dependencies_sha256": "cc8d38645fd8f65fe03ab787d501a682cb3dc279ab231dd3d8896bd3e0038a2c", "empty_context_closure": { "body_proof_depth": 36, "body_proof_nodes": 61, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 197, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "bc59b8c39cfce44b3beef19b9717d275dbb189d1e5d45645f644061759ff5a6e", "status": "checked" }, "enrollment_index": 931, "enrollment_origin": "bertrand_b2_valuation_multiplication", "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=197]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_add_mul]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "8fc312cd66ec76fcb66d0d9b6bb10aa9a1259cdba8fb24816b026c3d3b197bce", "membership": "alpha_only", "name": "power_divides_add_mul", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro e", "intro f", "intro s", "intro a", "intro b", "intro hsum", "intro hleft", "intro hright", "cases hleft", "cases hleft_witness", "cases hleft_witness_right", "cases hright", "cases hright_witness", "cases hright_witness_right", "have htotal : exists r. (exists ff_b_bpd_add_mul_total_witness ff_c_bpd_add_mul_total_witness. ((forall ff_i_bpd_add_mul_total_witness_repeat. (exists ff_lt_bpd_add_mul_total_witness_repeat_bound. ff_lt_bpd_add_mul_total_witness_repeat_bound + S ff_i_bpd_add_mul_total_witness_repeat = s) -> (((exists ff_h_bpd_add_mul_total_witness_repeat_decoded. ff_h_bpd_add_mul_total_witness_repeat_decoded + S (p) = S ((S (ff_i_bpd_add_mul_total_witness_repeat)) * ff_c_bpd_add_mul_total_witness)) /\\ exists ff_q_bpd_add_mul_total_witness_repeat_decoded. ff_b_bpd_add_mul_total_witness = ff_q_bpd_add_mul_total_witness_repeat_decoded * S ((S (ff_i_bpd_add_mul_total_witness_repeat)) * ff_c_bpd_add_mul_total_witness) + (p)))) /\\ (exists ff_u_bpd_add_mul_total_witness_product ff_v_bpd_add_mul_total_witness_product. ((((exists ff_h_bpd_add_mul_total_witness_product_start. ff_h_bpd_add_mul_total_witness_product_start + S (1) = S ((S (0)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_start. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_start * S ((S (0)) * ff_v_bpd_add_mul_total_witness_product) + (1))) /\\ ((((exists ff_h_bpd_add_mul_total_witness_product_terminal. ff_h_bpd_add_mul_total_witness_product_terminal + S (r) = S ((S (s)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_terminal. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_terminal * S ((S (s)) * ff_v_bpd_add_mul_total_witness_product) + (r))) /\\ forall ff_i_bpd_add_mul_total_witness_product. (exists ff_lt_bpd_add_mul_total_witness_product_bound. ff_lt_bpd_add_mul_total_witness_product_bound + S ff_i_bpd_add_mul_total_witness_product = s) -> exists ff_p_bpd_add_mul_total_witness_product ff_r_bpd_add_mul_total_witness_product ff_s_bpd_add_mul_total_witness_product. ((((exists ff_h_bpd_add_mul_total_witness_product_factor. ff_h_bpd_add_mul_total_witness_product_factor + S (ff_p_bpd_add_mul_total_witness_product) = S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_c_bpd_add_mul_total_witness)) /\\ exists ff_q_bpd_add_mul_total_witness_product_factor. ff_b_bpd_add_mul_total_witness = ff_q_bpd_add_mul_total_witness_product_factor * S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_c_bpd_add_mul_total_witness) + (ff_p_bpd_add_mul_total_witness_product))) /\\ ((((exists ff_h_bpd_add_mul_total_witness_product_partial. ff_h_bpd_add_mul_total_witness_product_partial + S (ff_r_bpd_add_mul_total_witness_product) = S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_partial. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_partial * S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product) + (ff_r_bpd_add_mul_total_witness_product))) /\\ ((((exists ff_h_bpd_add_mul_total_witness_product_successor. ff_h_bpd_add_mul_total_witness_product_successor + S (ff_s_bpd_add_mul_total_witness_product) = S ((S (S ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_successor. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_successor * S ((S (S ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product) + (ff_s_bpd_add_mul_total_witness_product))) /\\ ff_s_bpd_add_mul_total_witness_product = ff_r_bpd_add_mul_total_witness_product * ff_p_bpd_add_mul_total_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists s", "exact pow_exists", "cases htotal", "have hpower_product : x4 = x * x2", "specialize pow_add p", "specialize pow_add e", "specialize pow_add f", "specialize pow_add s", "specialize pow_add x", "specialize pow_add x2", "specialize pow_add x4", "apply pow_add", "exact hsum", "exact hleft_witness_left", "exact hright_witness_left", "exact htotal_witness", "exists x4", "split", "exact htotal_witness", "exists x1 * x3", "trans (x * x1) * (x2 * x3)", "congr", "exact hleft_witness_right_witness", "exact hright_witness_right_witness", "trans (x * x2) * (x1 * x3)", "apply mul_shuffle_four", "congr", "symm", "exact hpower_product", "refl" ], "script_sha256": "ed66ced275461010ccdecf03cf82f9c2175322176aaf079445e69080eccd4953", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p e f s a b. s = e + f -> (exists bpv_result_add_mul_left. ((exists ff_b_add_mul_left_power ff_c_add_mul_left_power. ((forall ff_i_add_mul_left_power_repeat. (exists ff_lt_add_mul_left_power_repeat_bound. ff_lt_add_mul_left_power_repeat_bound + S ff_i_add_mul_left_power_repeat = e) -> (((exists ff_h_add_mul_left_power_repeat_decoded. ff_h_add_mul_left_power_repeat_decoded + S (p) = S ((S (ff_i_add_mul_left_power_repeat)) * ff_c_add_mul_left_power)) /\\ exists ff_q_add_mul_left_power_repeat_decoded. ff_b_add_mul_left_power = ff_q_add_mul_left_power_repeat_decoded * S ((S (ff_i_add_mul_left_power_repeat)) * ff_c_add_mul_left_power) + (p)))) /\\ (exists ff_u_add_mul_left_power_product ff_v_add_mul_left_power_product. ((((exists ff_h_add_mul_left_power_product_start. ff_h_add_mul_left_power_product_start + S (1) = S ((S (0)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_start. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_start * S ((S (0)) * ff_v_add_mul_left_power_product) + (1))) /\\ ((((exists ff_h_add_mul_left_power_product_terminal. ff_h_add_mul_left_power_product_terminal + S (bpv_result_add_mul_left) = S ((S (e)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_terminal. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_terminal * S ((S (e)) * ff_v_add_mul_left_power_product) + (bpv_result_add_mul_left))) /\\ forall ff_i_add_mul_left_power_product. (exists ff_lt_add_mul_left_power_product_bound. ff_lt_add_mul_left_power_product_bound + S ff_i_add_mul_left_power_product = e) -> exists ff_p_add_mul_left_power_product ff_r_add_mul_left_power_product ff_s_add_mul_left_power_product. ((((exists ff_h_add_mul_left_power_product_factor. ff_h_add_mul_left_power_product_factor + S (ff_p_add_mul_left_power_product) = S ((S (ff_i_add_mul_left_power_product)) * ff_c_add_mul_left_power)) /\\ exists ff_q_add_mul_left_power_product_factor. ff_b_add_mul_left_power = ff_q_add_mul_left_power_product_factor * S ((S (ff_i_add_mul_left_power_product)) * ff_c_add_mul_left_power) + (ff_p_add_mul_left_power_product))) /\\ ((((exists ff_h_add_mul_left_power_product_partial. ff_h_add_mul_left_power_product_partial + S (ff_r_add_mul_left_power_product) = S ((S (ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_partial. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_partial * S ((S (ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product) + (ff_r_add_mul_left_power_product))) /\\ ((((exists ff_h_add_mul_left_power_product_successor. ff_h_add_mul_left_power_product_successor + S (ff_s_add_mul_left_power_product) = S ((S (S ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_successor. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_successor * S ((S (S ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product) + (ff_s_add_mul_left_power_product))) /\\ ff_s_add_mul_left_power_product = ff_r_add_mul_left_power_product * ff_p_add_mul_left_power_product)))))))) /\\ (exists bpv_factor_add_mul_left_divides. a = bpv_result_add_mul_left * bpv_factor_add_mul_left_divides))) -> (exists bpv_result_add_mul_right. ((exists ff_b_add_mul_right_power ff_c_add_mul_right_power. ((forall ff_i_add_mul_right_power_repeat. (exists ff_lt_add_mul_right_power_repeat_bound. ff_lt_add_mul_right_power_repeat_bound + S ff_i_add_mul_right_power_repeat = f) -> (((exists ff_h_add_mul_right_power_repeat_decoded. ff_h_add_mul_right_power_repeat_decoded + S (p) = S ((S (ff_i_add_mul_right_power_repeat)) * ff_c_add_mul_right_power)) /\\ exists ff_q_add_mul_right_power_repeat_decoded. ff_b_add_mul_right_power = ff_q_add_mul_right_power_repeat_decoded * S ((S (ff_i_add_mul_right_power_repeat)) * ff_c_add_mul_right_power) + (p)))) /\\ (exists ff_u_add_mul_right_power_product ff_v_add_mul_right_power_product. ((((exists ff_h_add_mul_right_power_product_start. ff_h_add_mul_right_power_product_start + S (1) = S ((S (0)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_start. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_start * S ((S (0)) * ff_v_add_mul_right_power_product) + (1))) /\\ ((((exists ff_h_add_mul_right_power_product_terminal. ff_h_add_mul_right_power_product_terminal + S (bpv_result_add_mul_right) = S ((S (f)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_terminal. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_terminal * S ((S (f)) * ff_v_add_mul_right_power_product) + (bpv_result_add_mul_right))) /\\ forall ff_i_add_mul_right_power_product. (exists ff_lt_add_mul_right_power_product_bound. ff_lt_add_mul_right_power_product_bound + S ff_i_add_mul_right_power_product = f) -> exists ff_p_add_mul_right_power_product ff_r_add_mul_right_power_product ff_s_add_mul_right_power_product. ((((exists ff_h_add_mul_right_power_product_factor. ff_h_add_mul_right_power_product_factor + S (ff_p_add_mul_right_power_product) = S ((S (ff_i_add_mul_right_power_product)) * ff_c_add_mul_right_power)) /\\ exists ff_q_add_mul_right_power_product_factor. ff_b_add_mul_right_power = ff_q_add_mul_right_power_product_factor * S ((S (ff_i_add_mul_right_power_product)) * ff_c_add_mul_right_power) + (ff_p_add_mul_right_power_product))) /\\ ((((exists ff_h_add_mul_right_power_product_partial. ff_h_add_mul_right_power_product_partial + S (ff_r_add_mul_right_power_product) = S ((S (ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_partial. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_partial * S ((S (ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product) + (ff_r_add_mul_right_power_product))) /\\ ((((exists ff_h_add_mul_right_power_product_successor. ff_h_add_mul_right_power_product_successor + S (ff_s_add_mul_right_power_product) = S ((S (S ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_successor. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_successor * S ((S (S ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product) + (ff_s_add_mul_right_power_product))) /\\ ff_s_add_mul_right_power_product = ff_r_add_mul_right_power_product * ff_p_add_mul_right_power_product)))))))) /\\ (exists bpv_factor_add_mul_right_divides. b = bpv_result_add_mul_right * bpv_factor_add_mul_right_divides))) -> (exists bpvi_result_add_mul_result. ((exists bpvi_b_add_mul_result_power bpvi_c_add_mul_result_power. ((forall bpvi_i_add_mul_result_power. (exists bpvi_repeat_gap_add_mul_result_power. bpvi_repeat_gap_add_mul_result_power + S bpvi_i_add_mul_result_power = s) -> (((exists bpvi_h_add_mul_result_power_repeat. bpvi_h_add_mul_result_power_repeat + S (p) = S ((S (bpvi_i_add_mul_result_power)) * bpvi_c_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_repeat. bpvi_b_add_mul_result_power = bpvi_q_add_mul_result_power_repeat * S ((S (bpvi_i_add_mul_result_power)) * bpvi_c_add_mul_result_power) + (p)))) /\\ (exists bpvi_u_add_mul_result_power bpvi_v_add_mul_result_power. ((((exists bpvi_h_add_mul_result_power_start. bpvi_h_add_mul_result_power_start + S (1) = S ((S (0)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_start. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_start * S ((S (0)) * bpvi_v_add_mul_result_power) + (1))) /\\ ((((exists bpvi_h_add_mul_result_power_terminal. bpvi_h_add_mul_result_power_terminal + S (bpvi_result_add_mul_result) = S ((S (s)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_terminal. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_terminal * S ((S (s)) * bpvi_v_add_mul_result_power) + (bpvi_result_add_mul_result))) /\\ forall bpvi_j_add_mul_result_power. (exists bpvi_product_gap_add_mul_result_power. bpvi_product_gap_add_mul_result_power + S bpvi_j_add_mul_result_power = s) -> exists bpvi_factor_add_mul_result_power bpvi_partial_add_mul_result_power bpvi_successor_add_mul_result_power. ((((exists bpvi_h_add_mul_result_power_factor. bpvi_h_add_mul_result_power_factor + S (bpvi_factor_add_mul_result_power) = S ((S (bpvi_j_add_mul_result_power)) * bpvi_c_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_factor. bpvi_b_add_mul_result_power = bpvi_q_add_mul_result_power_factor * S ((S (bpvi_j_add_mul_result_power)) * bpvi_c_add_mul_result_power) + (bpvi_factor_add_mul_result_power))) /\\ ((((exists bpvi_h_add_mul_result_power_partial. bpvi_h_add_mul_result_power_partial + S (bpvi_partial_add_mul_result_power) = S ((S (bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_partial. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_partial * S ((S (bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power) + (bpvi_partial_add_mul_result_power))) /\\ ((((exists bpvi_h_add_mul_result_power_successor. bpvi_h_add_mul_result_power_successor + S (bpvi_successor_add_mul_result_power) = S ((S (S bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_successor. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_successor * S ((S (S bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power) + (bpvi_successor_add_mul_result_power))) /\\ bpvi_successor_add_mul_result_power = bpvi_partial_add_mul_result_power * bpvi_factor_add_mul_result_power)))))))) /\\ exists bpvi_divisor_factor_add_mul_result. a * b = bpvi_result_add_mul_result * bpvi_divisor_factor_add_mul_result))", "statement_sha256": "bc59b8c39cfce44b3beef19b9717d275dbb189d1e5d45645f644061759ff5a6e", "summary": "Multiplying power divisors adds their exponents.", "summary_sha256": "3327a63642a709a62e3e12ba530676b2a7e3b2e115dabf4f01394709380e318b" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_exists", "pow_add", "mul_shuffle_four" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=197]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_add_mul]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_divides_add_mul", "parent_alpha_version": "v34", "proof_bundle_node_id": 211, "reference_route": "jordan-totient/checkpoint.html#theorem-power_divides_add_mul", "script": [ "intro p", "intro e", "intro f", "intro s", "intro a", "intro b", "intro hsum", "intro hleft", "intro hright", "cases hleft", "cases hleft_witness", "cases hleft_witness_right", "cases hright", "cases hright_witness", "cases hright_witness_right", "have htotal : exists r. (exists ff_b_bpd_add_mul_total_witness ff_c_bpd_add_mul_total_witness. ((forall ff_i_bpd_add_mul_total_witness_repeat. (exists ff_lt_bpd_add_mul_total_witness_repeat_bound. ff_lt_bpd_add_mul_total_witness_repeat_bound + S ff_i_bpd_add_mul_total_witness_repeat = s) -> (((exists ff_h_bpd_add_mul_total_witness_repeat_decoded. ff_h_bpd_add_mul_total_witness_repeat_decoded + S (p) = S ((S (ff_i_bpd_add_mul_total_witness_repeat)) * ff_c_bpd_add_mul_total_witness)) /\\ exists ff_q_bpd_add_mul_total_witness_repeat_decoded. ff_b_bpd_add_mul_total_witness = ff_q_bpd_add_mul_total_witness_repeat_decoded * S ((S (ff_i_bpd_add_mul_total_witness_repeat)) * ff_c_bpd_add_mul_total_witness) + (p)))) /\\ (exists ff_u_bpd_add_mul_total_witness_product ff_v_bpd_add_mul_total_witness_product. ((((exists ff_h_bpd_add_mul_total_witness_product_start. ff_h_bpd_add_mul_total_witness_product_start + S (1) = S ((S (0)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_start. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_start * S ((S (0)) * ff_v_bpd_add_mul_total_witness_product) + (1))) /\\ ((((exists ff_h_bpd_add_mul_total_witness_product_terminal. ff_h_bpd_add_mul_total_witness_product_terminal + S (r) = S ((S (s)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_terminal. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_terminal * S ((S (s)) * ff_v_bpd_add_mul_total_witness_product) + (r))) /\\ forall ff_i_bpd_add_mul_total_witness_product. (exists ff_lt_bpd_add_mul_total_witness_product_bound. ff_lt_bpd_add_mul_total_witness_product_bound + S ff_i_bpd_add_mul_total_witness_product = s) -> exists ff_p_bpd_add_mul_total_witness_product ff_r_bpd_add_mul_total_witness_product ff_s_bpd_add_mul_total_witness_product. ((((exists ff_h_bpd_add_mul_total_witness_product_factor. ff_h_bpd_add_mul_total_witness_product_factor + S (ff_p_bpd_add_mul_total_witness_product) = S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_c_bpd_add_mul_total_witness)) /\\ exists ff_q_bpd_add_mul_total_witness_product_factor. ff_b_bpd_add_mul_total_witness = ff_q_bpd_add_mul_total_witness_product_factor * S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_c_bpd_add_mul_total_witness) + (ff_p_bpd_add_mul_total_witness_product))) /\\ ((((exists ff_h_bpd_add_mul_total_witness_product_partial. ff_h_bpd_add_mul_total_witness_product_partial + S (ff_r_bpd_add_mul_total_witness_product) = S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_partial. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_partial * S ((S (ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product) + (ff_r_bpd_add_mul_total_witness_product))) /\\ ((((exists ff_h_bpd_add_mul_total_witness_product_successor. ff_h_bpd_add_mul_total_witness_product_successor + S (ff_s_bpd_add_mul_total_witness_product) = S ((S (S ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product)) /\\ exists ff_q_bpd_add_mul_total_witness_product_successor. ff_u_bpd_add_mul_total_witness_product = ff_q_bpd_add_mul_total_witness_product_successor * S ((S (S ff_i_bpd_add_mul_total_witness_product)) * ff_v_bpd_add_mul_total_witness_product) + (ff_s_bpd_add_mul_total_witness_product))) /\\ ff_s_bpd_add_mul_total_witness_product = ff_r_bpd_add_mul_total_witness_product * ff_p_bpd_add_mul_total_witness_product))))))))", "specialize pow_exists p", "specialize pow_exists s", "exact pow_exists", "cases htotal", "have hpower_product : x4 = x * x2", "specialize pow_add p", "specialize pow_add e", "specialize pow_add f", "specialize pow_add s", "specialize pow_add x", "specialize pow_add x2", "specialize pow_add x4", "apply pow_add", "exact hsum", "exact hleft_witness_left", "exact hright_witness_left", "exact htotal_witness", "exists x4", "split", "exact htotal_witness", "exists x1 * x3", "trans (x * x1) * (x2 * x3)", "congr", "exact hleft_witness_right_witness", "exact hright_witness_right_witness", "trans (x * x2) * (x1 * x3)", "apply mul_shuffle_four", "congr", "symm", "exact hpower_product", "refl" ], "script_sha256": "ed66ced275461010ccdecf03cf82f9c2175322176aaf079445e69080eccd4953", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p e f s a b. s = e + f -> (exists bpv_result_add_mul_left. ((exists ff_b_add_mul_left_power ff_c_add_mul_left_power. ((forall ff_i_add_mul_left_power_repeat. (exists ff_lt_add_mul_left_power_repeat_bound. ff_lt_add_mul_left_power_repeat_bound + S ff_i_add_mul_left_power_repeat = e) -> (((exists ff_h_add_mul_left_power_repeat_decoded. ff_h_add_mul_left_power_repeat_decoded + S (p) = S ((S (ff_i_add_mul_left_power_repeat)) * ff_c_add_mul_left_power)) /\\ exists ff_q_add_mul_left_power_repeat_decoded. ff_b_add_mul_left_power = ff_q_add_mul_left_power_repeat_decoded * S ((S (ff_i_add_mul_left_power_repeat)) * ff_c_add_mul_left_power) + (p)))) /\\ (exists ff_u_add_mul_left_power_product ff_v_add_mul_left_power_product. ((((exists ff_h_add_mul_left_power_product_start. ff_h_add_mul_left_power_product_start + S (1) = S ((S (0)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_start. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_start * S ((S (0)) * ff_v_add_mul_left_power_product) + (1))) /\\ ((((exists ff_h_add_mul_left_power_product_terminal. ff_h_add_mul_left_power_product_terminal + S (bpv_result_add_mul_left) = S ((S (e)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_terminal. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_terminal * S ((S (e)) * ff_v_add_mul_left_power_product) + (bpv_result_add_mul_left))) /\\ forall ff_i_add_mul_left_power_product. (exists ff_lt_add_mul_left_power_product_bound. ff_lt_add_mul_left_power_product_bound + S ff_i_add_mul_left_power_product = e) -> exists ff_p_add_mul_left_power_product ff_r_add_mul_left_power_product ff_s_add_mul_left_power_product. ((((exists ff_h_add_mul_left_power_product_factor. ff_h_add_mul_left_power_product_factor + S (ff_p_add_mul_left_power_product) = S ((S (ff_i_add_mul_left_power_product)) * ff_c_add_mul_left_power)) /\\ exists ff_q_add_mul_left_power_product_factor. ff_b_add_mul_left_power = ff_q_add_mul_left_power_product_factor * S ((S (ff_i_add_mul_left_power_product)) * ff_c_add_mul_left_power) + (ff_p_add_mul_left_power_product))) /\\ ((((exists ff_h_add_mul_left_power_product_partial. ff_h_add_mul_left_power_product_partial + S (ff_r_add_mul_left_power_product) = S ((S (ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_partial. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_partial * S ((S (ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product) + (ff_r_add_mul_left_power_product))) /\\ ((((exists ff_h_add_mul_left_power_product_successor. ff_h_add_mul_left_power_product_successor + S (ff_s_add_mul_left_power_product) = S ((S (S ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product)) /\\ exists ff_q_add_mul_left_power_product_successor. ff_u_add_mul_left_power_product = ff_q_add_mul_left_power_product_successor * S ((S (S ff_i_add_mul_left_power_product)) * ff_v_add_mul_left_power_product) + (ff_s_add_mul_left_power_product))) /\\ ff_s_add_mul_left_power_product = ff_r_add_mul_left_power_product * ff_p_add_mul_left_power_product)))))))) /\\ (exists bpv_factor_add_mul_left_divides. a = bpv_result_add_mul_left * bpv_factor_add_mul_left_divides))) -> (exists bpv_result_add_mul_right. ((exists ff_b_add_mul_right_power ff_c_add_mul_right_power. ((forall ff_i_add_mul_right_power_repeat. (exists ff_lt_add_mul_right_power_repeat_bound. ff_lt_add_mul_right_power_repeat_bound + S ff_i_add_mul_right_power_repeat = f) -> (((exists ff_h_add_mul_right_power_repeat_decoded. ff_h_add_mul_right_power_repeat_decoded + S (p) = S ((S (ff_i_add_mul_right_power_repeat)) * ff_c_add_mul_right_power)) /\\ exists ff_q_add_mul_right_power_repeat_decoded. ff_b_add_mul_right_power = ff_q_add_mul_right_power_repeat_decoded * S ((S (ff_i_add_mul_right_power_repeat)) * ff_c_add_mul_right_power) + (p)))) /\\ (exists ff_u_add_mul_right_power_product ff_v_add_mul_right_power_product. ((((exists ff_h_add_mul_right_power_product_start. ff_h_add_mul_right_power_product_start + S (1) = S ((S (0)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_start. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_start * S ((S (0)) * ff_v_add_mul_right_power_product) + (1))) /\\ ((((exists ff_h_add_mul_right_power_product_terminal. ff_h_add_mul_right_power_product_terminal + S (bpv_result_add_mul_right) = S ((S (f)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_terminal. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_terminal * S ((S (f)) * ff_v_add_mul_right_power_product) + (bpv_result_add_mul_right))) /\\ forall ff_i_add_mul_right_power_product. (exists ff_lt_add_mul_right_power_product_bound. ff_lt_add_mul_right_power_product_bound + S ff_i_add_mul_right_power_product = f) -> exists ff_p_add_mul_right_power_product ff_r_add_mul_right_power_product ff_s_add_mul_right_power_product. ((((exists ff_h_add_mul_right_power_product_factor. ff_h_add_mul_right_power_product_factor + S (ff_p_add_mul_right_power_product) = S ((S (ff_i_add_mul_right_power_product)) * ff_c_add_mul_right_power)) /\\ exists ff_q_add_mul_right_power_product_factor. ff_b_add_mul_right_power = ff_q_add_mul_right_power_product_factor * S ((S (ff_i_add_mul_right_power_product)) * ff_c_add_mul_right_power) + (ff_p_add_mul_right_power_product))) /\\ ((((exists ff_h_add_mul_right_power_product_partial. ff_h_add_mul_right_power_product_partial + S (ff_r_add_mul_right_power_product) = S ((S (ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_partial. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_partial * S ((S (ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product) + (ff_r_add_mul_right_power_product))) /\\ ((((exists ff_h_add_mul_right_power_product_successor. ff_h_add_mul_right_power_product_successor + S (ff_s_add_mul_right_power_product) = S ((S (S ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product)) /\\ exists ff_q_add_mul_right_power_product_successor. ff_u_add_mul_right_power_product = ff_q_add_mul_right_power_product_successor * S ((S (S ff_i_add_mul_right_power_product)) * ff_v_add_mul_right_power_product) + (ff_s_add_mul_right_power_product))) /\\ ff_s_add_mul_right_power_product = ff_r_add_mul_right_power_product * ff_p_add_mul_right_power_product)))))))) /\\ (exists bpv_factor_add_mul_right_divides. b = bpv_result_add_mul_right * bpv_factor_add_mul_right_divides))) -> (exists bpvi_result_add_mul_result. ((exists bpvi_b_add_mul_result_power bpvi_c_add_mul_result_power. ((forall bpvi_i_add_mul_result_power. (exists bpvi_repeat_gap_add_mul_result_power. bpvi_repeat_gap_add_mul_result_power + S bpvi_i_add_mul_result_power = s) -> (((exists bpvi_h_add_mul_result_power_repeat. bpvi_h_add_mul_result_power_repeat + S (p) = S ((S (bpvi_i_add_mul_result_power)) * bpvi_c_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_repeat. bpvi_b_add_mul_result_power = bpvi_q_add_mul_result_power_repeat * S ((S (bpvi_i_add_mul_result_power)) * bpvi_c_add_mul_result_power) + (p)))) /\\ (exists bpvi_u_add_mul_result_power bpvi_v_add_mul_result_power. ((((exists bpvi_h_add_mul_result_power_start. bpvi_h_add_mul_result_power_start + S (1) = S ((S (0)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_start. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_start * S ((S (0)) * bpvi_v_add_mul_result_power) + (1))) /\\ ((((exists bpvi_h_add_mul_result_power_terminal. bpvi_h_add_mul_result_power_terminal + S (bpvi_result_add_mul_result) = S ((S (s)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_terminal. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_terminal * S ((S (s)) * bpvi_v_add_mul_result_power) + (bpvi_result_add_mul_result))) /\\ forall bpvi_j_add_mul_result_power. (exists bpvi_product_gap_add_mul_result_power. bpvi_product_gap_add_mul_result_power + S bpvi_j_add_mul_result_power = s) -> exists bpvi_factor_add_mul_result_power bpvi_partial_add_mul_result_power bpvi_successor_add_mul_result_power. ((((exists bpvi_h_add_mul_result_power_factor. bpvi_h_add_mul_result_power_factor + S (bpvi_factor_add_mul_result_power) = S ((S (bpvi_j_add_mul_result_power)) * bpvi_c_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_factor. bpvi_b_add_mul_result_power = bpvi_q_add_mul_result_power_factor * S ((S (bpvi_j_add_mul_result_power)) * bpvi_c_add_mul_result_power) + (bpvi_factor_add_mul_result_power))) /\\ ((((exists bpvi_h_add_mul_result_power_partial. bpvi_h_add_mul_result_power_partial + S (bpvi_partial_add_mul_result_power) = S ((S (bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_partial. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_partial * S ((S (bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power) + (bpvi_partial_add_mul_result_power))) /\\ ((((exists bpvi_h_add_mul_result_power_successor. bpvi_h_add_mul_result_power_successor + S (bpvi_successor_add_mul_result_power) = S ((S (S bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power)) /\\ exists bpvi_q_add_mul_result_power_successor. bpvi_u_add_mul_result_power = bpvi_q_add_mul_result_power_successor * S ((S (S bpvi_j_add_mul_result_power)) * bpvi_v_add_mul_result_power) + (bpvi_successor_add_mul_result_power))) /\\ bpvi_successor_add_mul_result_power = bpvi_partial_add_mul_result_power * bpvi_factor_add_mul_result_power)))))))) /\\ exists bpvi_divisor_factor_add_mul_result. a * b = bpvi_result_add_mul_result * bpvi_divisor_factor_add_mul_result))", "statement_sha256": "bc59b8c39cfce44b3beef19b9717d275dbb189d1e5d45645f644061759ff5a6e" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_divides_successor_of_cofactor", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 198, "bundle_sha256": 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"script": [ "intro p", "intro e", "intro a", "intro r", "intro q", "intro hr", "intro ha", "intro hq", "cases hq", "have hsuccessor : exists s. (exists bpvi_b_bpd_successor_witness bpvi_c_bpd_successor_witness. ((forall bpvi_i_bpd_successor_witness. (exists bpvi_repeat_gap_bpd_successor_witness. bpvi_repeat_gap_bpd_successor_witness + S bpvi_i_bpd_successor_witness = S e) -> (((exists bpvi_h_bpd_successor_witness_repeat. bpvi_h_bpd_successor_witness_repeat + S (p) = S ((S (bpvi_i_bpd_successor_witness)) * bpvi_c_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_repeat. bpvi_b_bpd_successor_witness = bpvi_q_bpd_successor_witness_repeat * S ((S (bpvi_i_bpd_successor_witness)) * bpvi_c_bpd_successor_witness) + (p)))) /\\ (exists bpvi_u_bpd_successor_witness bpvi_v_bpd_successor_witness. ((((exists bpvi_h_bpd_successor_witness_start. bpvi_h_bpd_successor_witness_start + S (1) = S ((S (0)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_start. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_start * S ((S (0)) * bpvi_v_bpd_successor_witness) + (1))) /\\ ((((exists bpvi_h_bpd_successor_witness_terminal. bpvi_h_bpd_successor_witness_terminal + S (s) = S ((S (S e)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_terminal. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_terminal * S ((S (S e)) * bpvi_v_bpd_successor_witness) + (s))) /\\ forall bpvi_j_bpd_successor_witness. (exists bpvi_product_gap_bpd_successor_witness. bpvi_product_gap_bpd_successor_witness + S bpvi_j_bpd_successor_witness = S e) -> exists bpvi_factor_bpd_successor_witness bpvi_partial_bpd_successor_witness bpvi_successor_bpd_successor_witness. ((((exists bpvi_h_bpd_successor_witness_factor. bpvi_h_bpd_successor_witness_factor + S (bpvi_factor_bpd_successor_witness) = S ((S (bpvi_j_bpd_successor_witness)) * bpvi_c_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_factor. bpvi_b_bpd_successor_witness = bpvi_q_bpd_successor_witness_factor * S ((S (bpvi_j_bpd_successor_witness)) * bpvi_c_bpd_successor_witness) + (bpvi_factor_bpd_successor_witness))) /\\ ((((exists bpvi_h_bpd_successor_witness_partial. bpvi_h_bpd_successor_witness_partial + S (bpvi_partial_bpd_successor_witness) = S ((S (bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_partial. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_partial * S ((S (bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness) + (bpvi_partial_bpd_successor_witness))) /\\ ((((exists bpvi_h_bpd_successor_witness_successor. bpvi_h_bpd_successor_witness_successor + S (bpvi_successor_bpd_successor_witness) = S ((S (S bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_successor. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_successor * S ((S (S bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness) + (bpvi_successor_bpd_successor_witness))) /\\ bpvi_successor_bpd_successor_witness = bpvi_partial_bpd_successor_witness * bpvi_factor_bpd_successor_witness))))))))", "specialize pow_exists p", "specialize pow_exists (S e)", "exact pow_exists", "cases hsuccessor", "have hs : x1 = r * p", "specialize pow_successor_pair_mul p", "specialize pow_successor_pair_mul e", "specialize pow_successor_pair_mul (S e)", "specialize pow_successor_pair_mul r", "specialize pow_successor_pair_mul x1", "apply pow_successor_pair_mul", "refl", "exact hr", "exact hsuccessor_witness", "exists x1", "split", "exact hsuccessor_witness", "exists x", "trans r * q", "exact ha", "rewrite hq_witness", "trans (r * p) * x", "symm", "apply mul_assoc", "congr", "symm", "exact hs", "refl" ], "script_sha256": "0a5d9f4172e877812c4b4b592f04926a16998e6d1577de9458650796f342ba9b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p e a r q. (exists ff_b_bpd_successor_prefix ff_c_bpd_successor_prefix. ((forall ff_i_bpd_successor_prefix_repeat. (exists ff_lt_bpd_successor_prefix_repeat_bound. ff_lt_bpd_successor_prefix_repeat_bound + S ff_i_bpd_successor_prefix_repeat = e) -> (((exists ff_h_bpd_successor_prefix_repeat_decoded. ff_h_bpd_successor_prefix_repeat_decoded + S (p) = S ((S (ff_i_bpd_successor_prefix_repeat)) * ff_c_bpd_successor_prefix)) /\\ exists ff_q_bpd_successor_prefix_repeat_decoded. ff_b_bpd_successor_prefix = ff_q_bpd_successor_prefix_repeat_decoded * S ((S (ff_i_bpd_successor_prefix_repeat)) * ff_c_bpd_successor_prefix) + (p)))) /\\ (exists ff_u_bpd_successor_prefix_product ff_v_bpd_successor_prefix_product. ((((exists ff_h_bpd_successor_prefix_product_start. ff_h_bpd_successor_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_start. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_start * S ((S (0)) * ff_v_bpd_successor_prefix_product) + (1))) /\\ ((((exists ff_h_bpd_successor_prefix_product_terminal. ff_h_bpd_successor_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_terminal. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_terminal * S ((S (e)) * ff_v_bpd_successor_prefix_product) + (r))) /\\ forall ff_i_bpd_successor_prefix_product. (exists ff_lt_bpd_successor_prefix_product_bound. ff_lt_bpd_successor_prefix_product_bound + S ff_i_bpd_successor_prefix_product = e) -> exists ff_p_bpd_successor_prefix_product ff_r_bpd_successor_prefix_product ff_s_bpd_successor_prefix_product. ((((exists ff_h_bpd_successor_prefix_product_factor. ff_h_bpd_successor_prefix_product_factor + S (ff_p_bpd_successor_prefix_product) = S ((S (ff_i_bpd_successor_prefix_product)) * ff_c_bpd_successor_prefix)) /\\ exists ff_q_bpd_successor_prefix_product_factor. ff_b_bpd_successor_prefix = ff_q_bpd_successor_prefix_product_factor * S ((S (ff_i_bpd_successor_prefix_product)) * ff_c_bpd_successor_prefix) + (ff_p_bpd_successor_prefix_product))) /\\ ((((exists ff_h_bpd_successor_prefix_product_partial. ff_h_bpd_successor_prefix_product_partial + S (ff_r_bpd_successor_prefix_product) = S ((S (ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_partial. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_partial * S ((S (ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product) + (ff_r_bpd_successor_prefix_product))) /\\ ((((exists ff_h_bpd_successor_prefix_product_successor. ff_h_bpd_successor_prefix_product_successor + S (ff_s_bpd_successor_prefix_product) = S ((S (S ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_successor. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_successor * S ((S (S ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product) + (ff_s_bpd_successor_prefix_product))) /\\ ff_s_bpd_successor_prefix_product = ff_r_bpd_successor_prefix_product * ff_p_bpd_successor_prefix_product)))))))) -> a = r * q -> (exists bpd_factor_successor_cofactor. q = (p) * bpd_factor_successor_cofactor) -> (exists bpvi_result_successor_result. ((exists bpvi_b_successor_result_power bpvi_c_successor_result_power. ((forall bpvi_i_successor_result_power. (exists bpvi_repeat_gap_successor_result_power. bpvi_repeat_gap_successor_result_power + S bpvi_i_successor_result_power = S e) -> (((exists bpvi_h_successor_result_power_repeat. bpvi_h_successor_result_power_repeat + S (p) = S ((S (bpvi_i_successor_result_power)) * bpvi_c_successor_result_power)) /\\ exists bpvi_q_successor_result_power_repeat. bpvi_b_successor_result_power = bpvi_q_successor_result_power_repeat * S ((S (bpvi_i_successor_result_power)) * bpvi_c_successor_result_power) + (p)))) /\\ (exists bpvi_u_successor_result_power bpvi_v_successor_result_power. ((((exists bpvi_h_successor_result_power_start. bpvi_h_successor_result_power_start + S (1) = S ((S (0)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_start. bpvi_u_successor_result_power = bpvi_q_successor_result_power_start * S ((S (0)) * bpvi_v_successor_result_power) + (1))) /\\ ((((exists bpvi_h_successor_result_power_terminal. bpvi_h_successor_result_power_terminal + S (bpvi_result_successor_result) = S ((S (S e)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_terminal. bpvi_u_successor_result_power = bpvi_q_successor_result_power_terminal * S ((S (S e)) * bpvi_v_successor_result_power) + (bpvi_result_successor_result))) /\\ forall bpvi_j_successor_result_power. (exists bpvi_product_gap_successor_result_power. bpvi_product_gap_successor_result_power + S bpvi_j_successor_result_power = S e) -> exists bpvi_factor_successor_result_power bpvi_partial_successor_result_power bpvi_successor_successor_result_power. ((((exists bpvi_h_successor_result_power_factor. bpvi_h_successor_result_power_factor + S (bpvi_factor_successor_result_power) = S ((S (bpvi_j_successor_result_power)) * bpvi_c_successor_result_power)) /\\ exists bpvi_q_successor_result_power_factor. bpvi_b_successor_result_power = bpvi_q_successor_result_power_factor * S ((S (bpvi_j_successor_result_power)) * bpvi_c_successor_result_power) + (bpvi_factor_successor_result_power))) /\\ ((((exists bpvi_h_successor_result_power_partial. bpvi_h_successor_result_power_partial + S (bpvi_partial_successor_result_power) = S ((S (bpvi_j_successor_result_power)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_partial. bpvi_u_successor_result_power = bpvi_q_successor_result_power_partial * S ((S (bpvi_j_successor_result_power)) * bpvi_v_successor_result_power) + (bpvi_partial_successor_result_power))) /\\ ((((exists bpvi_h_successor_result_power_successor. bpvi_h_successor_result_power_successor + S (bpvi_successor_successor_result_power) = S ((S (S bpvi_j_successor_result_power)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_successor. bpvi_u_successor_result_power = bpvi_q_successor_result_power_successor * S ((S (S bpvi_j_successor_result_power)) * bpvi_v_successor_result_power) + (bpvi_successor_successor_result_power))) /\\ bpvi_successor_successor_result_power = bpvi_partial_successor_result_power * bpvi_factor_successor_result_power)))))))) /\\ exists bpvi_divisor_factor_successor_result. a = bpvi_result_successor_result * bpvi_divisor_factor_successor_result))", "statement_sha256": "ca4fd229a6bde9c593b7948b0abf14786a80db9bd0c24343770abcef342abc23", "summary": "Divisibility of a power cofactor by its base raises the exponent by one.", "summary_sha256": "7ebb64116bb6db98ba7552743f5aca7223fbba79a23b65bbd4fba0ef8e084e15" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_exists", "pow_successor_pair_mul", "mul_assoc" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=198]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_divides_successor_of_cofactor]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_divides_successor_of_cofactor", "parent_alpha_version": "v34", "proof_bundle_node_id": 212, "reference_route": "jordan-totient/checkpoint.html#theorem-power_divides_successor_of_cofactor", "script": [ "intro p", "intro e", "intro a", "intro r", "intro q", "intro hr", "intro ha", "intro hq", "cases hq", "have hsuccessor : exists s. (exists bpvi_b_bpd_successor_witness bpvi_c_bpd_successor_witness. ((forall bpvi_i_bpd_successor_witness. (exists bpvi_repeat_gap_bpd_successor_witness. bpvi_repeat_gap_bpd_successor_witness + S bpvi_i_bpd_successor_witness = S e) -> (((exists bpvi_h_bpd_successor_witness_repeat. bpvi_h_bpd_successor_witness_repeat + S (p) = S ((S (bpvi_i_bpd_successor_witness)) * bpvi_c_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_repeat. bpvi_b_bpd_successor_witness = bpvi_q_bpd_successor_witness_repeat * S ((S (bpvi_i_bpd_successor_witness)) * bpvi_c_bpd_successor_witness) + (p)))) /\\ (exists bpvi_u_bpd_successor_witness bpvi_v_bpd_successor_witness. ((((exists bpvi_h_bpd_successor_witness_start. bpvi_h_bpd_successor_witness_start + S (1) = S ((S (0)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_start. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_start * S ((S (0)) * bpvi_v_bpd_successor_witness) + (1))) /\\ ((((exists bpvi_h_bpd_successor_witness_terminal. bpvi_h_bpd_successor_witness_terminal + S (s) = S ((S (S e)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_terminal. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_terminal * S ((S (S e)) * bpvi_v_bpd_successor_witness) + (s))) /\\ forall bpvi_j_bpd_successor_witness. (exists bpvi_product_gap_bpd_successor_witness. bpvi_product_gap_bpd_successor_witness + S bpvi_j_bpd_successor_witness = S e) -> exists bpvi_factor_bpd_successor_witness bpvi_partial_bpd_successor_witness bpvi_successor_bpd_successor_witness. ((((exists bpvi_h_bpd_successor_witness_factor. bpvi_h_bpd_successor_witness_factor + S (bpvi_factor_bpd_successor_witness) = S ((S (bpvi_j_bpd_successor_witness)) * bpvi_c_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_factor. bpvi_b_bpd_successor_witness = bpvi_q_bpd_successor_witness_factor * S ((S (bpvi_j_bpd_successor_witness)) * bpvi_c_bpd_successor_witness) + (bpvi_factor_bpd_successor_witness))) /\\ ((((exists bpvi_h_bpd_successor_witness_partial. bpvi_h_bpd_successor_witness_partial + S (bpvi_partial_bpd_successor_witness) = S ((S (bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_partial. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_partial * S ((S (bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness) + (bpvi_partial_bpd_successor_witness))) /\\ ((((exists bpvi_h_bpd_successor_witness_successor. bpvi_h_bpd_successor_witness_successor + S (bpvi_successor_bpd_successor_witness) = S ((S (S bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness)) /\\ exists bpvi_q_bpd_successor_witness_successor. bpvi_u_bpd_successor_witness = bpvi_q_bpd_successor_witness_successor * S ((S (S bpvi_j_bpd_successor_witness)) * bpvi_v_bpd_successor_witness) + (bpvi_successor_bpd_successor_witness))) /\\ bpvi_successor_bpd_successor_witness = bpvi_partial_bpd_successor_witness * bpvi_factor_bpd_successor_witness))))))))", "specialize pow_exists p", "specialize pow_exists (S e)", "exact pow_exists", "cases hsuccessor", "have hs : x1 = r * p", "specialize pow_successor_pair_mul p", "specialize pow_successor_pair_mul e", "specialize pow_successor_pair_mul (S e)", "specialize pow_successor_pair_mul r", "specialize pow_successor_pair_mul x1", "apply pow_successor_pair_mul", "refl", "exact hr", "exact hsuccessor_witness", "exists x1", "split", "exact hsuccessor_witness", "exists x", "trans r * q", "exact ha", "rewrite hq_witness", "trans (r * p) * x", "symm", "apply mul_assoc", "congr", "symm", "exact hs", "refl" ], "script_sha256": "0a5d9f4172e877812c4b4b592f04926a16998e6d1577de9458650796f342ba9b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p e a r q. (exists ff_b_bpd_successor_prefix ff_c_bpd_successor_prefix. ((forall ff_i_bpd_successor_prefix_repeat. (exists ff_lt_bpd_successor_prefix_repeat_bound. ff_lt_bpd_successor_prefix_repeat_bound + S ff_i_bpd_successor_prefix_repeat = e) -> (((exists ff_h_bpd_successor_prefix_repeat_decoded. ff_h_bpd_successor_prefix_repeat_decoded + S (p) = S ((S (ff_i_bpd_successor_prefix_repeat)) * ff_c_bpd_successor_prefix)) /\\ exists ff_q_bpd_successor_prefix_repeat_decoded. ff_b_bpd_successor_prefix = ff_q_bpd_successor_prefix_repeat_decoded * S ((S (ff_i_bpd_successor_prefix_repeat)) * ff_c_bpd_successor_prefix) + (p)))) /\\ (exists ff_u_bpd_successor_prefix_product ff_v_bpd_successor_prefix_product. ((((exists ff_h_bpd_successor_prefix_product_start. ff_h_bpd_successor_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_start. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_start * S ((S (0)) * ff_v_bpd_successor_prefix_product) + (1))) /\\ ((((exists ff_h_bpd_successor_prefix_product_terminal. ff_h_bpd_successor_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_terminal. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_terminal * S ((S (e)) * ff_v_bpd_successor_prefix_product) + (r))) /\\ forall ff_i_bpd_successor_prefix_product. (exists ff_lt_bpd_successor_prefix_product_bound. ff_lt_bpd_successor_prefix_product_bound + S ff_i_bpd_successor_prefix_product = e) -> exists ff_p_bpd_successor_prefix_product ff_r_bpd_successor_prefix_product ff_s_bpd_successor_prefix_product. ((((exists ff_h_bpd_successor_prefix_product_factor. ff_h_bpd_successor_prefix_product_factor + S (ff_p_bpd_successor_prefix_product) = S ((S (ff_i_bpd_successor_prefix_product)) * ff_c_bpd_successor_prefix)) /\\ exists ff_q_bpd_successor_prefix_product_factor. ff_b_bpd_successor_prefix = ff_q_bpd_successor_prefix_product_factor * S ((S (ff_i_bpd_successor_prefix_product)) * ff_c_bpd_successor_prefix) + (ff_p_bpd_successor_prefix_product))) /\\ ((((exists ff_h_bpd_successor_prefix_product_partial. ff_h_bpd_successor_prefix_product_partial + S (ff_r_bpd_successor_prefix_product) = S ((S (ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_partial. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_partial * S ((S (ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product) + (ff_r_bpd_successor_prefix_product))) /\\ ((((exists ff_h_bpd_successor_prefix_product_successor. ff_h_bpd_successor_prefix_product_successor + S (ff_s_bpd_successor_prefix_product) = S ((S (S ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product)) /\\ exists ff_q_bpd_successor_prefix_product_successor. ff_u_bpd_successor_prefix_product = ff_q_bpd_successor_prefix_product_successor * S ((S (S ff_i_bpd_successor_prefix_product)) * ff_v_bpd_successor_prefix_product) + (ff_s_bpd_successor_prefix_product))) /\\ ff_s_bpd_successor_prefix_product = ff_r_bpd_successor_prefix_product * ff_p_bpd_successor_prefix_product)))))))) -> a = r * q -> (exists bpd_factor_successor_cofactor. q = (p) * bpd_factor_successor_cofactor) -> (exists bpvi_result_successor_result. ((exists bpvi_b_successor_result_power bpvi_c_successor_result_power. ((forall bpvi_i_successor_result_power. (exists bpvi_repeat_gap_successor_result_power. bpvi_repeat_gap_successor_result_power + S bpvi_i_successor_result_power = S e) -> (((exists bpvi_h_successor_result_power_repeat. bpvi_h_successor_result_power_repeat + S (p) = S ((S (bpvi_i_successor_result_power)) * bpvi_c_successor_result_power)) /\\ exists bpvi_q_successor_result_power_repeat. bpvi_b_successor_result_power = bpvi_q_successor_result_power_repeat * S ((S (bpvi_i_successor_result_power)) * bpvi_c_successor_result_power) + (p)))) /\\ (exists bpvi_u_successor_result_power bpvi_v_successor_result_power. ((((exists bpvi_h_successor_result_power_start. bpvi_h_successor_result_power_start + S (1) = S ((S (0)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_start. bpvi_u_successor_result_power = bpvi_q_successor_result_power_start * S ((S (0)) * bpvi_v_successor_result_power) + (1))) /\\ ((((exists bpvi_h_successor_result_power_terminal. bpvi_h_successor_result_power_terminal + S (bpvi_result_successor_result) = S ((S (S e)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_terminal. bpvi_u_successor_result_power = bpvi_q_successor_result_power_terminal * S ((S (S e)) * bpvi_v_successor_result_power) + (bpvi_result_successor_result))) /\\ forall bpvi_j_successor_result_power. (exists bpvi_product_gap_successor_result_power. bpvi_product_gap_successor_result_power + S bpvi_j_successor_result_power = S e) -> exists bpvi_factor_successor_result_power bpvi_partial_successor_result_power bpvi_successor_successor_result_power. ((((exists bpvi_h_successor_result_power_factor. bpvi_h_successor_result_power_factor + S (bpvi_factor_successor_result_power) = S ((S (bpvi_j_successor_result_power)) * bpvi_c_successor_result_power)) /\\ exists bpvi_q_successor_result_power_factor. bpvi_b_successor_result_power = bpvi_q_successor_result_power_factor * S ((S (bpvi_j_successor_result_power)) * bpvi_c_successor_result_power) + (bpvi_factor_successor_result_power))) /\\ ((((exists bpvi_h_successor_result_power_partial. bpvi_h_successor_result_power_partial + S (bpvi_partial_successor_result_power) = S ((S (bpvi_j_successor_result_power)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_partial. bpvi_u_successor_result_power = bpvi_q_successor_result_power_partial * S ((S (bpvi_j_successor_result_power)) * bpvi_v_successor_result_power) + (bpvi_partial_successor_result_power))) /\\ ((((exists bpvi_h_successor_result_power_successor. bpvi_h_successor_result_power_successor + S (bpvi_successor_successor_result_power) = S ((S (S bpvi_j_successor_result_power)) * bpvi_v_successor_result_power)) /\\ exists bpvi_q_successor_result_power_successor. bpvi_u_successor_result_power = bpvi_q_successor_result_power_successor * S ((S (S bpvi_j_successor_result_power)) * bpvi_v_successor_result_power) + (bpvi_successor_successor_result_power))) /\\ bpvi_successor_successor_result_power = bpvi_partial_successor_result_power * bpvi_factor_successor_result_power)))))))) /\\ exists bpvi_divisor_factor_successor_result. a = bpvi_result_successor_result * bpvi_divisor_factor_successor_result))", "statement_sha256": "ca4fd229a6bde9c593b7948b0abf14786a80db9bd0c24343770abcef342abc23" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_power_successor_cancel_cofactor", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 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((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> (exists ff_b_bpd_cancel_prefix ff_c_bpd_cancel_prefix. ((forall ff_i_bpd_cancel_prefix_repeat. (exists ff_lt_bpd_cancel_prefix_repeat_bound. ff_lt_bpd_cancel_prefix_repeat_bound + S ff_i_bpd_cancel_prefix_repeat = e) -> (((exists ff_h_bpd_cancel_prefix_repeat_decoded. ff_h_bpd_cancel_prefix_repeat_decoded + S (p) = S ((S (ff_i_bpd_cancel_prefix_repeat)) * ff_c_bpd_cancel_prefix)) /\\ exists ff_q_bpd_cancel_prefix_repeat_decoded. ff_b_bpd_cancel_prefix = ff_q_bpd_cancel_prefix_repeat_decoded * S ((S (ff_i_bpd_cancel_prefix_repeat)) * ff_c_bpd_cancel_prefix) + (p)))) /\\ (exists ff_u_bpd_cancel_prefix_product ff_v_bpd_cancel_prefix_product. ((((exists ff_h_bpd_cancel_prefix_product_start. ff_h_bpd_cancel_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_start. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_start * S ((S (0)) * ff_v_bpd_cancel_prefix_product) + (1))) /\\ ((((exists ff_h_bpd_cancel_prefix_product_terminal. ff_h_bpd_cancel_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_terminal. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_terminal * S ((S (e)) * ff_v_bpd_cancel_prefix_product) + (r))) /\\ forall ff_i_bpd_cancel_prefix_product. (exists ff_lt_bpd_cancel_prefix_product_bound. ff_lt_bpd_cancel_prefix_product_bound + S ff_i_bpd_cancel_prefix_product = e) -> exists ff_p_bpd_cancel_prefix_product ff_r_bpd_cancel_prefix_product ff_s_bpd_cancel_prefix_product. ((((exists ff_h_bpd_cancel_prefix_product_factor. ff_h_bpd_cancel_prefix_product_factor + S (ff_p_bpd_cancel_prefix_product) = S ((S (ff_i_bpd_cancel_prefix_product)) * ff_c_bpd_cancel_prefix)) /\\ exists ff_q_bpd_cancel_prefix_product_factor. ff_b_bpd_cancel_prefix = ff_q_bpd_cancel_prefix_product_factor * S ((S (ff_i_bpd_cancel_prefix_product)) * ff_c_bpd_cancel_prefix) + (ff_p_bpd_cancel_prefix_product))) /\\ ((((exists ff_h_bpd_cancel_prefix_product_partial. ff_h_bpd_cancel_prefix_product_partial + S (ff_r_bpd_cancel_prefix_product) = S ((S (ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_partial. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_partial * S ((S (ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product) + (ff_r_bpd_cancel_prefix_product))) /\\ ((((exists ff_h_bpd_cancel_prefix_product_successor. ff_h_bpd_cancel_prefix_product_successor + S (ff_s_bpd_cancel_prefix_product) = S ((S (S ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_successor. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_successor * S ((S (S ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product) + (ff_s_bpd_cancel_prefix_product))) /\\ ff_s_bpd_cancel_prefix_product = ff_r_bpd_cancel_prefix_product * ff_p_bpd_cancel_prefix_product)))))))) -> a = r * q -> (exists bpvi_result_cancel_successor. ((exists bpvi_b_cancel_successor_power bpvi_c_cancel_successor_power. ((forall bpvi_i_cancel_successor_power. (exists bpvi_repeat_gap_cancel_successor_power. bpvi_repeat_gap_cancel_successor_power + S bpvi_i_cancel_successor_power = S e) -> (((exists bpvi_h_cancel_successor_power_repeat. bpvi_h_cancel_successor_power_repeat + S (p) = S ((S (bpvi_i_cancel_successor_power)) * bpvi_c_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_repeat. bpvi_b_cancel_successor_power = bpvi_q_cancel_successor_power_repeat * S ((S (bpvi_i_cancel_successor_power)) * bpvi_c_cancel_successor_power) + (p)))) /\\ (exists bpvi_u_cancel_successor_power bpvi_v_cancel_successor_power. ((((exists bpvi_h_cancel_successor_power_start. bpvi_h_cancel_successor_power_start + S (1) = S ((S (0)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_start. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_start * S ((S (0)) * bpvi_v_cancel_successor_power) + (1))) /\\ ((((exists bpvi_h_cancel_successor_power_terminal. bpvi_h_cancel_successor_power_terminal + S (bpvi_result_cancel_successor) = S ((S (S e)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_terminal. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_terminal * S ((S (S e)) * bpvi_v_cancel_successor_power) + (bpvi_result_cancel_successor))) /\\ forall bpvi_j_cancel_successor_power. (exists bpvi_product_gap_cancel_successor_power. bpvi_product_gap_cancel_successor_power + S bpvi_j_cancel_successor_power = S e) -> exists bpvi_factor_cancel_successor_power bpvi_partial_cancel_successor_power bpvi_successor_cancel_successor_power. ((((exists bpvi_h_cancel_successor_power_factor. bpvi_h_cancel_successor_power_factor + S (bpvi_factor_cancel_successor_power) = S ((S (bpvi_j_cancel_successor_power)) * bpvi_c_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_factor. bpvi_b_cancel_successor_power = bpvi_q_cancel_successor_power_factor * S ((S (bpvi_j_cancel_successor_power)) * bpvi_c_cancel_successor_power) + (bpvi_factor_cancel_successor_power))) /\\ ((((exists bpvi_h_cancel_successor_power_partial. bpvi_h_cancel_successor_power_partial + S (bpvi_partial_cancel_successor_power) = S ((S (bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_partial. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_partial * S ((S (bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power) + (bpvi_partial_cancel_successor_power))) /\\ ((((exists bpvi_h_cancel_successor_power_successor. bpvi_h_cancel_successor_power_successor + S (bpvi_successor_cancel_successor_power) = S ((S (S bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_successor. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_successor * S ((S (S bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power) + (bpvi_successor_cancel_successor_power))) /\\ bpvi_successor_cancel_successor_power = bpvi_partial_cancel_successor_power * bpvi_factor_cancel_successor_power)))))))) /\\ exists bpvi_divisor_factor_cancel_successor. a = bpvi_result_cancel_successor * bpvi_divisor_factor_cancel_successor)) -> (exists bpd_factor_cancel_cofactor_result. q = (p) * bpd_factor_cancel_cofactor_result)", "statement_sha256": "e31164bb3d38f8b8cc3de74b17f6e9cf2d0b7a00525c2b0c183d55f23a4a6e1b", "summary": "A successor power divisor cancels to a prime divisor of the exact cofactor.", "summary_sha256": "731f2c9b753029898813c1b4fbb22c756fb27aade94e729f8cb2c68c3e859bbc" }, "canonical_theorem_route": null, 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((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> (exists ff_b_bpd_cancel_prefix ff_c_bpd_cancel_prefix. ((forall ff_i_bpd_cancel_prefix_repeat. (exists ff_lt_bpd_cancel_prefix_repeat_bound. ff_lt_bpd_cancel_prefix_repeat_bound + S ff_i_bpd_cancel_prefix_repeat = e) -> (((exists ff_h_bpd_cancel_prefix_repeat_decoded. ff_h_bpd_cancel_prefix_repeat_decoded + S (p) = S ((S (ff_i_bpd_cancel_prefix_repeat)) * ff_c_bpd_cancel_prefix)) /\\ exists ff_q_bpd_cancel_prefix_repeat_decoded. ff_b_bpd_cancel_prefix = ff_q_bpd_cancel_prefix_repeat_decoded * S ((S (ff_i_bpd_cancel_prefix_repeat)) * ff_c_bpd_cancel_prefix) + (p)))) /\\ (exists ff_u_bpd_cancel_prefix_product ff_v_bpd_cancel_prefix_product. ((((exists ff_h_bpd_cancel_prefix_product_start. ff_h_bpd_cancel_prefix_product_start + S (1) = S ((S (0)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_start. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_start * S ((S (0)) * ff_v_bpd_cancel_prefix_product) + (1))) /\\ ((((exists ff_h_bpd_cancel_prefix_product_terminal. ff_h_bpd_cancel_prefix_product_terminal + S (r) = S ((S (e)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_terminal. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_terminal * S ((S (e)) * ff_v_bpd_cancel_prefix_product) + (r))) /\\ forall ff_i_bpd_cancel_prefix_product. (exists ff_lt_bpd_cancel_prefix_product_bound. ff_lt_bpd_cancel_prefix_product_bound + S ff_i_bpd_cancel_prefix_product = e) -> exists ff_p_bpd_cancel_prefix_product ff_r_bpd_cancel_prefix_product ff_s_bpd_cancel_prefix_product. ((((exists ff_h_bpd_cancel_prefix_product_factor. ff_h_bpd_cancel_prefix_product_factor + S (ff_p_bpd_cancel_prefix_product) = S ((S (ff_i_bpd_cancel_prefix_product)) * ff_c_bpd_cancel_prefix)) /\\ exists ff_q_bpd_cancel_prefix_product_factor. ff_b_bpd_cancel_prefix = ff_q_bpd_cancel_prefix_product_factor * S ((S (ff_i_bpd_cancel_prefix_product)) * ff_c_bpd_cancel_prefix) + (ff_p_bpd_cancel_prefix_product))) /\\ ((((exists ff_h_bpd_cancel_prefix_product_partial. ff_h_bpd_cancel_prefix_product_partial + S (ff_r_bpd_cancel_prefix_product) = S ((S (ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_partial. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_partial * S ((S (ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product) + (ff_r_bpd_cancel_prefix_product))) /\\ ((((exists ff_h_bpd_cancel_prefix_product_successor. ff_h_bpd_cancel_prefix_product_successor + S (ff_s_bpd_cancel_prefix_product) = S ((S (S ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product)) /\\ exists ff_q_bpd_cancel_prefix_product_successor. ff_u_bpd_cancel_prefix_product = ff_q_bpd_cancel_prefix_product_successor * S ((S (S ff_i_bpd_cancel_prefix_product)) * ff_v_bpd_cancel_prefix_product) + (ff_s_bpd_cancel_prefix_product))) /\\ ff_s_bpd_cancel_prefix_product = ff_r_bpd_cancel_prefix_product * ff_p_bpd_cancel_prefix_product)))))))) -> a = r * q -> (exists bpvi_result_cancel_successor. ((exists bpvi_b_cancel_successor_power bpvi_c_cancel_successor_power. ((forall bpvi_i_cancel_successor_power. (exists bpvi_repeat_gap_cancel_successor_power. bpvi_repeat_gap_cancel_successor_power + S bpvi_i_cancel_successor_power = S e) -> (((exists bpvi_h_cancel_successor_power_repeat. bpvi_h_cancel_successor_power_repeat + S (p) = S ((S (bpvi_i_cancel_successor_power)) * bpvi_c_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_repeat. bpvi_b_cancel_successor_power = bpvi_q_cancel_successor_power_repeat * S ((S (bpvi_i_cancel_successor_power)) * bpvi_c_cancel_successor_power) + (p)))) /\\ (exists bpvi_u_cancel_successor_power bpvi_v_cancel_successor_power. ((((exists bpvi_h_cancel_successor_power_start. bpvi_h_cancel_successor_power_start + S (1) = S ((S (0)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_start. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_start * S ((S (0)) * bpvi_v_cancel_successor_power) + (1))) /\\ ((((exists bpvi_h_cancel_successor_power_terminal. bpvi_h_cancel_successor_power_terminal + S (bpvi_result_cancel_successor) = S ((S (S e)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_terminal. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_terminal * S ((S (S e)) * bpvi_v_cancel_successor_power) + (bpvi_result_cancel_successor))) /\\ forall bpvi_j_cancel_successor_power. (exists bpvi_product_gap_cancel_successor_power. bpvi_product_gap_cancel_successor_power + S bpvi_j_cancel_successor_power = S e) -> exists bpvi_factor_cancel_successor_power bpvi_partial_cancel_successor_power bpvi_successor_cancel_successor_power. ((((exists bpvi_h_cancel_successor_power_factor. bpvi_h_cancel_successor_power_factor + S (bpvi_factor_cancel_successor_power) = S ((S (bpvi_j_cancel_successor_power)) * bpvi_c_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_factor. bpvi_b_cancel_successor_power = bpvi_q_cancel_successor_power_factor * S ((S (bpvi_j_cancel_successor_power)) * bpvi_c_cancel_successor_power) + (bpvi_factor_cancel_successor_power))) /\\ ((((exists bpvi_h_cancel_successor_power_partial. bpvi_h_cancel_successor_power_partial + S (bpvi_partial_cancel_successor_power) = S ((S (bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_partial. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_partial * S ((S (bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power) + (bpvi_partial_cancel_successor_power))) /\\ ((((exists bpvi_h_cancel_successor_power_successor. bpvi_h_cancel_successor_power_successor + S (bpvi_successor_cancel_successor_power) = S ((S (S bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power)) /\\ exists bpvi_q_cancel_successor_power_successor. bpvi_u_cancel_successor_power = bpvi_q_cancel_successor_power_successor * S ((S (S bpvi_j_cancel_successor_power)) * bpvi_v_cancel_successor_power) + (bpvi_successor_cancel_successor_power))) /\\ bpvi_successor_cancel_successor_power = bpvi_partial_cancel_successor_power * bpvi_factor_cancel_successor_power)))))))) /\\ exists bpvi_divisor_factor_cancel_successor. a = bpvi_result_cancel_successor * bpvi_divisor_factor_cancel_successor)) -> (exists bpd_factor_cancel_cofactor_result. q = (p) * bpd_factor_cancel_cofactor_result)", "statement_sha256": "e31164bb3d38f8b8cc3de74b17f6e9cf2d0b7a00525c2b0c183d55f23a4a6e1b" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_nondivisor_mul", "canonical_catalog_record": { 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"body_proof_nodes": 23, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 200, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "f1950ba21ba465ab263a9788abfdaf46dec2d234613142e4661daa115ca460ec", "status": "checked" }, "enrollment_index": 934, "enrollment_origin": "bertrand_b2_valuation_multiplication", "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", 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"bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro a", "intro b", "intro hp", "intro ha", "intro hb", "intro hab", "have hsplit : (exists u. a = p * u) \\/ exists v. b = p * v", "specialize euclid_prime_dvd_product p", "specialize euclid_prime_dvd_product a", "specialize euclid_prime_dvd_product b", "apply euclid_prime_dvd_product", "exact hp", "exact hab", "cases hsplit", "apply ha", "exact hsplit_left", "apply hb", "exact hsplit_right" ], "script_sha256": "dce0d19388ed88545957ab48c0efe59f12683a0b72db71c16e93fe4e7c797ef2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p a b. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(exists bpd_factor_nondivisor_left. a = (p) * bpd_factor_nondivisor_left) -> ~(exists bpd_factor_nondivisor_right. b = (p) * bpd_factor_nondivisor_right) -> ~(exists bpd_factor_nondivisor_product. a * b = (p) * bpd_factor_nondivisor_product)", "statement_sha256": "f1950ba21ba465ab263a9788abfdaf46dec2d234613142e4661daa115ca460ec", "summary": "A prime dividing neither factor does not divide their product.", "summary_sha256": "cf988ceee0db1258170abcd686214a2dcb78233d45969fd631c59f7c07e2dd71" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "euclid_prime_dvd_product" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=200]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_nondivisor_mul]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_nondivisor_mul", "parent_alpha_version": "v34", "proof_bundle_node_id": 214, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_nondivisor_mul", "script": [ "intro p", "intro a", "intro b", "intro hp", "intro ha", "intro hb", "intro hab", "have hsplit : (exists u. a = p * u) \\/ exists v. b = p * v", "specialize euclid_prime_dvd_product p", "specialize euclid_prime_dvd_product a", "specialize euclid_prime_dvd_product b", "apply euclid_prime_dvd_product", "exact hp", "exact hab", "cases hsplit", "apply ha", "exact hsplit_left", "apply hb", "exact hsplit_right" ], "script_sha256": "dce0d19388ed88545957ab48c0efe59f12683a0b72db71c16e93fe4e7c797ef2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p a b. 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281, "bundle_node_id": 201, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "52ad2dcbb6081c8f9a5380f905c5427c2a1168219875f958bdb9c14065ea6c44", "status": "checked" }, "enrollment_index": 935, "enrollment_origin": "bertrand_b2_valuation_multiplication", "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", 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"selector": "nodes[id=201]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_exact_cofactor]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "0527e467953d1db7dabe457e01da022e0fd445a90875bbcf1a8670781c137ceb", "membership": "alpha_only", "name": "power_valuation_exact_cofactor", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro a", "intro e", "intro hp", "intro ha", "intro hvaluation", "have hcharacterization : (exists bpv_result_bpd_exact_selected. ((exists ff_b_bpd_exact_selected_power ff_c_bpd_exact_selected_power. ((forall ff_i_bpd_exact_selected_power_repeat. (exists ff_lt_bpd_exact_selected_power_repeat_bound. ff_lt_bpd_exact_selected_power_repeat_bound + S ff_i_bpd_exact_selected_power_repeat = e) -> (((exists ff_h_bpd_exact_selected_power_repeat_decoded. ff_h_bpd_exact_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_exact_selected_power_repeat)) * ff_c_bpd_exact_selected_power)) /\\ exists ff_q_bpd_exact_selected_power_repeat_decoded. ff_b_bpd_exact_selected_power = ff_q_bpd_exact_selected_power_repeat_decoded * S ((S (ff_i_bpd_exact_selected_power_repeat)) * ff_c_bpd_exact_selected_power) + (p)))) /\\ (exists ff_u_bpd_exact_selected_power_product ff_v_bpd_exact_selected_power_product. ((((exists ff_h_bpd_exact_selected_power_product_start. ff_h_bpd_exact_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_start. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_start * S ((S (0)) * ff_v_bpd_exact_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_exact_selected_power_product_terminal. ff_h_bpd_exact_selected_power_product_terminal + S (bpv_result_bpd_exact_selected) = S ((S (e)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_terminal. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_exact_selected_power_product) + (bpv_result_bpd_exact_selected))) /\\ forall ff_i_bpd_exact_selected_power_product. (exists ff_lt_bpd_exact_selected_power_product_bound. ff_lt_bpd_exact_selected_power_product_bound + S ff_i_bpd_exact_selected_power_product = e) -> exists ff_p_bpd_exact_selected_power_product ff_r_bpd_exact_selected_power_product ff_s_bpd_exact_selected_power_product. ((((exists ff_h_bpd_exact_selected_power_product_factor. ff_h_bpd_exact_selected_power_product_factor + S (ff_p_bpd_exact_selected_power_product) = S ((S (ff_i_bpd_exact_selected_power_product)) * ff_c_bpd_exact_selected_power)) /\\ exists ff_q_bpd_exact_selected_power_product_factor. ff_b_bpd_exact_selected_power = ff_q_bpd_exact_selected_power_product_factor * S ((S (ff_i_bpd_exact_selected_power_product)) * ff_c_bpd_exact_selected_power) + (ff_p_bpd_exact_selected_power_product))) /\\ ((((exists ff_h_bpd_exact_selected_power_product_partial. ff_h_bpd_exact_selected_power_product_partial + S (ff_r_bpd_exact_selected_power_product) = S ((S (ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_partial. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_partial * S ((S (ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product) + (ff_r_bpd_exact_selected_power_product))) /\\ ((((exists ff_h_bpd_exact_selected_power_product_successor. ff_h_bpd_exact_selected_power_product_successor + S (ff_s_bpd_exact_selected_power_product) = S ((S (S ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_successor. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_successor * S ((S (S ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product) + (ff_s_bpd_exact_selected_power_product))) /\\ ff_s_bpd_exact_selected_power_product = ff_r_bpd_exact_selected_power_product * ff_p_bpd_exact_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_exact_selected_divides. a = bpv_result_bpd_exact_selected * bpv_factor_bpd_exact_selected_divides))) /\\ ~(exists bpvi_result_bpd_exact_successor. ((exists bpvi_b_bpd_exact_successor_power bpvi_c_bpd_exact_successor_power. ((forall bpvi_i_bpd_exact_successor_power. (exists bpvi_repeat_gap_bpd_exact_successor_power. bpvi_repeat_gap_bpd_exact_successor_power + S bpvi_i_bpd_exact_successor_power = S e) -> (((exists bpvi_h_bpd_exact_successor_power_repeat. bpvi_h_bpd_exact_successor_power_repeat + S (p) = S ((S (bpvi_i_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_repeat. bpvi_b_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_repeat * S ((S (bpvi_i_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power) + (p)))) /\\ (exists bpvi_u_bpd_exact_successor_power bpvi_v_bpd_exact_successor_power. ((((exists bpvi_h_bpd_exact_successor_power_start. bpvi_h_bpd_exact_successor_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_start. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_start * S ((S (0)) * bpvi_v_bpd_exact_successor_power) + (1))) /\\ ((((exists bpvi_h_bpd_exact_successor_power_terminal. bpvi_h_bpd_exact_successor_power_terminal + S (bpvi_result_bpd_exact_successor) = S ((S (S e)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_terminal. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_terminal * S ((S (S e)) * bpvi_v_bpd_exact_successor_power) + (bpvi_result_bpd_exact_successor))) /\\ forall bpvi_j_bpd_exact_successor_power. (exists bpvi_product_gap_bpd_exact_successor_power. bpvi_product_gap_bpd_exact_successor_power + S bpvi_j_bpd_exact_successor_power = S e) -> exists bpvi_factor_bpd_exact_successor_power bpvi_partial_bpd_exact_successor_power bpvi_successor_bpd_exact_successor_power. ((((exists bpvi_h_bpd_exact_successor_power_factor. bpvi_h_bpd_exact_successor_power_factor + S (bpvi_factor_bpd_exact_successor_power) = S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_factor. bpvi_b_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_factor * S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power) + (bpvi_factor_bpd_exact_successor_power))) /\\ ((((exists bpvi_h_bpd_exact_successor_power_partial. bpvi_h_bpd_exact_successor_power_partial + S (bpvi_partial_bpd_exact_successor_power) = S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_partial. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_partial * S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power) + (bpvi_partial_bpd_exact_successor_power))) /\\ ((((exists bpvi_h_bpd_exact_successor_power_successor. bpvi_h_bpd_exact_successor_power_successor + S (bpvi_successor_bpd_exact_successor_power) = S ((S (S bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_successor. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_successor * S ((S (S bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power) + (bpvi_successor_bpd_exact_successor_power))) /\\ bpvi_successor_bpd_exact_successor_power = bpvi_partial_bpd_exact_successor_power * bpvi_factor_bpd_exact_successor_power)))))))) /\\ exists bpvi_divisor_factor_bpd_exact_successor. a = bpvi_result_bpd_exact_successor * bpvi_divisor_factor_bpd_exact_successor))", "specialize power_valuation_selected_and_successor_not_divides p", "specialize power_valuation_selected_and_successor_not_divides a", "specialize power_valuation_selected_and_successor_not_divides e", "apply power_valuation_selected_and_successor_not_divides", "exact hp", "exact ha", "exact hvaluation", "cases hcharacterization", "cases hcharacterization_left", "cases hcharacterization_left_witness", "cases hcharacterization_left_witness_right", "exists x", "exists x1", "split", "exact hcharacterization_left_witness_left", "split", "exact hcharacterization_left_witness_right_witness", "split", "intro hcofactor_zero", "apply ha", "trans x * x1", "exact hcharacterization_left_witness_right_witness", "rewrite hcofactor_zero", "apply PA5", "intro hcofactor_prime", "apply hcharacterization_right", "specialize power_divides_successor_of_cofactor p", "specialize power_divides_successor_of_cofactor e", "specialize power_divides_successor_of_cofactor a", "specialize power_divides_successor_of_cofactor x", "specialize power_divides_successor_of_cofactor x1", "apply power_divides_successor_of_cofactor", "exact hcharacterization_left_witness_left", "exact hcharacterization_left_witness_right_witness", "exact hcofactor_prime" ], "script_sha256": "cee5b7cb13f1910c6baef3e36c1e977848a63f9d140ab6fd5f9cef5d7f106949", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p a e. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (exists bpd_result_valuation_exact bpd_cofactor_valuation_exact. ((exists ff_b_valuation_exact_power ff_c_valuation_exact_power. ((forall ff_i_valuation_exact_power_repeat. (exists ff_lt_valuation_exact_power_repeat_bound. ff_lt_valuation_exact_power_repeat_bound + S ff_i_valuation_exact_power_repeat = e) -> (((exists ff_h_valuation_exact_power_repeat_decoded. ff_h_valuation_exact_power_repeat_decoded + S (p) = S ((S (ff_i_valuation_exact_power_repeat)) * ff_c_valuation_exact_power)) /\\ exists ff_q_valuation_exact_power_repeat_decoded. ff_b_valuation_exact_power = ff_q_valuation_exact_power_repeat_decoded * S ((S (ff_i_valuation_exact_power_repeat)) * ff_c_valuation_exact_power) + (p)))) /\\ (exists ff_u_valuation_exact_power_product ff_v_valuation_exact_power_product. ((((exists ff_h_valuation_exact_power_product_start. ff_h_valuation_exact_power_product_start + S (1) = S ((S (0)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_start. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_start * S ((S (0)) * ff_v_valuation_exact_power_product) + (1))) /\\ ((((exists ff_h_valuation_exact_power_product_terminal. ff_h_valuation_exact_power_product_terminal + S (bpd_result_valuation_exact) = S ((S (e)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_terminal. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_terminal * S ((S (e)) * ff_v_valuation_exact_power_product) + (bpd_result_valuation_exact))) /\\ forall ff_i_valuation_exact_power_product. (exists ff_lt_valuation_exact_power_product_bound. ff_lt_valuation_exact_power_product_bound + S ff_i_valuation_exact_power_product = e) -> exists ff_p_valuation_exact_power_product ff_r_valuation_exact_power_product ff_s_valuation_exact_power_product. ((((exists ff_h_valuation_exact_power_product_factor. ff_h_valuation_exact_power_product_factor + S (ff_p_valuation_exact_power_product) = S ((S (ff_i_valuation_exact_power_product)) * ff_c_valuation_exact_power)) /\\ exists ff_q_valuation_exact_power_product_factor. ff_b_valuation_exact_power = ff_q_valuation_exact_power_product_factor * S ((S (ff_i_valuation_exact_power_product)) * ff_c_valuation_exact_power) + (ff_p_valuation_exact_power_product))) /\\ ((((exists ff_h_valuation_exact_power_product_partial. ff_h_valuation_exact_power_product_partial + S (ff_r_valuation_exact_power_product) = S ((S (ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_partial. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_partial * S ((S (ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product) + (ff_r_valuation_exact_power_product))) /\\ ((((exists ff_h_valuation_exact_power_product_successor. ff_h_valuation_exact_power_product_successor + S (ff_s_valuation_exact_power_product) = S ((S (S ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_successor. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_successor * S ((S (S ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product) + (ff_s_valuation_exact_power_product))) /\\ ff_s_valuation_exact_power_product = ff_r_valuation_exact_power_product * ff_p_valuation_exact_power_product)))))))) /\\ ((a = bpd_result_valuation_exact * bpd_cofactor_valuation_exact) /\\ ((~(bpd_cofactor_valuation_exact = 0)) /\\ (~(exists bpd_factor_valuation_exact_prime. bpd_cofactor_valuation_exact = (p) * bpd_factor_valuation_exact_prime))))))", "statement_sha256": "52ad2dcbb6081c8f9a5380f905c5427c2a1168219875f958bdb9c14065ea6c44", "summary": "A prime valuation extracts a nonzero cofactor not divisible by its prime.", "summary_sha256": "1b6c576f5c7dfaa631c8524e906eb73011bf2d31f61e212acfcee6dbdeab9a10" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_selected_and_successor_not_divides", "power_divides_successor_of_cofactor" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=201]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_exact_cofactor]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_exact_cofactor", "parent_alpha_version": "v34", "proof_bundle_node_id": 215, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_exact_cofactor", "script": [ "intro p", "intro a", "intro e", "intro hp", "intro ha", "intro hvaluation", "have hcharacterization : (exists bpv_result_bpd_exact_selected. ((exists ff_b_bpd_exact_selected_power ff_c_bpd_exact_selected_power. ((forall ff_i_bpd_exact_selected_power_repeat. (exists ff_lt_bpd_exact_selected_power_repeat_bound. ff_lt_bpd_exact_selected_power_repeat_bound + S ff_i_bpd_exact_selected_power_repeat = e) -> (((exists ff_h_bpd_exact_selected_power_repeat_decoded. ff_h_bpd_exact_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_exact_selected_power_repeat)) * ff_c_bpd_exact_selected_power)) /\\ exists ff_q_bpd_exact_selected_power_repeat_decoded. ff_b_bpd_exact_selected_power = ff_q_bpd_exact_selected_power_repeat_decoded * S ((S (ff_i_bpd_exact_selected_power_repeat)) * ff_c_bpd_exact_selected_power) + (p)))) /\\ (exists ff_u_bpd_exact_selected_power_product ff_v_bpd_exact_selected_power_product. ((((exists ff_h_bpd_exact_selected_power_product_start. ff_h_bpd_exact_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_start. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_start * S ((S (0)) * ff_v_bpd_exact_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_exact_selected_power_product_terminal. ff_h_bpd_exact_selected_power_product_terminal + S (bpv_result_bpd_exact_selected) = S ((S (e)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_terminal. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_exact_selected_power_product) + (bpv_result_bpd_exact_selected))) /\\ forall ff_i_bpd_exact_selected_power_product. (exists ff_lt_bpd_exact_selected_power_product_bound. ff_lt_bpd_exact_selected_power_product_bound + S ff_i_bpd_exact_selected_power_product = e) -> exists ff_p_bpd_exact_selected_power_product ff_r_bpd_exact_selected_power_product ff_s_bpd_exact_selected_power_product. ((((exists ff_h_bpd_exact_selected_power_product_factor. ff_h_bpd_exact_selected_power_product_factor + S (ff_p_bpd_exact_selected_power_product) = S ((S (ff_i_bpd_exact_selected_power_product)) * ff_c_bpd_exact_selected_power)) /\\ exists ff_q_bpd_exact_selected_power_product_factor. ff_b_bpd_exact_selected_power = ff_q_bpd_exact_selected_power_product_factor * S ((S (ff_i_bpd_exact_selected_power_product)) * ff_c_bpd_exact_selected_power) + (ff_p_bpd_exact_selected_power_product))) /\\ ((((exists ff_h_bpd_exact_selected_power_product_partial. ff_h_bpd_exact_selected_power_product_partial + S (ff_r_bpd_exact_selected_power_product) = S ((S (ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_partial. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_partial * S ((S (ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product) + (ff_r_bpd_exact_selected_power_product))) /\\ ((((exists ff_h_bpd_exact_selected_power_product_successor. ff_h_bpd_exact_selected_power_product_successor + S (ff_s_bpd_exact_selected_power_product) = S ((S (S ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product)) /\\ exists ff_q_bpd_exact_selected_power_product_successor. ff_u_bpd_exact_selected_power_product = ff_q_bpd_exact_selected_power_product_successor * S ((S (S ff_i_bpd_exact_selected_power_product)) * ff_v_bpd_exact_selected_power_product) + (ff_s_bpd_exact_selected_power_product))) /\\ ff_s_bpd_exact_selected_power_product = ff_r_bpd_exact_selected_power_product * ff_p_bpd_exact_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_exact_selected_divides. a = bpv_result_bpd_exact_selected * bpv_factor_bpd_exact_selected_divides))) /\\ ~(exists bpvi_result_bpd_exact_successor. ((exists bpvi_b_bpd_exact_successor_power bpvi_c_bpd_exact_successor_power. ((forall bpvi_i_bpd_exact_successor_power. (exists bpvi_repeat_gap_bpd_exact_successor_power. bpvi_repeat_gap_bpd_exact_successor_power + S bpvi_i_bpd_exact_successor_power = S e) -> (((exists bpvi_h_bpd_exact_successor_power_repeat. bpvi_h_bpd_exact_successor_power_repeat + S (p) = S ((S (bpvi_i_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_repeat. bpvi_b_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_repeat * S ((S (bpvi_i_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power) + (p)))) /\\ (exists bpvi_u_bpd_exact_successor_power bpvi_v_bpd_exact_successor_power. ((((exists bpvi_h_bpd_exact_successor_power_start. bpvi_h_bpd_exact_successor_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_start. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_start * S ((S (0)) * bpvi_v_bpd_exact_successor_power) + (1))) /\\ ((((exists bpvi_h_bpd_exact_successor_power_terminal. bpvi_h_bpd_exact_successor_power_terminal + S (bpvi_result_bpd_exact_successor) = S ((S (S e)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_terminal. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_terminal * S ((S (S e)) * bpvi_v_bpd_exact_successor_power) + (bpvi_result_bpd_exact_successor))) /\\ forall bpvi_j_bpd_exact_successor_power. (exists bpvi_product_gap_bpd_exact_successor_power. bpvi_product_gap_bpd_exact_successor_power + S bpvi_j_bpd_exact_successor_power = S e) -> exists bpvi_factor_bpd_exact_successor_power bpvi_partial_bpd_exact_successor_power bpvi_successor_bpd_exact_successor_power. ((((exists bpvi_h_bpd_exact_successor_power_factor. bpvi_h_bpd_exact_successor_power_factor + S (bpvi_factor_bpd_exact_successor_power) = S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_factor. bpvi_b_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_factor * S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_c_bpd_exact_successor_power) + (bpvi_factor_bpd_exact_successor_power))) /\\ ((((exists bpvi_h_bpd_exact_successor_power_partial. bpvi_h_bpd_exact_successor_power_partial + S (bpvi_partial_bpd_exact_successor_power) = S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_partial. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_partial * S ((S (bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power) + (bpvi_partial_bpd_exact_successor_power))) /\\ ((((exists bpvi_h_bpd_exact_successor_power_successor. bpvi_h_bpd_exact_successor_power_successor + S (bpvi_successor_bpd_exact_successor_power) = S ((S (S bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power)) /\\ exists bpvi_q_bpd_exact_successor_power_successor. bpvi_u_bpd_exact_successor_power = bpvi_q_bpd_exact_successor_power_successor * S ((S (S bpvi_j_bpd_exact_successor_power)) * bpvi_v_bpd_exact_successor_power) + (bpvi_successor_bpd_exact_successor_power))) /\\ bpvi_successor_bpd_exact_successor_power = bpvi_partial_bpd_exact_successor_power * bpvi_factor_bpd_exact_successor_power)))))))) /\\ exists bpvi_divisor_factor_bpd_exact_successor. a = bpvi_result_bpd_exact_successor * bpvi_divisor_factor_bpd_exact_successor))", "specialize power_valuation_selected_and_successor_not_divides p", "specialize power_valuation_selected_and_successor_not_divides a", "specialize power_valuation_selected_and_successor_not_divides e", "apply power_valuation_selected_and_successor_not_divides", "exact hp", "exact ha", "exact hvaluation", "cases hcharacterization", "cases hcharacterization_left", "cases hcharacterization_left_witness", "cases hcharacterization_left_witness_right", "exists x", "exists x1", "split", "exact hcharacterization_left_witness_left", "split", "exact hcharacterization_left_witness_right_witness", "split", "intro hcofactor_zero", "apply ha", "trans x * x1", "exact hcharacterization_left_witness_right_witness", "rewrite hcofactor_zero", "apply PA5", "intro hcofactor_prime", "apply hcharacterization_right", "specialize power_divides_successor_of_cofactor p", "specialize power_divides_successor_of_cofactor e", "specialize power_divides_successor_of_cofactor a", "specialize power_divides_successor_of_cofactor x", "specialize power_divides_successor_of_cofactor x1", "apply power_divides_successor_of_cofactor", "exact hcharacterization_left_witness_left", "exact hcharacterization_left_witness_right_witness", "exact hcofactor_prime" ], "script_sha256": "cee5b7cb13f1910c6baef3e36c1e977848a63f9d140ab6fd5f9cef5d7f106949", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p a e. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (exists bpd_result_valuation_exact bpd_cofactor_valuation_exact. ((exists ff_b_valuation_exact_power ff_c_valuation_exact_power. ((forall ff_i_valuation_exact_power_repeat. (exists ff_lt_valuation_exact_power_repeat_bound. ff_lt_valuation_exact_power_repeat_bound + S ff_i_valuation_exact_power_repeat = e) -> (((exists ff_h_valuation_exact_power_repeat_decoded. ff_h_valuation_exact_power_repeat_decoded + S (p) = S ((S (ff_i_valuation_exact_power_repeat)) * ff_c_valuation_exact_power)) /\\ exists ff_q_valuation_exact_power_repeat_decoded. ff_b_valuation_exact_power = ff_q_valuation_exact_power_repeat_decoded * S ((S (ff_i_valuation_exact_power_repeat)) * ff_c_valuation_exact_power) + (p)))) /\\ (exists ff_u_valuation_exact_power_product ff_v_valuation_exact_power_product. ((((exists ff_h_valuation_exact_power_product_start. ff_h_valuation_exact_power_product_start + S (1) = S ((S (0)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_start. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_start * S ((S (0)) * ff_v_valuation_exact_power_product) + (1))) /\\ ((((exists ff_h_valuation_exact_power_product_terminal. ff_h_valuation_exact_power_product_terminal + S (bpd_result_valuation_exact) = S ((S (e)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_terminal. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_terminal * S ((S (e)) * ff_v_valuation_exact_power_product) + (bpd_result_valuation_exact))) /\\ forall ff_i_valuation_exact_power_product. (exists ff_lt_valuation_exact_power_product_bound. ff_lt_valuation_exact_power_product_bound + S ff_i_valuation_exact_power_product = e) -> exists ff_p_valuation_exact_power_product ff_r_valuation_exact_power_product ff_s_valuation_exact_power_product. ((((exists ff_h_valuation_exact_power_product_factor. ff_h_valuation_exact_power_product_factor + S (ff_p_valuation_exact_power_product) = S ((S (ff_i_valuation_exact_power_product)) * ff_c_valuation_exact_power)) /\\ exists ff_q_valuation_exact_power_product_factor. ff_b_valuation_exact_power = ff_q_valuation_exact_power_product_factor * S ((S (ff_i_valuation_exact_power_product)) * ff_c_valuation_exact_power) + (ff_p_valuation_exact_power_product))) /\\ ((((exists ff_h_valuation_exact_power_product_partial. ff_h_valuation_exact_power_product_partial + S (ff_r_valuation_exact_power_product) = S ((S (ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_partial. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_partial * S ((S (ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product) + (ff_r_valuation_exact_power_product))) /\\ ((((exists ff_h_valuation_exact_power_product_successor. ff_h_valuation_exact_power_product_successor + S (ff_s_valuation_exact_power_product) = S ((S (S ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product)) /\\ exists ff_q_valuation_exact_power_product_successor. ff_u_valuation_exact_power_product = ff_q_valuation_exact_power_product_successor * S ((S (S ff_i_valuation_exact_power_product)) * ff_v_valuation_exact_power_product) + (ff_s_valuation_exact_power_product))) /\\ ff_s_valuation_exact_power_product = ff_r_valuation_exact_power_product * ff_p_valuation_exact_power_product)))))))) /\\ ((a = bpd_result_valuation_exact * bpd_cofactor_valuation_exact) /\\ ((~(bpd_cofactor_valuation_exact = 0)) /\\ (~(exists bpd_factor_valuation_exact_prime. bpd_cofactor_valuation_exact = (p) * bpd_factor_valuation_exact_prime))))))", "statement_sha256": "52ad2dcbb6081c8f9a5380f905c5427c2a1168219875f958bdb9c14065ea6c44" }, { "admitted_to_alpha": true, "alpha_checked_use": true, 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"exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_mul_successor_not_divides]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "05e9ace8cd1cff9a45452595f6d4ee7e12034830e49564725889c6a486e6ec2e", "membership": "alpha_only", "name": "power_valuation_mul_successor_not_divides", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro a", "intro b", "intro e", "intro f", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "have hleft : exists bpd_result_bpd_mul_left_exact bpd_cofactor_bpd_mul_left_exact. ((exists ff_b_bpd_mul_left_exact_power ff_c_bpd_mul_left_exact_power. ((forall ff_i_bpd_mul_left_exact_power_repeat. (exists ff_lt_bpd_mul_left_exact_power_repeat_bound. ff_lt_bpd_mul_left_exact_power_repeat_bound + S ff_i_bpd_mul_left_exact_power_repeat = e) -> (((exists ff_h_bpd_mul_left_exact_power_repeat_decoded. ff_h_bpd_mul_left_exact_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_left_exact_power_repeat)) * ff_c_bpd_mul_left_exact_power)) /\\ exists ff_q_bpd_mul_left_exact_power_repeat_decoded. ff_b_bpd_mul_left_exact_power = ff_q_bpd_mul_left_exact_power_repeat_decoded * S ((S (ff_i_bpd_mul_left_exact_power_repeat)) * ff_c_bpd_mul_left_exact_power) + (p)))) /\\ (exists ff_u_bpd_mul_left_exact_power_product ff_v_bpd_mul_left_exact_power_product. ((((exists ff_h_bpd_mul_left_exact_power_product_start. ff_h_bpd_mul_left_exact_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_start. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_start * S ((S (0)) * ff_v_bpd_mul_left_exact_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_left_exact_power_product_terminal. ff_h_bpd_mul_left_exact_power_product_terminal + S (bpd_result_bpd_mul_left_exact) = S ((S (e)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_terminal. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_terminal * S ((S (e)) * ff_v_bpd_mul_left_exact_power_product) + (bpd_result_bpd_mul_left_exact))) /\\ forall ff_i_bpd_mul_left_exact_power_product. (exists ff_lt_bpd_mul_left_exact_power_product_bound. ff_lt_bpd_mul_left_exact_power_product_bound + S ff_i_bpd_mul_left_exact_power_product = e) -> exists ff_p_bpd_mul_left_exact_power_product ff_r_bpd_mul_left_exact_power_product ff_s_bpd_mul_left_exact_power_product. ((((exists ff_h_bpd_mul_left_exact_power_product_factor. ff_h_bpd_mul_left_exact_power_product_factor + S (ff_p_bpd_mul_left_exact_power_product) = S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_c_bpd_mul_left_exact_power)) /\\ exists ff_q_bpd_mul_left_exact_power_product_factor. ff_b_bpd_mul_left_exact_power = ff_q_bpd_mul_left_exact_power_product_factor * S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_c_bpd_mul_left_exact_power) + (ff_p_bpd_mul_left_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_left_exact_power_product_partial. ff_h_bpd_mul_left_exact_power_product_partial + S (ff_r_bpd_mul_left_exact_power_product) = S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_partial. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_partial * S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product) + (ff_r_bpd_mul_left_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_left_exact_power_product_successor. ff_h_bpd_mul_left_exact_power_product_successor + S (ff_s_bpd_mul_left_exact_power_product) = S ((S (S ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_successor. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_successor * S ((S (S ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product) + (ff_s_bpd_mul_left_exact_power_product))) /\\ ff_s_bpd_mul_left_exact_power_product = ff_r_bpd_mul_left_exact_power_product * ff_p_bpd_mul_left_exact_power_product)))))))) /\\ ((a = bpd_result_bpd_mul_left_exact * bpd_cofactor_bpd_mul_left_exact) /\\ ((~(bpd_cofactor_bpd_mul_left_exact = 0)) /\\ (~(exists bpd_factor_bpd_mul_left_exact_prime. bpd_cofactor_bpd_mul_left_exact = (p) * bpd_factor_bpd_mul_left_exact_prime)))))", "specialize power_valuation_exact_cofactor p", "specialize power_valuation_exact_cofactor a", "specialize power_valuation_exact_cofactor e", "apply power_valuation_exact_cofactor", "exact hp", "exact ha", "exact hvaluation_a", "cases hleft", "cases hleft_witness", "cases hleft_witness_witness", "cases hleft_witness_witness_right", "cases hleft_witness_witness_right_right", "have hright : exists bpd_result_bpd_mul_right_exact bpd_cofactor_bpd_mul_right_exact. ((exists ff_b_bpd_mul_right_exact_power ff_c_bpd_mul_right_exact_power. ((forall ff_i_bpd_mul_right_exact_power_repeat. (exists ff_lt_bpd_mul_right_exact_power_repeat_bound. ff_lt_bpd_mul_right_exact_power_repeat_bound + S ff_i_bpd_mul_right_exact_power_repeat = f) -> (((exists ff_h_bpd_mul_right_exact_power_repeat_decoded. ff_h_bpd_mul_right_exact_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_right_exact_power_repeat)) * ff_c_bpd_mul_right_exact_power)) /\\ exists ff_q_bpd_mul_right_exact_power_repeat_decoded. ff_b_bpd_mul_right_exact_power = ff_q_bpd_mul_right_exact_power_repeat_decoded * S ((S (ff_i_bpd_mul_right_exact_power_repeat)) * ff_c_bpd_mul_right_exact_power) + (p)))) /\\ (exists ff_u_bpd_mul_right_exact_power_product ff_v_bpd_mul_right_exact_power_product. ((((exists ff_h_bpd_mul_right_exact_power_product_start. ff_h_bpd_mul_right_exact_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_start. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_start * S ((S (0)) * ff_v_bpd_mul_right_exact_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_right_exact_power_product_terminal. ff_h_bpd_mul_right_exact_power_product_terminal + S (bpd_result_bpd_mul_right_exact) = S ((S (f)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_terminal. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_terminal * S ((S (f)) * ff_v_bpd_mul_right_exact_power_product) + (bpd_result_bpd_mul_right_exact))) /\\ forall ff_i_bpd_mul_right_exact_power_product. (exists ff_lt_bpd_mul_right_exact_power_product_bound. ff_lt_bpd_mul_right_exact_power_product_bound + S ff_i_bpd_mul_right_exact_power_product = f) -> exists ff_p_bpd_mul_right_exact_power_product ff_r_bpd_mul_right_exact_power_product ff_s_bpd_mul_right_exact_power_product. ((((exists ff_h_bpd_mul_right_exact_power_product_factor. ff_h_bpd_mul_right_exact_power_product_factor + S (ff_p_bpd_mul_right_exact_power_product) = S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_c_bpd_mul_right_exact_power)) /\\ exists ff_q_bpd_mul_right_exact_power_product_factor. ff_b_bpd_mul_right_exact_power = ff_q_bpd_mul_right_exact_power_product_factor * S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_c_bpd_mul_right_exact_power) + (ff_p_bpd_mul_right_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_right_exact_power_product_partial. ff_h_bpd_mul_right_exact_power_product_partial + S (ff_r_bpd_mul_right_exact_power_product) = S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_partial. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_partial * S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product) + (ff_r_bpd_mul_right_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_right_exact_power_product_successor. ff_h_bpd_mul_right_exact_power_product_successor + S (ff_s_bpd_mul_right_exact_power_product) = S ((S (S ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_successor. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_successor * S ((S (S ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product) + (ff_s_bpd_mul_right_exact_power_product))) /\\ ff_s_bpd_mul_right_exact_power_product = ff_r_bpd_mul_right_exact_power_product * ff_p_bpd_mul_right_exact_power_product)))))))) /\\ ((b = bpd_result_bpd_mul_right_exact * bpd_cofactor_bpd_mul_right_exact) /\\ ((~(bpd_cofactor_bpd_mul_right_exact = 0)) /\\ (~(exists bpd_factor_bpd_mul_right_exact_prime. bpd_cofactor_bpd_mul_right_exact = (p) * bpd_factor_bpd_mul_right_exact_prime)))))", "specialize power_valuation_exact_cofactor p", "specialize power_valuation_exact_cofactor b", "specialize power_valuation_exact_cofactor f", "apply power_valuation_exact_cofactor", "exact hp", "exact hb", "exact hvaluation_b", "cases hright", "cases hright_witness", "cases hright_witness_witness", "cases hright_witness_witness_right", "cases hright_witness_witness_right_right", "have hsum_power : exists t. (exists bpvi_b_bpd_mul_sum_power bpvi_c_bpd_mul_sum_power. ((forall bpvi_i_bpd_mul_sum_power. (exists bpvi_repeat_gap_bpd_mul_sum_power. bpvi_repeat_gap_bpd_mul_sum_power + S bpvi_i_bpd_mul_sum_power = e + f) -> (((exists bpvi_h_bpd_mul_sum_power_repeat. bpvi_h_bpd_mul_sum_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_repeat. bpvi_b_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_repeat * S ((S (bpvi_i_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_sum_power bpvi_v_bpd_mul_sum_power. ((((exists bpvi_h_bpd_mul_sum_power_start. bpvi_h_bpd_mul_sum_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_start. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_start * S ((S (0)) * bpvi_v_bpd_mul_sum_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_sum_power_terminal. bpvi_h_bpd_mul_sum_power_terminal + S (t) = S ((S (e + f)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_terminal. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_terminal * S ((S (e + f)) * bpvi_v_bpd_mul_sum_power) + (t))) /\\ forall bpvi_j_bpd_mul_sum_power. (exists bpvi_product_gap_bpd_mul_sum_power. bpvi_product_gap_bpd_mul_sum_power + S bpvi_j_bpd_mul_sum_power = e + f) -> exists bpvi_factor_bpd_mul_sum_power bpvi_partial_bpd_mul_sum_power bpvi_successor_bpd_mul_sum_power. ((((exists bpvi_h_bpd_mul_sum_power_factor. bpvi_h_bpd_mul_sum_power_factor + S (bpvi_factor_bpd_mul_sum_power) = S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_factor. bpvi_b_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_factor * S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power) + (bpvi_factor_bpd_mul_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_sum_power_partial. bpvi_h_bpd_mul_sum_power_partial + S (bpvi_partial_bpd_mul_sum_power) = S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_partial. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_partial * S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power) + (bpvi_partial_bpd_mul_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_sum_power_successor. bpvi_h_bpd_mul_sum_power_successor + S (bpvi_successor_bpd_mul_sum_power) = S ((S (S bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_successor. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_successor * S ((S (S bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power) + (bpvi_successor_bpd_mul_sum_power))) /\\ bpvi_successor_bpd_mul_sum_power = bpvi_partial_bpd_mul_sum_power * bpvi_factor_bpd_mul_sum_power))))))))", "specialize pow_exists p", "specialize pow_exists (e + f)", "exact pow_exists", "cases hsum_power", "have hpower_product : x4 = x * x2", "specialize pow_add p", "specialize pow_add e", "specialize pow_add f", "specialize pow_add (e + f)", "specialize pow_add x", "specialize pow_add x2", "specialize pow_add x4", "apply pow_add", "refl", "exact hleft_witness_witness_left", "exact hright_witness_witness_left", "exact hsum_power_witness", "have hproduct_eq : a * b = x4 * (x1 * x3)", "trans (x * x1) * (x2 * x3)", "congr", "exact hleft_witness_witness_right_left", "exact hright_witness_witness_right_left", "trans (x * x2) * (x1 * x3)", "apply mul_shuffle_four", "congr", "symm", "exact hpower_product", "refl", "have hcofactor_nondiv : ~(exists u. x1 * x3 = p * u)", "intro hcofactor_div", "specialize prime_nondivisor_mul p", "specialize prime_nondivisor_mul x1", "specialize prime_nondivisor_mul x3", "apply prime_nondivisor_mul", "exact hp", "exact hleft_witness_witness_right_right_right", "exact hright_witness_witness_right_right_right", "exact hcofactor_div", "intro hsuccessor", "apply hcofactor_nondiv", "specialize prime_power_successor_cancel_cofactor p", "specialize prime_power_successor_cancel_cofactor (e + f)", "specialize prime_power_successor_cancel_cofactor (a * b)", "specialize prime_power_successor_cancel_cofactor x4", "specialize prime_power_successor_cancel_cofactor (x1 * x3)", "apply prime_power_successor_cancel_cofactor", "exact hp", "exact hsum_power_witness", "exact hproduct_eq", "exact hsuccessor" ], "script_sha256": "f6bc4b2fec1c1961442a0e8bcde838fcc9f1ef559a8fbf5eaabe44d46fa8c90b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p a b e f. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> ~(exists bpvi_result_valuation_mul_successor. ((exists bpvi_b_valuation_mul_successor_power bpvi_c_valuation_mul_successor_power. ((forall bpvi_i_valuation_mul_successor_power. (exists bpvi_repeat_gap_valuation_mul_successor_power. bpvi_repeat_gap_valuation_mul_successor_power + S bpvi_i_valuation_mul_successor_power = S (e + f)) -> (((exists bpvi_h_valuation_mul_successor_power_repeat. bpvi_h_valuation_mul_successor_power_repeat + S (p) = S ((S (bpvi_i_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_repeat. bpvi_b_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_repeat * S ((S (bpvi_i_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power) + (p)))) /\\ (exists bpvi_u_valuation_mul_successor_power bpvi_v_valuation_mul_successor_power. ((((exists bpvi_h_valuation_mul_successor_power_start. bpvi_h_valuation_mul_successor_power_start + S (1) = S ((S (0)) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_start. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_start * S ((S (0)) * bpvi_v_valuation_mul_successor_power) + (1))) /\\ ((((exists bpvi_h_valuation_mul_successor_power_terminal. bpvi_h_valuation_mul_successor_power_terminal + S (bpvi_result_valuation_mul_successor) = S ((S (S (e + f))) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_terminal. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_terminal * S ((S (S (e + f))) * bpvi_v_valuation_mul_successor_power) + (bpvi_result_valuation_mul_successor))) /\\ forall bpvi_j_valuation_mul_successor_power. (exists bpvi_product_gap_valuation_mul_successor_power. bpvi_product_gap_valuation_mul_successor_power + S bpvi_j_valuation_mul_successor_power = S (e + f)) -> exists bpvi_factor_valuation_mul_successor_power bpvi_partial_valuation_mul_successor_power bpvi_successor_valuation_mul_successor_power. ((((exists bpvi_h_valuation_mul_successor_power_factor. bpvi_h_valuation_mul_successor_power_factor + S (bpvi_factor_valuation_mul_successor_power) = S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_factor. bpvi_b_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_factor * S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power) + (bpvi_factor_valuation_mul_successor_power))) /\\ ((((exists bpvi_h_valuation_mul_successor_power_partial. bpvi_h_valuation_mul_successor_power_partial + S (bpvi_partial_valuation_mul_successor_power) = S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_partial. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_partial * S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power) + (bpvi_partial_valuation_mul_successor_power))) /\\ ((((exists bpvi_h_valuation_mul_successor_power_successor. bpvi_h_valuation_mul_successor_power_successor + S (bpvi_successor_valuation_mul_successor_power) = S ((S (S bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_successor. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_successor * S ((S (S bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power) + (bpvi_successor_valuation_mul_successor_power))) /\\ bpvi_successor_valuation_mul_successor_power = bpvi_partial_valuation_mul_successor_power * bpvi_factor_valuation_mul_successor_power)))))))) /\\ exists bpvi_divisor_factor_valuation_mul_successor. a * b = bpvi_result_valuation_mul_successor * bpvi_divisor_factor_valuation_mul_successor))", "statement_sha256": "901e81718280314d7721bbae7ddede7b1f9995a78e6c80b696347d12d2b9e4a9", "summary": "The product of exact prime-power valuations has no next power divisor.", "summary_sha256": "60dd015f1c1f0bf906b38cbff9e02037fc001c72ddfdb70ce8a34bc0c12db3ab" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_exact_cofactor", "pow_exists", "pow_add", "mul_shuffle_four", "prime_nondivisor_mul", "prime_power_successor_cancel_cofactor" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": 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hvaluation_b", "have hleft : exists bpd_result_bpd_mul_left_exact bpd_cofactor_bpd_mul_left_exact. ((exists ff_b_bpd_mul_left_exact_power ff_c_bpd_mul_left_exact_power. ((forall ff_i_bpd_mul_left_exact_power_repeat. (exists ff_lt_bpd_mul_left_exact_power_repeat_bound. ff_lt_bpd_mul_left_exact_power_repeat_bound + S ff_i_bpd_mul_left_exact_power_repeat = e) -> (((exists ff_h_bpd_mul_left_exact_power_repeat_decoded. ff_h_bpd_mul_left_exact_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_left_exact_power_repeat)) * ff_c_bpd_mul_left_exact_power)) /\\ exists ff_q_bpd_mul_left_exact_power_repeat_decoded. ff_b_bpd_mul_left_exact_power = ff_q_bpd_mul_left_exact_power_repeat_decoded * S ((S (ff_i_bpd_mul_left_exact_power_repeat)) * ff_c_bpd_mul_left_exact_power) + (p)))) /\\ (exists ff_u_bpd_mul_left_exact_power_product ff_v_bpd_mul_left_exact_power_product. ((((exists ff_h_bpd_mul_left_exact_power_product_start. ff_h_bpd_mul_left_exact_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_start. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_start * S ((S (0)) * ff_v_bpd_mul_left_exact_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_left_exact_power_product_terminal. ff_h_bpd_mul_left_exact_power_product_terminal + S (bpd_result_bpd_mul_left_exact) = S ((S (e)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_terminal. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_terminal * S ((S (e)) * ff_v_bpd_mul_left_exact_power_product) + (bpd_result_bpd_mul_left_exact))) /\\ forall ff_i_bpd_mul_left_exact_power_product. (exists ff_lt_bpd_mul_left_exact_power_product_bound. ff_lt_bpd_mul_left_exact_power_product_bound + S ff_i_bpd_mul_left_exact_power_product = e) -> exists ff_p_bpd_mul_left_exact_power_product ff_r_bpd_mul_left_exact_power_product ff_s_bpd_mul_left_exact_power_product. ((((exists ff_h_bpd_mul_left_exact_power_product_factor. ff_h_bpd_mul_left_exact_power_product_factor + S (ff_p_bpd_mul_left_exact_power_product) = S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_c_bpd_mul_left_exact_power)) /\\ exists ff_q_bpd_mul_left_exact_power_product_factor. ff_b_bpd_mul_left_exact_power = ff_q_bpd_mul_left_exact_power_product_factor * S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_c_bpd_mul_left_exact_power) + (ff_p_bpd_mul_left_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_left_exact_power_product_partial. ff_h_bpd_mul_left_exact_power_product_partial + S (ff_r_bpd_mul_left_exact_power_product) = S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_partial. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_partial * S ((S (ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product) + (ff_r_bpd_mul_left_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_left_exact_power_product_successor. ff_h_bpd_mul_left_exact_power_product_successor + S (ff_s_bpd_mul_left_exact_power_product) = S ((S (S ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product)) /\\ exists ff_q_bpd_mul_left_exact_power_product_successor. ff_u_bpd_mul_left_exact_power_product = ff_q_bpd_mul_left_exact_power_product_successor * S ((S (S ff_i_bpd_mul_left_exact_power_product)) * ff_v_bpd_mul_left_exact_power_product) + (ff_s_bpd_mul_left_exact_power_product))) /\\ ff_s_bpd_mul_left_exact_power_product = ff_r_bpd_mul_left_exact_power_product * ff_p_bpd_mul_left_exact_power_product)))))))) /\\ ((a = bpd_result_bpd_mul_left_exact * bpd_cofactor_bpd_mul_left_exact) /\\ ((~(bpd_cofactor_bpd_mul_left_exact = 0)) /\\ (~(exists bpd_factor_bpd_mul_left_exact_prime. bpd_cofactor_bpd_mul_left_exact = (p) * bpd_factor_bpd_mul_left_exact_prime)))))", "specialize power_valuation_exact_cofactor p", "specialize power_valuation_exact_cofactor a", "specialize power_valuation_exact_cofactor e", "apply power_valuation_exact_cofactor", "exact hp", "exact ha", "exact hvaluation_a", "cases hleft", "cases hleft_witness", "cases hleft_witness_witness", "cases hleft_witness_witness_right", "cases hleft_witness_witness_right_right", "have hright : exists bpd_result_bpd_mul_right_exact bpd_cofactor_bpd_mul_right_exact. ((exists ff_b_bpd_mul_right_exact_power ff_c_bpd_mul_right_exact_power. ((forall ff_i_bpd_mul_right_exact_power_repeat. (exists ff_lt_bpd_mul_right_exact_power_repeat_bound. ff_lt_bpd_mul_right_exact_power_repeat_bound + S ff_i_bpd_mul_right_exact_power_repeat = f) -> (((exists ff_h_bpd_mul_right_exact_power_repeat_decoded. ff_h_bpd_mul_right_exact_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_right_exact_power_repeat)) * ff_c_bpd_mul_right_exact_power)) /\\ exists ff_q_bpd_mul_right_exact_power_repeat_decoded. ff_b_bpd_mul_right_exact_power = ff_q_bpd_mul_right_exact_power_repeat_decoded * S ((S (ff_i_bpd_mul_right_exact_power_repeat)) * ff_c_bpd_mul_right_exact_power) + (p)))) /\\ (exists ff_u_bpd_mul_right_exact_power_product ff_v_bpd_mul_right_exact_power_product. ((((exists ff_h_bpd_mul_right_exact_power_product_start. ff_h_bpd_mul_right_exact_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_start. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_start * S ((S (0)) * ff_v_bpd_mul_right_exact_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_right_exact_power_product_terminal. ff_h_bpd_mul_right_exact_power_product_terminal + S (bpd_result_bpd_mul_right_exact) = S ((S (f)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_terminal. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_terminal * S ((S (f)) * ff_v_bpd_mul_right_exact_power_product) + (bpd_result_bpd_mul_right_exact))) /\\ forall ff_i_bpd_mul_right_exact_power_product. (exists ff_lt_bpd_mul_right_exact_power_product_bound. ff_lt_bpd_mul_right_exact_power_product_bound + S ff_i_bpd_mul_right_exact_power_product = f) -> exists ff_p_bpd_mul_right_exact_power_product ff_r_bpd_mul_right_exact_power_product ff_s_bpd_mul_right_exact_power_product. ((((exists ff_h_bpd_mul_right_exact_power_product_factor. ff_h_bpd_mul_right_exact_power_product_factor + S (ff_p_bpd_mul_right_exact_power_product) = S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_c_bpd_mul_right_exact_power)) /\\ exists ff_q_bpd_mul_right_exact_power_product_factor. ff_b_bpd_mul_right_exact_power = ff_q_bpd_mul_right_exact_power_product_factor * S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_c_bpd_mul_right_exact_power) + (ff_p_bpd_mul_right_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_right_exact_power_product_partial. ff_h_bpd_mul_right_exact_power_product_partial + S (ff_r_bpd_mul_right_exact_power_product) = S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_partial. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_partial * S ((S (ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product) + (ff_r_bpd_mul_right_exact_power_product))) /\\ ((((exists ff_h_bpd_mul_right_exact_power_product_successor. ff_h_bpd_mul_right_exact_power_product_successor + S (ff_s_bpd_mul_right_exact_power_product) = S ((S (S ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product)) /\\ exists ff_q_bpd_mul_right_exact_power_product_successor. ff_u_bpd_mul_right_exact_power_product = ff_q_bpd_mul_right_exact_power_product_successor * S ((S (S ff_i_bpd_mul_right_exact_power_product)) * ff_v_bpd_mul_right_exact_power_product) + (ff_s_bpd_mul_right_exact_power_product))) /\\ ff_s_bpd_mul_right_exact_power_product = ff_r_bpd_mul_right_exact_power_product * ff_p_bpd_mul_right_exact_power_product)))))))) /\\ ((b = bpd_result_bpd_mul_right_exact * bpd_cofactor_bpd_mul_right_exact) /\\ ((~(bpd_cofactor_bpd_mul_right_exact = 0)) /\\ (~(exists bpd_factor_bpd_mul_right_exact_prime. bpd_cofactor_bpd_mul_right_exact = (p) * bpd_factor_bpd_mul_right_exact_prime)))))", "specialize power_valuation_exact_cofactor p", "specialize power_valuation_exact_cofactor b", "specialize power_valuation_exact_cofactor f", "apply power_valuation_exact_cofactor", "exact hp", "exact hb", "exact hvaluation_b", "cases hright", "cases hright_witness", "cases hright_witness_witness", "cases hright_witness_witness_right", "cases hright_witness_witness_right_right", "have hsum_power : exists t. (exists bpvi_b_bpd_mul_sum_power bpvi_c_bpd_mul_sum_power. ((forall bpvi_i_bpd_mul_sum_power. (exists bpvi_repeat_gap_bpd_mul_sum_power. bpvi_repeat_gap_bpd_mul_sum_power + S bpvi_i_bpd_mul_sum_power = e + f) -> (((exists bpvi_h_bpd_mul_sum_power_repeat. bpvi_h_bpd_mul_sum_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_repeat. bpvi_b_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_repeat * S ((S (bpvi_i_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_sum_power bpvi_v_bpd_mul_sum_power. ((((exists bpvi_h_bpd_mul_sum_power_start. bpvi_h_bpd_mul_sum_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_start. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_start * S ((S (0)) * bpvi_v_bpd_mul_sum_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_sum_power_terminal. bpvi_h_bpd_mul_sum_power_terminal + S (t) = S ((S (e + f)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_terminal. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_terminal * S ((S (e + f)) * bpvi_v_bpd_mul_sum_power) + (t))) /\\ forall bpvi_j_bpd_mul_sum_power. (exists bpvi_product_gap_bpd_mul_sum_power. bpvi_product_gap_bpd_mul_sum_power + S bpvi_j_bpd_mul_sum_power = e + f) -> exists bpvi_factor_bpd_mul_sum_power bpvi_partial_bpd_mul_sum_power bpvi_successor_bpd_mul_sum_power. ((((exists bpvi_h_bpd_mul_sum_power_factor. bpvi_h_bpd_mul_sum_power_factor + S (bpvi_factor_bpd_mul_sum_power) = S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_factor. bpvi_b_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_factor * S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_c_bpd_mul_sum_power) + (bpvi_factor_bpd_mul_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_sum_power_partial. bpvi_h_bpd_mul_sum_power_partial + S (bpvi_partial_bpd_mul_sum_power) = S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_partial. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_partial * S ((S (bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power) + (bpvi_partial_bpd_mul_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_sum_power_successor. bpvi_h_bpd_mul_sum_power_successor + S (bpvi_successor_bpd_mul_sum_power) = S ((S (S bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power)) /\\ exists bpvi_q_bpd_mul_sum_power_successor. bpvi_u_bpd_mul_sum_power = bpvi_q_bpd_mul_sum_power_successor * S ((S (S bpvi_j_bpd_mul_sum_power)) * bpvi_v_bpd_mul_sum_power) + (bpvi_successor_bpd_mul_sum_power))) /\\ bpvi_successor_bpd_mul_sum_power = bpvi_partial_bpd_mul_sum_power * bpvi_factor_bpd_mul_sum_power))))))))", "specialize pow_exists p", "specialize pow_exists (e + f)", "exact pow_exists", "cases hsum_power", "have hpower_product : x4 = x * x2", "specialize pow_add p", "specialize pow_add e", "specialize pow_add f", "specialize pow_add (e + f)", "specialize pow_add x", "specialize pow_add x2", "specialize pow_add x4", "apply pow_add", "refl", "exact hleft_witness_witness_left", "exact hright_witness_witness_left", "exact hsum_power_witness", "have hproduct_eq : a * b = x4 * (x1 * x3)", "trans (x * x1) * (x2 * x3)", "congr", "exact hleft_witness_witness_right_left", "exact hright_witness_witness_right_left", "trans (x * x2) * (x1 * x3)", "apply mul_shuffle_four", "congr", "symm", "exact hpower_product", "refl", "have hcofactor_nondiv : ~(exists u. x1 * x3 = p * u)", "intro hcofactor_div", "specialize prime_nondivisor_mul p", "specialize prime_nondivisor_mul x1", "specialize prime_nondivisor_mul x3", "apply prime_nondivisor_mul", "exact hp", "exact hleft_witness_witness_right_right_right", "exact hright_witness_witness_right_right_right", "exact hcofactor_div", "intro hsuccessor", "apply hcofactor_nondiv", "specialize prime_power_successor_cancel_cofactor p", "specialize prime_power_successor_cancel_cofactor (e + f)", "specialize prime_power_successor_cancel_cofactor (a * b)", "specialize prime_power_successor_cancel_cofactor x4", "specialize prime_power_successor_cancel_cofactor (x1 * x3)", "apply prime_power_successor_cancel_cofactor", "exact hp", "exact hsum_power_witness", "exact hproduct_eq", "exact hsuccessor" ], "script_sha256": "f6bc4b2fec1c1961442a0e8bcde838fcc9f1ef559a8fbf5eaabe44d46fa8c90b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p a b e f. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> ~(exists bpvi_result_valuation_mul_successor. ((exists bpvi_b_valuation_mul_successor_power bpvi_c_valuation_mul_successor_power. ((forall bpvi_i_valuation_mul_successor_power. (exists bpvi_repeat_gap_valuation_mul_successor_power. bpvi_repeat_gap_valuation_mul_successor_power + S bpvi_i_valuation_mul_successor_power = S (e + f)) -> (((exists bpvi_h_valuation_mul_successor_power_repeat. bpvi_h_valuation_mul_successor_power_repeat + S (p) = S ((S (bpvi_i_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_repeat. bpvi_b_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_repeat * S ((S (bpvi_i_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power) + (p)))) /\\ (exists bpvi_u_valuation_mul_successor_power bpvi_v_valuation_mul_successor_power. ((((exists bpvi_h_valuation_mul_successor_power_start. bpvi_h_valuation_mul_successor_power_start + S (1) = S ((S (0)) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_start. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_start * S ((S (0)) * bpvi_v_valuation_mul_successor_power) + (1))) /\\ ((((exists bpvi_h_valuation_mul_successor_power_terminal. bpvi_h_valuation_mul_successor_power_terminal + S (bpvi_result_valuation_mul_successor) = S ((S (S (e + f))) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_terminal. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_terminal * S ((S (S (e + f))) * bpvi_v_valuation_mul_successor_power) + (bpvi_result_valuation_mul_successor))) /\\ forall bpvi_j_valuation_mul_successor_power. (exists bpvi_product_gap_valuation_mul_successor_power. bpvi_product_gap_valuation_mul_successor_power + S bpvi_j_valuation_mul_successor_power = S (e + f)) -> exists bpvi_factor_valuation_mul_successor_power bpvi_partial_valuation_mul_successor_power bpvi_successor_valuation_mul_successor_power. ((((exists bpvi_h_valuation_mul_successor_power_factor. bpvi_h_valuation_mul_successor_power_factor + S (bpvi_factor_valuation_mul_successor_power) = S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_factor. bpvi_b_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_factor * S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_c_valuation_mul_successor_power) + (bpvi_factor_valuation_mul_successor_power))) /\\ ((((exists bpvi_h_valuation_mul_successor_power_partial. bpvi_h_valuation_mul_successor_power_partial + S (bpvi_partial_valuation_mul_successor_power) = S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_partial. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_partial * S ((S (bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power) + (bpvi_partial_valuation_mul_successor_power))) /\\ ((((exists bpvi_h_valuation_mul_successor_power_successor. bpvi_h_valuation_mul_successor_power_successor + S (bpvi_successor_valuation_mul_successor_power) = S ((S (S bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power)) /\\ exists bpvi_q_valuation_mul_successor_power_successor. bpvi_u_valuation_mul_successor_power = bpvi_q_valuation_mul_successor_power_successor * S ((S (S bpvi_j_valuation_mul_successor_power)) * bpvi_v_valuation_mul_successor_power) + (bpvi_successor_valuation_mul_successor_power))) /\\ bpvi_successor_valuation_mul_successor_power = bpvi_partial_valuation_mul_successor_power * bpvi_factor_valuation_mul_successor_power)))))))) /\\ exists bpvi_divisor_factor_valuation_mul_successor. a * b = bpvi_result_valuation_mul_successor * bpvi_divisor_factor_valuation_mul_successor))", "statement_sha256": "901e81718280314d7721bbae7ddede7b1f9995a78e6c80b696347d12d2b9e4a9" }, { "admitted_to_alpha": true, 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"exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_mul_lower]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "a1aed8f8822ab646921ca5cf3a5aa7c9eff8082fe9399f200b98924b32960147", "membership": "alpha_only", "name": "power_valuation_mul_lower", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro a", "intro b", "intro e", "intro f", "intro g", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "intro hvaluation_product", "have hleft : exists bpv_result_bpd_mul_lower_left. ((exists ff_b_bpd_mul_lower_left_power ff_c_bpd_mul_lower_left_power. ((forall ff_i_bpd_mul_lower_left_power_repeat. (exists ff_lt_bpd_mul_lower_left_power_repeat_bound. ff_lt_bpd_mul_lower_left_power_repeat_bound + S ff_i_bpd_mul_lower_left_power_repeat = e) -> (((exists ff_h_bpd_mul_lower_left_power_repeat_decoded. ff_h_bpd_mul_lower_left_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_lower_left_power_repeat)) * ff_c_bpd_mul_lower_left_power)) /\\ exists ff_q_bpd_mul_lower_left_power_repeat_decoded. ff_b_bpd_mul_lower_left_power = ff_q_bpd_mul_lower_left_power_repeat_decoded * S ((S (ff_i_bpd_mul_lower_left_power_repeat)) * ff_c_bpd_mul_lower_left_power) + (p)))) /\\ (exists ff_u_bpd_mul_lower_left_power_product ff_v_bpd_mul_lower_left_power_product. ((((exists ff_h_bpd_mul_lower_left_power_product_start. ff_h_bpd_mul_lower_left_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_start. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_start * S ((S (0)) * ff_v_bpd_mul_lower_left_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_lower_left_power_product_terminal. ff_h_bpd_mul_lower_left_power_product_terminal + S (bpv_result_bpd_mul_lower_left) = S ((S (e)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_terminal. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_terminal * S ((S (e)) * ff_v_bpd_mul_lower_left_power_product) + (bpv_result_bpd_mul_lower_left))) /\\ forall ff_i_bpd_mul_lower_left_power_product. (exists ff_lt_bpd_mul_lower_left_power_product_bound. ff_lt_bpd_mul_lower_left_power_product_bound + S ff_i_bpd_mul_lower_left_power_product = e) -> exists ff_p_bpd_mul_lower_left_power_product ff_r_bpd_mul_lower_left_power_product ff_s_bpd_mul_lower_left_power_product. ((((exists ff_h_bpd_mul_lower_left_power_product_factor. ff_h_bpd_mul_lower_left_power_product_factor + S (ff_p_bpd_mul_lower_left_power_product) = S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_c_bpd_mul_lower_left_power)) /\\ exists ff_q_bpd_mul_lower_left_power_product_factor. ff_b_bpd_mul_lower_left_power = ff_q_bpd_mul_lower_left_power_product_factor * S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_c_bpd_mul_lower_left_power) + (ff_p_bpd_mul_lower_left_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_left_power_product_partial. ff_h_bpd_mul_lower_left_power_product_partial + S (ff_r_bpd_mul_lower_left_power_product) = S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_partial. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_partial * S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product) + (ff_r_bpd_mul_lower_left_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_left_power_product_successor. ff_h_bpd_mul_lower_left_power_product_successor + S (ff_s_bpd_mul_lower_left_power_product) = S ((S (S ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_successor. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_successor * S ((S (S ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product) + (ff_s_bpd_mul_lower_left_power_product))) /\\ ff_s_bpd_mul_lower_left_power_product = ff_r_bpd_mul_lower_left_power_product * ff_p_bpd_mul_lower_left_power_product)))))))) /\\ (exists bpv_factor_bpd_mul_lower_left_divides. a = bpv_result_bpd_mul_lower_left * bpv_factor_bpd_mul_lower_left_divides))", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides a", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation_a", "have hright : exists bpv_result_bpd_mul_lower_right. ((exists ff_b_bpd_mul_lower_right_power ff_c_bpd_mul_lower_right_power. ((forall ff_i_bpd_mul_lower_right_power_repeat. (exists ff_lt_bpd_mul_lower_right_power_repeat_bound. ff_lt_bpd_mul_lower_right_power_repeat_bound + S ff_i_bpd_mul_lower_right_power_repeat = f) -> (((exists ff_h_bpd_mul_lower_right_power_repeat_decoded. ff_h_bpd_mul_lower_right_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_lower_right_power_repeat)) * ff_c_bpd_mul_lower_right_power)) /\\ exists ff_q_bpd_mul_lower_right_power_repeat_decoded. ff_b_bpd_mul_lower_right_power = ff_q_bpd_mul_lower_right_power_repeat_decoded * S ((S (ff_i_bpd_mul_lower_right_power_repeat)) * ff_c_bpd_mul_lower_right_power) + (p)))) /\\ (exists ff_u_bpd_mul_lower_right_power_product ff_v_bpd_mul_lower_right_power_product. ((((exists ff_h_bpd_mul_lower_right_power_product_start. ff_h_bpd_mul_lower_right_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_start. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_start * S ((S (0)) * ff_v_bpd_mul_lower_right_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_lower_right_power_product_terminal. ff_h_bpd_mul_lower_right_power_product_terminal + S (bpv_result_bpd_mul_lower_right) = S ((S (f)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_terminal. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_terminal * S ((S (f)) * ff_v_bpd_mul_lower_right_power_product) + (bpv_result_bpd_mul_lower_right))) /\\ forall ff_i_bpd_mul_lower_right_power_product. (exists ff_lt_bpd_mul_lower_right_power_product_bound. ff_lt_bpd_mul_lower_right_power_product_bound + S ff_i_bpd_mul_lower_right_power_product = f) -> exists ff_p_bpd_mul_lower_right_power_product ff_r_bpd_mul_lower_right_power_product ff_s_bpd_mul_lower_right_power_product. ((((exists ff_h_bpd_mul_lower_right_power_product_factor. ff_h_bpd_mul_lower_right_power_product_factor + S (ff_p_bpd_mul_lower_right_power_product) = S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_c_bpd_mul_lower_right_power)) /\\ exists ff_q_bpd_mul_lower_right_power_product_factor. ff_b_bpd_mul_lower_right_power = ff_q_bpd_mul_lower_right_power_product_factor * S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_c_bpd_mul_lower_right_power) + (ff_p_bpd_mul_lower_right_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_right_power_product_partial. ff_h_bpd_mul_lower_right_power_product_partial + S (ff_r_bpd_mul_lower_right_power_product) = S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_partial. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_partial * S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product) + (ff_r_bpd_mul_lower_right_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_right_power_product_successor. ff_h_bpd_mul_lower_right_power_product_successor + S (ff_s_bpd_mul_lower_right_power_product) = S ((S (S ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_successor. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_successor * S ((S (S ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product) + (ff_s_bpd_mul_lower_right_power_product))) /\\ ff_s_bpd_mul_lower_right_power_product = ff_r_bpd_mul_lower_right_power_product * ff_p_bpd_mul_lower_right_power_product)))))))) /\\ (exists bpv_factor_bpd_mul_lower_right_divides. b = bpv_result_bpd_mul_lower_right * bpv_factor_bpd_mul_lower_right_divides))", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides b", "specialize power_valuation_power_divides f", "apply power_valuation_power_divides", "exact hvaluation_b", "have hsum : exists bpvi_result_bpd_mul_lower_sum. ((exists bpvi_b_bpd_mul_lower_sum_power bpvi_c_bpd_mul_lower_sum_power. ((forall bpvi_i_bpd_mul_lower_sum_power. (exists bpvi_repeat_gap_bpd_mul_lower_sum_power. bpvi_repeat_gap_bpd_mul_lower_sum_power + S bpvi_i_bpd_mul_lower_sum_power = e + f) -> (((exists bpvi_h_bpd_mul_lower_sum_power_repeat. bpvi_h_bpd_mul_lower_sum_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_repeat. bpvi_b_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_repeat * S ((S (bpvi_i_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_lower_sum_power bpvi_v_bpd_mul_lower_sum_power. ((((exists bpvi_h_bpd_mul_lower_sum_power_start. bpvi_h_bpd_mul_lower_sum_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_start. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_start * S ((S (0)) * bpvi_v_bpd_mul_lower_sum_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_lower_sum_power_terminal. bpvi_h_bpd_mul_lower_sum_power_terminal + S (bpvi_result_bpd_mul_lower_sum) = S ((S (e + f)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_terminal. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_terminal * S ((S (e + f)) * bpvi_v_bpd_mul_lower_sum_power) + (bpvi_result_bpd_mul_lower_sum))) /\\ forall bpvi_j_bpd_mul_lower_sum_power. (exists bpvi_product_gap_bpd_mul_lower_sum_power. bpvi_product_gap_bpd_mul_lower_sum_power + S bpvi_j_bpd_mul_lower_sum_power = e + f) -> exists bpvi_factor_bpd_mul_lower_sum_power bpvi_partial_bpd_mul_lower_sum_power bpvi_successor_bpd_mul_lower_sum_power. ((((exists bpvi_h_bpd_mul_lower_sum_power_factor. bpvi_h_bpd_mul_lower_sum_power_factor + S (bpvi_factor_bpd_mul_lower_sum_power) = S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_factor. bpvi_b_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_factor * S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power) + (bpvi_factor_bpd_mul_lower_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_lower_sum_power_partial. bpvi_h_bpd_mul_lower_sum_power_partial + S (bpvi_partial_bpd_mul_lower_sum_power) = S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_partial. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_partial * S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power) + (bpvi_partial_bpd_mul_lower_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_lower_sum_power_successor. bpvi_h_bpd_mul_lower_sum_power_successor + S (bpvi_successor_bpd_mul_lower_sum_power) = S ((S (S bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_successor. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_successor * S ((S (S bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power) + (bpvi_successor_bpd_mul_lower_sum_power))) /\\ bpvi_successor_bpd_mul_lower_sum_power = bpvi_partial_bpd_mul_lower_sum_power * bpvi_factor_bpd_mul_lower_sum_power)))))))) /\\ exists bpvi_divisor_factor_bpd_mul_lower_sum. a * b = bpvi_result_bpd_mul_lower_sum * bpvi_divisor_factor_bpd_mul_lower_sum)", "specialize power_divides_add_mul p", "specialize power_divides_add_mul e", "specialize power_divides_add_mul f", "specialize power_divides_add_mul (e + f)", "specialize power_divides_add_mul a", "specialize power_divides_add_mul b", "apply power_divides_add_mul", "refl", "exact hleft", "exact hright", "have hproduct0 : ~(a * b = 0)", "intro hproductzero", "specialize mul_ne_zero a", "specialize mul_ne_zero b", "apply mul_ne_zero", "exact ha", "exact hb", "exact hproductzero", "have hsum_bound : exists k. k + (e + f) = a * b", "specialize prime_power_divides_exponent_le_value p", "specialize prime_power_divides_exponent_le_value (e + f)", "specialize prime_power_divides_exponent_le_value (a * b)", "apply prime_power_divides_exponent_le_value", "exact hp", "exact hproduct0", "exact hsum", "specialize power_valuation_dominates p", "specialize power_valuation_dominates (a * b)", "specialize power_valuation_dominates g", "specialize power_valuation_dominates (e + f)", "apply power_valuation_dominates", "exact hvaluation_product", "exact hsum_bound", "exact hsum" ], "script_sha256": "4034a9aa6ebfc72f1215f9d171d3141bc75e01a8a2c44dbc031f79bb1ffd5146", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p a b e f g. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> (exists bpd_gap_valuation_mul_lower. bpd_gap_valuation_mul_lower + (e + f) = (g))", "statement_sha256": "25a7880e19a3694c80eba908f2af9e8e0bb827c380227e3468d8302af7fd32f9", "summary": "The valuation of a nonzero product is at least the sum of factor valuations.", "summary_sha256": "3fce1906a27c883e63305c1aaf62757a8401450cf1a410abb9976afb92d69be5" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_power_divides", "power_divides_add_mul", "mul_ne_zero", "prime_power_divides_exponent_le_value", "power_valuation_dominates" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=203]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_mul_lower]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_mul_lower", "parent_alpha_version": "v34", "proof_bundle_node_id": 217, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_mul_lower", "script": [ "intro p", "intro a", "intro b", "intro e", "intro f", "intro g", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "intro hvaluation_product", "have hleft : exists bpv_result_bpd_mul_lower_left. ((exists ff_b_bpd_mul_lower_left_power ff_c_bpd_mul_lower_left_power. ((forall ff_i_bpd_mul_lower_left_power_repeat. (exists ff_lt_bpd_mul_lower_left_power_repeat_bound. ff_lt_bpd_mul_lower_left_power_repeat_bound + S ff_i_bpd_mul_lower_left_power_repeat = e) -> (((exists ff_h_bpd_mul_lower_left_power_repeat_decoded. ff_h_bpd_mul_lower_left_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_lower_left_power_repeat)) * ff_c_bpd_mul_lower_left_power)) /\\ exists ff_q_bpd_mul_lower_left_power_repeat_decoded. ff_b_bpd_mul_lower_left_power = ff_q_bpd_mul_lower_left_power_repeat_decoded * S ((S (ff_i_bpd_mul_lower_left_power_repeat)) * ff_c_bpd_mul_lower_left_power) + (p)))) /\\ (exists ff_u_bpd_mul_lower_left_power_product ff_v_bpd_mul_lower_left_power_product. ((((exists ff_h_bpd_mul_lower_left_power_product_start. ff_h_bpd_mul_lower_left_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_start. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_start * S ((S (0)) * ff_v_bpd_mul_lower_left_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_lower_left_power_product_terminal. ff_h_bpd_mul_lower_left_power_product_terminal + S (bpv_result_bpd_mul_lower_left) = S ((S (e)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_terminal. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_terminal * S ((S (e)) * ff_v_bpd_mul_lower_left_power_product) + (bpv_result_bpd_mul_lower_left))) /\\ forall ff_i_bpd_mul_lower_left_power_product. (exists ff_lt_bpd_mul_lower_left_power_product_bound. ff_lt_bpd_mul_lower_left_power_product_bound + S ff_i_bpd_mul_lower_left_power_product = e) -> exists ff_p_bpd_mul_lower_left_power_product ff_r_bpd_mul_lower_left_power_product ff_s_bpd_mul_lower_left_power_product. ((((exists ff_h_bpd_mul_lower_left_power_product_factor. ff_h_bpd_mul_lower_left_power_product_factor + S (ff_p_bpd_mul_lower_left_power_product) = S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_c_bpd_mul_lower_left_power)) /\\ exists ff_q_bpd_mul_lower_left_power_product_factor. ff_b_bpd_mul_lower_left_power = ff_q_bpd_mul_lower_left_power_product_factor * S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_c_bpd_mul_lower_left_power) + (ff_p_bpd_mul_lower_left_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_left_power_product_partial. ff_h_bpd_mul_lower_left_power_product_partial + S (ff_r_bpd_mul_lower_left_power_product) = S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_partial. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_partial * S ((S (ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product) + (ff_r_bpd_mul_lower_left_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_left_power_product_successor. ff_h_bpd_mul_lower_left_power_product_successor + S (ff_s_bpd_mul_lower_left_power_product) = S ((S (S ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product)) /\\ exists ff_q_bpd_mul_lower_left_power_product_successor. ff_u_bpd_mul_lower_left_power_product = ff_q_bpd_mul_lower_left_power_product_successor * S ((S (S ff_i_bpd_mul_lower_left_power_product)) * ff_v_bpd_mul_lower_left_power_product) + (ff_s_bpd_mul_lower_left_power_product))) /\\ ff_s_bpd_mul_lower_left_power_product = ff_r_bpd_mul_lower_left_power_product * ff_p_bpd_mul_lower_left_power_product)))))))) /\\ (exists bpv_factor_bpd_mul_lower_left_divides. a = bpv_result_bpd_mul_lower_left * bpv_factor_bpd_mul_lower_left_divides))", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides a", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation_a", "have hright : exists bpv_result_bpd_mul_lower_right. ((exists ff_b_bpd_mul_lower_right_power ff_c_bpd_mul_lower_right_power. ((forall ff_i_bpd_mul_lower_right_power_repeat. (exists ff_lt_bpd_mul_lower_right_power_repeat_bound. ff_lt_bpd_mul_lower_right_power_repeat_bound + S ff_i_bpd_mul_lower_right_power_repeat = f) -> (((exists ff_h_bpd_mul_lower_right_power_repeat_decoded. ff_h_bpd_mul_lower_right_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_mul_lower_right_power_repeat)) * ff_c_bpd_mul_lower_right_power)) /\\ exists ff_q_bpd_mul_lower_right_power_repeat_decoded. ff_b_bpd_mul_lower_right_power = ff_q_bpd_mul_lower_right_power_repeat_decoded * S ((S (ff_i_bpd_mul_lower_right_power_repeat)) * ff_c_bpd_mul_lower_right_power) + (p)))) /\\ (exists ff_u_bpd_mul_lower_right_power_product ff_v_bpd_mul_lower_right_power_product. ((((exists ff_h_bpd_mul_lower_right_power_product_start. ff_h_bpd_mul_lower_right_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_start. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_start * S ((S (0)) * ff_v_bpd_mul_lower_right_power_product) + (1))) /\\ ((((exists ff_h_bpd_mul_lower_right_power_product_terminal. ff_h_bpd_mul_lower_right_power_product_terminal + S (bpv_result_bpd_mul_lower_right) = S ((S (f)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_terminal. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_terminal * S ((S (f)) * ff_v_bpd_mul_lower_right_power_product) + (bpv_result_bpd_mul_lower_right))) /\\ forall ff_i_bpd_mul_lower_right_power_product. (exists ff_lt_bpd_mul_lower_right_power_product_bound. ff_lt_bpd_mul_lower_right_power_product_bound + S ff_i_bpd_mul_lower_right_power_product = f) -> exists ff_p_bpd_mul_lower_right_power_product ff_r_bpd_mul_lower_right_power_product ff_s_bpd_mul_lower_right_power_product. ((((exists ff_h_bpd_mul_lower_right_power_product_factor. ff_h_bpd_mul_lower_right_power_product_factor + S (ff_p_bpd_mul_lower_right_power_product) = S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_c_bpd_mul_lower_right_power)) /\\ exists ff_q_bpd_mul_lower_right_power_product_factor. ff_b_bpd_mul_lower_right_power = ff_q_bpd_mul_lower_right_power_product_factor * S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_c_bpd_mul_lower_right_power) + (ff_p_bpd_mul_lower_right_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_right_power_product_partial. ff_h_bpd_mul_lower_right_power_product_partial + S (ff_r_bpd_mul_lower_right_power_product) = S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_partial. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_partial * S ((S (ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product) + (ff_r_bpd_mul_lower_right_power_product))) /\\ ((((exists ff_h_bpd_mul_lower_right_power_product_successor. ff_h_bpd_mul_lower_right_power_product_successor + S (ff_s_bpd_mul_lower_right_power_product) = S ((S (S ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product)) /\\ exists ff_q_bpd_mul_lower_right_power_product_successor. ff_u_bpd_mul_lower_right_power_product = ff_q_bpd_mul_lower_right_power_product_successor * S ((S (S ff_i_bpd_mul_lower_right_power_product)) * ff_v_bpd_mul_lower_right_power_product) + (ff_s_bpd_mul_lower_right_power_product))) /\\ ff_s_bpd_mul_lower_right_power_product = ff_r_bpd_mul_lower_right_power_product * ff_p_bpd_mul_lower_right_power_product)))))))) /\\ (exists bpv_factor_bpd_mul_lower_right_divides. b = bpv_result_bpd_mul_lower_right * bpv_factor_bpd_mul_lower_right_divides))", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides b", "specialize power_valuation_power_divides f", "apply power_valuation_power_divides", "exact hvaluation_b", "have hsum : exists bpvi_result_bpd_mul_lower_sum. ((exists bpvi_b_bpd_mul_lower_sum_power bpvi_c_bpd_mul_lower_sum_power. ((forall bpvi_i_bpd_mul_lower_sum_power. (exists bpvi_repeat_gap_bpd_mul_lower_sum_power. bpvi_repeat_gap_bpd_mul_lower_sum_power + S bpvi_i_bpd_mul_lower_sum_power = e + f) -> (((exists bpvi_h_bpd_mul_lower_sum_power_repeat. bpvi_h_bpd_mul_lower_sum_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_repeat. bpvi_b_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_repeat * S ((S (bpvi_i_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_lower_sum_power bpvi_v_bpd_mul_lower_sum_power. ((((exists bpvi_h_bpd_mul_lower_sum_power_start. bpvi_h_bpd_mul_lower_sum_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_start. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_start * S ((S (0)) * bpvi_v_bpd_mul_lower_sum_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_lower_sum_power_terminal. bpvi_h_bpd_mul_lower_sum_power_terminal + S (bpvi_result_bpd_mul_lower_sum) = S ((S (e + f)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_terminal. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_terminal * S ((S (e + f)) * bpvi_v_bpd_mul_lower_sum_power) + (bpvi_result_bpd_mul_lower_sum))) /\\ forall bpvi_j_bpd_mul_lower_sum_power. (exists bpvi_product_gap_bpd_mul_lower_sum_power. bpvi_product_gap_bpd_mul_lower_sum_power + S bpvi_j_bpd_mul_lower_sum_power = e + f) -> exists bpvi_factor_bpd_mul_lower_sum_power bpvi_partial_bpd_mul_lower_sum_power bpvi_successor_bpd_mul_lower_sum_power. ((((exists bpvi_h_bpd_mul_lower_sum_power_factor. bpvi_h_bpd_mul_lower_sum_power_factor + S (bpvi_factor_bpd_mul_lower_sum_power) = S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_factor. bpvi_b_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_factor * S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_c_bpd_mul_lower_sum_power) + (bpvi_factor_bpd_mul_lower_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_lower_sum_power_partial. bpvi_h_bpd_mul_lower_sum_power_partial + S (bpvi_partial_bpd_mul_lower_sum_power) = S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_partial. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_partial * S ((S (bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power) + (bpvi_partial_bpd_mul_lower_sum_power))) /\\ ((((exists bpvi_h_bpd_mul_lower_sum_power_successor. bpvi_h_bpd_mul_lower_sum_power_successor + S (bpvi_successor_bpd_mul_lower_sum_power) = S ((S (S bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power)) /\\ exists bpvi_q_bpd_mul_lower_sum_power_successor. bpvi_u_bpd_mul_lower_sum_power = bpvi_q_bpd_mul_lower_sum_power_successor * S ((S (S bpvi_j_bpd_mul_lower_sum_power)) * bpvi_v_bpd_mul_lower_sum_power) + (bpvi_successor_bpd_mul_lower_sum_power))) /\\ bpvi_successor_bpd_mul_lower_sum_power = bpvi_partial_bpd_mul_lower_sum_power * bpvi_factor_bpd_mul_lower_sum_power)))))))) /\\ exists bpvi_divisor_factor_bpd_mul_lower_sum. a * b = bpvi_result_bpd_mul_lower_sum * bpvi_divisor_factor_bpd_mul_lower_sum)", "specialize power_divides_add_mul p", "specialize power_divides_add_mul e", "specialize power_divides_add_mul f", "specialize power_divides_add_mul (e + f)", "specialize power_divides_add_mul a", "specialize power_divides_add_mul b", "apply power_divides_add_mul", "refl", "exact hleft", "exact hright", "have hproduct0 : ~(a * b = 0)", "intro hproductzero", "specialize mul_ne_zero a", "specialize mul_ne_zero b", "apply mul_ne_zero", "exact ha", "exact hb", "exact hproductzero", "have hsum_bound : exists k. k + (e + f) = a * b", "specialize prime_power_divides_exponent_le_value p", "specialize prime_power_divides_exponent_le_value (e + f)", "specialize prime_power_divides_exponent_le_value (a * b)", "apply prime_power_divides_exponent_le_value", "exact hp", "exact hproduct0", "exact hsum", "specialize power_valuation_dominates p", "specialize power_valuation_dominates (a * b)", "specialize power_valuation_dominates g", "specialize power_valuation_dominates (e + f)", "apply power_valuation_dominates", "exact hvaluation_product", "exact hsum_bound", "exact hsum" ], "script_sha256": "4034a9aa6ebfc72f1215f9d171d3141bc75e01a8a2c44dbc031f79bb1ffd5146", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p a b e f g. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> (exists bpd_gap_valuation_mul_lower. bpd_gap_valuation_mul_lower + (e + f) = (g))", "statement_sha256": "25a7880e19a3694c80eba908f2af9e8e0bb827c380227e3468d8302af7fd32f9" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_valuation_mul_upper", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 204, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": 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p", "intro a", "intro b", "intro e", "intro f", "intro g", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "intro hvaluation_product", "have horder : (exists k. k + g = e + f) \\/ exists k. k + S (e + f) = g", "specialize le_or_lt g", "specialize le_or_lt (e + f)", "exact le_or_lt", "cases horder", "exact horder_left", "exfalso", "have hhigh : exists bpvi_result_bpd_mul_upper_high. ((exists bpvi_b_bpd_mul_upper_high_power bpvi_c_bpd_mul_upper_high_power. ((forall bpvi_i_bpd_mul_upper_high_power. (exists bpvi_repeat_gap_bpd_mul_upper_high_power. bpvi_repeat_gap_bpd_mul_upper_high_power + S bpvi_i_bpd_mul_upper_high_power = g) -> (((exists bpvi_h_bpd_mul_upper_high_power_repeat. bpvi_h_bpd_mul_upper_high_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_repeat. bpvi_b_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_repeat * S ((S (bpvi_i_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_upper_high_power bpvi_v_bpd_mul_upper_high_power. ((((exists bpvi_h_bpd_mul_upper_high_power_start. bpvi_h_bpd_mul_upper_high_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_start. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_start * S ((S (0)) * bpvi_v_bpd_mul_upper_high_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_upper_high_power_terminal. bpvi_h_bpd_mul_upper_high_power_terminal + S (bpvi_result_bpd_mul_upper_high) = S ((S (g)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_terminal. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_terminal * S ((S (g)) * bpvi_v_bpd_mul_upper_high_power) + (bpvi_result_bpd_mul_upper_high))) /\\ forall bpvi_j_bpd_mul_upper_high_power. (exists bpvi_product_gap_bpd_mul_upper_high_power. bpvi_product_gap_bpd_mul_upper_high_power + S bpvi_j_bpd_mul_upper_high_power = g) -> exists bpvi_factor_bpd_mul_upper_high_power bpvi_partial_bpd_mul_upper_high_power bpvi_successor_bpd_mul_upper_high_power. ((((exists bpvi_h_bpd_mul_upper_high_power_factor. bpvi_h_bpd_mul_upper_high_power_factor + S (bpvi_factor_bpd_mul_upper_high_power) = S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_factor. bpvi_b_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_factor * S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power) + (bpvi_factor_bpd_mul_upper_high_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_high_power_partial. bpvi_h_bpd_mul_upper_high_power_partial + S (bpvi_partial_bpd_mul_upper_high_power) = S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_partial. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_partial * S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power) + (bpvi_partial_bpd_mul_upper_high_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_high_power_successor. bpvi_h_bpd_mul_upper_high_power_successor + S (bpvi_successor_bpd_mul_upper_high_power) = S ((S (S bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_successor. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_successor * S ((S (S bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power) + (bpvi_successor_bpd_mul_upper_high_power))) /\\ bpvi_successor_bpd_mul_upper_high_power = bpvi_partial_bpd_mul_upper_high_power * bpvi_factor_bpd_mul_upper_high_power)))))))) /\\ exists bpvi_divisor_factor_bpd_mul_upper_high. a * b = bpvi_result_bpd_mul_upper_high * bpvi_divisor_factor_bpd_mul_upper_high)", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides (a * b)", "specialize power_valuation_power_divides g", "apply power_valuation_power_divides", "exact hvaluation_product", "have hsuccessor : exists bpvi_result_bpd_mul_upper_successor. ((exists bpvi_b_bpd_mul_upper_successor_power bpvi_c_bpd_mul_upper_successor_power. ((forall bpvi_i_bpd_mul_upper_successor_power. (exists bpvi_repeat_gap_bpd_mul_upper_successor_power. bpvi_repeat_gap_bpd_mul_upper_successor_power + S bpvi_i_bpd_mul_upper_successor_power = S (e + f)) -> (((exists bpvi_h_bpd_mul_upper_successor_power_repeat. bpvi_h_bpd_mul_upper_successor_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_repeat. bpvi_b_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_repeat * S ((S (bpvi_i_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_upper_successor_power bpvi_v_bpd_mul_upper_successor_power. ((((exists bpvi_h_bpd_mul_upper_successor_power_start. bpvi_h_bpd_mul_upper_successor_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_start. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_start * S ((S (0)) * bpvi_v_bpd_mul_upper_successor_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_upper_successor_power_terminal. bpvi_h_bpd_mul_upper_successor_power_terminal + S (bpvi_result_bpd_mul_upper_successor) = S ((S (S (e + f))) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_terminal. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_terminal * S ((S (S (e + f))) * bpvi_v_bpd_mul_upper_successor_power) + (bpvi_result_bpd_mul_upper_successor))) /\\ forall bpvi_j_bpd_mul_upper_successor_power. (exists bpvi_product_gap_bpd_mul_upper_successor_power. bpvi_product_gap_bpd_mul_upper_successor_power + S bpvi_j_bpd_mul_upper_successor_power = S (e + f)) -> exists bpvi_factor_bpd_mul_upper_successor_power bpvi_partial_bpd_mul_upper_successor_power bpvi_successor_bpd_mul_upper_successor_power. ((((exists bpvi_h_bpd_mul_upper_successor_power_factor. bpvi_h_bpd_mul_upper_successor_power_factor + S (bpvi_factor_bpd_mul_upper_successor_power) = S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_factor. bpvi_b_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_factor * S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power) + (bpvi_factor_bpd_mul_upper_successor_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_successor_power_partial. bpvi_h_bpd_mul_upper_successor_power_partial + S (bpvi_partial_bpd_mul_upper_successor_power) = S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_partial. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_partial * S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power) + (bpvi_partial_bpd_mul_upper_successor_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_successor_power_successor. bpvi_h_bpd_mul_upper_successor_power_successor + S (bpvi_successor_bpd_mul_upper_successor_power) = S ((S (S bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_successor. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_successor * S ((S (S bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power) + (bpvi_successor_bpd_mul_upper_successor_power))) /\\ bpvi_successor_bpd_mul_upper_successor_power = bpvi_partial_bpd_mul_upper_successor_power * bpvi_factor_bpd_mul_upper_successor_power)))))))) /\\ exists bpvi_divisor_factor_bpd_mul_upper_successor. a * b = bpvi_result_bpd_mul_upper_successor * bpvi_divisor_factor_bpd_mul_upper_successor)", "specialize power_divides_exponent_antitone p", "specialize power_divides_exponent_antitone (S (e + f))", "specialize power_divides_exponent_antitone g", "specialize power_divides_exponent_antitone (a * b)", "apply power_divides_exponent_antitone", "exact horder_right", "exact hhigh", "specialize power_valuation_mul_successor_not_divides p", "specialize power_valuation_mul_successor_not_divides a", "specialize power_valuation_mul_successor_not_divides b", "specialize power_valuation_mul_successor_not_divides e", "specialize power_valuation_mul_successor_not_divides f", "apply power_valuation_mul_successor_not_divides", "exact hp", "exact ha", "exact hb", "exact hvaluation_a", "exact hvaluation_b", "exact hsuccessor" ], "script_sha256": "17e639e443bdaa2e0895144f688ef88ea992432745a18522845564e4dabb8546", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p a b e f g. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> (exists bpd_gap_valuation_mul_upper. bpd_gap_valuation_mul_upper + (g) = (e + f))", "statement_sha256": "b6e66d62b853f86becf7f4a9ded9633e55acaf1821cad08f3202b8e6f001185e", "summary": "The valuation of a nonzero product is at most the sum of factor valuations.", "summary_sha256": "5f1dfc5685a57a7c94465d5f993d6c627b0cad2ebe8fdf4f5b7592ea63a7410b" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "le_or_lt", "power_valuation_power_divides", "power_divides_exponent_antitone", "power_valuation_mul_successor_not_divides" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=204]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_mul_upper]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_mul_upper", "parent_alpha_version": "v34", "proof_bundle_node_id": 218, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_mul_upper", "script": [ "intro p", "intro a", "intro b", "intro e", "intro f", "intro g", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "intro hvaluation_product", "have horder : (exists k. k + g = e + f) \\/ exists k. k + S (e + f) = g", "specialize le_or_lt g", "specialize le_or_lt (e + f)", "exact le_or_lt", "cases horder", "exact horder_left", "exfalso", "have hhigh : exists bpvi_result_bpd_mul_upper_high. ((exists bpvi_b_bpd_mul_upper_high_power bpvi_c_bpd_mul_upper_high_power. ((forall bpvi_i_bpd_mul_upper_high_power. (exists bpvi_repeat_gap_bpd_mul_upper_high_power. bpvi_repeat_gap_bpd_mul_upper_high_power + S bpvi_i_bpd_mul_upper_high_power = g) -> (((exists bpvi_h_bpd_mul_upper_high_power_repeat. bpvi_h_bpd_mul_upper_high_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_repeat. bpvi_b_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_repeat * S ((S (bpvi_i_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_upper_high_power bpvi_v_bpd_mul_upper_high_power. ((((exists bpvi_h_bpd_mul_upper_high_power_start. bpvi_h_bpd_mul_upper_high_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_start. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_start * S ((S (0)) * bpvi_v_bpd_mul_upper_high_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_upper_high_power_terminal. bpvi_h_bpd_mul_upper_high_power_terminal + S (bpvi_result_bpd_mul_upper_high) = S ((S (g)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_terminal. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_terminal * S ((S (g)) * bpvi_v_bpd_mul_upper_high_power) + (bpvi_result_bpd_mul_upper_high))) /\\ forall bpvi_j_bpd_mul_upper_high_power. (exists bpvi_product_gap_bpd_mul_upper_high_power. bpvi_product_gap_bpd_mul_upper_high_power + S bpvi_j_bpd_mul_upper_high_power = g) -> exists bpvi_factor_bpd_mul_upper_high_power bpvi_partial_bpd_mul_upper_high_power bpvi_successor_bpd_mul_upper_high_power. ((((exists bpvi_h_bpd_mul_upper_high_power_factor. bpvi_h_bpd_mul_upper_high_power_factor + S (bpvi_factor_bpd_mul_upper_high_power) = S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_factor. bpvi_b_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_factor * S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_c_bpd_mul_upper_high_power) + (bpvi_factor_bpd_mul_upper_high_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_high_power_partial. bpvi_h_bpd_mul_upper_high_power_partial + S (bpvi_partial_bpd_mul_upper_high_power) = S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_partial. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_partial * S ((S (bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power) + (bpvi_partial_bpd_mul_upper_high_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_high_power_successor. bpvi_h_bpd_mul_upper_high_power_successor + S (bpvi_successor_bpd_mul_upper_high_power) = S ((S (S bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power)) /\\ exists bpvi_q_bpd_mul_upper_high_power_successor. bpvi_u_bpd_mul_upper_high_power = bpvi_q_bpd_mul_upper_high_power_successor * S ((S (S bpvi_j_bpd_mul_upper_high_power)) * bpvi_v_bpd_mul_upper_high_power) + (bpvi_successor_bpd_mul_upper_high_power))) /\\ bpvi_successor_bpd_mul_upper_high_power = bpvi_partial_bpd_mul_upper_high_power * bpvi_factor_bpd_mul_upper_high_power)))))))) /\\ exists bpvi_divisor_factor_bpd_mul_upper_high. a * b = bpvi_result_bpd_mul_upper_high * bpvi_divisor_factor_bpd_mul_upper_high)", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides (a * b)", "specialize power_valuation_power_divides g", "apply power_valuation_power_divides", "exact hvaluation_product", "have hsuccessor : exists bpvi_result_bpd_mul_upper_successor. ((exists bpvi_b_bpd_mul_upper_successor_power bpvi_c_bpd_mul_upper_successor_power. ((forall bpvi_i_bpd_mul_upper_successor_power. (exists bpvi_repeat_gap_bpd_mul_upper_successor_power. bpvi_repeat_gap_bpd_mul_upper_successor_power + S bpvi_i_bpd_mul_upper_successor_power = S (e + f)) -> (((exists bpvi_h_bpd_mul_upper_successor_power_repeat. bpvi_h_bpd_mul_upper_successor_power_repeat + S (p) = S ((S (bpvi_i_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_repeat. bpvi_b_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_repeat * S ((S (bpvi_i_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power) + (p)))) /\\ (exists bpvi_u_bpd_mul_upper_successor_power bpvi_v_bpd_mul_upper_successor_power. ((((exists bpvi_h_bpd_mul_upper_successor_power_start. bpvi_h_bpd_mul_upper_successor_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_start. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_start * S ((S (0)) * bpvi_v_bpd_mul_upper_successor_power) + (1))) /\\ ((((exists bpvi_h_bpd_mul_upper_successor_power_terminal. bpvi_h_bpd_mul_upper_successor_power_terminal + S (bpvi_result_bpd_mul_upper_successor) = S ((S (S (e + f))) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_terminal. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_terminal * S ((S (S (e + f))) * bpvi_v_bpd_mul_upper_successor_power) + (bpvi_result_bpd_mul_upper_successor))) /\\ forall bpvi_j_bpd_mul_upper_successor_power. (exists bpvi_product_gap_bpd_mul_upper_successor_power. bpvi_product_gap_bpd_mul_upper_successor_power + S bpvi_j_bpd_mul_upper_successor_power = S (e + f)) -> exists bpvi_factor_bpd_mul_upper_successor_power bpvi_partial_bpd_mul_upper_successor_power bpvi_successor_bpd_mul_upper_successor_power. ((((exists bpvi_h_bpd_mul_upper_successor_power_factor. bpvi_h_bpd_mul_upper_successor_power_factor + S (bpvi_factor_bpd_mul_upper_successor_power) = S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_factor. bpvi_b_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_factor * S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_c_bpd_mul_upper_successor_power) + (bpvi_factor_bpd_mul_upper_successor_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_successor_power_partial. bpvi_h_bpd_mul_upper_successor_power_partial + S (bpvi_partial_bpd_mul_upper_successor_power) = S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_partial. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_partial * S ((S (bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power) + (bpvi_partial_bpd_mul_upper_successor_power))) /\\ ((((exists bpvi_h_bpd_mul_upper_successor_power_successor. bpvi_h_bpd_mul_upper_successor_power_successor + S (bpvi_successor_bpd_mul_upper_successor_power) = S ((S (S bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power)) /\\ exists bpvi_q_bpd_mul_upper_successor_power_successor. bpvi_u_bpd_mul_upper_successor_power = bpvi_q_bpd_mul_upper_successor_power_successor * S ((S (S bpvi_j_bpd_mul_upper_successor_power)) * bpvi_v_bpd_mul_upper_successor_power) + (bpvi_successor_bpd_mul_upper_successor_power))) /\\ bpvi_successor_bpd_mul_upper_successor_power = bpvi_partial_bpd_mul_upper_successor_power * bpvi_factor_bpd_mul_upper_successor_power)))))))) /\\ exists bpvi_divisor_factor_bpd_mul_upper_successor. a * b = bpvi_result_bpd_mul_upper_successor * bpvi_divisor_factor_bpd_mul_upper_successor)", "specialize power_divides_exponent_antitone p", "specialize power_divides_exponent_antitone (S (e + f))", "specialize power_divides_exponent_antitone g", "specialize power_divides_exponent_antitone (a * b)", "apply power_divides_exponent_antitone", "exact horder_right", "exact hhigh", "specialize power_valuation_mul_successor_not_divides p", "specialize power_valuation_mul_successor_not_divides a", "specialize power_valuation_mul_successor_not_divides b", "specialize power_valuation_mul_successor_not_divides e", "specialize power_valuation_mul_successor_not_divides f", "apply power_valuation_mul_successor_not_divides", "exact hp", "exact ha", "exact hb", "exact hvaluation_a", "exact hvaluation_b", "exact hsuccessor" ], "script_sha256": "17e639e443bdaa2e0895144f688ef88ea992432745a18522845564e4dabb8546", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p a b e f g. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> (exists bpd_gap_valuation_mul_upper. bpd_gap_valuation_mul_upper + (g) = (e + f))", "statement_sha256": "b6e66d62b853f86becf7f4a9ded9633e55acaf1821cad08f3202b8e6f001185e" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_power_valuation_mul", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 205, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "dff7338eebcee2722d31c897e9cc13ceb0a9462fa1080d098d1551af2da77ea3" }, "bertrand_v4_evidence_bundle_sha256": "33cf482795260e76f81ecbb5704b824f0d4cd09433b0360cd3611460750b266d", "body_checked": true, "body_receipt": { "command_count": 45, "dependency_count": 3, "dne_command_count": 0, "name": "prime_power_valuation_mul", "proof_depth": 30, "proof_edges": 57, "proof_nodes": 58, "proof_objects": 58, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "power_valuation_mul_lower", "power_valuation_mul_upper", "le_antisymm" ], "dependencies_sha256": "556d533316f79cad8b7b6886bd02eed28ce68a470ed0bcb72b6ce6b84e88dc54", "empty_context_closure": { "body_proof_depth": 30, "body_proof_nodes": 58, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 205, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "6fd025a8b94441961ffc3bdf7820ba071ac5bf0ac9a638244392fbc73af7844b", "status": "checked" }, "enrollment_index": 939, "enrollment_origin": "bertrand_b2_valuation_multiplication", "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b2b0302b661bf9267b3c0a248e5a36239c10852e391d1f007a0a2f9c6983004a", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_power_divisibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "1cd6b31379737efb3d889318e1c40beffcc14f77432a1b18cb74e80a5d29d199", "kind": "sealed_alpha_v3_parent", "path": "artifacts/peano-library/alpha/catalog-v3.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=205]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_power_valuation_mul]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "be73076c05f0d9523d48aaba915282f58bac14fa43815bcea4b661c4dad4ae7a", "membership": "alpha_only", "name": "prime_power_valuation_mul", "proof_tag": null, "provenance": [ "bertrand_b2_valuation_multiplication" ], "script": [ "intro p", "intro a", "intro b", "intro e", "intro f", "intro g", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "intro hvaluation_product", "have hlower : exists k. k + (e + f) = g", "specialize power_valuation_mul_lower p", "specialize power_valuation_mul_lower a", "specialize power_valuation_mul_lower b", "specialize power_valuation_mul_lower e", "specialize power_valuation_mul_lower f", "specialize power_valuation_mul_lower g", "apply power_valuation_mul_lower", "exact hp", "exact ha", "exact hb", "exact hvaluation_a", "exact hvaluation_b", "exact hvaluation_product", "have hupper : exists k. k + g = e + f", "specialize power_valuation_mul_upper p", "specialize power_valuation_mul_upper a", "specialize power_valuation_mul_upper b", "specialize power_valuation_mul_upper e", "specialize power_valuation_mul_upper f", "specialize power_valuation_mul_upper g", "apply power_valuation_mul_upper", "exact hp", "exact ha", "exact hb", "exact hvaluation_a", "exact hvaluation_b", "exact hvaluation_product", "specialize le_antisymm g", "specialize le_antisymm (e + f)", "apply le_antisymm", "exact hupper", "exact hlower" ], "script_sha256": "2b7251591b78d4753c3e97192f6585e54e6324adb5a92b557647ac7fd37e08cf", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "statement": "forall p a b e f g. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> g = e + f", "statement_sha256": "6fd025a8b94441961ffc3bdf7820ba071ac5bf0ac9a638244392fbc73af7844b", "summary": "Prime-power valuation is additive on nonzero products.", "summary_sha256": "859238914e3f6b5acab89917e43e7c7be303d013c8a092da8c11ddaff92e2fe8" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_mul_lower", "power_valuation_mul_upper", "le_antisymm" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963", "kind": 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"is_inherited_source_alias": false, "name": "prime_power_valuation_mul", "parent_alpha_version": "v34", "proof_bundle_node_id": 219, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_power_valuation_mul", "script": [ "intro p", "intro a", "intro b", "intro e", "intro f", "intro g", "intro hp", "intro ha", "intro hb", "intro hvaluation_a", "intro hvaluation_b", "intro hvaluation_product", "have hlower : exists k. k + (e + f) = g", "specialize power_valuation_mul_lower p", "specialize power_valuation_mul_lower a", "specialize power_valuation_mul_lower b", "specialize power_valuation_mul_lower e", "specialize power_valuation_mul_lower f", "specialize power_valuation_mul_lower g", "apply power_valuation_mul_lower", "exact hp", "exact ha", "exact hb", "exact hvaluation_a", "exact hvaluation_b", "exact hvaluation_product", "have hupper : exists k. k + g = e + f", "specialize power_valuation_mul_upper p", "specialize power_valuation_mul_upper a", "specialize power_valuation_mul_upper b", "specialize power_valuation_mul_upper e", "specialize power_valuation_mul_upper f", "specialize power_valuation_mul_upper g", "apply power_valuation_mul_upper", "exact hp", "exact ha", "exact hb", "exact hvaluation_a", "exact hvaluation_b", "exact hvaluation_product", "specialize le_antisymm g", "specialize le_antisymm (e + f)", "apply le_antisymm", "exact hupper", "exact hlower" ], "script_sha256": "2b7251591b78d4753c3e97192f6585e54e6324adb5a92b557647ac7fd37e08cf", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_power_divisibility_candidate.py", "sha256": "d3b0f53bd9e7de7c77b1fe2e80cdbedf001b9ff4c6b02c6aaa7f0e5aa5953963" }, "stable_member": false, "statement": "forall p a b e f g. ((~(p = 1) /\\ forall frm_prime_left_bpd_prime frm_prime_right_bpd_prime. p = frm_prime_left_bpd_prime * frm_prime_right_bpd_prime -> frm_prime_left_bpd_prime = 1 \\/ frm_prime_right_bpd_prime = 1)) -> ~(a = 0) -> ~(b = 0) -> (((exists bpv_gap_bpd_valuation_a_exponent_bound. bpv_gap_bpd_valuation_a_exponent_bound + e = a) /\\ (exists bpv_result_bpd_valuation_a_selected. ((exists ff_b_bpd_valuation_a_selected_power ff_c_bpd_valuation_a_selected_power. ((forall ff_i_bpd_valuation_a_selected_power_repeat. (exists ff_lt_bpd_valuation_a_selected_power_repeat_bound. ff_lt_bpd_valuation_a_selected_power_repeat_bound + S ff_i_bpd_valuation_a_selected_power_repeat = e) -> (((exists ff_h_bpd_valuation_a_selected_power_repeat_decoded. ff_h_bpd_valuation_a_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_repeat_decoded. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_selected_power_repeat)) * ff_c_bpd_valuation_a_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_selected_power_product ff_v_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_start. ff_h_bpd_valuation_a_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_start. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_terminal. ff_h_bpd_valuation_a_selected_power_product_terminal + S (bpv_result_bpd_valuation_a_selected) = S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_terminal. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_terminal * S ((S (e)) * ff_v_bpd_valuation_a_selected_power_product) + (bpv_result_bpd_valuation_a_selected))) /\\ forall ff_i_bpd_valuation_a_selected_power_product. (exists ff_lt_bpd_valuation_a_selected_power_product_bound. ff_lt_bpd_valuation_a_selected_power_product_bound + S ff_i_bpd_valuation_a_selected_power_product = e) -> exists ff_p_bpd_valuation_a_selected_power_product ff_r_bpd_valuation_a_selected_power_product ff_s_bpd_valuation_a_selected_power_product. ((((exists ff_h_bpd_valuation_a_selected_power_product_factor. ff_h_bpd_valuation_a_selected_power_product_factor + S (ff_p_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_factor. ff_b_bpd_valuation_a_selected_power = ff_q_bpd_valuation_a_selected_power_product_factor * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_c_bpd_valuation_a_selected_power) + (ff_p_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_partial. ff_h_bpd_valuation_a_selected_power_product_partial + S (ff_r_bpd_valuation_a_selected_power_product) = S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_partial. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_partial * S ((S (ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_r_bpd_valuation_a_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_selected_power_product_successor. ff_h_bpd_valuation_a_selected_power_product_successor + S (ff_s_bpd_valuation_a_selected_power_product) = S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product)) /\\ exists ff_q_bpd_valuation_a_selected_power_product_successor. ff_u_bpd_valuation_a_selected_power_product = ff_q_bpd_valuation_a_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_a_selected_power_product)) * ff_v_bpd_valuation_a_selected_power_product) + (ff_s_bpd_valuation_a_selected_power_product))) /\\ ff_s_bpd_valuation_a_selected_power_product = ff_r_bpd_valuation_a_selected_power_product * ff_p_bpd_valuation_a_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_selected_divides. a = bpv_result_bpd_valuation_a_selected * bpv_factor_bpd_valuation_a_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_a. (exists bpv_gap_bpd_valuation_a_candidate_bound. bpv_gap_bpd_valuation_a_candidate_bound + bpv_candidate_bpd_valuation_a = a) -> (exists bpv_result_bpd_valuation_a_candidate. ((exists ff_b_bpd_valuation_a_candidate_power ff_c_bpd_valuation_a_candidate_power. ((forall ff_i_bpd_valuation_a_candidate_power_repeat. (exists ff_lt_bpd_valuation_a_candidate_power_repeat_bound. ff_lt_bpd_valuation_a_candidate_power_repeat_bound + S ff_i_bpd_valuation_a_candidate_power_repeat = bpv_candidate_bpd_valuation_a) -> (((exists ff_h_bpd_valuation_a_candidate_power_repeat_decoded. ff_h_bpd_valuation_a_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_repeat_decoded. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_a_candidate_power_repeat)) * ff_c_bpd_valuation_a_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_a_candidate_power_product ff_v_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_start. ff_h_bpd_valuation_a_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_start. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_a_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_terminal. ff_h_bpd_valuation_a_candidate_power_product_terminal + S (bpv_result_bpd_valuation_a_candidate) = S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_terminal. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_a)) * ff_v_bpd_valuation_a_candidate_power_product) + (bpv_result_bpd_valuation_a_candidate))) /\\ forall ff_i_bpd_valuation_a_candidate_power_product. (exists ff_lt_bpd_valuation_a_candidate_power_product_bound. ff_lt_bpd_valuation_a_candidate_power_product_bound + S ff_i_bpd_valuation_a_candidate_power_product = bpv_candidate_bpd_valuation_a) -> exists ff_p_bpd_valuation_a_candidate_power_product ff_r_bpd_valuation_a_candidate_power_product ff_s_bpd_valuation_a_candidate_power_product. ((((exists ff_h_bpd_valuation_a_candidate_power_product_factor. ff_h_bpd_valuation_a_candidate_power_product_factor + S (ff_p_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_factor. ff_b_bpd_valuation_a_candidate_power = ff_q_bpd_valuation_a_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_c_bpd_valuation_a_candidate_power) + (ff_p_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_partial. ff_h_bpd_valuation_a_candidate_power_product_partial + S (ff_r_bpd_valuation_a_candidate_power_product) = S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_partial. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_r_bpd_valuation_a_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_a_candidate_power_product_successor. ff_h_bpd_valuation_a_candidate_power_product_successor + S (ff_s_bpd_valuation_a_candidate_power_product) = S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product)) /\\ exists ff_q_bpd_valuation_a_candidate_power_product_successor. ff_u_bpd_valuation_a_candidate_power_product = ff_q_bpd_valuation_a_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_a_candidate_power_product)) * ff_v_bpd_valuation_a_candidate_power_product) + (ff_s_bpd_valuation_a_candidate_power_product))) /\\ ff_s_bpd_valuation_a_candidate_power_product = ff_r_bpd_valuation_a_candidate_power_product * ff_p_bpd_valuation_a_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_a_candidate_divides. a = bpv_result_bpd_valuation_a_candidate * bpv_factor_bpd_valuation_a_candidate_divides))) -> (exists bpv_gap_bpd_valuation_a_maximal. bpv_gap_bpd_valuation_a_maximal + bpv_candidate_bpd_valuation_a = e)) -> (((exists bpv_gap_bpd_valuation_b_exponent_bound. bpv_gap_bpd_valuation_b_exponent_bound + f = b) /\\ (exists bpv_result_bpd_valuation_b_selected. ((exists ff_b_bpd_valuation_b_selected_power ff_c_bpd_valuation_b_selected_power. ((forall ff_i_bpd_valuation_b_selected_power_repeat. (exists ff_lt_bpd_valuation_b_selected_power_repeat_bound. ff_lt_bpd_valuation_b_selected_power_repeat_bound + S ff_i_bpd_valuation_b_selected_power_repeat = f) -> (((exists ff_h_bpd_valuation_b_selected_power_repeat_decoded. ff_h_bpd_valuation_b_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_repeat_decoded. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_selected_power_repeat)) * ff_c_bpd_valuation_b_selected_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_selected_power_product ff_v_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_start. ff_h_bpd_valuation_b_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_start. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_selected_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_terminal. ff_h_bpd_valuation_b_selected_power_product_terminal + S (bpv_result_bpd_valuation_b_selected) = S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_terminal. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_terminal * S ((S (f)) * ff_v_bpd_valuation_b_selected_power_product) + (bpv_result_bpd_valuation_b_selected))) /\\ forall ff_i_bpd_valuation_b_selected_power_product. (exists ff_lt_bpd_valuation_b_selected_power_product_bound. ff_lt_bpd_valuation_b_selected_power_product_bound + S ff_i_bpd_valuation_b_selected_power_product = f) -> exists ff_p_bpd_valuation_b_selected_power_product ff_r_bpd_valuation_b_selected_power_product ff_s_bpd_valuation_b_selected_power_product. ((((exists ff_h_bpd_valuation_b_selected_power_product_factor. ff_h_bpd_valuation_b_selected_power_product_factor + S (ff_p_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_factor. ff_b_bpd_valuation_b_selected_power = ff_q_bpd_valuation_b_selected_power_product_factor * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_c_bpd_valuation_b_selected_power) + (ff_p_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_partial. ff_h_bpd_valuation_b_selected_power_product_partial + S (ff_r_bpd_valuation_b_selected_power_product) = S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_partial. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_partial * S ((S (ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_r_bpd_valuation_b_selected_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_selected_power_product_successor. ff_h_bpd_valuation_b_selected_power_product_successor + S (ff_s_bpd_valuation_b_selected_power_product) = S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product)) /\\ exists ff_q_bpd_valuation_b_selected_power_product_successor. ff_u_bpd_valuation_b_selected_power_product = ff_q_bpd_valuation_b_selected_power_product_successor * S ((S (S ff_i_bpd_valuation_b_selected_power_product)) * ff_v_bpd_valuation_b_selected_power_product) + (ff_s_bpd_valuation_b_selected_power_product))) /\\ ff_s_bpd_valuation_b_selected_power_product = ff_r_bpd_valuation_b_selected_power_product * ff_p_bpd_valuation_b_selected_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_selected_divides. b = bpv_result_bpd_valuation_b_selected * bpv_factor_bpd_valuation_b_selected_divides)))) /\\ forall bpv_candidate_bpd_valuation_b. (exists bpv_gap_bpd_valuation_b_candidate_bound. bpv_gap_bpd_valuation_b_candidate_bound + bpv_candidate_bpd_valuation_b = b) -> (exists bpv_result_bpd_valuation_b_candidate. ((exists ff_b_bpd_valuation_b_candidate_power ff_c_bpd_valuation_b_candidate_power. ((forall ff_i_bpd_valuation_b_candidate_power_repeat. (exists ff_lt_bpd_valuation_b_candidate_power_repeat_bound. ff_lt_bpd_valuation_b_candidate_power_repeat_bound + S ff_i_bpd_valuation_b_candidate_power_repeat = bpv_candidate_bpd_valuation_b) -> (((exists ff_h_bpd_valuation_b_candidate_power_repeat_decoded. ff_h_bpd_valuation_b_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_repeat_decoded. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_repeat_decoded * S ((S (ff_i_bpd_valuation_b_candidate_power_repeat)) * ff_c_bpd_valuation_b_candidate_power) + (p)))) /\\ (exists ff_u_bpd_valuation_b_candidate_power_product ff_v_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_start. ff_h_bpd_valuation_b_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_start. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_start * S ((S (0)) * ff_v_bpd_valuation_b_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_terminal. ff_h_bpd_valuation_b_candidate_power_product_terminal + S (bpv_result_bpd_valuation_b_candidate) = S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_terminal. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_terminal * S ((S (bpv_candidate_bpd_valuation_b)) * ff_v_bpd_valuation_b_candidate_power_product) + (bpv_result_bpd_valuation_b_candidate))) /\\ forall ff_i_bpd_valuation_b_candidate_power_product. (exists ff_lt_bpd_valuation_b_candidate_power_product_bound. ff_lt_bpd_valuation_b_candidate_power_product_bound + S ff_i_bpd_valuation_b_candidate_power_product = bpv_candidate_bpd_valuation_b) -> exists ff_p_bpd_valuation_b_candidate_power_product ff_r_bpd_valuation_b_candidate_power_product ff_s_bpd_valuation_b_candidate_power_product. ((((exists ff_h_bpd_valuation_b_candidate_power_product_factor. ff_h_bpd_valuation_b_candidate_power_product_factor + S (ff_p_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_factor. ff_b_bpd_valuation_b_candidate_power = ff_q_bpd_valuation_b_candidate_power_product_factor * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_c_bpd_valuation_b_candidate_power) + (ff_p_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_partial. ff_h_bpd_valuation_b_candidate_power_product_partial + S (ff_r_bpd_valuation_b_candidate_power_product) = S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_partial. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_partial * S ((S (ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_r_bpd_valuation_b_candidate_power_product))) /\\ ((((exists ff_h_bpd_valuation_b_candidate_power_product_successor. ff_h_bpd_valuation_b_candidate_power_product_successor + S (ff_s_bpd_valuation_b_candidate_power_product) = S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product)) /\\ exists ff_q_bpd_valuation_b_candidate_power_product_successor. ff_u_bpd_valuation_b_candidate_power_product = ff_q_bpd_valuation_b_candidate_power_product_successor * S ((S (S ff_i_bpd_valuation_b_candidate_power_product)) * ff_v_bpd_valuation_b_candidate_power_product) + (ff_s_bpd_valuation_b_candidate_power_product))) /\\ ff_s_bpd_valuation_b_candidate_power_product = ff_r_bpd_valuation_b_candidate_power_product * ff_p_bpd_valuation_b_candidate_power_product)))))))) /\\ (exists bpv_factor_bpd_valuation_b_candidate_divides. b = bpv_result_bpd_valuation_b_candidate * bpv_factor_bpd_valuation_b_candidate_divides))) -> (exists bpv_gap_bpd_valuation_b_maximal. bpv_gap_bpd_valuation_b_maximal + bpv_candidate_bpd_valuation_b = f)) -> (((exists bpd_gap_bpd_valuation_product_selected_bound. bpd_gap_bpd_valuation_product_selected_bound + (g) = (a * b)) /\\ (exists bpvi_result_bpd_valuation_product_selected. ((exists bpvi_b_bpd_valuation_product_selected_power bpvi_c_bpd_valuation_product_selected_power. ((forall bpvi_i_bpd_valuation_product_selected_power. (exists bpvi_repeat_gap_bpd_valuation_product_selected_power. bpvi_repeat_gap_bpd_valuation_product_selected_power + S bpvi_i_bpd_valuation_product_selected_power = g) -> (((exists bpvi_h_bpd_valuation_product_selected_power_repeat. bpvi_h_bpd_valuation_product_selected_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_repeat. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_repeat * S ((S (bpvi_i_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_selected_power bpvi_v_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_start. bpvi_h_bpd_valuation_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_start. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_selected_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_terminal. bpvi_h_bpd_valuation_product_selected_power_terminal + S (bpvi_result_bpd_valuation_product_selected) = S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_terminal. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_terminal * S ((S (g)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_result_bpd_valuation_product_selected))) /\\ forall bpvi_j_bpd_valuation_product_selected_power. (exists bpvi_product_gap_bpd_valuation_product_selected_power. bpvi_product_gap_bpd_valuation_product_selected_power + S bpvi_j_bpd_valuation_product_selected_power = g) -> exists bpvi_factor_bpd_valuation_product_selected_power bpvi_partial_bpd_valuation_product_selected_power bpvi_successor_bpd_valuation_product_selected_power. ((((exists bpvi_h_bpd_valuation_product_selected_power_factor. bpvi_h_bpd_valuation_product_selected_power_factor + S (bpvi_factor_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_factor. bpvi_b_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_factor * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_c_bpd_valuation_product_selected_power) + (bpvi_factor_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_partial. bpvi_h_bpd_valuation_product_selected_power_partial + S (bpvi_partial_bpd_valuation_product_selected_power) = S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_partial. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_partial * S ((S (bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_partial_bpd_valuation_product_selected_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_selected_power_successor. bpvi_h_bpd_valuation_product_selected_power_successor + S (bpvi_successor_bpd_valuation_product_selected_power) = S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power)) /\\ exists bpvi_q_bpd_valuation_product_selected_power_successor. bpvi_u_bpd_valuation_product_selected_power = bpvi_q_bpd_valuation_product_selected_power_successor * S ((S (S bpvi_j_bpd_valuation_product_selected_power)) * bpvi_v_bpd_valuation_product_selected_power) + (bpvi_successor_bpd_valuation_product_selected_power))) /\\ bpvi_successor_bpd_valuation_product_selected_power = bpvi_partial_bpd_valuation_product_selected_power * bpvi_factor_bpd_valuation_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_selected. a * b = bpvi_result_bpd_valuation_product_selected * bpvi_divisor_factor_bpd_valuation_product_selected))) /\\ forall bpd_candidate_bpd_valuation_product. (exists bpd_gap_bpd_valuation_product_candidate_bound. bpd_gap_bpd_valuation_product_candidate_bound + (bpd_candidate_bpd_valuation_product) = (a * b)) -> (exists bpvi_result_bpd_valuation_product_candidate. ((exists bpvi_b_bpd_valuation_product_candidate_power bpvi_c_bpd_valuation_product_candidate_power. ((forall bpvi_i_bpd_valuation_product_candidate_power. (exists bpvi_repeat_gap_bpd_valuation_product_candidate_power. bpvi_repeat_gap_bpd_valuation_product_candidate_power + S bpvi_i_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> (((exists bpvi_h_bpd_valuation_product_candidate_power_repeat. bpvi_h_bpd_valuation_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_repeat. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_repeat * S ((S (bpvi_i_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (p)))) /\\ (exists bpvi_u_bpd_valuation_product_candidate_power bpvi_v_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_start. bpvi_h_bpd_valuation_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_start. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_start * S ((S (0)) * bpvi_v_bpd_valuation_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_terminal. bpvi_h_bpd_valuation_product_candidate_power_terminal + S (bpvi_result_bpd_valuation_product_candidate) = S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_terminal. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_terminal * S ((S (bpd_candidate_bpd_valuation_product)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_result_bpd_valuation_product_candidate))) /\\ forall bpvi_j_bpd_valuation_product_candidate_power. (exists bpvi_product_gap_bpd_valuation_product_candidate_power. bpvi_product_gap_bpd_valuation_product_candidate_power + S bpvi_j_bpd_valuation_product_candidate_power = bpd_candidate_bpd_valuation_product) -> exists bpvi_factor_bpd_valuation_product_candidate_power bpvi_partial_bpd_valuation_product_candidate_power bpvi_successor_bpd_valuation_product_candidate_power. ((((exists bpvi_h_bpd_valuation_product_candidate_power_factor. bpvi_h_bpd_valuation_product_candidate_power_factor + S (bpvi_factor_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_factor. bpvi_b_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_factor * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_c_bpd_valuation_product_candidate_power) + (bpvi_factor_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_partial. bpvi_h_bpd_valuation_product_candidate_power_partial + S (bpvi_partial_bpd_valuation_product_candidate_power) = S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_partial. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_partial * S ((S (bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_partial_bpd_valuation_product_candidate_power))) /\\ ((((exists bpvi_h_bpd_valuation_product_candidate_power_successor. bpvi_h_bpd_valuation_product_candidate_power_successor + S (bpvi_successor_bpd_valuation_product_candidate_power) = S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power)) /\\ exists bpvi_q_bpd_valuation_product_candidate_power_successor. bpvi_u_bpd_valuation_product_candidate_power = bpvi_q_bpd_valuation_product_candidate_power_successor * S ((S (S bpvi_j_bpd_valuation_product_candidate_power)) * bpvi_v_bpd_valuation_product_candidate_power) + (bpvi_successor_bpd_valuation_product_candidate_power))) /\\ bpvi_successor_bpd_valuation_product_candidate_power = bpvi_partial_bpd_valuation_product_candidate_power * bpvi_factor_bpd_valuation_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_bpd_valuation_product_candidate. a * b = bpvi_result_bpd_valuation_product_candidate * bpvi_divisor_factor_bpd_valuation_product_candidate)) -> (exists bpd_gap_bpd_valuation_product_maximal. bpd_gap_bpd_valuation_product_maximal + (bpd_candidate_bpd_valuation_product) = (g))) -> g = e + f", "statement_sha256": "6fd025a8b94441961ffc3bdf7820ba071ac5bf0ac9a638244392fbc73af7844b" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_power_valuation_one_zero", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 207, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", 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"research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "814fe3e6f732c3317abe83a5c245fd339156ce333d8b7780546b5fc92a1822a9", "status": "checked" }, "enrollment_index": 966, "enrollment_origin": "bertrand", "evidence_links": [ { "document_sha256": "0a2c0f617d401b4e60f11870f4ce5c23cea3c042c0308031aa21a331b5508af9", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_factorial_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d7e481694a74a2b481bc4928664caa5bf0a2e466cc88e1dfdf394a299c003ae0", "kind": "bertrand_executable_audit", "path": 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"7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_power_valuation_one_zero]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "f6bb3411b7742a69620d91173e76afd67601afce790057e3a633aae6d05cab49", "membership": "alpha_only", "name": "prime_power_valuation_one_zero", "proof_tag": null, "provenance": [ "bertrand" ], "script": [ "intro p", "intro one", "intro e", "intro hone", "intro hp", "intro hvaluation", "cases hp", "have hselected : exists bpv_result_bfv_one_selected. ((exists ff_b_bfv_one_selected_power ff_c_bfv_one_selected_power. ((forall ff_i_bfv_one_selected_power_repeat. (exists ff_lt_bfv_one_selected_power_repeat_bound. ff_lt_bfv_one_selected_power_repeat_bound + S ff_i_bfv_one_selected_power_repeat = e) -> (((exists ff_h_bfv_one_selected_power_repeat_decoded. ff_h_bfv_one_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_repeat_decoded. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_repeat_decoded * S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power) + (p)))) /\\ (exists ff_u_bfv_one_selected_power_product ff_v_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_start. ff_h_bfv_one_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_start. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_start * S ((S (0)) * ff_v_bfv_one_selected_power_product) + (1))) /\\ ((((exists ff_h_bfv_one_selected_power_product_terminal. ff_h_bfv_one_selected_power_product_terminal + S (bpv_result_bfv_one_selected) = S ((S (e)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_terminal. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_one_selected_power_product) + (bpv_result_bfv_one_selected))) /\\ forall ff_i_bfv_one_selected_power_product. (exists ff_lt_bfv_one_selected_power_product_bound. ff_lt_bfv_one_selected_power_product_bound + S ff_i_bfv_one_selected_power_product = e) -> exists ff_p_bfv_one_selected_power_product ff_r_bfv_one_selected_power_product ff_s_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_factor. ff_h_bfv_one_selected_power_product_factor + S (ff_p_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_product_factor. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_product_factor * S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power) + (ff_p_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_partial. ff_h_bfv_one_selected_power_product_partial + S (ff_r_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_partial. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_partial * S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_r_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_successor. ff_h_bfv_one_selected_power_product_successor + S (ff_s_bfv_one_selected_power_product) = S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_successor. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_successor * S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_s_bfv_one_selected_power_product))) /\\ ff_s_bfv_one_selected_power_product = ff_r_bfv_one_selected_power_product * ff_p_bfv_one_selected_power_product)))))))) /\\ (exists bpv_factor_bfv_one_selected_divides. one = bpv_result_bfv_one_selected * bpv_factor_bfv_one_selected_divides))", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides one", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation", "cases hselected", "cases hselected_witness", "cases hselected_witness_right", "specialize zero_or_succ e", "cases zero_or_succ", "exact zero_or_succ_left", "cases zero_or_succ_right", "have hstep : exists R. (exists ff_b_bfv_one_prefix ff_c_bfv_one_prefix. ((forall ff_i_bfv_one_prefix_repeat. (exists ff_lt_bfv_one_prefix_repeat_bound. ff_lt_bfv_one_prefix_repeat_bound + S ff_i_bfv_one_prefix_repeat = x2) -> (((exists ff_h_bfv_one_prefix_repeat_decoded. ff_h_bfv_one_prefix_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_prefix_repeat)) * ff_c_bfv_one_prefix)) /\\ exists ff_q_bfv_one_prefix_repeat_decoded. ff_b_bfv_one_prefix = ff_q_bfv_one_prefix_repeat_decoded * S ((S (ff_i_bfv_one_prefix_repeat)) * ff_c_bfv_one_prefix) + (p)))) /\\ (exists ff_u_bfv_one_prefix_product ff_v_bfv_one_prefix_product. ((((exists ff_h_bfv_one_prefix_product_start. ff_h_bfv_one_prefix_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_start. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_start * S ((S (0)) * ff_v_bfv_one_prefix_product) + (1))) /\\ ((((exists ff_h_bfv_one_prefix_product_terminal. ff_h_bfv_one_prefix_product_terminal + S (R) = S ((S (x2)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_terminal. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_terminal * S ((S (x2)) * ff_v_bfv_one_prefix_product) + (R))) /\\ forall ff_i_bfv_one_prefix_product. (exists ff_lt_bfv_one_prefix_product_bound. ff_lt_bfv_one_prefix_product_bound + S ff_i_bfv_one_prefix_product = x2) -> exists ff_p_bfv_one_prefix_product ff_r_bfv_one_prefix_product ff_s_bfv_one_prefix_product. ((((exists ff_h_bfv_one_prefix_product_factor. ff_h_bfv_one_prefix_product_factor + S (ff_p_bfv_one_prefix_product) = S ((S (ff_i_bfv_one_prefix_product)) * ff_c_bfv_one_prefix)) /\\ exists ff_q_bfv_one_prefix_product_factor. ff_b_bfv_one_prefix = ff_q_bfv_one_prefix_product_factor * S ((S (ff_i_bfv_one_prefix_product)) * ff_c_bfv_one_prefix) + (ff_p_bfv_one_prefix_product))) /\\ ((((exists ff_h_bfv_one_prefix_product_partial. ff_h_bfv_one_prefix_product_partial + S (ff_r_bfv_one_prefix_product) = S ((S (ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_partial. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_partial * S ((S (ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product) + (ff_r_bfv_one_prefix_product))) /\\ ((((exists ff_h_bfv_one_prefix_product_successor. ff_h_bfv_one_prefix_product_successor + S (ff_s_bfv_one_prefix_product) = S ((S (S ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_successor. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_successor * S ((S (S ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product) + (ff_s_bfv_one_prefix_product))) /\\ ff_s_bfv_one_prefix_product = ff_r_bfv_one_prefix_product * ff_p_bfv_one_prefix_product)))))))) /\\ x = R * p", "specialize pow_successor_decompose p", "specialize pow_successor_decompose x2", "specialize pow_successor_decompose e", "specialize pow_successor_decompose x", "apply pow_successor_decompose", "exact zero_or_succ_right_witness", "exact hselected_witness_left", "cases hstep", "cases hstep_witness", "have hresult_one : x = 1", "specialize mul_eq_one_components x", "specialize mul_eq_one_components x1", "have hresult_parts : x = 1 /\\ x1 = 1", "apply mul_eq_one_components", "symm", "trans one", "symm", "exact hone", "exact hselected_witness_right_witness", "cases hresult_parts", "exact hresult_parts_left", "have hprime_one : p = 1", "specialize mul_eq_one_components x3", "specialize mul_eq_one_components p", "have hstep_parts : x3 = 1 /\\ p = 1", "apply mul_eq_one_components", "trans x", "symm", "exact hstep_witness_right", "exact hresult_one", "cases hstep_parts", "exact hstep_parts_right", "exfalso", "apply hp_left", "exact hprime_one" ], "script_sha256": "1a27cf3b6e8f966ef29beca763c80cc287f2ac45bbc3efb495899718f3e0b437", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_factorial_valuation_candidate.py", "sha256": "0a2c0f617d401b4e60f11870f4ce5c23cea3c042c0308031aa21a331b5508af9" }, "statement": "forall p one e. one = 1 -> ((~(p = 1) /\\ forall frm_prime_left_bfv_prime frm_prime_right_bfv_prime. p = frm_prime_left_bfv_prime * frm_prime_right_bfv_prime -> frm_prime_left_bfv_prime = 1 \\/ frm_prime_right_bfv_prime = 1)) -> (((exists bpv_gap_bfv_one_exponent_bound. bpv_gap_bfv_one_exponent_bound + e = one) /\\ (exists bpv_result_bfv_one_selected. ((exists ff_b_bfv_one_selected_power ff_c_bfv_one_selected_power. ((forall ff_i_bfv_one_selected_power_repeat. (exists ff_lt_bfv_one_selected_power_repeat_bound. ff_lt_bfv_one_selected_power_repeat_bound + S ff_i_bfv_one_selected_power_repeat = e) -> (((exists ff_h_bfv_one_selected_power_repeat_decoded. ff_h_bfv_one_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_repeat_decoded. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_repeat_decoded * S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power) + (p)))) /\\ (exists ff_u_bfv_one_selected_power_product ff_v_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_start. ff_h_bfv_one_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_start. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_start * S ((S (0)) * ff_v_bfv_one_selected_power_product) + (1))) /\\ ((((exists ff_h_bfv_one_selected_power_product_terminal. ff_h_bfv_one_selected_power_product_terminal + S (bpv_result_bfv_one_selected) = S ((S (e)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_terminal. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_one_selected_power_product) + (bpv_result_bfv_one_selected))) /\\ forall ff_i_bfv_one_selected_power_product. (exists ff_lt_bfv_one_selected_power_product_bound. ff_lt_bfv_one_selected_power_product_bound + S ff_i_bfv_one_selected_power_product = e) -> exists ff_p_bfv_one_selected_power_product ff_r_bfv_one_selected_power_product ff_s_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_factor. ff_h_bfv_one_selected_power_product_factor + S (ff_p_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_product_factor. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_product_factor * S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power) + (ff_p_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_partial. ff_h_bfv_one_selected_power_product_partial + S (ff_r_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_partial. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_partial * S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_r_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_successor. ff_h_bfv_one_selected_power_product_successor + S (ff_s_bfv_one_selected_power_product) = S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_successor. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_successor * S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_s_bfv_one_selected_power_product))) /\\ ff_s_bfv_one_selected_power_product = ff_r_bfv_one_selected_power_product * ff_p_bfv_one_selected_power_product)))))))) /\\ (exists bpv_factor_bfv_one_selected_divides. one = bpv_result_bfv_one_selected * bpv_factor_bfv_one_selected_divides)))) /\\ forall bpv_candidate_bfv_one. (exists bpv_gap_bfv_one_candidate_bound. bpv_gap_bfv_one_candidate_bound + bpv_candidate_bfv_one = one) -> (exists bpv_result_bfv_one_candidate. ((exists ff_b_bfv_one_candidate_power ff_c_bfv_one_candidate_power. ((forall ff_i_bfv_one_candidate_power_repeat. (exists ff_lt_bfv_one_candidate_power_repeat_bound. ff_lt_bfv_one_candidate_power_repeat_bound + S ff_i_bfv_one_candidate_power_repeat = bpv_candidate_bfv_one) -> (((exists ff_h_bfv_one_candidate_power_repeat_decoded. ff_h_bfv_one_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_candidate_power_repeat)) * ff_c_bfv_one_candidate_power)) /\\ exists ff_q_bfv_one_candidate_power_repeat_decoded. ff_b_bfv_one_candidate_power = ff_q_bfv_one_candidate_power_repeat_decoded * S ((S (ff_i_bfv_one_candidate_power_repeat)) * ff_c_bfv_one_candidate_power) + (p)))) /\\ (exists ff_u_bfv_one_candidate_power_product ff_v_bfv_one_candidate_power_product. ((((exists ff_h_bfv_one_candidate_power_product_start. ff_h_bfv_one_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_start. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_start * S ((S (0)) * ff_v_bfv_one_candidate_power_product) + (1))) /\\ ((((exists ff_h_bfv_one_candidate_power_product_terminal. ff_h_bfv_one_candidate_power_product_terminal + S (bpv_result_bfv_one_candidate) = S ((S (bpv_candidate_bfv_one)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_terminal. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_one)) * ff_v_bfv_one_candidate_power_product) + (bpv_result_bfv_one_candidate))) /\\ forall ff_i_bfv_one_candidate_power_product. (exists ff_lt_bfv_one_candidate_power_product_bound. ff_lt_bfv_one_candidate_power_product_bound + S ff_i_bfv_one_candidate_power_product = bpv_candidate_bfv_one) -> exists ff_p_bfv_one_candidate_power_product ff_r_bfv_one_candidate_power_product ff_s_bfv_one_candidate_power_product. ((((exists ff_h_bfv_one_candidate_power_product_factor. ff_h_bfv_one_candidate_power_product_factor + S (ff_p_bfv_one_candidate_power_product) = S ((S (ff_i_bfv_one_candidate_power_product)) * ff_c_bfv_one_candidate_power)) /\\ exists ff_q_bfv_one_candidate_power_product_factor. ff_b_bfv_one_candidate_power = ff_q_bfv_one_candidate_power_product_factor * S ((S (ff_i_bfv_one_candidate_power_product)) * ff_c_bfv_one_candidate_power) + (ff_p_bfv_one_candidate_power_product))) /\\ ((((exists ff_h_bfv_one_candidate_power_product_partial. ff_h_bfv_one_candidate_power_product_partial + S (ff_r_bfv_one_candidate_power_product) = S ((S (ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_partial. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_partial * S ((S (ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product) + (ff_r_bfv_one_candidate_power_product))) /\\ ((((exists ff_h_bfv_one_candidate_power_product_successor. ff_h_bfv_one_candidate_power_product_successor + S (ff_s_bfv_one_candidate_power_product) = S ((S (S ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_successor. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_successor * S ((S (S ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product) + (ff_s_bfv_one_candidate_power_product))) /\\ ff_s_bfv_one_candidate_power_product = ff_r_bfv_one_candidate_power_product * ff_p_bfv_one_candidate_power_product)))))))) /\\ (exists bpv_factor_bfv_one_candidate_divides. one = bpv_result_bfv_one_candidate * bpv_factor_bfv_one_candidate_divides))) -> (exists bpv_gap_bfv_one_maximal. bpv_gap_bfv_one_maximal + bpv_candidate_bfv_one = e)) -> e = 0", "statement_sha256": "814fe3e6f732c3317abe83a5c245fd339156ce333d8b7780546b5fc92a1822a9", "summary": "At a prime base, the bounded valuation of one has exponent zero.", "summary_sha256": "ba83ec555cfc2023f0a1dff37dace852f2daf317fb3afc15832e2c1aa6127f73" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_power_divides", "zero_or_succ", "pow_successor_decompose", "mul_eq_one_components" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "0a2c0f617d401b4e60f11870f4ce5c23cea3c042c0308031aa21a331b5508af9", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_factorial_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "d7e481694a74a2b481bc4928664caa5bf0a2e466cc88e1dfdf394a299c003ae0", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_factorial_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "0b8bf90d53878150272ed3949c6316568d83d857b2e392622bfb8a7b65af8a0b", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-postulate-campaign-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "16e2b99de69487e7439521b25ee070b208d6a7436df48f60801d5628a3678f1a", "kind": "sealed_alpha_v4_parent", "path": "artifacts/peano-library/alpha/catalog-v4.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=207]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_power_valuation_one_zero]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_power_valuation_one_zero", "parent_alpha_version": "v34", "proof_bundle_node_id": 220, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_power_valuation_one_zero", "script": [ "intro p", "intro one", "intro e", "intro hone", "intro hp", "intro hvaluation", "cases hp", "have hselected : exists bpv_result_bfv_one_selected. ((exists ff_b_bfv_one_selected_power ff_c_bfv_one_selected_power. ((forall ff_i_bfv_one_selected_power_repeat. (exists ff_lt_bfv_one_selected_power_repeat_bound. ff_lt_bfv_one_selected_power_repeat_bound + S ff_i_bfv_one_selected_power_repeat = e) -> (((exists ff_h_bfv_one_selected_power_repeat_decoded. ff_h_bfv_one_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_repeat_decoded. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_repeat_decoded * S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power) + (p)))) /\\ (exists ff_u_bfv_one_selected_power_product ff_v_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_start. ff_h_bfv_one_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_start. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_start * S ((S (0)) * ff_v_bfv_one_selected_power_product) + (1))) /\\ ((((exists ff_h_bfv_one_selected_power_product_terminal. ff_h_bfv_one_selected_power_product_terminal + S (bpv_result_bfv_one_selected) = S ((S (e)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_terminal. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_one_selected_power_product) + (bpv_result_bfv_one_selected))) /\\ forall ff_i_bfv_one_selected_power_product. (exists ff_lt_bfv_one_selected_power_product_bound. ff_lt_bfv_one_selected_power_product_bound + S ff_i_bfv_one_selected_power_product = e) -> exists ff_p_bfv_one_selected_power_product ff_r_bfv_one_selected_power_product ff_s_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_factor. ff_h_bfv_one_selected_power_product_factor + S (ff_p_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_product_factor. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_product_factor * S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power) + (ff_p_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_partial. ff_h_bfv_one_selected_power_product_partial + S (ff_r_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_partial. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_partial * S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_r_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_successor. ff_h_bfv_one_selected_power_product_successor + S (ff_s_bfv_one_selected_power_product) = S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_successor. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_successor * S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_s_bfv_one_selected_power_product))) /\\ ff_s_bfv_one_selected_power_product = ff_r_bfv_one_selected_power_product * ff_p_bfv_one_selected_power_product)))))))) /\\ (exists bpv_factor_bfv_one_selected_divides. one = bpv_result_bfv_one_selected * bpv_factor_bfv_one_selected_divides))", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides one", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation", "cases hselected", "cases hselected_witness", "cases hselected_witness_right", "specialize zero_or_succ e", "cases zero_or_succ", "exact zero_or_succ_left", "cases zero_or_succ_right", "have hstep : exists R. (exists ff_b_bfv_one_prefix ff_c_bfv_one_prefix. ((forall ff_i_bfv_one_prefix_repeat. (exists ff_lt_bfv_one_prefix_repeat_bound. ff_lt_bfv_one_prefix_repeat_bound + S ff_i_bfv_one_prefix_repeat = x2) -> (((exists ff_h_bfv_one_prefix_repeat_decoded. ff_h_bfv_one_prefix_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_prefix_repeat)) * ff_c_bfv_one_prefix)) /\\ exists ff_q_bfv_one_prefix_repeat_decoded. ff_b_bfv_one_prefix = ff_q_bfv_one_prefix_repeat_decoded * S ((S (ff_i_bfv_one_prefix_repeat)) * ff_c_bfv_one_prefix) + (p)))) /\\ (exists ff_u_bfv_one_prefix_product ff_v_bfv_one_prefix_product. ((((exists ff_h_bfv_one_prefix_product_start. ff_h_bfv_one_prefix_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_start. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_start * S ((S (0)) * ff_v_bfv_one_prefix_product) + (1))) /\\ ((((exists ff_h_bfv_one_prefix_product_terminal. ff_h_bfv_one_prefix_product_terminal + S (R) = S ((S (x2)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_terminal. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_terminal * S ((S (x2)) * ff_v_bfv_one_prefix_product) + (R))) /\\ forall ff_i_bfv_one_prefix_product. (exists ff_lt_bfv_one_prefix_product_bound. ff_lt_bfv_one_prefix_product_bound + S ff_i_bfv_one_prefix_product = x2) -> exists ff_p_bfv_one_prefix_product ff_r_bfv_one_prefix_product ff_s_bfv_one_prefix_product. ((((exists ff_h_bfv_one_prefix_product_factor. ff_h_bfv_one_prefix_product_factor + S (ff_p_bfv_one_prefix_product) = S ((S (ff_i_bfv_one_prefix_product)) * ff_c_bfv_one_prefix)) /\\ exists ff_q_bfv_one_prefix_product_factor. ff_b_bfv_one_prefix = ff_q_bfv_one_prefix_product_factor * S ((S (ff_i_bfv_one_prefix_product)) * ff_c_bfv_one_prefix) + (ff_p_bfv_one_prefix_product))) /\\ ((((exists ff_h_bfv_one_prefix_product_partial. ff_h_bfv_one_prefix_product_partial + S (ff_r_bfv_one_prefix_product) = S ((S (ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_partial. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_partial * S ((S (ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product) + (ff_r_bfv_one_prefix_product))) /\\ ((((exists ff_h_bfv_one_prefix_product_successor. ff_h_bfv_one_prefix_product_successor + S (ff_s_bfv_one_prefix_product) = S ((S (S ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product)) /\\ exists ff_q_bfv_one_prefix_product_successor. ff_u_bfv_one_prefix_product = ff_q_bfv_one_prefix_product_successor * S ((S (S ff_i_bfv_one_prefix_product)) * ff_v_bfv_one_prefix_product) + (ff_s_bfv_one_prefix_product))) /\\ ff_s_bfv_one_prefix_product = ff_r_bfv_one_prefix_product * ff_p_bfv_one_prefix_product)))))))) /\\ x = R * p", "specialize pow_successor_decompose p", "specialize pow_successor_decompose x2", "specialize pow_successor_decompose e", "specialize pow_successor_decompose x", "apply pow_successor_decompose", "exact zero_or_succ_right_witness", "exact hselected_witness_left", "cases hstep", "cases hstep_witness", "have hresult_one : x = 1", "specialize mul_eq_one_components x", "specialize mul_eq_one_components x1", "have hresult_parts : x = 1 /\\ x1 = 1", "apply mul_eq_one_components", "symm", "trans one", "symm", "exact hone", "exact hselected_witness_right_witness", "cases hresult_parts", "exact hresult_parts_left", "have hprime_one : p = 1", "specialize mul_eq_one_components x3", "specialize mul_eq_one_components p", "have hstep_parts : x3 = 1 /\\ p = 1", "apply mul_eq_one_components", "trans x", "symm", "exact hstep_witness_right", "exact hresult_one", "cases hstep_parts", "exact hstep_parts_right", "exfalso", "apply hp_left", "exact hprime_one" ], "script_sha256": "1a27cf3b6e8f966ef29beca763c80cc287f2ac45bbc3efb495899718f3e0b437", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_factorial_valuation_candidate.py", "sha256": "0a2c0f617d401b4e60f11870f4ce5c23cea3c042c0308031aa21a331b5508af9" }, "stable_member": false, "statement": "forall p one e. one = 1 -> ((~(p = 1) /\\ forall frm_prime_left_bfv_prime frm_prime_right_bfv_prime. p = frm_prime_left_bfv_prime * frm_prime_right_bfv_prime -> frm_prime_left_bfv_prime = 1 \\/ frm_prime_right_bfv_prime = 1)) -> (((exists bpv_gap_bfv_one_exponent_bound. bpv_gap_bfv_one_exponent_bound + e = one) /\\ (exists bpv_result_bfv_one_selected. ((exists ff_b_bfv_one_selected_power ff_c_bfv_one_selected_power. ((forall ff_i_bfv_one_selected_power_repeat. (exists ff_lt_bfv_one_selected_power_repeat_bound. ff_lt_bfv_one_selected_power_repeat_bound + S ff_i_bfv_one_selected_power_repeat = e) -> (((exists ff_h_bfv_one_selected_power_repeat_decoded. ff_h_bfv_one_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_repeat_decoded. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_repeat_decoded * S ((S (ff_i_bfv_one_selected_power_repeat)) * ff_c_bfv_one_selected_power) + (p)))) /\\ (exists ff_u_bfv_one_selected_power_product ff_v_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_start. ff_h_bfv_one_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_start. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_start * S ((S (0)) * ff_v_bfv_one_selected_power_product) + (1))) /\\ ((((exists ff_h_bfv_one_selected_power_product_terminal. ff_h_bfv_one_selected_power_product_terminal + S (bpv_result_bfv_one_selected) = S ((S (e)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_terminal. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_terminal * S ((S (e)) * ff_v_bfv_one_selected_power_product) + (bpv_result_bfv_one_selected))) /\\ forall ff_i_bfv_one_selected_power_product. (exists ff_lt_bfv_one_selected_power_product_bound. ff_lt_bfv_one_selected_power_product_bound + S ff_i_bfv_one_selected_power_product = e) -> exists ff_p_bfv_one_selected_power_product ff_r_bfv_one_selected_power_product ff_s_bfv_one_selected_power_product. ((((exists ff_h_bfv_one_selected_power_product_factor. ff_h_bfv_one_selected_power_product_factor + S (ff_p_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power)) /\\ exists ff_q_bfv_one_selected_power_product_factor. ff_b_bfv_one_selected_power = ff_q_bfv_one_selected_power_product_factor * S ((S (ff_i_bfv_one_selected_power_product)) * ff_c_bfv_one_selected_power) + (ff_p_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_partial. ff_h_bfv_one_selected_power_product_partial + S (ff_r_bfv_one_selected_power_product) = S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_partial. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_partial * S ((S (ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_r_bfv_one_selected_power_product))) /\\ ((((exists ff_h_bfv_one_selected_power_product_successor. ff_h_bfv_one_selected_power_product_successor + S (ff_s_bfv_one_selected_power_product) = S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product)) /\\ exists ff_q_bfv_one_selected_power_product_successor. ff_u_bfv_one_selected_power_product = ff_q_bfv_one_selected_power_product_successor * S ((S (S ff_i_bfv_one_selected_power_product)) * ff_v_bfv_one_selected_power_product) + (ff_s_bfv_one_selected_power_product))) /\\ ff_s_bfv_one_selected_power_product = ff_r_bfv_one_selected_power_product * ff_p_bfv_one_selected_power_product)))))))) /\\ (exists bpv_factor_bfv_one_selected_divides. one = bpv_result_bfv_one_selected * bpv_factor_bfv_one_selected_divides)))) /\\ forall bpv_candidate_bfv_one. (exists bpv_gap_bfv_one_candidate_bound. bpv_gap_bfv_one_candidate_bound + bpv_candidate_bfv_one = one) -> (exists bpv_result_bfv_one_candidate. ((exists ff_b_bfv_one_candidate_power ff_c_bfv_one_candidate_power. ((forall ff_i_bfv_one_candidate_power_repeat. (exists ff_lt_bfv_one_candidate_power_repeat_bound. ff_lt_bfv_one_candidate_power_repeat_bound + S ff_i_bfv_one_candidate_power_repeat = bpv_candidate_bfv_one) -> (((exists ff_h_bfv_one_candidate_power_repeat_decoded. ff_h_bfv_one_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bfv_one_candidate_power_repeat)) * ff_c_bfv_one_candidate_power)) /\\ exists ff_q_bfv_one_candidate_power_repeat_decoded. ff_b_bfv_one_candidate_power = ff_q_bfv_one_candidate_power_repeat_decoded * S ((S (ff_i_bfv_one_candidate_power_repeat)) * ff_c_bfv_one_candidate_power) + (p)))) /\\ (exists ff_u_bfv_one_candidate_power_product ff_v_bfv_one_candidate_power_product. ((((exists ff_h_bfv_one_candidate_power_product_start. ff_h_bfv_one_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_start. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_start * S ((S (0)) * ff_v_bfv_one_candidate_power_product) + (1))) /\\ ((((exists ff_h_bfv_one_candidate_power_product_terminal. ff_h_bfv_one_candidate_power_product_terminal + S (bpv_result_bfv_one_candidate) = S ((S (bpv_candidate_bfv_one)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_terminal. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_terminal * S ((S (bpv_candidate_bfv_one)) * ff_v_bfv_one_candidate_power_product) + (bpv_result_bfv_one_candidate))) /\\ forall ff_i_bfv_one_candidate_power_product. (exists ff_lt_bfv_one_candidate_power_product_bound. ff_lt_bfv_one_candidate_power_product_bound + S ff_i_bfv_one_candidate_power_product = bpv_candidate_bfv_one) -> exists ff_p_bfv_one_candidate_power_product ff_r_bfv_one_candidate_power_product ff_s_bfv_one_candidate_power_product. ((((exists ff_h_bfv_one_candidate_power_product_factor. ff_h_bfv_one_candidate_power_product_factor + S (ff_p_bfv_one_candidate_power_product) = S ((S (ff_i_bfv_one_candidate_power_product)) * ff_c_bfv_one_candidate_power)) /\\ exists ff_q_bfv_one_candidate_power_product_factor. ff_b_bfv_one_candidate_power = ff_q_bfv_one_candidate_power_product_factor * S ((S (ff_i_bfv_one_candidate_power_product)) * ff_c_bfv_one_candidate_power) + (ff_p_bfv_one_candidate_power_product))) /\\ ((((exists ff_h_bfv_one_candidate_power_product_partial. ff_h_bfv_one_candidate_power_product_partial + S (ff_r_bfv_one_candidate_power_product) = S ((S (ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_partial. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_partial * S ((S (ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product) + (ff_r_bfv_one_candidate_power_product))) /\\ ((((exists ff_h_bfv_one_candidate_power_product_successor. ff_h_bfv_one_candidate_power_product_successor + S (ff_s_bfv_one_candidate_power_product) = S ((S (S ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product)) /\\ exists ff_q_bfv_one_candidate_power_product_successor. ff_u_bfv_one_candidate_power_product = ff_q_bfv_one_candidate_power_product_successor * S ((S (S ff_i_bfv_one_candidate_power_product)) * ff_v_bfv_one_candidate_power_product) + (ff_s_bfv_one_candidate_power_product))) /\\ ff_s_bfv_one_candidate_power_product = ff_r_bfv_one_candidate_power_product * ff_p_bfv_one_candidate_power_product)))))))) /\\ (exists bpv_factor_bfv_one_candidate_divides. one = bpv_result_bfv_one_candidate * bpv_factor_bfv_one_candidate_divides))) -> (exists bpv_gap_bfv_one_maximal. bpv_gap_bfv_one_maximal + bpv_candidate_bfv_one = e)) -> e = 0", "statement_sha256": "814fe3e6f732c3317abe83a5c245fd339156ce333d8b7780546b5fc92a1822a9" }, { "admitted_to_alpha": true, 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"theorems[name=prime_power_divides_exponent_le_valuation]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "dc20b69712d0144d260bc670866ca7932bfadf480bae8069f72b0046b08f57ac", "membership": "alpha_only", "name": "prime_power_divides_exponent_le_valuation", "proof_tag": null, "provenance": [ "bertrand" ], "script": [ "intro p", "intro a", "intro f", "intro k", "intro hp", "intro ha", "intro hvaluation", "intro hdivides", "have hvalue_bound : exists gap. gap + k = a", "specialize prime_power_divides_exponent_le_value p", "specialize prime_power_divides_exponent_le_value k", "specialize prime_power_divides_exponent_le_value a", "apply prime_power_divides_exponent_le_value", "exact hp", "exact ha", "exact hdivides", "specialize power_valuation_dominates p", "specialize power_valuation_dominates a", "specialize power_valuation_dominates f", "specialize power_valuation_dominates k", "apply power_valuation_dominates", "exact hvaluation", "exact hvalue_bound", "exact hdivides" ], "script_sha256": "345d4effe42a7a04f0581183828a383d670de18f35769c0e7e26bc1ab9dce1cf", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_legendre_valuation_bridge_candidate.py", "sha256": "6af2e6ad82bc47120cbf6f9d6b5dace8a2f20a45968d3dad88c6003c4637c89d" }, "statement": "forall p a f k. ((~(p = 1) /\\ forall frm_prime_left_blvb_prime frm_prime_right_blvb_prime. p = frm_prime_left_blvb_prime * frm_prime_right_blvb_prime -> frm_prime_left_blvb_prime = 1 \\/ frm_prime_right_blvb_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_blvb_valuation_exponent_bound. bpv_gap_blvb_valuation_exponent_bound + f = a) /\\ (exists bpv_result_blvb_valuation_selected. ((exists ff_b_blvb_valuation_selected_power ff_c_blvb_valuation_selected_power. ((forall ff_i_blvb_valuation_selected_power_repeat. (exists ff_lt_blvb_valuation_selected_power_repeat_bound. ff_lt_blvb_valuation_selected_power_repeat_bound + S ff_i_blvb_valuation_selected_power_repeat = f) -> (((exists ff_h_blvb_valuation_selected_power_repeat_decoded. ff_h_blvb_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_valuation_selected_power_repeat)) * ff_c_blvb_valuation_selected_power)) /\\ exists ff_q_blvb_valuation_selected_power_repeat_decoded. ff_b_blvb_valuation_selected_power = ff_q_blvb_valuation_selected_power_repeat_decoded * S ((S (ff_i_blvb_valuation_selected_power_repeat)) * ff_c_blvb_valuation_selected_power) + (p)))) /\\ (exists ff_u_blvb_valuation_selected_power_product ff_v_blvb_valuation_selected_power_product. ((((exists ff_h_blvb_valuation_selected_power_product_start. ff_h_blvb_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_start. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_start * S ((S (0)) * ff_v_blvb_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_blvb_valuation_selected_power_product_terminal. ff_h_blvb_valuation_selected_power_product_terminal + S (bpv_result_blvb_valuation_selected) = S ((S (f)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_terminal. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_terminal * S ((S (f)) * ff_v_blvb_valuation_selected_power_product) + (bpv_result_blvb_valuation_selected))) /\\ forall ff_i_blvb_valuation_selected_power_product. (exists ff_lt_blvb_valuation_selected_power_product_bound. ff_lt_blvb_valuation_selected_power_product_bound + S ff_i_blvb_valuation_selected_power_product = f) -> exists ff_p_blvb_valuation_selected_power_product ff_r_blvb_valuation_selected_power_product ff_s_blvb_valuation_selected_power_product. ((((exists ff_h_blvb_valuation_selected_power_product_factor. ff_h_blvb_valuation_selected_power_product_factor + S (ff_p_blvb_valuation_selected_power_product) = S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_c_blvb_valuation_selected_power)) /\\ exists ff_q_blvb_valuation_selected_power_product_factor. ff_b_blvb_valuation_selected_power = ff_q_blvb_valuation_selected_power_product_factor * S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_c_blvb_valuation_selected_power) + (ff_p_blvb_valuation_selected_power_product))) /\\ ((((exists ff_h_blvb_valuation_selected_power_product_partial. ff_h_blvb_valuation_selected_power_product_partial + S (ff_r_blvb_valuation_selected_power_product) = S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_partial. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_partial * S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product) + (ff_r_blvb_valuation_selected_power_product))) /\\ ((((exists ff_h_blvb_valuation_selected_power_product_successor. ff_h_blvb_valuation_selected_power_product_successor + S (ff_s_blvb_valuation_selected_power_product) = S ((S (S ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_successor. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_successor * S ((S (S ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product) + (ff_s_blvb_valuation_selected_power_product))) /\\ ff_s_blvb_valuation_selected_power_product = ff_r_blvb_valuation_selected_power_product * ff_p_blvb_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_blvb_valuation_selected_divides. a = bpv_result_blvb_valuation_selected * bpv_factor_blvb_valuation_selected_divides)))) /\\ forall bpv_candidate_blvb_valuation. (exists bpv_gap_blvb_valuation_candidate_bound. bpv_gap_blvb_valuation_candidate_bound + bpv_candidate_blvb_valuation = a) -> (exists bpv_result_blvb_valuation_candidate. ((exists ff_b_blvb_valuation_candidate_power ff_c_blvb_valuation_candidate_power. ((forall ff_i_blvb_valuation_candidate_power_repeat. (exists ff_lt_blvb_valuation_candidate_power_repeat_bound. ff_lt_blvb_valuation_candidate_power_repeat_bound + S ff_i_blvb_valuation_candidate_power_repeat = bpv_candidate_blvb_valuation) -> (((exists ff_h_blvb_valuation_candidate_power_repeat_decoded. ff_h_blvb_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_valuation_candidate_power_repeat)) * ff_c_blvb_valuation_candidate_power)) /\\ exists ff_q_blvb_valuation_candidate_power_repeat_decoded. ff_b_blvb_valuation_candidate_power = ff_q_blvb_valuation_candidate_power_repeat_decoded * S ((S (ff_i_blvb_valuation_candidate_power_repeat)) * ff_c_blvb_valuation_candidate_power) + (p)))) /\\ (exists ff_u_blvb_valuation_candidate_power_product ff_v_blvb_valuation_candidate_power_product. ((((exists ff_h_blvb_valuation_candidate_power_product_start. ff_h_blvb_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_start. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_start * S ((S (0)) * ff_v_blvb_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_blvb_valuation_candidate_power_product_terminal. ff_h_blvb_valuation_candidate_power_product_terminal + S (bpv_result_blvb_valuation_candidate) = S ((S (bpv_candidate_blvb_valuation)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_terminal. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_blvb_valuation)) * ff_v_blvb_valuation_candidate_power_product) + (bpv_result_blvb_valuation_candidate))) /\\ forall ff_i_blvb_valuation_candidate_power_product. (exists ff_lt_blvb_valuation_candidate_power_product_bound. ff_lt_blvb_valuation_candidate_power_product_bound + S ff_i_blvb_valuation_candidate_power_product = bpv_candidate_blvb_valuation) -> exists ff_p_blvb_valuation_candidate_power_product ff_r_blvb_valuation_candidate_power_product ff_s_blvb_valuation_candidate_power_product. ((((exists ff_h_blvb_valuation_candidate_power_product_factor. ff_h_blvb_valuation_candidate_power_product_factor + S (ff_p_blvb_valuation_candidate_power_product) = S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_c_blvb_valuation_candidate_power)) /\\ exists ff_q_blvb_valuation_candidate_power_product_factor. ff_b_blvb_valuation_candidate_power = ff_q_blvb_valuation_candidate_power_product_factor * S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_c_blvb_valuation_candidate_power) + (ff_p_blvb_valuation_candidate_power_product))) /\\ ((((exists ff_h_blvb_valuation_candidate_power_product_partial. ff_h_blvb_valuation_candidate_power_product_partial + S (ff_r_blvb_valuation_candidate_power_product) = S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_partial. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_partial * S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product) + (ff_r_blvb_valuation_candidate_power_product))) /\\ ((((exists ff_h_blvb_valuation_candidate_power_product_successor. ff_h_blvb_valuation_candidate_power_product_successor + S (ff_s_blvb_valuation_candidate_power_product) = S ((S (S ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_successor. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_successor * S ((S (S ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product) + (ff_s_blvb_valuation_candidate_power_product))) /\\ ff_s_blvb_valuation_candidate_power_product = ff_r_blvb_valuation_candidate_power_product * ff_p_blvb_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_blvb_valuation_candidate_divides. a = bpv_result_blvb_valuation_candidate * bpv_factor_blvb_valuation_candidate_divides))) -> (exists bpv_gap_blvb_valuation_maximal. bpv_gap_blvb_valuation_maximal + bpv_candidate_blvb_valuation = f)) -> (exists bpv_result_blvb_candidate_divides. ((exists ff_b_blvb_candidate_divides_power ff_c_blvb_candidate_divides_power. ((forall ff_i_blvb_candidate_divides_power_repeat. (exists ff_lt_blvb_candidate_divides_power_repeat_bound. ff_lt_blvb_candidate_divides_power_repeat_bound + S ff_i_blvb_candidate_divides_power_repeat = k) -> (((exists ff_h_blvb_candidate_divides_power_repeat_decoded. ff_h_blvb_candidate_divides_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_candidate_divides_power_repeat)) * ff_c_blvb_candidate_divides_power)) /\\ exists ff_q_blvb_candidate_divides_power_repeat_decoded. ff_b_blvb_candidate_divides_power = ff_q_blvb_candidate_divides_power_repeat_decoded * S ((S (ff_i_blvb_candidate_divides_power_repeat)) * ff_c_blvb_candidate_divides_power) + (p)))) /\\ (exists ff_u_blvb_candidate_divides_power_product ff_v_blvb_candidate_divides_power_product. ((((exists ff_h_blvb_candidate_divides_power_product_start. ff_h_blvb_candidate_divides_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_start. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_start * S ((S (0)) * ff_v_blvb_candidate_divides_power_product) + (1))) /\\ ((((exists ff_h_blvb_candidate_divides_power_product_terminal. ff_h_blvb_candidate_divides_power_product_terminal + S (bpv_result_blvb_candidate_divides) = S ((S (k)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_terminal. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_terminal * S ((S (k)) * ff_v_blvb_candidate_divides_power_product) + (bpv_result_blvb_candidate_divides))) /\\ forall ff_i_blvb_candidate_divides_power_product. (exists ff_lt_blvb_candidate_divides_power_product_bound. ff_lt_blvb_candidate_divides_power_product_bound + S ff_i_blvb_candidate_divides_power_product = k) -> exists ff_p_blvb_candidate_divides_power_product ff_r_blvb_candidate_divides_power_product ff_s_blvb_candidate_divides_power_product. ((((exists ff_h_blvb_candidate_divides_power_product_factor. ff_h_blvb_candidate_divides_power_product_factor + S (ff_p_blvb_candidate_divides_power_product) = S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_c_blvb_candidate_divides_power)) /\\ exists ff_q_blvb_candidate_divides_power_product_factor. ff_b_blvb_candidate_divides_power = ff_q_blvb_candidate_divides_power_product_factor * S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_c_blvb_candidate_divides_power) + (ff_p_blvb_candidate_divides_power_product))) /\\ ((((exists ff_h_blvb_candidate_divides_power_product_partial. ff_h_blvb_candidate_divides_power_product_partial + S (ff_r_blvb_candidate_divides_power_product) = S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_partial. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_partial * S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product) + (ff_r_blvb_candidate_divides_power_product))) /\\ ((((exists ff_h_blvb_candidate_divides_power_product_successor. ff_h_blvb_candidate_divides_power_product_successor + S (ff_s_blvb_candidate_divides_power_product) = S ((S (S ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_successor. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_successor * S ((S (S ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product) + (ff_s_blvb_candidate_divides_power_product))) /\\ ff_s_blvb_candidate_divides_power_product = ff_r_blvb_candidate_divides_power_product * ff_p_blvb_candidate_divides_power_product)))))))) /\\ (exists bpv_factor_blvb_candidate_divides_divides. a = bpv_result_blvb_candidate_divides * bpv_factor_blvb_candidate_divides_divides))) -> 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((~(p = 1) /\\ forall frm_prime_left_blvb_prime frm_prime_right_blvb_prime. p = frm_prime_left_blvb_prime * frm_prime_right_blvb_prime -> frm_prime_left_blvb_prime = 1 \\/ frm_prime_right_blvb_prime = 1)) -> ~(a = 0) -> (((exists bpv_gap_blvb_valuation_exponent_bound. bpv_gap_blvb_valuation_exponent_bound + f = a) /\\ (exists bpv_result_blvb_valuation_selected. ((exists ff_b_blvb_valuation_selected_power ff_c_blvb_valuation_selected_power. ((forall ff_i_blvb_valuation_selected_power_repeat. (exists ff_lt_blvb_valuation_selected_power_repeat_bound. ff_lt_blvb_valuation_selected_power_repeat_bound + S ff_i_blvb_valuation_selected_power_repeat = f) -> (((exists ff_h_blvb_valuation_selected_power_repeat_decoded. ff_h_blvb_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_valuation_selected_power_repeat)) * ff_c_blvb_valuation_selected_power)) /\\ exists ff_q_blvb_valuation_selected_power_repeat_decoded. ff_b_blvb_valuation_selected_power = ff_q_blvb_valuation_selected_power_repeat_decoded * S ((S (ff_i_blvb_valuation_selected_power_repeat)) * ff_c_blvb_valuation_selected_power) + (p)))) /\\ (exists ff_u_blvb_valuation_selected_power_product ff_v_blvb_valuation_selected_power_product. ((((exists ff_h_blvb_valuation_selected_power_product_start. ff_h_blvb_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_start. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_start * S ((S (0)) * ff_v_blvb_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_blvb_valuation_selected_power_product_terminal. ff_h_blvb_valuation_selected_power_product_terminal + S (bpv_result_blvb_valuation_selected) = S ((S (f)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_terminal. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_terminal * S ((S (f)) * ff_v_blvb_valuation_selected_power_product) + (bpv_result_blvb_valuation_selected))) /\\ forall ff_i_blvb_valuation_selected_power_product. (exists ff_lt_blvb_valuation_selected_power_product_bound. ff_lt_blvb_valuation_selected_power_product_bound + S ff_i_blvb_valuation_selected_power_product = f) -> exists ff_p_blvb_valuation_selected_power_product ff_r_blvb_valuation_selected_power_product ff_s_blvb_valuation_selected_power_product. ((((exists ff_h_blvb_valuation_selected_power_product_factor. ff_h_blvb_valuation_selected_power_product_factor + S (ff_p_blvb_valuation_selected_power_product) = S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_c_blvb_valuation_selected_power)) /\\ exists ff_q_blvb_valuation_selected_power_product_factor. ff_b_blvb_valuation_selected_power = ff_q_blvb_valuation_selected_power_product_factor * S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_c_blvb_valuation_selected_power) + (ff_p_blvb_valuation_selected_power_product))) /\\ ((((exists ff_h_blvb_valuation_selected_power_product_partial. ff_h_blvb_valuation_selected_power_product_partial + S (ff_r_blvb_valuation_selected_power_product) = S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_partial. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_partial * S ((S (ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product) + (ff_r_blvb_valuation_selected_power_product))) /\\ ((((exists ff_h_blvb_valuation_selected_power_product_successor. ff_h_blvb_valuation_selected_power_product_successor + S (ff_s_blvb_valuation_selected_power_product) = S ((S (S ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product)) /\\ exists ff_q_blvb_valuation_selected_power_product_successor. ff_u_blvb_valuation_selected_power_product = ff_q_blvb_valuation_selected_power_product_successor * S ((S (S ff_i_blvb_valuation_selected_power_product)) * ff_v_blvb_valuation_selected_power_product) + (ff_s_blvb_valuation_selected_power_product))) /\\ ff_s_blvb_valuation_selected_power_product = ff_r_blvb_valuation_selected_power_product * ff_p_blvb_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_blvb_valuation_selected_divides. a = bpv_result_blvb_valuation_selected * bpv_factor_blvb_valuation_selected_divides)))) /\\ forall bpv_candidate_blvb_valuation. (exists bpv_gap_blvb_valuation_candidate_bound. bpv_gap_blvb_valuation_candidate_bound + bpv_candidate_blvb_valuation = a) -> (exists bpv_result_blvb_valuation_candidate. ((exists ff_b_blvb_valuation_candidate_power ff_c_blvb_valuation_candidate_power. ((forall ff_i_blvb_valuation_candidate_power_repeat. (exists ff_lt_blvb_valuation_candidate_power_repeat_bound. ff_lt_blvb_valuation_candidate_power_repeat_bound + S ff_i_blvb_valuation_candidate_power_repeat = bpv_candidate_blvb_valuation) -> (((exists ff_h_blvb_valuation_candidate_power_repeat_decoded. ff_h_blvb_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_valuation_candidate_power_repeat)) * ff_c_blvb_valuation_candidate_power)) /\\ exists ff_q_blvb_valuation_candidate_power_repeat_decoded. ff_b_blvb_valuation_candidate_power = ff_q_blvb_valuation_candidate_power_repeat_decoded * S ((S (ff_i_blvb_valuation_candidate_power_repeat)) * ff_c_blvb_valuation_candidate_power) + (p)))) /\\ (exists ff_u_blvb_valuation_candidate_power_product ff_v_blvb_valuation_candidate_power_product. ((((exists ff_h_blvb_valuation_candidate_power_product_start. ff_h_blvb_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_start. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_start * S ((S (0)) * ff_v_blvb_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_blvb_valuation_candidate_power_product_terminal. ff_h_blvb_valuation_candidate_power_product_terminal + S (bpv_result_blvb_valuation_candidate) = S ((S (bpv_candidate_blvb_valuation)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_terminal. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_blvb_valuation)) * ff_v_blvb_valuation_candidate_power_product) + (bpv_result_blvb_valuation_candidate))) /\\ forall ff_i_blvb_valuation_candidate_power_product. (exists ff_lt_blvb_valuation_candidate_power_product_bound. ff_lt_blvb_valuation_candidate_power_product_bound + S ff_i_blvb_valuation_candidate_power_product = bpv_candidate_blvb_valuation) -> exists ff_p_blvb_valuation_candidate_power_product ff_r_blvb_valuation_candidate_power_product ff_s_blvb_valuation_candidate_power_product. ((((exists ff_h_blvb_valuation_candidate_power_product_factor. ff_h_blvb_valuation_candidate_power_product_factor + S (ff_p_blvb_valuation_candidate_power_product) = S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_c_blvb_valuation_candidate_power)) /\\ exists ff_q_blvb_valuation_candidate_power_product_factor. ff_b_blvb_valuation_candidate_power = ff_q_blvb_valuation_candidate_power_product_factor * S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_c_blvb_valuation_candidate_power) + (ff_p_blvb_valuation_candidate_power_product))) /\\ ((((exists ff_h_blvb_valuation_candidate_power_product_partial. ff_h_blvb_valuation_candidate_power_product_partial + S (ff_r_blvb_valuation_candidate_power_product) = S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_partial. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_partial * S ((S (ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product) + (ff_r_blvb_valuation_candidate_power_product))) /\\ ((((exists ff_h_blvb_valuation_candidate_power_product_successor. ff_h_blvb_valuation_candidate_power_product_successor + S (ff_s_blvb_valuation_candidate_power_product) = S ((S (S ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product)) /\\ exists ff_q_blvb_valuation_candidate_power_product_successor. ff_u_blvb_valuation_candidate_power_product = ff_q_blvb_valuation_candidate_power_product_successor * S ((S (S ff_i_blvb_valuation_candidate_power_product)) * ff_v_blvb_valuation_candidate_power_product) + (ff_s_blvb_valuation_candidate_power_product))) /\\ ff_s_blvb_valuation_candidate_power_product = ff_r_blvb_valuation_candidate_power_product * ff_p_blvb_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_blvb_valuation_candidate_divides. a = bpv_result_blvb_valuation_candidate * bpv_factor_blvb_valuation_candidate_divides))) -> (exists bpv_gap_blvb_valuation_maximal. bpv_gap_blvb_valuation_maximal + bpv_candidate_blvb_valuation = f)) -> (exists bpv_result_blvb_candidate_divides. ((exists ff_b_blvb_candidate_divides_power ff_c_blvb_candidate_divides_power. ((forall ff_i_blvb_candidate_divides_power_repeat. (exists ff_lt_blvb_candidate_divides_power_repeat_bound. ff_lt_blvb_candidate_divides_power_repeat_bound + S ff_i_blvb_candidate_divides_power_repeat = k) -> (((exists ff_h_blvb_candidate_divides_power_repeat_decoded. ff_h_blvb_candidate_divides_power_repeat_decoded + S (p) = S ((S (ff_i_blvb_candidate_divides_power_repeat)) * ff_c_blvb_candidate_divides_power)) /\\ exists ff_q_blvb_candidate_divides_power_repeat_decoded. ff_b_blvb_candidate_divides_power = ff_q_blvb_candidate_divides_power_repeat_decoded * S ((S (ff_i_blvb_candidate_divides_power_repeat)) * ff_c_blvb_candidate_divides_power) + (p)))) /\\ (exists ff_u_blvb_candidate_divides_power_product ff_v_blvb_candidate_divides_power_product. ((((exists ff_h_blvb_candidate_divides_power_product_start. ff_h_blvb_candidate_divides_power_product_start + S (1) = S ((S (0)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_start. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_start * S ((S (0)) * ff_v_blvb_candidate_divides_power_product) + (1))) /\\ ((((exists ff_h_blvb_candidate_divides_power_product_terminal. ff_h_blvb_candidate_divides_power_product_terminal + S (bpv_result_blvb_candidate_divides) = S ((S (k)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_terminal. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_terminal * S ((S (k)) * ff_v_blvb_candidate_divides_power_product) + (bpv_result_blvb_candidate_divides))) /\\ forall ff_i_blvb_candidate_divides_power_product. (exists ff_lt_blvb_candidate_divides_power_product_bound. ff_lt_blvb_candidate_divides_power_product_bound + S ff_i_blvb_candidate_divides_power_product = k) -> exists ff_p_blvb_candidate_divides_power_product ff_r_blvb_candidate_divides_power_product ff_s_blvb_candidate_divides_power_product. ((((exists ff_h_blvb_candidate_divides_power_product_factor. ff_h_blvb_candidate_divides_power_product_factor + S (ff_p_blvb_candidate_divides_power_product) = S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_c_blvb_candidate_divides_power)) /\\ exists ff_q_blvb_candidate_divides_power_product_factor. ff_b_blvb_candidate_divides_power = ff_q_blvb_candidate_divides_power_product_factor * S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_c_blvb_candidate_divides_power) + (ff_p_blvb_candidate_divides_power_product))) /\\ ((((exists ff_h_blvb_candidate_divides_power_product_partial. ff_h_blvb_candidate_divides_power_product_partial + S (ff_r_blvb_candidate_divides_power_product) = S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_partial. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_partial * S ((S (ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product) + (ff_r_blvb_candidate_divides_power_product))) /\\ ((((exists ff_h_blvb_candidate_divides_power_product_successor. ff_h_blvb_candidate_divides_power_product_successor + S (ff_s_blvb_candidate_divides_power_product) = S ((S (S ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product)) /\\ exists ff_q_blvb_candidate_divides_power_product_successor. ff_u_blvb_candidate_divides_power_product = ff_q_blvb_candidate_divides_power_product_successor * S ((S (S ff_i_blvb_candidate_divides_power_product)) * ff_v_blvb_candidate_divides_power_product) + (ff_s_blvb_candidate_divides_power_product))) /\\ ff_s_blvb_candidate_divides_power_product = ff_r_blvb_candidate_divides_power_product * ff_p_blvb_candidate_divides_power_product)))))))) /\\ (exists bpv_factor_blvb_candidate_divides_divides. a = bpv_result_blvb_candidate_divides * bpv_factor_blvb_candidate_divides_divides))) -> (exists bpv_gap_blvb_candidate_bound. bpv_gap_blvb_candidate_bound + k = f)", "statement_sha256": "8f6cb78d3d5cfc70d371d3c729258e4064101bc04b457029713a10e09d12a56e" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_valuation_nonzero_exponent_divides_base", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 257, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "a31c3a553106e0a9cc25bf7e3d07f0a425524ca005a5015e5bc6f60d916ef866" }, "bertrand_v11_evidence_bundle_sha256": "5d9ef2e55100b48bf80efa98edae05c0d3eff3a5bf4e099cebf5b09cb44c43c2", "body_checked": true, "body_receipt": { "command_count": 34, "dependency_count": 4, "dne_command_count": 0, "name": "power_valuation_nonzero_exponent_divides_base", "proof_depth": 17, "proof_edges": 38, "proof_nodes": 39, "proof_objects": 39, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "one_le_of_ne_zero", "power_valuation_power_divides", "power_divides_exponent_antitone", "pow_one" ], "dependencies_sha256": "67394c44f8d491b2a0cc0368998ca65c0a527813517076468c4f5c4d8e964d7b", "empty_context_closure": { "body_proof_depth": 17, "body_proof_nodes": 39, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 257, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", 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"original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_nonzero_exponent_divides_base]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "b5afdf046c030d9035a5f1d0c3c27ce93a4eb9d936c0439839e3bdf4d769a014", "membership": "alpha_only", "name": "power_valuation_nonzero_exponent_divides_base", "proof_tag": null, "provenance": [ "bertrand" ], "script": [ "intro p", "intro c", "intro e", "intro hvaluation", "intro hexponent", "have hone : exists bpr_le_gap_bpvnedb_one_bound. bpr_le_gap_bpvnedb_one_bound + (1) = (e)", "specialize one_le_of_ne_zero e", "apply one_le_of_ne_zero", "exact hexponent", "have hselected : exists bpvi_result_bpvnedb_selected. ((exists bpvi_b_bpvnedb_selected_power bpvi_c_bpvnedb_selected_power. ((forall bpvi_i_bpvnedb_selected_power. (exists bpvi_repeat_gap_bpvnedb_selected_power. bpvi_repeat_gap_bpvnedb_selected_power + S bpvi_i_bpvnedb_selected_power = e) -> (((exists bpvi_h_bpvnedb_selected_power_repeat. bpvi_h_bpvnedb_selected_power_repeat + S (p) = S ((S (bpvi_i_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_repeat. bpvi_b_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_repeat * S ((S (bpvi_i_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power) + (p)))) /\\ (exists bpvi_u_bpvnedb_selected_power bpvi_v_bpvnedb_selected_power. ((((exists bpvi_h_bpvnedb_selected_power_start. bpvi_h_bpvnedb_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_start. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_start * S ((S (0)) * bpvi_v_bpvnedb_selected_power) + (1))) /\\ ((((exists bpvi_h_bpvnedb_selected_power_terminal. bpvi_h_bpvnedb_selected_power_terminal + S (bpvi_result_bpvnedb_selected) = S ((S (e)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_terminal. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_terminal * S ((S (e)) * bpvi_v_bpvnedb_selected_power) + (bpvi_result_bpvnedb_selected))) /\\ forall bpvi_j_bpvnedb_selected_power. (exists bpvi_product_gap_bpvnedb_selected_power. bpvi_product_gap_bpvnedb_selected_power + S bpvi_j_bpvnedb_selected_power = e) -> exists bpvi_factor_bpvnedb_selected_power bpvi_partial_bpvnedb_selected_power bpvi_successor_bpvnedb_selected_power. ((((exists bpvi_h_bpvnedb_selected_power_factor. bpvi_h_bpvnedb_selected_power_factor + S (bpvi_factor_bpvnedb_selected_power) = S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_factor. bpvi_b_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_factor * S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power) + (bpvi_factor_bpvnedb_selected_power))) /\\ ((((exists bpvi_h_bpvnedb_selected_power_partial. bpvi_h_bpvnedb_selected_power_partial + S (bpvi_partial_bpvnedb_selected_power) = S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_partial. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_partial * S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power) + (bpvi_partial_bpvnedb_selected_power))) /\\ ((((exists bpvi_h_bpvnedb_selected_power_successor. bpvi_h_bpvnedb_selected_power_successor + S (bpvi_successor_bpvnedb_selected_power) = S ((S (S bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_successor. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_successor * S ((S (S bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power) + (bpvi_successor_bpvnedb_selected_power))) /\\ bpvi_successor_bpvnedb_selected_power = bpvi_partial_bpvnedb_selected_power * bpvi_factor_bpvnedb_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpvnedb_selected. c = bpvi_result_bpvnedb_selected * bpvi_divisor_factor_bpvnedb_selected)", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides c", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation", "have hunit : exists bpvi_result_bpvnedb_unit. ((exists bpvi_b_bpvnedb_unit_power bpvi_c_bpvnedb_unit_power. ((forall bpvi_i_bpvnedb_unit_power. (exists bpvi_repeat_gap_bpvnedb_unit_power. bpvi_repeat_gap_bpvnedb_unit_power + S bpvi_i_bpvnedb_unit_power = 1) -> (((exists bpvi_h_bpvnedb_unit_power_repeat. bpvi_h_bpvnedb_unit_power_repeat + S (p) = S ((S (bpvi_i_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_repeat. bpvi_b_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_repeat * S ((S (bpvi_i_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power) + (p)))) /\\ (exists bpvi_u_bpvnedb_unit_power bpvi_v_bpvnedb_unit_power. ((((exists bpvi_h_bpvnedb_unit_power_start. bpvi_h_bpvnedb_unit_power_start + S (1) = S ((S (0)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_start. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_start * S ((S (0)) * bpvi_v_bpvnedb_unit_power) + (1))) /\\ ((((exists bpvi_h_bpvnedb_unit_power_terminal. bpvi_h_bpvnedb_unit_power_terminal + S (bpvi_result_bpvnedb_unit) = S ((S (1)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_terminal. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_terminal * S ((S (1)) * bpvi_v_bpvnedb_unit_power) + (bpvi_result_bpvnedb_unit))) /\\ forall bpvi_j_bpvnedb_unit_power. (exists bpvi_product_gap_bpvnedb_unit_power. bpvi_product_gap_bpvnedb_unit_power + S bpvi_j_bpvnedb_unit_power = 1) -> exists bpvi_factor_bpvnedb_unit_power bpvi_partial_bpvnedb_unit_power bpvi_successor_bpvnedb_unit_power. ((((exists bpvi_h_bpvnedb_unit_power_factor. bpvi_h_bpvnedb_unit_power_factor + S (bpvi_factor_bpvnedb_unit_power) = S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_factor. bpvi_b_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_factor * S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power) + (bpvi_factor_bpvnedb_unit_power))) /\\ ((((exists bpvi_h_bpvnedb_unit_power_partial. bpvi_h_bpvnedb_unit_power_partial + S (bpvi_partial_bpvnedb_unit_power) = S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_partial. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_partial * S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power) + (bpvi_partial_bpvnedb_unit_power))) /\\ ((((exists bpvi_h_bpvnedb_unit_power_successor. bpvi_h_bpvnedb_unit_power_successor + S (bpvi_successor_bpvnedb_unit_power) = S ((S (S bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_successor. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_successor * S ((S (S bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power) + (bpvi_successor_bpvnedb_unit_power))) /\\ bpvi_successor_bpvnedb_unit_power = bpvi_partial_bpvnedb_unit_power * bpvi_factor_bpvnedb_unit_power)))))))) /\\ exists bpvi_divisor_factor_bpvnedb_unit. c = bpvi_result_bpvnedb_unit * bpvi_divisor_factor_bpvnedb_unit)", "specialize power_divides_exponent_antitone p", "specialize power_divides_exponent_antitone 1", "specialize power_divides_exponent_antitone e", "specialize power_divides_exponent_antitone c", "apply power_divides_exponent_antitone", "exact hone", "exact hselected", "cases hunit", "cases hunit_witness", "have hvalue : x = p", "specialize pow_one p", "specialize pow_one 1", "specialize pow_one x", "apply pow_one", "refl", "exact hunit_witness_left", "rewrite hvalue at hunit_witness_right", "exact hunit_witness_right" ], "script_sha256": "3d3830d1b3938fcf707315a92a3f1f001b80b2fbbcb0ae3684d341f9a6c9f9b2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_prime_support_candidate.py", "sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab" }, "statement": "forall p c e. (((exists bpv_gap_bpvnedb_source_exponent_bound. bpv_gap_bpvnedb_source_exponent_bound + e = c) /\\ (exists bpv_result_bpvnedb_source_selected. ((exists ff_b_bpvnedb_source_selected_power ff_c_bpvnedb_source_selected_power. ((forall ff_i_bpvnedb_source_selected_power_repeat. (exists ff_lt_bpvnedb_source_selected_power_repeat_bound. ff_lt_bpvnedb_source_selected_power_repeat_bound + S ff_i_bpvnedb_source_selected_power_repeat = e) -> (((exists ff_h_bpvnedb_source_selected_power_repeat_decoded. ff_h_bpvnedb_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvnedb_source_selected_power_repeat)) * ff_c_bpvnedb_source_selected_power)) /\\ exists ff_q_bpvnedb_source_selected_power_repeat_decoded. ff_b_bpvnedb_source_selected_power = ff_q_bpvnedb_source_selected_power_repeat_decoded * S ((S (ff_i_bpvnedb_source_selected_power_repeat)) * ff_c_bpvnedb_source_selected_power) + (p)))) /\\ (exists ff_u_bpvnedb_source_selected_power_product ff_v_bpvnedb_source_selected_power_product. ((((exists ff_h_bpvnedb_source_selected_power_product_start. ff_h_bpvnedb_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_start. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_start * S ((S (0)) * ff_v_bpvnedb_source_selected_power_product) + (1))) /\\ ((((exists ff_h_bpvnedb_source_selected_power_product_terminal. ff_h_bpvnedb_source_selected_power_product_terminal + S (bpv_result_bpvnedb_source_selected) = S ((S (e)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_terminal. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_terminal * S ((S (e)) * ff_v_bpvnedb_source_selected_power_product) + (bpv_result_bpvnedb_source_selected))) /\\ forall ff_i_bpvnedb_source_selected_power_product. (exists ff_lt_bpvnedb_source_selected_power_product_bound. ff_lt_bpvnedb_source_selected_power_product_bound + S ff_i_bpvnedb_source_selected_power_product = e) -> exists ff_p_bpvnedb_source_selected_power_product ff_r_bpvnedb_source_selected_power_product ff_s_bpvnedb_source_selected_power_product. ((((exists ff_h_bpvnedb_source_selected_power_product_factor. ff_h_bpvnedb_source_selected_power_product_factor + S (ff_p_bpvnedb_source_selected_power_product) = S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_c_bpvnedb_source_selected_power)) /\\ exists ff_q_bpvnedb_source_selected_power_product_factor. ff_b_bpvnedb_source_selected_power = ff_q_bpvnedb_source_selected_power_product_factor * S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_c_bpvnedb_source_selected_power) + (ff_p_bpvnedb_source_selected_power_product))) /\\ ((((exists ff_h_bpvnedb_source_selected_power_product_partial. ff_h_bpvnedb_source_selected_power_product_partial + S (ff_r_bpvnedb_source_selected_power_product) = S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_partial. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_partial * S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product) + (ff_r_bpvnedb_source_selected_power_product))) /\\ ((((exists ff_h_bpvnedb_source_selected_power_product_successor. ff_h_bpvnedb_source_selected_power_product_successor + S (ff_s_bpvnedb_source_selected_power_product) = S ((S (S ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_successor. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_successor * S ((S (S ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product) + (ff_s_bpvnedb_source_selected_power_product))) /\\ ff_s_bpvnedb_source_selected_power_product = ff_r_bpvnedb_source_selected_power_product * ff_p_bpvnedb_source_selected_power_product)))))))) /\\ (exists bpv_factor_bpvnedb_source_selected_divides. c = bpv_result_bpvnedb_source_selected * bpv_factor_bpvnedb_source_selected_divides)))) /\\ forall bpv_candidate_bpvnedb_source. (exists bpv_gap_bpvnedb_source_candidate_bound. bpv_gap_bpvnedb_source_candidate_bound + bpv_candidate_bpvnedb_source = c) -> (exists bpv_result_bpvnedb_source_candidate. ((exists ff_b_bpvnedb_source_candidate_power ff_c_bpvnedb_source_candidate_power. ((forall ff_i_bpvnedb_source_candidate_power_repeat. (exists ff_lt_bpvnedb_source_candidate_power_repeat_bound. ff_lt_bpvnedb_source_candidate_power_repeat_bound + S ff_i_bpvnedb_source_candidate_power_repeat = bpv_candidate_bpvnedb_source) -> (((exists ff_h_bpvnedb_source_candidate_power_repeat_decoded. ff_h_bpvnedb_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvnedb_source_candidate_power_repeat)) * ff_c_bpvnedb_source_candidate_power)) /\\ exists ff_q_bpvnedb_source_candidate_power_repeat_decoded. ff_b_bpvnedb_source_candidate_power = ff_q_bpvnedb_source_candidate_power_repeat_decoded * S ((S (ff_i_bpvnedb_source_candidate_power_repeat)) * ff_c_bpvnedb_source_candidate_power) + (p)))) /\\ (exists ff_u_bpvnedb_source_candidate_power_product ff_v_bpvnedb_source_candidate_power_product. ((((exists ff_h_bpvnedb_source_candidate_power_product_start. ff_h_bpvnedb_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_start. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_start * S ((S (0)) * ff_v_bpvnedb_source_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpvnedb_source_candidate_power_product_terminal. ff_h_bpvnedb_source_candidate_power_product_terminal + S (bpv_result_bpvnedb_source_candidate) = S ((S (bpv_candidate_bpvnedb_source)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_terminal. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_terminal * S ((S (bpv_candidate_bpvnedb_source)) * ff_v_bpvnedb_source_candidate_power_product) + (bpv_result_bpvnedb_source_candidate))) /\\ forall ff_i_bpvnedb_source_candidate_power_product. (exists ff_lt_bpvnedb_source_candidate_power_product_bound. ff_lt_bpvnedb_source_candidate_power_product_bound + S ff_i_bpvnedb_source_candidate_power_product = bpv_candidate_bpvnedb_source) -> exists ff_p_bpvnedb_source_candidate_power_product ff_r_bpvnedb_source_candidate_power_product ff_s_bpvnedb_source_candidate_power_product. ((((exists ff_h_bpvnedb_source_candidate_power_product_factor. ff_h_bpvnedb_source_candidate_power_product_factor + S (ff_p_bpvnedb_source_candidate_power_product) = S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_c_bpvnedb_source_candidate_power)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_factor. ff_b_bpvnedb_source_candidate_power = ff_q_bpvnedb_source_candidate_power_product_factor * S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_c_bpvnedb_source_candidate_power) + (ff_p_bpvnedb_source_candidate_power_product))) /\\ ((((exists ff_h_bpvnedb_source_candidate_power_product_partial. ff_h_bpvnedb_source_candidate_power_product_partial + S (ff_r_bpvnedb_source_candidate_power_product) = S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_partial. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_partial * S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product) + (ff_r_bpvnedb_source_candidate_power_product))) /\\ ((((exists ff_h_bpvnedb_source_candidate_power_product_successor. ff_h_bpvnedb_source_candidate_power_product_successor + S (ff_s_bpvnedb_source_candidate_power_product) = S ((S (S ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_successor. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_successor * S ((S (S ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product) + (ff_s_bpvnedb_source_candidate_power_product))) /\\ ff_s_bpvnedb_source_candidate_power_product = ff_r_bpvnedb_source_candidate_power_product * ff_p_bpvnedb_source_candidate_power_product)))))))) /\\ (exists bpv_factor_bpvnedb_source_candidate_divides. c = bpv_result_bpvnedb_source_candidate * bpv_factor_bpvnedb_source_candidate_divides))) -> (exists bpv_gap_bpvnedb_source_maximal. bpv_gap_bpvnedb_source_maximal + bpv_candidate_bpvnedb_source = e)) -> ~(e = 0) -> (exists bpr_quotient_bpvnedb_result. c = (p) * bpr_quotient_bpvnedb_result)", "statement_sha256": "22a1f82a7ec4d135267f9db5347c627faa73621d5dd3471895a3f824640c22a3", "summary": "A nonzero valuation exponent exposes the base as a divisor.", "summary_sha256": "9646d7324f14022dd921c9d8c0936f9587fef7754df9c4490ecd487304c798af" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "one_le_of_ne_zero", "power_valuation_power_divides", "power_divides_exponent_antitone", "pow_one" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab", "kind": "bertrand_dependency_curried_body", "path": 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"49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=257]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_nonzero_exponent_divides_base]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_nonzero_exponent_divides_base", "parent_alpha_version": "v34", "proof_bundle_node_id": 222, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_nonzero_exponent_divides_base", "script": [ "intro p", "intro c", "intro e", "intro hvaluation", "intro hexponent", "have hone : exists bpr_le_gap_bpvnedb_one_bound. bpr_le_gap_bpvnedb_one_bound + (1) = (e)", "specialize one_le_of_ne_zero e", "apply one_le_of_ne_zero", "exact hexponent", "have hselected : exists bpvi_result_bpvnedb_selected. ((exists bpvi_b_bpvnedb_selected_power bpvi_c_bpvnedb_selected_power. ((forall bpvi_i_bpvnedb_selected_power. (exists bpvi_repeat_gap_bpvnedb_selected_power. bpvi_repeat_gap_bpvnedb_selected_power + S bpvi_i_bpvnedb_selected_power = e) -> (((exists bpvi_h_bpvnedb_selected_power_repeat. bpvi_h_bpvnedb_selected_power_repeat + S (p) = S ((S (bpvi_i_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_repeat. bpvi_b_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_repeat * S ((S (bpvi_i_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power) + (p)))) /\\ (exists bpvi_u_bpvnedb_selected_power bpvi_v_bpvnedb_selected_power. ((((exists bpvi_h_bpvnedb_selected_power_start. bpvi_h_bpvnedb_selected_power_start + S (1) = S ((S (0)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_start. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_start * S ((S (0)) * bpvi_v_bpvnedb_selected_power) + (1))) /\\ ((((exists bpvi_h_bpvnedb_selected_power_terminal. bpvi_h_bpvnedb_selected_power_terminal + S (bpvi_result_bpvnedb_selected) = S ((S (e)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_terminal. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_terminal * S ((S (e)) * bpvi_v_bpvnedb_selected_power) + (bpvi_result_bpvnedb_selected))) /\\ forall bpvi_j_bpvnedb_selected_power. (exists bpvi_product_gap_bpvnedb_selected_power. bpvi_product_gap_bpvnedb_selected_power + S bpvi_j_bpvnedb_selected_power = e) -> exists bpvi_factor_bpvnedb_selected_power bpvi_partial_bpvnedb_selected_power bpvi_successor_bpvnedb_selected_power. ((((exists bpvi_h_bpvnedb_selected_power_factor. bpvi_h_bpvnedb_selected_power_factor + S (bpvi_factor_bpvnedb_selected_power) = S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_factor. bpvi_b_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_factor * S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_c_bpvnedb_selected_power) + (bpvi_factor_bpvnedb_selected_power))) /\\ ((((exists bpvi_h_bpvnedb_selected_power_partial. bpvi_h_bpvnedb_selected_power_partial + S (bpvi_partial_bpvnedb_selected_power) = S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_partial. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_partial * S ((S (bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power) + (bpvi_partial_bpvnedb_selected_power))) /\\ ((((exists bpvi_h_bpvnedb_selected_power_successor. bpvi_h_bpvnedb_selected_power_successor + S (bpvi_successor_bpvnedb_selected_power) = S ((S (S bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power)) /\\ exists bpvi_q_bpvnedb_selected_power_successor. bpvi_u_bpvnedb_selected_power = bpvi_q_bpvnedb_selected_power_successor * S ((S (S bpvi_j_bpvnedb_selected_power)) * bpvi_v_bpvnedb_selected_power) + (bpvi_successor_bpvnedb_selected_power))) /\\ bpvi_successor_bpvnedb_selected_power = bpvi_partial_bpvnedb_selected_power * bpvi_factor_bpvnedb_selected_power)))))))) /\\ exists bpvi_divisor_factor_bpvnedb_selected. c = bpvi_result_bpvnedb_selected * bpvi_divisor_factor_bpvnedb_selected)", "specialize power_valuation_power_divides p", "specialize power_valuation_power_divides c", "specialize power_valuation_power_divides e", "apply power_valuation_power_divides", "exact hvaluation", "have hunit : exists bpvi_result_bpvnedb_unit. ((exists bpvi_b_bpvnedb_unit_power bpvi_c_bpvnedb_unit_power. ((forall bpvi_i_bpvnedb_unit_power. (exists bpvi_repeat_gap_bpvnedb_unit_power. bpvi_repeat_gap_bpvnedb_unit_power + S bpvi_i_bpvnedb_unit_power = 1) -> (((exists bpvi_h_bpvnedb_unit_power_repeat. bpvi_h_bpvnedb_unit_power_repeat + S (p) = S ((S (bpvi_i_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_repeat. bpvi_b_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_repeat * S ((S (bpvi_i_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power) + (p)))) /\\ (exists bpvi_u_bpvnedb_unit_power bpvi_v_bpvnedb_unit_power. ((((exists bpvi_h_bpvnedb_unit_power_start. bpvi_h_bpvnedb_unit_power_start + S (1) = S ((S (0)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_start. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_start * S ((S (0)) * bpvi_v_bpvnedb_unit_power) + (1))) /\\ ((((exists bpvi_h_bpvnedb_unit_power_terminal. bpvi_h_bpvnedb_unit_power_terminal + S (bpvi_result_bpvnedb_unit) = S ((S (1)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_terminal. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_terminal * S ((S (1)) * bpvi_v_bpvnedb_unit_power) + (bpvi_result_bpvnedb_unit))) /\\ forall bpvi_j_bpvnedb_unit_power. (exists bpvi_product_gap_bpvnedb_unit_power. bpvi_product_gap_bpvnedb_unit_power + S bpvi_j_bpvnedb_unit_power = 1) -> exists bpvi_factor_bpvnedb_unit_power bpvi_partial_bpvnedb_unit_power bpvi_successor_bpvnedb_unit_power. ((((exists bpvi_h_bpvnedb_unit_power_factor. bpvi_h_bpvnedb_unit_power_factor + S (bpvi_factor_bpvnedb_unit_power) = S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_factor. bpvi_b_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_factor * S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_c_bpvnedb_unit_power) + (bpvi_factor_bpvnedb_unit_power))) /\\ ((((exists bpvi_h_bpvnedb_unit_power_partial. bpvi_h_bpvnedb_unit_power_partial + S (bpvi_partial_bpvnedb_unit_power) = S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_partial. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_partial * S ((S (bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power) + (bpvi_partial_bpvnedb_unit_power))) /\\ ((((exists bpvi_h_bpvnedb_unit_power_successor. bpvi_h_bpvnedb_unit_power_successor + S (bpvi_successor_bpvnedb_unit_power) = S ((S (S bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power)) /\\ exists bpvi_q_bpvnedb_unit_power_successor. bpvi_u_bpvnedb_unit_power = bpvi_q_bpvnedb_unit_power_successor * S ((S (S bpvi_j_bpvnedb_unit_power)) * bpvi_v_bpvnedb_unit_power) + (bpvi_successor_bpvnedb_unit_power))) /\\ bpvi_successor_bpvnedb_unit_power = bpvi_partial_bpvnedb_unit_power * bpvi_factor_bpvnedb_unit_power)))))))) /\\ exists bpvi_divisor_factor_bpvnedb_unit. c = bpvi_result_bpvnedb_unit * bpvi_divisor_factor_bpvnedb_unit)", "specialize power_divides_exponent_antitone p", "specialize power_divides_exponent_antitone 1", "specialize power_divides_exponent_antitone e", "specialize power_divides_exponent_antitone c", "apply power_divides_exponent_antitone", "exact hone", "exact hselected", "cases hunit", "cases hunit_witness", "have hvalue : x = p", "specialize pow_one p", "specialize pow_one 1", "specialize pow_one x", "apply pow_one", "refl", "exact hunit_witness_left", "rewrite hvalue at hunit_witness_right", "exact hunit_witness_right" ], "script_sha256": "3d3830d1b3938fcf707315a92a3f1f001b80b2fbbcb0ae3684d341f9a6c9f9b2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_prime_support_candidate.py", "sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab" }, "stable_member": false, "statement": "forall p c e. (((exists bpv_gap_bpvnedb_source_exponent_bound. bpv_gap_bpvnedb_source_exponent_bound + e = c) /\\ (exists bpv_result_bpvnedb_source_selected. ((exists ff_b_bpvnedb_source_selected_power ff_c_bpvnedb_source_selected_power. ((forall ff_i_bpvnedb_source_selected_power_repeat. (exists ff_lt_bpvnedb_source_selected_power_repeat_bound. ff_lt_bpvnedb_source_selected_power_repeat_bound + S ff_i_bpvnedb_source_selected_power_repeat = e) -> (((exists ff_h_bpvnedb_source_selected_power_repeat_decoded. ff_h_bpvnedb_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpvnedb_source_selected_power_repeat)) * ff_c_bpvnedb_source_selected_power)) /\\ exists ff_q_bpvnedb_source_selected_power_repeat_decoded. ff_b_bpvnedb_source_selected_power = ff_q_bpvnedb_source_selected_power_repeat_decoded * S ((S (ff_i_bpvnedb_source_selected_power_repeat)) * ff_c_bpvnedb_source_selected_power) + (p)))) /\\ (exists ff_u_bpvnedb_source_selected_power_product ff_v_bpvnedb_source_selected_power_product. ((((exists ff_h_bpvnedb_source_selected_power_product_start. ff_h_bpvnedb_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_start. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_start * S ((S (0)) * ff_v_bpvnedb_source_selected_power_product) + (1))) /\\ ((((exists ff_h_bpvnedb_source_selected_power_product_terminal. ff_h_bpvnedb_source_selected_power_product_terminal + S (bpv_result_bpvnedb_source_selected) = S ((S (e)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_terminal. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_terminal * S ((S (e)) * ff_v_bpvnedb_source_selected_power_product) + (bpv_result_bpvnedb_source_selected))) /\\ forall ff_i_bpvnedb_source_selected_power_product. (exists ff_lt_bpvnedb_source_selected_power_product_bound. ff_lt_bpvnedb_source_selected_power_product_bound + S ff_i_bpvnedb_source_selected_power_product = e) -> exists ff_p_bpvnedb_source_selected_power_product ff_r_bpvnedb_source_selected_power_product ff_s_bpvnedb_source_selected_power_product. ((((exists ff_h_bpvnedb_source_selected_power_product_factor. ff_h_bpvnedb_source_selected_power_product_factor + S (ff_p_bpvnedb_source_selected_power_product) = S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_c_bpvnedb_source_selected_power)) /\\ exists ff_q_bpvnedb_source_selected_power_product_factor. ff_b_bpvnedb_source_selected_power = ff_q_bpvnedb_source_selected_power_product_factor * S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_c_bpvnedb_source_selected_power) + (ff_p_bpvnedb_source_selected_power_product))) /\\ ((((exists ff_h_bpvnedb_source_selected_power_product_partial. ff_h_bpvnedb_source_selected_power_product_partial + S (ff_r_bpvnedb_source_selected_power_product) = S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_partial. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_partial * S ((S (ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product) + (ff_r_bpvnedb_source_selected_power_product))) /\\ ((((exists ff_h_bpvnedb_source_selected_power_product_successor. ff_h_bpvnedb_source_selected_power_product_successor + S (ff_s_bpvnedb_source_selected_power_product) = S ((S (S ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product)) /\\ exists ff_q_bpvnedb_source_selected_power_product_successor. ff_u_bpvnedb_source_selected_power_product = ff_q_bpvnedb_source_selected_power_product_successor * S ((S (S ff_i_bpvnedb_source_selected_power_product)) * ff_v_bpvnedb_source_selected_power_product) + (ff_s_bpvnedb_source_selected_power_product))) /\\ ff_s_bpvnedb_source_selected_power_product = ff_r_bpvnedb_source_selected_power_product * ff_p_bpvnedb_source_selected_power_product)))))))) /\\ (exists bpv_factor_bpvnedb_source_selected_divides. c = bpv_result_bpvnedb_source_selected * bpv_factor_bpvnedb_source_selected_divides)))) /\\ forall bpv_candidate_bpvnedb_source. (exists bpv_gap_bpvnedb_source_candidate_bound. bpv_gap_bpvnedb_source_candidate_bound + bpv_candidate_bpvnedb_source = c) -> (exists bpv_result_bpvnedb_source_candidate. ((exists ff_b_bpvnedb_source_candidate_power ff_c_bpvnedb_source_candidate_power. ((forall ff_i_bpvnedb_source_candidate_power_repeat. (exists ff_lt_bpvnedb_source_candidate_power_repeat_bound. ff_lt_bpvnedb_source_candidate_power_repeat_bound + S ff_i_bpvnedb_source_candidate_power_repeat = bpv_candidate_bpvnedb_source) -> (((exists ff_h_bpvnedb_source_candidate_power_repeat_decoded. ff_h_bpvnedb_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpvnedb_source_candidate_power_repeat)) * ff_c_bpvnedb_source_candidate_power)) /\\ exists ff_q_bpvnedb_source_candidate_power_repeat_decoded. ff_b_bpvnedb_source_candidate_power = ff_q_bpvnedb_source_candidate_power_repeat_decoded * S ((S (ff_i_bpvnedb_source_candidate_power_repeat)) * ff_c_bpvnedb_source_candidate_power) + (p)))) /\\ (exists ff_u_bpvnedb_source_candidate_power_product ff_v_bpvnedb_source_candidate_power_product. ((((exists ff_h_bpvnedb_source_candidate_power_product_start. ff_h_bpvnedb_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_start. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_start * S ((S (0)) * ff_v_bpvnedb_source_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpvnedb_source_candidate_power_product_terminal. ff_h_bpvnedb_source_candidate_power_product_terminal + S (bpv_result_bpvnedb_source_candidate) = S ((S (bpv_candidate_bpvnedb_source)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_terminal. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_terminal * S ((S (bpv_candidate_bpvnedb_source)) * ff_v_bpvnedb_source_candidate_power_product) + (bpv_result_bpvnedb_source_candidate))) /\\ forall ff_i_bpvnedb_source_candidate_power_product. (exists ff_lt_bpvnedb_source_candidate_power_product_bound. ff_lt_bpvnedb_source_candidate_power_product_bound + S ff_i_bpvnedb_source_candidate_power_product = bpv_candidate_bpvnedb_source) -> exists ff_p_bpvnedb_source_candidate_power_product ff_r_bpvnedb_source_candidate_power_product ff_s_bpvnedb_source_candidate_power_product. ((((exists ff_h_bpvnedb_source_candidate_power_product_factor. ff_h_bpvnedb_source_candidate_power_product_factor + S (ff_p_bpvnedb_source_candidate_power_product) = S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_c_bpvnedb_source_candidate_power)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_factor. ff_b_bpvnedb_source_candidate_power = ff_q_bpvnedb_source_candidate_power_product_factor * S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_c_bpvnedb_source_candidate_power) + (ff_p_bpvnedb_source_candidate_power_product))) /\\ ((((exists ff_h_bpvnedb_source_candidate_power_product_partial. ff_h_bpvnedb_source_candidate_power_product_partial + S (ff_r_bpvnedb_source_candidate_power_product) = S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_partial. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_partial * S ((S (ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product) + (ff_r_bpvnedb_source_candidate_power_product))) /\\ ((((exists ff_h_bpvnedb_source_candidate_power_product_successor. ff_h_bpvnedb_source_candidate_power_product_successor + S (ff_s_bpvnedb_source_candidate_power_product) = S ((S (S ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product)) /\\ exists ff_q_bpvnedb_source_candidate_power_product_successor. ff_u_bpvnedb_source_candidate_power_product = ff_q_bpvnedb_source_candidate_power_product_successor * S ((S (S ff_i_bpvnedb_source_candidate_power_product)) * ff_v_bpvnedb_source_candidate_power_product) + (ff_s_bpvnedb_source_candidate_power_product))) /\\ ff_s_bpvnedb_source_candidate_power_product = ff_r_bpvnedb_source_candidate_power_product * ff_p_bpvnedb_source_candidate_power_product)))))))) /\\ (exists bpv_factor_bpvnedb_source_candidate_divides. c = bpv_result_bpvnedb_source_candidate * bpv_factor_bpvnedb_source_candidate_divides))) -> (exists bpv_gap_bpvnedb_source_maximal. bpv_gap_bpvnedb_source_maximal + bpv_candidate_bpvnedb_source = e)) -> ~(e = 0) -> (exists bpr_quotient_bpvnedb_result. c = (p) * bpr_quotient_bpvnedb_result)", "statement_sha256": "22a1f82a7ec4d135267f9db5347c627faa73621d5dd3471895a3f824640c22a3" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_divisor_power_valuation_nonzero", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 258, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "a1ff9d346a85d3464c289c09801cf6f2ffff8554e47ecc506300838ec7dd0b0e" }, "bertrand_v11_evidence_bundle_sha256": "caf94160f87f7372ef4d3a23327cd689a81ee1da422fca7a7a29abcae82345a6", "body_checked": true, "body_receipt": { "command_count": 43, "dependency_count": 4, "dne_command_count": 0, "name": "prime_divisor_power_valuation_nonzero", "proof_depth": 27, "proof_edges": 49, "proof_nodes": 50, "proof_objects": 50, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "pow_exists", "pow_one", "prime_power_divides_exponent_le_valuation", "le_zero" ], "dependencies_sha256": "ef31942ec31fae06a89ade1d4b99207143d62dde8c789397409d07e7da49a422", "empty_context_closure": { "body_proof_depth": 27, "body_proof_nodes": 50, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 258, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "8000af01e004415d265e86042a506b3c8a0cea2554309eadd883a00b6295ebbe", "status": "checked" }, "enrollment_index": 1121, "enrollment_origin": "bertrand", "evidence_links": [ { "document_sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_prime_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "774339f3ba92f9d50d945b3cb553adee649ec36d8110df5b818fc6b6f2f625c7", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_central_binom_prime_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "709a4ad357529d7f41ec086db1fd27fc9e4277f1ed0680532a9cb20d1ad02de9", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-central-prime-support-tranche-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "46bd50c19b694470542f53f1ef7f61d1ee8fab1f08ad5573ca3534da29053dc3", "kind": "sealed_alpha_v10_parent", "path": "artifacts/peano-library/alpha/catalog-v10.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=258]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=prime_divisor_power_valuation_nonzero]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "db6c68e6c6db4d6316b79fd9ad32cfcbae952e06a16bd821ce076f9cdaa96aa2", "membership": "alpha_only", "name": "prime_divisor_power_valuation_nonzero", "proof_tag": null, "provenance": [ "bertrand" ], "script": [ "intro p", "intro c", "intro e", "intro hp", "intro hc", "intro hvaluation", "intro hdivides", "have hpower : exists x. (exists bpvi_b_bpdvpn_unit_power bpvi_c_bpdvpn_unit_power. ((forall bpvi_i_bpdvpn_unit_power. (exists bpvi_repeat_gap_bpdvpn_unit_power. bpvi_repeat_gap_bpdvpn_unit_power + S bpvi_i_bpdvpn_unit_power = 1) -> (((exists bpvi_h_bpdvpn_unit_power_repeat. bpvi_h_bpdvpn_unit_power_repeat + S (p) = S ((S (bpvi_i_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_repeat. bpvi_b_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_repeat * S ((S (bpvi_i_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power) + (p)))) /\\ (exists bpvi_u_bpdvpn_unit_power bpvi_v_bpdvpn_unit_power. ((((exists bpvi_h_bpdvpn_unit_power_start. bpvi_h_bpdvpn_unit_power_start + S (1) = S ((S (0)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_start. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_start * S ((S (0)) * bpvi_v_bpdvpn_unit_power) + (1))) /\\ ((((exists bpvi_h_bpdvpn_unit_power_terminal. bpvi_h_bpdvpn_unit_power_terminal + S (x) = S ((S (1)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_terminal. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_terminal * S ((S (1)) * bpvi_v_bpdvpn_unit_power) + (x))) /\\ forall bpvi_j_bpdvpn_unit_power. (exists bpvi_product_gap_bpdvpn_unit_power. bpvi_product_gap_bpdvpn_unit_power + S bpvi_j_bpdvpn_unit_power = 1) -> exists bpvi_factor_bpdvpn_unit_power bpvi_partial_bpdvpn_unit_power bpvi_successor_bpdvpn_unit_power. ((((exists bpvi_h_bpdvpn_unit_power_factor. bpvi_h_bpdvpn_unit_power_factor + S (bpvi_factor_bpdvpn_unit_power) = S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_factor. bpvi_b_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_factor * S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power) + (bpvi_factor_bpdvpn_unit_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_power_partial. bpvi_h_bpdvpn_unit_power_partial + S (bpvi_partial_bpdvpn_unit_power) = S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_partial. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_partial * S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power) + (bpvi_partial_bpdvpn_unit_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_power_successor. bpvi_h_bpdvpn_unit_power_successor + S (bpvi_successor_bpdvpn_unit_power) = S ((S (S bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_successor. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_successor * S ((S (S bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power) + (bpvi_successor_bpdvpn_unit_power))) /\\ bpvi_successor_bpdvpn_unit_power = bpvi_partial_bpdvpn_unit_power * bpvi_factor_bpdvpn_unit_power))))))))", "specialize pow_exists p", "specialize pow_exists 1", "exact pow_exists", "cases hpower", "have hvalue : x = p", "specialize pow_one p", "specialize pow_one 1", "specialize pow_one x", "apply pow_one", "refl", "exact hpower_witness", "have hunit : exists bpvi_result_bpdvpn_unit_divides. ((exists bpvi_b_bpdvpn_unit_divides_power bpvi_c_bpdvpn_unit_divides_power. ((forall bpvi_i_bpdvpn_unit_divides_power. (exists bpvi_repeat_gap_bpdvpn_unit_divides_power. bpvi_repeat_gap_bpdvpn_unit_divides_power + S bpvi_i_bpdvpn_unit_divides_power = 1) -> (((exists bpvi_h_bpdvpn_unit_divides_power_repeat. bpvi_h_bpdvpn_unit_divides_power_repeat + S (p) = S ((S (bpvi_i_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_repeat. bpvi_b_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_repeat * S ((S (bpvi_i_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power) + (p)))) /\\ (exists bpvi_u_bpdvpn_unit_divides_power bpvi_v_bpdvpn_unit_divides_power. ((((exists bpvi_h_bpdvpn_unit_divides_power_start. bpvi_h_bpdvpn_unit_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_start. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_start * S ((S (0)) * bpvi_v_bpdvpn_unit_divides_power) + (1))) /\\ ((((exists bpvi_h_bpdvpn_unit_divides_power_terminal. bpvi_h_bpdvpn_unit_divides_power_terminal + S (bpvi_result_bpdvpn_unit_divides) = S ((S (1)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_terminal. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_terminal * S ((S (1)) * bpvi_v_bpdvpn_unit_divides_power) + (bpvi_result_bpdvpn_unit_divides))) /\\ forall bpvi_j_bpdvpn_unit_divides_power. (exists bpvi_product_gap_bpdvpn_unit_divides_power. bpvi_product_gap_bpdvpn_unit_divides_power + S bpvi_j_bpdvpn_unit_divides_power = 1) -> exists bpvi_factor_bpdvpn_unit_divides_power bpvi_partial_bpdvpn_unit_divides_power bpvi_successor_bpdvpn_unit_divides_power. ((((exists bpvi_h_bpdvpn_unit_divides_power_factor. bpvi_h_bpdvpn_unit_divides_power_factor + S (bpvi_factor_bpdvpn_unit_divides_power) = S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_factor. bpvi_b_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_factor * S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power) + (bpvi_factor_bpdvpn_unit_divides_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_divides_power_partial. bpvi_h_bpdvpn_unit_divides_power_partial + S (bpvi_partial_bpdvpn_unit_divides_power) = S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_partial. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_partial * S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power) + (bpvi_partial_bpdvpn_unit_divides_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_divides_power_successor. bpvi_h_bpdvpn_unit_divides_power_successor + S (bpvi_successor_bpdvpn_unit_divides_power) = S ((S (S bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_successor. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_successor * S ((S (S bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power) + (bpvi_successor_bpdvpn_unit_divides_power))) /\\ bpvi_successor_bpdvpn_unit_divides_power = bpvi_partial_bpdvpn_unit_divides_power * bpvi_factor_bpdvpn_unit_divides_power)))))))) /\\ exists bpvi_divisor_factor_bpdvpn_unit_divides. c = bpvi_result_bpdvpn_unit_divides * bpvi_divisor_factor_bpdvpn_unit_divides)", "exists x", "split", "exact hpower_witness", "rewrite hvalue", "exact hdivides", "have hbound : exists bpr_le_gap_bpdvpn_bound. bpr_le_gap_bpdvpn_bound + (1) = (e)", "specialize prime_power_divides_exponent_le_valuation p", "specialize prime_power_divides_exponent_le_valuation c", "specialize prime_power_divides_exponent_le_valuation e", "specialize prime_power_divides_exponent_le_valuation 1", "apply prime_power_divides_exponent_le_valuation", "exact hp", "exact hc", "exact hvaluation", "exact hunit", "intro heq", "rewrite heq at hbound", "have hone_zero : 1 = 0", "specialize le_zero 1", "apply le_zero", "exact hbound", "apply PA1", "exact hone_zero" ], "script_sha256": "89a23d06682f85d78c49249be2402ff4652e3b50b0fe69083ac1ca05a8fb2a5a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_prime_support_candidate.py", "sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab" }, "statement": "forall p c e. ((~(p = 1) /\\ forall bpr_left_bpdvpn_prime bpr_right_bpdvpn_prime. p = bpr_left_bpdvpn_prime * bpr_right_bpdvpn_prime -> bpr_left_bpdvpn_prime = 1 \\/ bpr_right_bpdvpn_prime = 1)) -> ~(c = 0) -> (((exists bpv_gap_bpdvpn_source_exponent_bound. bpv_gap_bpdvpn_source_exponent_bound + e = c) /\\ (exists bpv_result_bpdvpn_source_selected. ((exists ff_b_bpdvpn_source_selected_power ff_c_bpdvpn_source_selected_power. ((forall ff_i_bpdvpn_source_selected_power_repeat. (exists ff_lt_bpdvpn_source_selected_power_repeat_bound. ff_lt_bpdvpn_source_selected_power_repeat_bound + S ff_i_bpdvpn_source_selected_power_repeat = e) -> (((exists ff_h_bpdvpn_source_selected_power_repeat_decoded. ff_h_bpdvpn_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpdvpn_source_selected_power_repeat)) * ff_c_bpdvpn_source_selected_power)) /\\ exists ff_q_bpdvpn_source_selected_power_repeat_decoded. ff_b_bpdvpn_source_selected_power = ff_q_bpdvpn_source_selected_power_repeat_decoded * S ((S (ff_i_bpdvpn_source_selected_power_repeat)) * ff_c_bpdvpn_source_selected_power) + (p)))) /\\ (exists ff_u_bpdvpn_source_selected_power_product ff_v_bpdvpn_source_selected_power_product. ((((exists ff_h_bpdvpn_source_selected_power_product_start. ff_h_bpdvpn_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_start. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_start * S ((S (0)) * ff_v_bpdvpn_source_selected_power_product) + (1))) /\\ ((((exists ff_h_bpdvpn_source_selected_power_product_terminal. ff_h_bpdvpn_source_selected_power_product_terminal + S (bpv_result_bpdvpn_source_selected) = S ((S (e)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_terminal. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_terminal * S ((S (e)) * ff_v_bpdvpn_source_selected_power_product) + (bpv_result_bpdvpn_source_selected))) /\\ forall ff_i_bpdvpn_source_selected_power_product. (exists ff_lt_bpdvpn_source_selected_power_product_bound. ff_lt_bpdvpn_source_selected_power_product_bound + S ff_i_bpdvpn_source_selected_power_product = e) -> exists ff_p_bpdvpn_source_selected_power_product ff_r_bpdvpn_source_selected_power_product ff_s_bpdvpn_source_selected_power_product. ((((exists ff_h_bpdvpn_source_selected_power_product_factor. ff_h_bpdvpn_source_selected_power_product_factor + S (ff_p_bpdvpn_source_selected_power_product) = S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_c_bpdvpn_source_selected_power)) /\\ exists ff_q_bpdvpn_source_selected_power_product_factor. ff_b_bpdvpn_source_selected_power = ff_q_bpdvpn_source_selected_power_product_factor * S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_c_bpdvpn_source_selected_power) + (ff_p_bpdvpn_source_selected_power_product))) /\\ ((((exists ff_h_bpdvpn_source_selected_power_product_partial. ff_h_bpdvpn_source_selected_power_product_partial + S (ff_r_bpdvpn_source_selected_power_product) = S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_partial. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_partial * S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product) + (ff_r_bpdvpn_source_selected_power_product))) /\\ ((((exists ff_h_bpdvpn_source_selected_power_product_successor. ff_h_bpdvpn_source_selected_power_product_successor + S (ff_s_bpdvpn_source_selected_power_product) = S ((S (S ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_successor. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_successor * S ((S (S ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product) + (ff_s_bpdvpn_source_selected_power_product))) /\\ ff_s_bpdvpn_source_selected_power_product = ff_r_bpdvpn_source_selected_power_product * ff_p_bpdvpn_source_selected_power_product)))))))) /\\ (exists bpv_factor_bpdvpn_source_selected_divides. c = bpv_result_bpdvpn_source_selected * bpv_factor_bpdvpn_source_selected_divides)))) /\\ forall bpv_candidate_bpdvpn_source. (exists bpv_gap_bpdvpn_source_candidate_bound. bpv_gap_bpdvpn_source_candidate_bound + bpv_candidate_bpdvpn_source = c) -> (exists bpv_result_bpdvpn_source_candidate. ((exists ff_b_bpdvpn_source_candidate_power ff_c_bpdvpn_source_candidate_power. ((forall ff_i_bpdvpn_source_candidate_power_repeat. (exists ff_lt_bpdvpn_source_candidate_power_repeat_bound. ff_lt_bpdvpn_source_candidate_power_repeat_bound + S ff_i_bpdvpn_source_candidate_power_repeat = bpv_candidate_bpdvpn_source) -> (((exists ff_h_bpdvpn_source_candidate_power_repeat_decoded. ff_h_bpdvpn_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpdvpn_source_candidate_power_repeat)) * ff_c_bpdvpn_source_candidate_power)) /\\ exists ff_q_bpdvpn_source_candidate_power_repeat_decoded. ff_b_bpdvpn_source_candidate_power = ff_q_bpdvpn_source_candidate_power_repeat_decoded * S ((S (ff_i_bpdvpn_source_candidate_power_repeat)) * ff_c_bpdvpn_source_candidate_power) + (p)))) /\\ (exists ff_u_bpdvpn_source_candidate_power_product ff_v_bpdvpn_source_candidate_power_product. ((((exists ff_h_bpdvpn_source_candidate_power_product_start. ff_h_bpdvpn_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_start. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_start * S ((S (0)) * ff_v_bpdvpn_source_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpdvpn_source_candidate_power_product_terminal. ff_h_bpdvpn_source_candidate_power_product_terminal + S (bpv_result_bpdvpn_source_candidate) = S ((S (bpv_candidate_bpdvpn_source)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_terminal. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_terminal * S ((S (bpv_candidate_bpdvpn_source)) * ff_v_bpdvpn_source_candidate_power_product) + (bpv_result_bpdvpn_source_candidate))) /\\ forall ff_i_bpdvpn_source_candidate_power_product. (exists ff_lt_bpdvpn_source_candidate_power_product_bound. ff_lt_bpdvpn_source_candidate_power_product_bound + S ff_i_bpdvpn_source_candidate_power_product = bpv_candidate_bpdvpn_source) -> exists ff_p_bpdvpn_source_candidate_power_product ff_r_bpdvpn_source_candidate_power_product ff_s_bpdvpn_source_candidate_power_product. ((((exists ff_h_bpdvpn_source_candidate_power_product_factor. ff_h_bpdvpn_source_candidate_power_product_factor + S (ff_p_bpdvpn_source_candidate_power_product) = S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_c_bpdvpn_source_candidate_power)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_factor. ff_b_bpdvpn_source_candidate_power = ff_q_bpdvpn_source_candidate_power_product_factor * S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_c_bpdvpn_source_candidate_power) + (ff_p_bpdvpn_source_candidate_power_product))) /\\ ((((exists ff_h_bpdvpn_source_candidate_power_product_partial. ff_h_bpdvpn_source_candidate_power_product_partial + S (ff_r_bpdvpn_source_candidate_power_product) = S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_partial. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_partial * S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product) + (ff_r_bpdvpn_source_candidate_power_product))) /\\ ((((exists ff_h_bpdvpn_source_candidate_power_product_successor. ff_h_bpdvpn_source_candidate_power_product_successor + S (ff_s_bpdvpn_source_candidate_power_product) = S ((S (S ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_successor. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_successor * S ((S (S ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product) + (ff_s_bpdvpn_source_candidate_power_product))) /\\ ff_s_bpdvpn_source_candidate_power_product = ff_r_bpdvpn_source_candidate_power_product * ff_p_bpdvpn_source_candidate_power_product)))))))) /\\ (exists bpv_factor_bpdvpn_source_candidate_divides. c = bpv_result_bpdvpn_source_candidate * bpv_factor_bpdvpn_source_candidate_divides))) -> (exists bpv_gap_bpdvpn_source_maximal. bpv_gap_bpdvpn_source_maximal + bpv_candidate_bpdvpn_source = e)) -> (exists bpr_quotient_bpdvpn_divides. c = (p) * bpr_quotient_bpdvpn_divides) -> ~(e = 0)", "statement_sha256": "8000af01e004415d265e86042a506b3c8a0cea2554309eadd883a00b6295ebbe", "summary": "A prime divisor forces its canonical valuation exponent nonzero.", "summary_sha256": "ad37df23aa0afbc84439f784f0ecf2d3ce6b8291f0146f977b74cf1c032ad7fb" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_exists", "pow_one", "prime_power_divides_exponent_le_valuation", "le_zero" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_prime_support_candidate.py", 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"prime_divisor_power_valuation_nonzero", "parent_alpha_version": "v34", "proof_bundle_node_id": 223, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_divisor_power_valuation_nonzero", "script": [ "intro p", "intro c", "intro e", "intro hp", "intro hc", "intro hvaluation", "intro hdivides", "have hpower : exists x. (exists bpvi_b_bpdvpn_unit_power bpvi_c_bpdvpn_unit_power. ((forall bpvi_i_bpdvpn_unit_power. (exists bpvi_repeat_gap_bpdvpn_unit_power. bpvi_repeat_gap_bpdvpn_unit_power + S bpvi_i_bpdvpn_unit_power = 1) -> (((exists bpvi_h_bpdvpn_unit_power_repeat. bpvi_h_bpdvpn_unit_power_repeat + S (p) = S ((S (bpvi_i_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_repeat. bpvi_b_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_repeat * S ((S (bpvi_i_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power) + (p)))) /\\ (exists bpvi_u_bpdvpn_unit_power bpvi_v_bpdvpn_unit_power. ((((exists bpvi_h_bpdvpn_unit_power_start. bpvi_h_bpdvpn_unit_power_start + S (1) = S ((S (0)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_start. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_start * S ((S (0)) * bpvi_v_bpdvpn_unit_power) + (1))) /\\ ((((exists bpvi_h_bpdvpn_unit_power_terminal. bpvi_h_bpdvpn_unit_power_terminal + S (x) = S ((S (1)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_terminal. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_terminal * S ((S (1)) * bpvi_v_bpdvpn_unit_power) + (x))) /\\ forall bpvi_j_bpdvpn_unit_power. (exists bpvi_product_gap_bpdvpn_unit_power. bpvi_product_gap_bpdvpn_unit_power + S bpvi_j_bpdvpn_unit_power = 1) -> exists bpvi_factor_bpdvpn_unit_power bpvi_partial_bpdvpn_unit_power bpvi_successor_bpdvpn_unit_power. ((((exists bpvi_h_bpdvpn_unit_power_factor. bpvi_h_bpdvpn_unit_power_factor + S (bpvi_factor_bpdvpn_unit_power) = S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_factor. bpvi_b_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_factor * S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_c_bpdvpn_unit_power) + (bpvi_factor_bpdvpn_unit_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_power_partial. bpvi_h_bpdvpn_unit_power_partial + S (bpvi_partial_bpdvpn_unit_power) = S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_partial. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_partial * S ((S (bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power) + (bpvi_partial_bpdvpn_unit_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_power_successor. bpvi_h_bpdvpn_unit_power_successor + S (bpvi_successor_bpdvpn_unit_power) = S ((S (S bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power)) /\\ exists bpvi_q_bpdvpn_unit_power_successor. bpvi_u_bpdvpn_unit_power = bpvi_q_bpdvpn_unit_power_successor * S ((S (S bpvi_j_bpdvpn_unit_power)) * bpvi_v_bpdvpn_unit_power) + (bpvi_successor_bpdvpn_unit_power))) /\\ bpvi_successor_bpdvpn_unit_power = bpvi_partial_bpdvpn_unit_power * bpvi_factor_bpdvpn_unit_power))))))))", "specialize pow_exists p", "specialize pow_exists 1", "exact pow_exists", "cases hpower", "have hvalue : x = p", "specialize pow_one p", "specialize pow_one 1", "specialize pow_one x", "apply pow_one", "refl", "exact hpower_witness", "have hunit : exists bpvi_result_bpdvpn_unit_divides. ((exists bpvi_b_bpdvpn_unit_divides_power bpvi_c_bpdvpn_unit_divides_power. ((forall bpvi_i_bpdvpn_unit_divides_power. (exists bpvi_repeat_gap_bpdvpn_unit_divides_power. bpvi_repeat_gap_bpdvpn_unit_divides_power + S bpvi_i_bpdvpn_unit_divides_power = 1) -> (((exists bpvi_h_bpdvpn_unit_divides_power_repeat. bpvi_h_bpdvpn_unit_divides_power_repeat + S (p) = S ((S (bpvi_i_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_repeat. bpvi_b_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_repeat * S ((S (bpvi_i_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power) + (p)))) /\\ (exists bpvi_u_bpdvpn_unit_divides_power bpvi_v_bpdvpn_unit_divides_power. ((((exists bpvi_h_bpdvpn_unit_divides_power_start. bpvi_h_bpdvpn_unit_divides_power_start + S (1) = S ((S (0)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_start. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_start * S ((S (0)) * bpvi_v_bpdvpn_unit_divides_power) + (1))) /\\ ((((exists bpvi_h_bpdvpn_unit_divides_power_terminal. bpvi_h_bpdvpn_unit_divides_power_terminal + S (bpvi_result_bpdvpn_unit_divides) = S ((S (1)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_terminal. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_terminal * S ((S (1)) * bpvi_v_bpdvpn_unit_divides_power) + (bpvi_result_bpdvpn_unit_divides))) /\\ forall bpvi_j_bpdvpn_unit_divides_power. (exists bpvi_product_gap_bpdvpn_unit_divides_power. bpvi_product_gap_bpdvpn_unit_divides_power + S bpvi_j_bpdvpn_unit_divides_power = 1) -> exists bpvi_factor_bpdvpn_unit_divides_power bpvi_partial_bpdvpn_unit_divides_power bpvi_successor_bpdvpn_unit_divides_power. ((((exists bpvi_h_bpdvpn_unit_divides_power_factor. bpvi_h_bpdvpn_unit_divides_power_factor + S (bpvi_factor_bpdvpn_unit_divides_power) = S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_factor. bpvi_b_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_factor * S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_c_bpdvpn_unit_divides_power) + (bpvi_factor_bpdvpn_unit_divides_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_divides_power_partial. bpvi_h_bpdvpn_unit_divides_power_partial + S (bpvi_partial_bpdvpn_unit_divides_power) = S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_partial. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_partial * S ((S (bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power) + (bpvi_partial_bpdvpn_unit_divides_power))) /\\ ((((exists bpvi_h_bpdvpn_unit_divides_power_successor. bpvi_h_bpdvpn_unit_divides_power_successor + S (bpvi_successor_bpdvpn_unit_divides_power) = S ((S (S bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power)) /\\ exists bpvi_q_bpdvpn_unit_divides_power_successor. bpvi_u_bpdvpn_unit_divides_power = bpvi_q_bpdvpn_unit_divides_power_successor * S ((S (S bpvi_j_bpdvpn_unit_divides_power)) * bpvi_v_bpdvpn_unit_divides_power) + (bpvi_successor_bpdvpn_unit_divides_power))) /\\ bpvi_successor_bpdvpn_unit_divides_power = bpvi_partial_bpdvpn_unit_divides_power * bpvi_factor_bpdvpn_unit_divides_power)))))))) /\\ exists bpvi_divisor_factor_bpdvpn_unit_divides. c = bpvi_result_bpdvpn_unit_divides * bpvi_divisor_factor_bpdvpn_unit_divides)", "exists x", "split", "exact hpower_witness", "rewrite hvalue", "exact hdivides", "have hbound : exists bpr_le_gap_bpdvpn_bound. bpr_le_gap_bpdvpn_bound + (1) = (e)", "specialize prime_power_divides_exponent_le_valuation p", "specialize prime_power_divides_exponent_le_valuation c", "specialize prime_power_divides_exponent_le_valuation e", "specialize prime_power_divides_exponent_le_valuation 1", "apply prime_power_divides_exponent_le_valuation", "exact hp", "exact hc", "exact hvaluation", "exact hunit", "intro heq", "rewrite heq at hbound", "have hone_zero : 1 = 0", "specialize le_zero 1", "apply le_zero", "exact hbound", "apply PA1", "exact hone_zero" ], "script_sha256": "89a23d06682f85d78c49249be2402ff4652e3b50b0fe69083ac1ca05a8fb2a5a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_prime_support_candidate.py", "sha256": "d48ed42c0b5289b1565947bb43dbcbe8389eed9aa196766ff90567cfc7fec7ab" }, "stable_member": false, "statement": "forall p c e. ((~(p = 1) /\\ forall bpr_left_bpdvpn_prime bpr_right_bpdvpn_prime. p = bpr_left_bpdvpn_prime * bpr_right_bpdvpn_prime -> bpr_left_bpdvpn_prime = 1 \\/ bpr_right_bpdvpn_prime = 1)) -> ~(c = 0) -> (((exists bpv_gap_bpdvpn_source_exponent_bound. bpv_gap_bpdvpn_source_exponent_bound + e = c) /\\ (exists bpv_result_bpdvpn_source_selected. ((exists ff_b_bpdvpn_source_selected_power ff_c_bpdvpn_source_selected_power. ((forall ff_i_bpdvpn_source_selected_power_repeat. (exists ff_lt_bpdvpn_source_selected_power_repeat_bound. ff_lt_bpdvpn_source_selected_power_repeat_bound + S ff_i_bpdvpn_source_selected_power_repeat = e) -> (((exists ff_h_bpdvpn_source_selected_power_repeat_decoded. ff_h_bpdvpn_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_bpdvpn_source_selected_power_repeat)) * ff_c_bpdvpn_source_selected_power)) /\\ exists ff_q_bpdvpn_source_selected_power_repeat_decoded. ff_b_bpdvpn_source_selected_power = ff_q_bpdvpn_source_selected_power_repeat_decoded * S ((S (ff_i_bpdvpn_source_selected_power_repeat)) * ff_c_bpdvpn_source_selected_power) + (p)))) /\\ (exists ff_u_bpdvpn_source_selected_power_product ff_v_bpdvpn_source_selected_power_product. ((((exists ff_h_bpdvpn_source_selected_power_product_start. ff_h_bpdvpn_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_start. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_start * S ((S (0)) * ff_v_bpdvpn_source_selected_power_product) + (1))) /\\ ((((exists ff_h_bpdvpn_source_selected_power_product_terminal. ff_h_bpdvpn_source_selected_power_product_terminal + S (bpv_result_bpdvpn_source_selected) = S ((S (e)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_terminal. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_terminal * S ((S (e)) * ff_v_bpdvpn_source_selected_power_product) + (bpv_result_bpdvpn_source_selected))) /\\ forall ff_i_bpdvpn_source_selected_power_product. (exists ff_lt_bpdvpn_source_selected_power_product_bound. ff_lt_bpdvpn_source_selected_power_product_bound + S ff_i_bpdvpn_source_selected_power_product = e) -> exists ff_p_bpdvpn_source_selected_power_product ff_r_bpdvpn_source_selected_power_product ff_s_bpdvpn_source_selected_power_product. ((((exists ff_h_bpdvpn_source_selected_power_product_factor. ff_h_bpdvpn_source_selected_power_product_factor + S (ff_p_bpdvpn_source_selected_power_product) = S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_c_bpdvpn_source_selected_power)) /\\ exists ff_q_bpdvpn_source_selected_power_product_factor. ff_b_bpdvpn_source_selected_power = ff_q_bpdvpn_source_selected_power_product_factor * S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_c_bpdvpn_source_selected_power) + (ff_p_bpdvpn_source_selected_power_product))) /\\ ((((exists ff_h_bpdvpn_source_selected_power_product_partial. ff_h_bpdvpn_source_selected_power_product_partial + S (ff_r_bpdvpn_source_selected_power_product) = S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_partial. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_partial * S ((S (ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product) + (ff_r_bpdvpn_source_selected_power_product))) /\\ ((((exists ff_h_bpdvpn_source_selected_power_product_successor. ff_h_bpdvpn_source_selected_power_product_successor + S (ff_s_bpdvpn_source_selected_power_product) = S ((S (S ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product)) /\\ exists ff_q_bpdvpn_source_selected_power_product_successor. ff_u_bpdvpn_source_selected_power_product = ff_q_bpdvpn_source_selected_power_product_successor * S ((S (S ff_i_bpdvpn_source_selected_power_product)) * ff_v_bpdvpn_source_selected_power_product) + (ff_s_bpdvpn_source_selected_power_product))) /\\ ff_s_bpdvpn_source_selected_power_product = ff_r_bpdvpn_source_selected_power_product * ff_p_bpdvpn_source_selected_power_product)))))))) /\\ (exists bpv_factor_bpdvpn_source_selected_divides. c = bpv_result_bpdvpn_source_selected * bpv_factor_bpdvpn_source_selected_divides)))) /\\ forall bpv_candidate_bpdvpn_source. (exists bpv_gap_bpdvpn_source_candidate_bound. bpv_gap_bpdvpn_source_candidate_bound + bpv_candidate_bpdvpn_source = c) -> (exists bpv_result_bpdvpn_source_candidate. ((exists ff_b_bpdvpn_source_candidate_power ff_c_bpdvpn_source_candidate_power. ((forall ff_i_bpdvpn_source_candidate_power_repeat. (exists ff_lt_bpdvpn_source_candidate_power_repeat_bound. ff_lt_bpdvpn_source_candidate_power_repeat_bound + S ff_i_bpdvpn_source_candidate_power_repeat = bpv_candidate_bpdvpn_source) -> (((exists ff_h_bpdvpn_source_candidate_power_repeat_decoded. ff_h_bpdvpn_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_bpdvpn_source_candidate_power_repeat)) * ff_c_bpdvpn_source_candidate_power)) /\\ exists ff_q_bpdvpn_source_candidate_power_repeat_decoded. ff_b_bpdvpn_source_candidate_power = ff_q_bpdvpn_source_candidate_power_repeat_decoded * S ((S (ff_i_bpdvpn_source_candidate_power_repeat)) * ff_c_bpdvpn_source_candidate_power) + (p)))) /\\ (exists ff_u_bpdvpn_source_candidate_power_product ff_v_bpdvpn_source_candidate_power_product. ((((exists ff_h_bpdvpn_source_candidate_power_product_start. ff_h_bpdvpn_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_start. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_start * S ((S (0)) * ff_v_bpdvpn_source_candidate_power_product) + (1))) /\\ ((((exists ff_h_bpdvpn_source_candidate_power_product_terminal. ff_h_bpdvpn_source_candidate_power_product_terminal + S (bpv_result_bpdvpn_source_candidate) = S ((S (bpv_candidate_bpdvpn_source)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_terminal. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_terminal * S ((S (bpv_candidate_bpdvpn_source)) * ff_v_bpdvpn_source_candidate_power_product) + (bpv_result_bpdvpn_source_candidate))) /\\ forall ff_i_bpdvpn_source_candidate_power_product. (exists ff_lt_bpdvpn_source_candidate_power_product_bound. ff_lt_bpdvpn_source_candidate_power_product_bound + S ff_i_bpdvpn_source_candidate_power_product = bpv_candidate_bpdvpn_source) -> exists ff_p_bpdvpn_source_candidate_power_product ff_r_bpdvpn_source_candidate_power_product ff_s_bpdvpn_source_candidate_power_product. ((((exists ff_h_bpdvpn_source_candidate_power_product_factor. ff_h_bpdvpn_source_candidate_power_product_factor + S (ff_p_bpdvpn_source_candidate_power_product) = S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_c_bpdvpn_source_candidate_power)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_factor. ff_b_bpdvpn_source_candidate_power = ff_q_bpdvpn_source_candidate_power_product_factor * S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_c_bpdvpn_source_candidate_power) + (ff_p_bpdvpn_source_candidate_power_product))) /\\ ((((exists ff_h_bpdvpn_source_candidate_power_product_partial. ff_h_bpdvpn_source_candidate_power_product_partial + S (ff_r_bpdvpn_source_candidate_power_product) = S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_partial. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_partial * S ((S (ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product) + (ff_r_bpdvpn_source_candidate_power_product))) /\\ ((((exists ff_h_bpdvpn_source_candidate_power_product_successor. ff_h_bpdvpn_source_candidate_power_product_successor + S (ff_s_bpdvpn_source_candidate_power_product) = S ((S (S ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product)) /\\ exists ff_q_bpdvpn_source_candidate_power_product_successor. ff_u_bpdvpn_source_candidate_power_product = ff_q_bpdvpn_source_candidate_power_product_successor * S ((S (S ff_i_bpdvpn_source_candidate_power_product)) * ff_v_bpdvpn_source_candidate_power_product) + (ff_s_bpdvpn_source_candidate_power_product))) /\\ ff_s_bpdvpn_source_candidate_power_product = ff_r_bpdvpn_source_candidate_power_product * ff_p_bpdvpn_source_candidate_power_product)))))))) /\\ (exists bpv_factor_bpdvpn_source_candidate_divides. c = bpv_result_bpdvpn_source_candidate * bpv_factor_bpdvpn_source_candidate_divides))) -> (exists bpv_gap_bpdvpn_source_maximal. bpv_gap_bpdvpn_source_maximal + bpv_candidate_bpdvpn_source = e)) -> (exists bpr_quotient_bpdvpn_divides. c = (p) * bpr_quotient_bpdvpn_divides) -> ~(e = 0)", "statement_sha256": "8000af01e004415d265e86042a506b3c8a0cea2554309eadd883a00b6295ebbe" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "power_valuation_value_eq_transport", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "kummer", "bundle_node_id": 262, "bundle_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "acaf8d5d0d9fa8f8b66c60edafdb4f37e93635493bf4bca0bb22dfc002794609" }, "bertrand_v12_evidence_bundle_sha256": "78644af75f1829a97979025720ceb11513e2cf0ffaaa23a6c3a9d491c60eaa06", "body_checked": true, "body_receipt": { "command_count": 11, "dependency_count": 0, "dne_command_count": 0, "name": "power_valuation_value_eq_transport", "proof_depth": 15, "proof_edges": 26, "proof_nodes": 27, "proof_objects": 27, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [], "dependencies_sha256": "01ba4719c80b6fe911b091a7c05124b64eeece964e09c058ef8f9805daca546b", "empty_context_closure": { "body_proof_depth": 15, "body_proof_nodes": 27, "bundle_campaign": "kummer", "bundle_dependency_edge_count": 779, "bundle_node_count": 281, "bundle_node_id": 262, "bundle_path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "bundle_root_id": 280, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "a714d85092357780e3e2b883a95f56b28f0110fbee763e6b1dd5955d42bbd0da", "status": "checked" }, "enrollment_index": 1176, "enrollment_origin": "bertrand", "evidence_links": [ { "document_sha256": "76ab449e7ae0dc58d7c99743e7df39e59d5619b8801387cd40a8cb242e2b79e8", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "4ec910f58563edc4f5cc31554e3f6bf92867015df25be1299873e6a285c1d02e", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_central_binom_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "aebab5f4cf6a63b67a0716c3dcd792a876f263bce6d371d25dcb4e3dbf78a8b3", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-b5-central-valuation-tranche-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "d992c4aeb37829838cefd668679c513c5d45f6304f9842dcbe825bb25563182c", "kind": "sealed_alpha_v11_parent", "path": "artifacts/peano-library/alpha/catalog-v11.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=262]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_value_eq_transport]" } ], "evidence_status": "alpha_closed", "logical_spec_sha256": "d66f3683015e02e783d350d7b1d2dcb672556c93a814b612230b0c5bd0d9a00a", "membership": "alpha_only", "name": "power_valuation_value_eq_transport", "proof_tag": null, "provenance": [ "bertrand" ], "script": [ "intro p", "intro a", "intro b", "intro e", "intro hvalue", "intro hsource", "rewrite hvalue at hsource", "rewrite hvalue at hsource", "rewrite hvalue at hsource", "rewrite hvalue at hsource", "exact hsource" ], "script_sha256": "0ef2220c34ded5fb94a79b90704acbf9a83f7a79cef0f126536d68481ca032cb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_valuation_candidate.py", "sha256": "76ab449e7ae0dc58d7c99743e7df39e59d5619b8801387cd40a8cb242e2b79e8" }, "statement": "forall p a b e. a = b -> (((exists bpv_gap_b5cvvet_source_exponent_bound. bpv_gap_b5cvvet_source_exponent_bound + e = a) /\\ (exists bpv_result_b5cvvet_source_selected. ((exists ff_b_b5cvvet_source_selected_power ff_c_b5cvvet_source_selected_power. ((forall ff_i_b5cvvet_source_selected_power_repeat. (exists ff_lt_b5cvvet_source_selected_power_repeat_bound. ff_lt_b5cvvet_source_selected_power_repeat_bound + S ff_i_b5cvvet_source_selected_power_repeat = e) -> (((exists ff_h_b5cvvet_source_selected_power_repeat_decoded. ff_h_b5cvvet_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_source_selected_power_repeat)) * ff_c_b5cvvet_source_selected_power)) /\\ exists ff_q_b5cvvet_source_selected_power_repeat_decoded. ff_b_b5cvvet_source_selected_power = ff_q_b5cvvet_source_selected_power_repeat_decoded * S ((S (ff_i_b5cvvet_source_selected_power_repeat)) * ff_c_b5cvvet_source_selected_power) + (p)))) /\\ (exists ff_u_b5cvvet_source_selected_power_product ff_v_b5cvvet_source_selected_power_product. ((((exists ff_h_b5cvvet_source_selected_power_product_start. ff_h_b5cvvet_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_start. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_start * S ((S (0)) * ff_v_b5cvvet_source_selected_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_source_selected_power_product_terminal. ff_h_b5cvvet_source_selected_power_product_terminal + S (bpv_result_b5cvvet_source_selected) = S ((S (e)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_terminal. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_terminal * S ((S (e)) * ff_v_b5cvvet_source_selected_power_product) + (bpv_result_b5cvvet_source_selected))) /\\ forall ff_i_b5cvvet_source_selected_power_product. (exists ff_lt_b5cvvet_source_selected_power_product_bound. ff_lt_b5cvvet_source_selected_power_product_bound + S ff_i_b5cvvet_source_selected_power_product = e) -> exists ff_p_b5cvvet_source_selected_power_product ff_r_b5cvvet_source_selected_power_product ff_s_b5cvvet_source_selected_power_product. ((((exists ff_h_b5cvvet_source_selected_power_product_factor. ff_h_b5cvvet_source_selected_power_product_factor + S (ff_p_b5cvvet_source_selected_power_product) = S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_c_b5cvvet_source_selected_power)) /\\ exists ff_q_b5cvvet_source_selected_power_product_factor. ff_b_b5cvvet_source_selected_power = ff_q_b5cvvet_source_selected_power_product_factor * S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_c_b5cvvet_source_selected_power) + (ff_p_b5cvvet_source_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_source_selected_power_product_partial. ff_h_b5cvvet_source_selected_power_product_partial + S (ff_r_b5cvvet_source_selected_power_product) = S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_partial. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_partial * S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product) + (ff_r_b5cvvet_source_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_source_selected_power_product_successor. ff_h_b5cvvet_source_selected_power_product_successor + S (ff_s_b5cvvet_source_selected_power_product) = S ((S (S ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_successor. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_successor * S ((S (S ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product) + (ff_s_b5cvvet_source_selected_power_product))) /\\ ff_s_b5cvvet_source_selected_power_product = ff_r_b5cvvet_source_selected_power_product * ff_p_b5cvvet_source_selected_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_source_selected_divides. a = bpv_result_b5cvvet_source_selected * bpv_factor_b5cvvet_source_selected_divides)))) /\\ forall bpv_candidate_b5cvvet_source. (exists bpv_gap_b5cvvet_source_candidate_bound. bpv_gap_b5cvvet_source_candidate_bound + bpv_candidate_b5cvvet_source = a) -> (exists bpv_result_b5cvvet_source_candidate. ((exists ff_b_b5cvvet_source_candidate_power ff_c_b5cvvet_source_candidate_power. ((forall ff_i_b5cvvet_source_candidate_power_repeat. (exists ff_lt_b5cvvet_source_candidate_power_repeat_bound. ff_lt_b5cvvet_source_candidate_power_repeat_bound + S ff_i_b5cvvet_source_candidate_power_repeat = bpv_candidate_b5cvvet_source) -> (((exists ff_h_b5cvvet_source_candidate_power_repeat_decoded. ff_h_b5cvvet_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_source_candidate_power_repeat)) * ff_c_b5cvvet_source_candidate_power)) /\\ exists ff_q_b5cvvet_source_candidate_power_repeat_decoded. ff_b_b5cvvet_source_candidate_power = ff_q_b5cvvet_source_candidate_power_repeat_decoded * S ((S (ff_i_b5cvvet_source_candidate_power_repeat)) * ff_c_b5cvvet_source_candidate_power) + (p)))) /\\ (exists ff_u_b5cvvet_source_candidate_power_product ff_v_b5cvvet_source_candidate_power_product. ((((exists ff_h_b5cvvet_source_candidate_power_product_start. ff_h_b5cvvet_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_start. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_start * S ((S (0)) * ff_v_b5cvvet_source_candidate_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_source_candidate_power_product_terminal. ff_h_b5cvvet_source_candidate_power_product_terminal + S (bpv_result_b5cvvet_source_candidate) = S ((S (bpv_candidate_b5cvvet_source)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_terminal. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvvet_source)) * ff_v_b5cvvet_source_candidate_power_product) + (bpv_result_b5cvvet_source_candidate))) /\\ forall ff_i_b5cvvet_source_candidate_power_product. (exists ff_lt_b5cvvet_source_candidate_power_product_bound. ff_lt_b5cvvet_source_candidate_power_product_bound + S ff_i_b5cvvet_source_candidate_power_product = bpv_candidate_b5cvvet_source) -> exists ff_p_b5cvvet_source_candidate_power_product ff_r_b5cvvet_source_candidate_power_product ff_s_b5cvvet_source_candidate_power_product. ((((exists ff_h_b5cvvet_source_candidate_power_product_factor. ff_h_b5cvvet_source_candidate_power_product_factor + S (ff_p_b5cvvet_source_candidate_power_product) = S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_c_b5cvvet_source_candidate_power)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_factor. ff_b_b5cvvet_source_candidate_power = ff_q_b5cvvet_source_candidate_power_product_factor * S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_c_b5cvvet_source_candidate_power) + (ff_p_b5cvvet_source_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_source_candidate_power_product_partial. ff_h_b5cvvet_source_candidate_power_product_partial + S (ff_r_b5cvvet_source_candidate_power_product) = S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_partial. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_partial * S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product) + (ff_r_b5cvvet_source_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_source_candidate_power_product_successor. ff_h_b5cvvet_source_candidate_power_product_successor + S (ff_s_b5cvvet_source_candidate_power_product) = S ((S (S ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_successor. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_successor * S ((S (S ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product) + (ff_s_b5cvvet_source_candidate_power_product))) /\\ ff_s_b5cvvet_source_candidate_power_product = ff_r_b5cvvet_source_candidate_power_product * ff_p_b5cvvet_source_candidate_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_source_candidate_divides. a = bpv_result_b5cvvet_source_candidate * bpv_factor_b5cvvet_source_candidate_divides))) -> (exists bpv_gap_b5cvvet_source_maximal. bpv_gap_b5cvvet_source_maximal + bpv_candidate_b5cvvet_source = e)) -> (((exists bpv_gap_b5cvvet_target_exponent_bound. bpv_gap_b5cvvet_target_exponent_bound + e = b) /\\ (exists bpv_result_b5cvvet_target_selected. ((exists ff_b_b5cvvet_target_selected_power ff_c_b5cvvet_target_selected_power. ((forall ff_i_b5cvvet_target_selected_power_repeat. (exists ff_lt_b5cvvet_target_selected_power_repeat_bound. ff_lt_b5cvvet_target_selected_power_repeat_bound + S ff_i_b5cvvet_target_selected_power_repeat = e) -> (((exists ff_h_b5cvvet_target_selected_power_repeat_decoded. ff_h_b5cvvet_target_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_target_selected_power_repeat)) * ff_c_b5cvvet_target_selected_power)) /\\ exists ff_q_b5cvvet_target_selected_power_repeat_decoded. ff_b_b5cvvet_target_selected_power = ff_q_b5cvvet_target_selected_power_repeat_decoded * S ((S (ff_i_b5cvvet_target_selected_power_repeat)) * ff_c_b5cvvet_target_selected_power) + (p)))) /\\ (exists ff_u_b5cvvet_target_selected_power_product ff_v_b5cvvet_target_selected_power_product. ((((exists ff_h_b5cvvet_target_selected_power_product_start. ff_h_b5cvvet_target_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_start. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_start * S ((S (0)) * ff_v_b5cvvet_target_selected_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_target_selected_power_product_terminal. ff_h_b5cvvet_target_selected_power_product_terminal + S (bpv_result_b5cvvet_target_selected) = S ((S (e)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_terminal. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_terminal * S ((S (e)) * ff_v_b5cvvet_target_selected_power_product) + (bpv_result_b5cvvet_target_selected))) /\\ forall ff_i_b5cvvet_target_selected_power_product. (exists ff_lt_b5cvvet_target_selected_power_product_bound. ff_lt_b5cvvet_target_selected_power_product_bound + S ff_i_b5cvvet_target_selected_power_product = e) -> exists ff_p_b5cvvet_target_selected_power_product ff_r_b5cvvet_target_selected_power_product ff_s_b5cvvet_target_selected_power_product. ((((exists ff_h_b5cvvet_target_selected_power_product_factor. ff_h_b5cvvet_target_selected_power_product_factor + S (ff_p_b5cvvet_target_selected_power_product) = S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_c_b5cvvet_target_selected_power)) /\\ exists ff_q_b5cvvet_target_selected_power_product_factor. ff_b_b5cvvet_target_selected_power = ff_q_b5cvvet_target_selected_power_product_factor * S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_c_b5cvvet_target_selected_power) + (ff_p_b5cvvet_target_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_target_selected_power_product_partial. ff_h_b5cvvet_target_selected_power_product_partial + S (ff_r_b5cvvet_target_selected_power_product) = S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_partial. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_partial * S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product) + (ff_r_b5cvvet_target_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_target_selected_power_product_successor. ff_h_b5cvvet_target_selected_power_product_successor + S (ff_s_b5cvvet_target_selected_power_product) = S ((S (S ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_successor. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_successor * S ((S (S ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product) + (ff_s_b5cvvet_target_selected_power_product))) /\\ ff_s_b5cvvet_target_selected_power_product = ff_r_b5cvvet_target_selected_power_product * ff_p_b5cvvet_target_selected_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_target_selected_divides. b = bpv_result_b5cvvet_target_selected * bpv_factor_b5cvvet_target_selected_divides)))) /\\ forall bpv_candidate_b5cvvet_target. (exists bpv_gap_b5cvvet_target_candidate_bound. bpv_gap_b5cvvet_target_candidate_bound + bpv_candidate_b5cvvet_target = b) -> (exists bpv_result_b5cvvet_target_candidate. ((exists ff_b_b5cvvet_target_candidate_power ff_c_b5cvvet_target_candidate_power. ((forall ff_i_b5cvvet_target_candidate_power_repeat. (exists ff_lt_b5cvvet_target_candidate_power_repeat_bound. ff_lt_b5cvvet_target_candidate_power_repeat_bound + S ff_i_b5cvvet_target_candidate_power_repeat = bpv_candidate_b5cvvet_target) -> (((exists ff_h_b5cvvet_target_candidate_power_repeat_decoded. ff_h_b5cvvet_target_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_target_candidate_power_repeat)) * ff_c_b5cvvet_target_candidate_power)) /\\ exists ff_q_b5cvvet_target_candidate_power_repeat_decoded. ff_b_b5cvvet_target_candidate_power = ff_q_b5cvvet_target_candidate_power_repeat_decoded * S ((S (ff_i_b5cvvet_target_candidate_power_repeat)) * ff_c_b5cvvet_target_candidate_power) + (p)))) /\\ (exists ff_u_b5cvvet_target_candidate_power_product ff_v_b5cvvet_target_candidate_power_product. ((((exists ff_h_b5cvvet_target_candidate_power_product_start. ff_h_b5cvvet_target_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_start. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_start * S ((S (0)) * ff_v_b5cvvet_target_candidate_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_target_candidate_power_product_terminal. ff_h_b5cvvet_target_candidate_power_product_terminal + S (bpv_result_b5cvvet_target_candidate) = S ((S (bpv_candidate_b5cvvet_target)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_terminal. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvvet_target)) * ff_v_b5cvvet_target_candidate_power_product) + (bpv_result_b5cvvet_target_candidate))) /\\ forall ff_i_b5cvvet_target_candidate_power_product. (exists ff_lt_b5cvvet_target_candidate_power_product_bound. ff_lt_b5cvvet_target_candidate_power_product_bound + S ff_i_b5cvvet_target_candidate_power_product = bpv_candidate_b5cvvet_target) -> exists ff_p_b5cvvet_target_candidate_power_product ff_r_b5cvvet_target_candidate_power_product ff_s_b5cvvet_target_candidate_power_product. ((((exists ff_h_b5cvvet_target_candidate_power_product_factor. ff_h_b5cvvet_target_candidate_power_product_factor + S (ff_p_b5cvvet_target_candidate_power_product) = S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_c_b5cvvet_target_candidate_power)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_factor. ff_b_b5cvvet_target_candidate_power = ff_q_b5cvvet_target_candidate_power_product_factor * S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_c_b5cvvet_target_candidate_power) + (ff_p_b5cvvet_target_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_target_candidate_power_product_partial. ff_h_b5cvvet_target_candidate_power_product_partial + S (ff_r_b5cvvet_target_candidate_power_product) = S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_partial. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_partial * S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product) + (ff_r_b5cvvet_target_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_target_candidate_power_product_successor. ff_h_b5cvvet_target_candidate_power_product_successor + S (ff_s_b5cvvet_target_candidate_power_product) = S ((S (S ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_successor. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_successor * S ((S (S ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product) + (ff_s_b5cvvet_target_candidate_power_product))) /\\ ff_s_b5cvvet_target_candidate_power_product = ff_r_b5cvvet_target_candidate_power_product * ff_p_b5cvvet_target_candidate_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_target_candidate_divides. b = bpv_result_b5cvvet_target_candidate * bpv_factor_b5cvvet_target_candidate_divides))) -> (exists bpv_gap_b5cvvet_target_maximal. bpv_gap_b5cvvet_target_maximal + bpv_candidate_b5cvvet_target = e))", "statement_sha256": "a714d85092357780e3e2b883a95f56b28f0110fbee763e6b1dd5955d42bbd0da", "summary": "Power valuation transports along equality of its valued number.", "summary_sha256": "7b80ce3d3ba4251387b19ed9dcdaffb695e40748b47c1a2b1f0ca82983fbdf86" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "76ab449e7ae0dc58d7c99743e7df39e59d5619b8801387cd40a8cb242e2b79e8", "kind": "bertrand_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_valuation_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "4ec910f58563edc4f5cc31554e3f6bf92867015df25be1299873e6a285c1d02e", "kind": "bertrand_executable_audit", "path": "peano-lab/py/tests/test_bertrand_central_binom_valuation_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "aebab5f4cf6a63b67a0716c3dcd792a876f263bce6d371d25dcb4e3dbf78a8b3", "kind": "bertrand_campaign_rfc", "path": "research/arithmetic-library/ha-bertrand-b5-central-valuation-tranche-rfc-v1.md", "role": "reviewed_campaign_contract", "selector": "document" }, { "document_sha256": "d992c4aeb37829838cefd668679c513c5d45f6304f9842dcbe825bb25563182c", "kind": "sealed_alpha_v11_parent", "path": "artifacts/peano-library/alpha/catalog-v11.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "49fd86708fe5b289d0159526285e73b2aea008c26e0eb41ae8a053c970d4210e", "kind": "kummer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/kummer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=262]" }, { "document_sha256": "7de5c5bc819c19dbf597dd10624d3da0b7ea48f6b5368b7fb749245e966f8893", "kind": "kummer_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/kummer-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=power_valuation_value_eq_transport]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "power_valuation_value_eq_transport", "parent_alpha_version": "v34", "proof_bundle_node_id": 224, "reference_route": "jordan-totient/checkpoint.html#theorem-power_valuation_value_eq_transport", "script": [ "intro p", "intro a", "intro b", "intro e", "intro hvalue", "intro hsource", "rewrite hvalue at hsource", "rewrite hvalue at hsource", "rewrite hvalue at hsource", "rewrite hvalue at hsource", "exact hsource" ], "script_sha256": "0ef2220c34ded5fb94a79b90704acbf9a83f7a79cef0f126536d68481ca032cb", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/bertrand_central_binom_valuation_candidate.py", "sha256": "76ab449e7ae0dc58d7c99743e7df39e59d5619b8801387cd40a8cb242e2b79e8" }, "stable_member": false, "statement": "forall p a b e. a = b -> (((exists bpv_gap_b5cvvet_source_exponent_bound. bpv_gap_b5cvvet_source_exponent_bound + e = a) /\\ (exists bpv_result_b5cvvet_source_selected. ((exists ff_b_b5cvvet_source_selected_power ff_c_b5cvvet_source_selected_power. ((forall ff_i_b5cvvet_source_selected_power_repeat. (exists ff_lt_b5cvvet_source_selected_power_repeat_bound. ff_lt_b5cvvet_source_selected_power_repeat_bound + S ff_i_b5cvvet_source_selected_power_repeat = e) -> (((exists ff_h_b5cvvet_source_selected_power_repeat_decoded. ff_h_b5cvvet_source_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_source_selected_power_repeat)) * ff_c_b5cvvet_source_selected_power)) /\\ exists ff_q_b5cvvet_source_selected_power_repeat_decoded. ff_b_b5cvvet_source_selected_power = ff_q_b5cvvet_source_selected_power_repeat_decoded * S ((S (ff_i_b5cvvet_source_selected_power_repeat)) * ff_c_b5cvvet_source_selected_power) + (p)))) /\\ (exists ff_u_b5cvvet_source_selected_power_product ff_v_b5cvvet_source_selected_power_product. ((((exists ff_h_b5cvvet_source_selected_power_product_start. ff_h_b5cvvet_source_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_start. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_start * S ((S (0)) * ff_v_b5cvvet_source_selected_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_source_selected_power_product_terminal. ff_h_b5cvvet_source_selected_power_product_terminal + S (bpv_result_b5cvvet_source_selected) = S ((S (e)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_terminal. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_terminal * S ((S (e)) * ff_v_b5cvvet_source_selected_power_product) + (bpv_result_b5cvvet_source_selected))) /\\ forall ff_i_b5cvvet_source_selected_power_product. (exists ff_lt_b5cvvet_source_selected_power_product_bound. ff_lt_b5cvvet_source_selected_power_product_bound + S ff_i_b5cvvet_source_selected_power_product = e) -> exists ff_p_b5cvvet_source_selected_power_product ff_r_b5cvvet_source_selected_power_product ff_s_b5cvvet_source_selected_power_product. ((((exists ff_h_b5cvvet_source_selected_power_product_factor. ff_h_b5cvvet_source_selected_power_product_factor + S (ff_p_b5cvvet_source_selected_power_product) = S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_c_b5cvvet_source_selected_power)) /\\ exists ff_q_b5cvvet_source_selected_power_product_factor. ff_b_b5cvvet_source_selected_power = ff_q_b5cvvet_source_selected_power_product_factor * S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_c_b5cvvet_source_selected_power) + (ff_p_b5cvvet_source_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_source_selected_power_product_partial. ff_h_b5cvvet_source_selected_power_product_partial + S (ff_r_b5cvvet_source_selected_power_product) = S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_partial. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_partial * S ((S (ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product) + (ff_r_b5cvvet_source_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_source_selected_power_product_successor. ff_h_b5cvvet_source_selected_power_product_successor + S (ff_s_b5cvvet_source_selected_power_product) = S ((S (S ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product)) /\\ exists ff_q_b5cvvet_source_selected_power_product_successor. ff_u_b5cvvet_source_selected_power_product = ff_q_b5cvvet_source_selected_power_product_successor * S ((S (S ff_i_b5cvvet_source_selected_power_product)) * ff_v_b5cvvet_source_selected_power_product) + (ff_s_b5cvvet_source_selected_power_product))) /\\ ff_s_b5cvvet_source_selected_power_product = ff_r_b5cvvet_source_selected_power_product * ff_p_b5cvvet_source_selected_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_source_selected_divides. a = bpv_result_b5cvvet_source_selected * bpv_factor_b5cvvet_source_selected_divides)))) /\\ forall bpv_candidate_b5cvvet_source. (exists bpv_gap_b5cvvet_source_candidate_bound. bpv_gap_b5cvvet_source_candidate_bound + bpv_candidate_b5cvvet_source = a) -> (exists bpv_result_b5cvvet_source_candidate. ((exists ff_b_b5cvvet_source_candidate_power ff_c_b5cvvet_source_candidate_power. ((forall ff_i_b5cvvet_source_candidate_power_repeat. (exists ff_lt_b5cvvet_source_candidate_power_repeat_bound. ff_lt_b5cvvet_source_candidate_power_repeat_bound + S ff_i_b5cvvet_source_candidate_power_repeat = bpv_candidate_b5cvvet_source) -> (((exists ff_h_b5cvvet_source_candidate_power_repeat_decoded. ff_h_b5cvvet_source_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_source_candidate_power_repeat)) * ff_c_b5cvvet_source_candidate_power)) /\\ exists ff_q_b5cvvet_source_candidate_power_repeat_decoded. ff_b_b5cvvet_source_candidate_power = ff_q_b5cvvet_source_candidate_power_repeat_decoded * S ((S (ff_i_b5cvvet_source_candidate_power_repeat)) * ff_c_b5cvvet_source_candidate_power) + (p)))) /\\ (exists ff_u_b5cvvet_source_candidate_power_product ff_v_b5cvvet_source_candidate_power_product. ((((exists ff_h_b5cvvet_source_candidate_power_product_start. ff_h_b5cvvet_source_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_start. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_start * S ((S (0)) * ff_v_b5cvvet_source_candidate_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_source_candidate_power_product_terminal. ff_h_b5cvvet_source_candidate_power_product_terminal + S (bpv_result_b5cvvet_source_candidate) = S ((S (bpv_candidate_b5cvvet_source)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_terminal. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvvet_source)) * ff_v_b5cvvet_source_candidate_power_product) + (bpv_result_b5cvvet_source_candidate))) /\\ forall ff_i_b5cvvet_source_candidate_power_product. (exists ff_lt_b5cvvet_source_candidate_power_product_bound. ff_lt_b5cvvet_source_candidate_power_product_bound + S ff_i_b5cvvet_source_candidate_power_product = bpv_candidate_b5cvvet_source) -> exists ff_p_b5cvvet_source_candidate_power_product ff_r_b5cvvet_source_candidate_power_product ff_s_b5cvvet_source_candidate_power_product. ((((exists ff_h_b5cvvet_source_candidate_power_product_factor. ff_h_b5cvvet_source_candidate_power_product_factor + S (ff_p_b5cvvet_source_candidate_power_product) = S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_c_b5cvvet_source_candidate_power)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_factor. ff_b_b5cvvet_source_candidate_power = ff_q_b5cvvet_source_candidate_power_product_factor * S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_c_b5cvvet_source_candidate_power) + (ff_p_b5cvvet_source_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_source_candidate_power_product_partial. ff_h_b5cvvet_source_candidate_power_product_partial + S (ff_r_b5cvvet_source_candidate_power_product) = S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_partial. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_partial * S ((S (ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product) + (ff_r_b5cvvet_source_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_source_candidate_power_product_successor. ff_h_b5cvvet_source_candidate_power_product_successor + S (ff_s_b5cvvet_source_candidate_power_product) = S ((S (S ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product)) /\\ exists ff_q_b5cvvet_source_candidate_power_product_successor. ff_u_b5cvvet_source_candidate_power_product = ff_q_b5cvvet_source_candidate_power_product_successor * S ((S (S ff_i_b5cvvet_source_candidate_power_product)) * ff_v_b5cvvet_source_candidate_power_product) + (ff_s_b5cvvet_source_candidate_power_product))) /\\ ff_s_b5cvvet_source_candidate_power_product = ff_r_b5cvvet_source_candidate_power_product * ff_p_b5cvvet_source_candidate_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_source_candidate_divides. a = bpv_result_b5cvvet_source_candidate * bpv_factor_b5cvvet_source_candidate_divides))) -> (exists bpv_gap_b5cvvet_source_maximal. bpv_gap_b5cvvet_source_maximal + bpv_candidate_b5cvvet_source = e)) -> (((exists bpv_gap_b5cvvet_target_exponent_bound. bpv_gap_b5cvvet_target_exponent_bound + e = b) /\\ (exists bpv_result_b5cvvet_target_selected. ((exists ff_b_b5cvvet_target_selected_power ff_c_b5cvvet_target_selected_power. ((forall ff_i_b5cvvet_target_selected_power_repeat. (exists ff_lt_b5cvvet_target_selected_power_repeat_bound. ff_lt_b5cvvet_target_selected_power_repeat_bound + S ff_i_b5cvvet_target_selected_power_repeat = e) -> (((exists ff_h_b5cvvet_target_selected_power_repeat_decoded. ff_h_b5cvvet_target_selected_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_target_selected_power_repeat)) * ff_c_b5cvvet_target_selected_power)) /\\ exists ff_q_b5cvvet_target_selected_power_repeat_decoded. ff_b_b5cvvet_target_selected_power = ff_q_b5cvvet_target_selected_power_repeat_decoded * S ((S (ff_i_b5cvvet_target_selected_power_repeat)) * ff_c_b5cvvet_target_selected_power) + (p)))) /\\ (exists ff_u_b5cvvet_target_selected_power_product ff_v_b5cvvet_target_selected_power_product. ((((exists ff_h_b5cvvet_target_selected_power_product_start. ff_h_b5cvvet_target_selected_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_start. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_start * S ((S (0)) * ff_v_b5cvvet_target_selected_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_target_selected_power_product_terminal. ff_h_b5cvvet_target_selected_power_product_terminal + S (bpv_result_b5cvvet_target_selected) = S ((S (e)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_terminal. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_terminal * S ((S (e)) * ff_v_b5cvvet_target_selected_power_product) + (bpv_result_b5cvvet_target_selected))) /\\ forall ff_i_b5cvvet_target_selected_power_product. (exists ff_lt_b5cvvet_target_selected_power_product_bound. ff_lt_b5cvvet_target_selected_power_product_bound + S ff_i_b5cvvet_target_selected_power_product = e) -> exists ff_p_b5cvvet_target_selected_power_product ff_r_b5cvvet_target_selected_power_product ff_s_b5cvvet_target_selected_power_product. ((((exists ff_h_b5cvvet_target_selected_power_product_factor. ff_h_b5cvvet_target_selected_power_product_factor + S (ff_p_b5cvvet_target_selected_power_product) = S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_c_b5cvvet_target_selected_power)) /\\ exists ff_q_b5cvvet_target_selected_power_product_factor. ff_b_b5cvvet_target_selected_power = ff_q_b5cvvet_target_selected_power_product_factor * S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_c_b5cvvet_target_selected_power) + (ff_p_b5cvvet_target_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_target_selected_power_product_partial. ff_h_b5cvvet_target_selected_power_product_partial + S (ff_r_b5cvvet_target_selected_power_product) = S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_partial. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_partial * S ((S (ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product) + (ff_r_b5cvvet_target_selected_power_product))) /\\ ((((exists ff_h_b5cvvet_target_selected_power_product_successor. ff_h_b5cvvet_target_selected_power_product_successor + S (ff_s_b5cvvet_target_selected_power_product) = S ((S (S ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product)) /\\ exists ff_q_b5cvvet_target_selected_power_product_successor. ff_u_b5cvvet_target_selected_power_product = ff_q_b5cvvet_target_selected_power_product_successor * S ((S (S ff_i_b5cvvet_target_selected_power_product)) * ff_v_b5cvvet_target_selected_power_product) + (ff_s_b5cvvet_target_selected_power_product))) /\\ ff_s_b5cvvet_target_selected_power_product = ff_r_b5cvvet_target_selected_power_product * ff_p_b5cvvet_target_selected_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_target_selected_divides. b = bpv_result_b5cvvet_target_selected * bpv_factor_b5cvvet_target_selected_divides)))) /\\ forall bpv_candidate_b5cvvet_target. (exists bpv_gap_b5cvvet_target_candidate_bound. bpv_gap_b5cvvet_target_candidate_bound + bpv_candidate_b5cvvet_target = b) -> (exists bpv_result_b5cvvet_target_candidate. ((exists ff_b_b5cvvet_target_candidate_power ff_c_b5cvvet_target_candidate_power. ((forall ff_i_b5cvvet_target_candidate_power_repeat. (exists ff_lt_b5cvvet_target_candidate_power_repeat_bound. ff_lt_b5cvvet_target_candidate_power_repeat_bound + S ff_i_b5cvvet_target_candidate_power_repeat = bpv_candidate_b5cvvet_target) -> (((exists ff_h_b5cvvet_target_candidate_power_repeat_decoded. ff_h_b5cvvet_target_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_b5cvvet_target_candidate_power_repeat)) * ff_c_b5cvvet_target_candidate_power)) /\\ exists ff_q_b5cvvet_target_candidate_power_repeat_decoded. ff_b_b5cvvet_target_candidate_power = ff_q_b5cvvet_target_candidate_power_repeat_decoded * S ((S (ff_i_b5cvvet_target_candidate_power_repeat)) * ff_c_b5cvvet_target_candidate_power) + (p)))) /\\ (exists ff_u_b5cvvet_target_candidate_power_product ff_v_b5cvvet_target_candidate_power_product. ((((exists ff_h_b5cvvet_target_candidate_power_product_start. ff_h_b5cvvet_target_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_start. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_start * S ((S (0)) * ff_v_b5cvvet_target_candidate_power_product) + (1))) /\\ ((((exists ff_h_b5cvvet_target_candidate_power_product_terminal. ff_h_b5cvvet_target_candidate_power_product_terminal + S (bpv_result_b5cvvet_target_candidate) = S ((S (bpv_candidate_b5cvvet_target)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_terminal. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_terminal * S ((S (bpv_candidate_b5cvvet_target)) * ff_v_b5cvvet_target_candidate_power_product) + (bpv_result_b5cvvet_target_candidate))) /\\ forall ff_i_b5cvvet_target_candidate_power_product. (exists ff_lt_b5cvvet_target_candidate_power_product_bound. ff_lt_b5cvvet_target_candidate_power_product_bound + S ff_i_b5cvvet_target_candidate_power_product = bpv_candidate_b5cvvet_target) -> exists ff_p_b5cvvet_target_candidate_power_product ff_r_b5cvvet_target_candidate_power_product ff_s_b5cvvet_target_candidate_power_product. ((((exists ff_h_b5cvvet_target_candidate_power_product_factor. ff_h_b5cvvet_target_candidate_power_product_factor + S (ff_p_b5cvvet_target_candidate_power_product) = S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_c_b5cvvet_target_candidate_power)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_factor. ff_b_b5cvvet_target_candidate_power = ff_q_b5cvvet_target_candidate_power_product_factor * S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_c_b5cvvet_target_candidate_power) + (ff_p_b5cvvet_target_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_target_candidate_power_product_partial. ff_h_b5cvvet_target_candidate_power_product_partial + S (ff_r_b5cvvet_target_candidate_power_product) = S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_partial. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_partial * S ((S (ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product) + (ff_r_b5cvvet_target_candidate_power_product))) /\\ ((((exists ff_h_b5cvvet_target_candidate_power_product_successor. ff_h_b5cvvet_target_candidate_power_product_successor + S (ff_s_b5cvvet_target_candidate_power_product) = S ((S (S ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product)) /\\ exists ff_q_b5cvvet_target_candidate_power_product_successor. ff_u_b5cvvet_target_candidate_power_product = ff_q_b5cvvet_target_candidate_power_product_successor * S ((S (S ff_i_b5cvvet_target_candidate_power_product)) * ff_v_b5cvvet_target_candidate_power_product) + (ff_s_b5cvvet_target_candidate_power_product))) /\\ ff_s_b5cvvet_target_candidate_power_product = ff_r_b5cvvet_target_candidate_power_product * ff_p_b5cvvet_target_candidate_power_product)))))))) /\\ (exists bpv_factor_b5cvvet_target_candidate_divides. b = bpv_result_b5cvvet_target_candidate * bpv_factor_b5cvvet_target_candidate_divides))) -> (exists bpv_gap_b5cvvet_target_maximal. bpv_gap_b5cvvet_target_maximal + bpv_candidate_b5cvvet_target = e))", "statement_sha256": "a714d85092357780e3e2b883a95f56b28f0110fbee763e6b1dd5955d42bbd0da" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "finite_bounded_into_oversized_not_injective", "canonical_catalog_record": { "alpha_v18_promotion": { "bundle_campaign": "four_square", "bundle_node_id": 281, "bundle_sha256": "dd8374b95184f95f28a296aba6682f8177538650c3cc2f8d94a8db723c9982f0", "parent_catalog_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "parent_evidence_status": "body_checked", "parent_row_sha256": "3f41c229313e7fec0a46e8712bdbfb2fd263d6d0024e8adf34c3cdc4ff6419c3" }, "body_checked": true, "body_receipt": { "command_count": 88, "dependency_count": 4, "dne_command_count": 0, "name": "finite_bounded_into_oversized_not_injective", "proof_depth": 33, "proof_edges": 116, "proof_nodes": 117, "proof_objects": 117, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "lt_to_le", "lt_of_lt_of_le", "finite_bounded_injective_surjective", "lt_irrefl_expanded" ], "dependencies_sha256": "bae96f4783a3702c401fbd7837b556e37face6da3286564ab034c3cc4ca54de8", "empty_context_closure": { "body_proof_depth": 33, "body_proof_nodes": 117, "bundle_campaign": "four_square", "bundle_dependency_edge_count": 1187, "bundle_node_count": 390, "bundle_node_id": 281, "bundle_path": "research/arithmetic-library/artifacts/four-square-proof-bundle-v1.json", "bundle_root_id": 389, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "dd8374b95184f95f28a296aba6682f8177538650c3cc2f8d94a8db723c9982f0", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": 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"32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=finite_bounded_into_oversized_not_injective]" } ], "evidence_status": "alpha_closed", "frontier_campaign": "four_square", "frontier_v13_evidence_bundle_sha256": "98ebc13cce85d60d3d8adb2d29b3b7041f81892d859eb607e05b0e4e00a360c6", "logical_spec_sha256": "640e3c38daaa03f582b6a5c20a3904637a0252986fd5f3c0aa362cfb544ca859", "membership": "alpha_only", "name": "finite_bounded_into_oversized_not_injective", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro b", "intro c", "intro l", "intro n", "intro hbounded", "intro hoverflow", "intro hinjective", "have hweak : exists k. k + n = l", "specialize lt_to_le n", "specialize lt_to_le l", "apply lt_to_le", "exact hoverflow", "have hsquarebounded : forall fp_i_ftsp_square_bounded. (exists fp_gap_ftsp_square_bounded_index. fp_gap_ftsp_square_bounded_index + S fp_i_ftsp_square_bounded = n) -> exists fp_value_ftsp_square_bounded. ((((exists ff_h_ftsp_square_bounded_entry. ff_h_ftsp_square_bounded_entry + S (fp_value_ftsp_square_bounded) = S ((S (fp_i_ftsp_square_bounded)) * c)) /\\ exists ff_q_ftsp_square_bounded_entry. b = ff_q_ftsp_square_bounded_entry * S ((S (fp_i_ftsp_square_bounded)) * c) + (fp_value_ftsp_square_bounded))) /\\ (exists fp_gap_ftsp_square_bounded_value. fp_gap_ftsp_square_bounded_value + S fp_value_ftsp_square_bounded = n))", "intro i", "intro hi", "have hlarge : exists k. k + S i = l", "specialize lt_of_lt_of_le i", "specialize lt_of_lt_of_le n", "specialize lt_of_lt_of_le l", "apply lt_of_lt_of_le", "exact hi", "exact hweak", "specialize hbounded i", "apply hbounded", "exact hlarge", "have hsquareinjective : forall fp_i_ftsp_square_injective fp_j_ftsp_square_injective fp_value_ftsp_square_injective. 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(exists fp_gap_ftsp_square_surjective_value. fp_gap_ftsp_square_surjective_value + S fp_value_ftsp_square_surjective = n) -> exists fp_i_ftsp_square_surjective. ((exists fp_gap_ftsp_square_surjective_index. fp_gap_ftsp_square_surjective_index + S fp_i_ftsp_square_surjective = n) /\\ (((exists ff_h_ftsp_square_surjective_entry. ff_h_ftsp_square_surjective_entry + S (fp_value_ftsp_square_surjective) = S ((S (fp_i_ftsp_square_surjective)) * c)) /\\ exists ff_q_ftsp_square_surjective_entry. b = ff_q_ftsp_square_surjective_entry * S ((S (fp_i_ftsp_square_surjective)) * c) + (fp_value_ftsp_square_surjective))))", "specialize finite_bounded_injective_surjective n", "specialize finite_bounded_injective_surjective b", "specialize finite_bounded_injective_surjective c", "apply finite_bounded_injective_surjective", "exact hsquarebounded", "exact hsquareinjective", "have hlast : exists v. ((((exists ff_h_ftsp_last_entry. ff_h_ftsp_last_entry + S (v) = S ((S (n)) * c)) /\\ exists ff_q_ftsp_last_entry. b = ff_q_ftsp_last_entry * S ((S (n)) * c) + (v))) /\\ (exists ftsp_gap_last_value. ftsp_gap_last_value + S (v) = n))", "specialize hbounded n", "apply hbounded", "exact hoverflow", "cases hlast", "cases hlast_witness", "have hearlier : exists i. ((exists ftsp_gap_earlier_index. ftsp_gap_earlier_index + S (i) = n) /\\ (((exists ff_h_ftsp_earlier_entry. ff_h_ftsp_earlier_entry + S (x) = S ((S (i)) * c)) /\\ exists ff_q_ftsp_earlier_entry. b = ff_q_ftsp_earlier_entry * S ((S (i)) * c) + (x))))", "specialize hsurjective x", "apply hsurjective", "exact hlast_witness_right", "cases hearlier", "cases hearlier_witness", "have hequal : x1 = n", "specialize hinjective x1", "specialize hinjective n", "specialize hinjective x", "apply hinjective", "specialize lt_of_lt_of_le x1", "specialize lt_of_lt_of_le n", "specialize lt_of_lt_of_le l", "apply lt_of_lt_of_le", "exact hearlier_witness_left", "exact hweak", "exact hoverflow", "exact hearlier_witness_right", "exact hlast_witness_left", "rewrite hequal at hearlier_witness_left", "specialize lt_irrefl_expanded n", "apply lt_irrefl_expanded", "exact hearlier_witness_left" ], "script_sha256": "f5ca72a87e8ffb9332022ebb22fa904f586daa8d166c901e4d79ff1566a383e9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/fermat_two_squares_pigeonhole_candidate.py", "sha256": "169fca06966de858a5c7dd85cb69f92586107c35e2d82216867752904692dac1" }, "statement": "forall b c l n. (forall fom_index_ftsp_full_bounded. (exists fom_gap_ftsp_full_bounded_index_bound. fom_gap_ftsp_full_bounded_index_bound + S (fom_index_ftsp_full_bounded) = l) -> exists fom_value_ftsp_full_bounded. ((((exists fom_beta_height_ftsp_full_bounded_entry. fom_beta_height_ftsp_full_bounded_entry + S (fom_value_ftsp_full_bounded) = S ((S (fom_index_ftsp_full_bounded)) * c)) /\\ exists fom_beta_quotient_ftsp_full_bounded_entry. b = fom_beta_quotient_ftsp_full_bounded_entry * S ((S (fom_index_ftsp_full_bounded)) * c) + (fom_value_ftsp_full_bounded))) /\\ (exists fom_gap_ftsp_full_bounded_value_bound. fom_gap_ftsp_full_bounded_value_bound + S (fom_value_ftsp_full_bounded) = n))) -> (exists ftsp_gap_domain_overflow. ftsp_gap_domain_overflow + S (n) = l) -> ~(forall fp_i_ftsp_full_injective fp_j_ftsp_full_injective fp_value_ftsp_full_injective. (exists fp_gap_ftsp_full_injective_i. fp_gap_ftsp_full_injective_i + S fp_i_ftsp_full_injective = l) -> (exists fp_gap_ftsp_full_injective_j. fp_gap_ftsp_full_injective_j + S fp_j_ftsp_full_injective = l) -> (((exists ff_h_ftsp_full_injective_left. ff_h_ftsp_full_injective_left + S (fp_value_ftsp_full_injective) = S ((S (fp_i_ftsp_full_injective)) * c)) /\\ exists ff_q_ftsp_full_injective_left. b = ff_q_ftsp_full_injective_left * S ((S (fp_i_ftsp_full_injective)) * c) + (fp_value_ftsp_full_injective))) -> (((exists ff_h_ftsp_full_injective_right. ff_h_ftsp_full_injective_right + S (fp_value_ftsp_full_injective) = S ((S (fp_j_ftsp_full_injective)) * c)) /\\ exists ff_q_ftsp_full_injective_right. b = ff_q_ftsp_full_injective_right * S ((S (fp_j_ftsp_full_injective)) * c) + (fp_value_ftsp_full_injective))) -> fp_i_ftsp_full_injective = fp_j_ftsp_full_injective)", "statement_sha256": "7c53db5855f093875477b5e6d26e2be3a2a0e0d035fdd5c7910f0ce46269d59a", "summary": "An explicitly bounded beta-coded map from a larger finite domain cannot be injective.", "summary_sha256": "84d636593098cdac815b5d1e98471cb3fd28defdeada5516645ec088174e4f08" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "lt_to_le", "lt_of_lt_of_le", "finite_bounded_injective_surjective", "lt_irrefl_expanded" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "169fca06966de858a5c7dd85cb69f92586107c35e2d82216867752904692dac1", "kind": "frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/fermat_two_squares_pigeonhole_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "27dd454ddd5c93542338b4cf48795cb21db413899d5a02d8ad3bfe271967a74e", "kind": "frontier_executable_audit", "path": "peano-lab/py/tests/test_fermat_two_squares_pigeonhole_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "963c818f646cfdecd2e6a35025d525775bc627e6e216cdbdd040241c8cd397aa", "kind": "frontier_campaign_rfc", "path": "research/arithmetic-library/fermat-two-squares-pigeonhole-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "825909e057492de87ef08208451c3475396ca009179c513457b05b57f7e2f109", "kind": "sealed_alpha_v12_parent", "path": "artifacts/peano-library/alpha/catalog-v12.json", "role": "exact_parent_catalog_bytes", "selector": "document" }, { "document_sha256": "dd8374b95184f95f28a296aba6682f8177538650c3cc2f8d94a8db723c9982f0", "kind": "four_square_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/four-square-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=281]" }, { "document_sha256": "caa717463ccd81e2bd8ac8a2449b4c3d7098b41f747fcd563701d0ac8d553eed", "kind": "four_square_ordinary_kernel_and_compiled_lean_receipt", "path": "research/arithmetic-library/four-square-complete-closure-receipt.md", "role": "original_kernel_and_independent_compiled_lean_verification", "selector": "document" }, { "document_sha256": "32acaae2a4dff14862469cf441e527ec1e1efbfff57974c246d603cd7a2e68d9", "kind": "sealed_alpha_v17_parent", "path": "artifacts/peano-library/alpha/catalog-v17.json", "role": "exact_immutable_pre_promotion_catalog_bytes", "selector": "theorems[name=finite_bounded_into_oversized_not_injective]" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "finite_bounded_into_oversized_not_injective", "parent_alpha_version": "v34", "proof_bundle_node_id": 225, "reference_route": "jordan-totient/checkpoint.html#theorem-finite_bounded_into_oversized_not_injective", "script": [ "intro b", "intro c", "intro l", "intro n", "intro hbounded", "intro hoverflow", "intro hinjective", "have hweak : exists k. k + n = l", "specialize lt_to_le n", "specialize lt_to_le l", "apply lt_to_le", "exact hoverflow", "have hsquarebounded : forall fp_i_ftsp_square_bounded. 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((((exists ff_h_ftsp_last_entry. ff_h_ftsp_last_entry + S (v) = S ((S (n)) * c)) /\\ exists ff_q_ftsp_last_entry. b = ff_q_ftsp_last_entry * S ((S (n)) * c) + (v))) /\\ (exists ftsp_gap_last_value. ftsp_gap_last_value + S (v) = n))", "specialize hbounded n", "apply hbounded", "exact hoverflow", "cases hlast", "cases hlast_witness", "have hearlier : exists i. ((exists ftsp_gap_earlier_index. ftsp_gap_earlier_index + S (i) = n) /\\ (((exists ff_h_ftsp_earlier_entry. ff_h_ftsp_earlier_entry + S (x) = S ((S (i)) * c)) /\\ exists ff_q_ftsp_earlier_entry. b = ff_q_ftsp_earlier_entry * S ((S (i)) * c) + (x))))", "specialize hsurjective x", "apply hsurjective", "exact hlast_witness_right", "cases hearlier", "cases hearlier_witness", "have hequal : x1 = n", "specialize hinjective x1", "specialize hinjective n", "specialize hinjective x", "apply hinjective", "specialize lt_of_lt_of_le x1", "specialize lt_of_lt_of_le n", "specialize lt_of_lt_of_le l", "apply lt_of_lt_of_le", "exact hearlier_witness_left", "exact hweak", "exact hoverflow", "exact hearlier_witness_right", "exact hlast_witness_left", "rewrite hequal at hearlier_witness_left", "specialize lt_irrefl_expanded n", "apply lt_irrefl_expanded", "exact hearlier_witness_left" ], "script_sha256": "f5ca72a87e8ffb9332022ebb22fa904f586daa8d166c901e4d79ff1566a383e9", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/fermat_two_squares_pigeonhole_candidate.py", "sha256": "169fca06966de858a5c7dd85cb69f92586107c35e2d82216867752904692dac1" }, "stable_member": false, "statement": "forall b c l n. 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((~(p = 1) /\\ forall frm_prime_left_kmcvznd_prime frm_prime_right_kmcvznd_prime. p = frm_prime_left_kmcvznd_prime * frm_prime_right_kmcvznd_prime -> frm_prime_left_kmcvznd_prime = 1 \\/ frm_prime_right_kmcvznd_prime = 1)) -> ~(c = 0) -> (((exists bpv_gap_kmcvznd_valuation_exponent_bound. bpv_gap_kmcvznd_valuation_exponent_bound + e = c) /\\ (exists bpv_result_kmcvznd_valuation_selected. ((exists ff_b_kmcvznd_valuation_selected_power ff_c_kmcvznd_valuation_selected_power. ((forall ff_i_kmcvznd_valuation_selected_power_repeat. (exists ff_lt_kmcvznd_valuation_selected_power_repeat_bound. ff_lt_kmcvznd_valuation_selected_power_repeat_bound + S ff_i_kmcvznd_valuation_selected_power_repeat = e) -> (((exists ff_h_kmcvznd_valuation_selected_power_repeat_decoded. ff_h_kmcvznd_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_kmcvznd_valuation_selected_power_repeat)) * ff_c_kmcvznd_valuation_selected_power)) /\\ exists ff_q_kmcvznd_valuation_selected_power_repeat_decoded. ff_b_kmcvznd_valuation_selected_power = ff_q_kmcvznd_valuation_selected_power_repeat_decoded * S ((S (ff_i_kmcvznd_valuation_selected_power_repeat)) * ff_c_kmcvznd_valuation_selected_power) + (p)))) /\\ (exists ff_u_kmcvznd_valuation_selected_power_product ff_v_kmcvznd_valuation_selected_power_product. ((((exists ff_h_kmcvznd_valuation_selected_power_product_start. ff_h_kmcvznd_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_start. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_start * S ((S (0)) * ff_v_kmcvznd_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_kmcvznd_valuation_selected_power_product_terminal. ff_h_kmcvznd_valuation_selected_power_product_terminal + S (bpv_result_kmcvznd_valuation_selected) = S ((S (e)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_terminal. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_kmcvznd_valuation_selected_power_product) + (bpv_result_kmcvznd_valuation_selected))) /\\ forall ff_i_kmcvznd_valuation_selected_power_product. (exists ff_lt_kmcvznd_valuation_selected_power_product_bound. ff_lt_kmcvznd_valuation_selected_power_product_bound + S ff_i_kmcvznd_valuation_selected_power_product = e) -> exists ff_p_kmcvznd_valuation_selected_power_product ff_r_kmcvznd_valuation_selected_power_product ff_s_kmcvznd_valuation_selected_power_product. ((((exists ff_h_kmcvznd_valuation_selected_power_product_factor. ff_h_kmcvznd_valuation_selected_power_product_factor + S (ff_p_kmcvznd_valuation_selected_power_product) = S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_c_kmcvznd_valuation_selected_power)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_factor. ff_b_kmcvznd_valuation_selected_power = ff_q_kmcvznd_valuation_selected_power_product_factor * S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_c_kmcvznd_valuation_selected_power) + (ff_p_kmcvznd_valuation_selected_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_selected_power_product_partial. ff_h_kmcvznd_valuation_selected_power_product_partial + S (ff_r_kmcvznd_valuation_selected_power_product) = S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_partial. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_partial * S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product) + (ff_r_kmcvznd_valuation_selected_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_selected_power_product_successor. ff_h_kmcvznd_valuation_selected_power_product_successor + S (ff_s_kmcvznd_valuation_selected_power_product) = S ((S (S ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_successor. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_successor * S ((S (S ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product) + (ff_s_kmcvznd_valuation_selected_power_product))) /\\ ff_s_kmcvznd_valuation_selected_power_product = ff_r_kmcvznd_valuation_selected_power_product * ff_p_kmcvznd_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_kmcvznd_valuation_selected_divides. c = bpv_result_kmcvznd_valuation_selected * bpv_factor_kmcvznd_valuation_selected_divides)))) /\\ forall bpv_candidate_kmcvznd_valuation. (exists bpv_gap_kmcvznd_valuation_candidate_bound. bpv_gap_kmcvznd_valuation_candidate_bound + bpv_candidate_kmcvznd_valuation = c) -> (exists bpv_result_kmcvznd_valuation_candidate. ((exists ff_b_kmcvznd_valuation_candidate_power ff_c_kmcvznd_valuation_candidate_power. ((forall ff_i_kmcvznd_valuation_candidate_power_repeat. (exists ff_lt_kmcvznd_valuation_candidate_power_repeat_bound. ff_lt_kmcvznd_valuation_candidate_power_repeat_bound + S ff_i_kmcvznd_valuation_candidate_power_repeat = bpv_candidate_kmcvznd_valuation) -> (((exists ff_h_kmcvznd_valuation_candidate_power_repeat_decoded. ff_h_kmcvznd_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_kmcvznd_valuation_candidate_power_repeat)) * ff_c_kmcvznd_valuation_candidate_power)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_repeat_decoded. ff_b_kmcvznd_valuation_candidate_power = ff_q_kmcvznd_valuation_candidate_power_repeat_decoded * S ((S (ff_i_kmcvznd_valuation_candidate_power_repeat)) * ff_c_kmcvznd_valuation_candidate_power) + (p)))) /\\ (exists ff_u_kmcvznd_valuation_candidate_power_product ff_v_kmcvznd_valuation_candidate_power_product. ((((exists ff_h_kmcvznd_valuation_candidate_power_product_start. ff_h_kmcvznd_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_start. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_start * S ((S (0)) * ff_v_kmcvznd_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_kmcvznd_valuation_candidate_power_product_terminal. ff_h_kmcvznd_valuation_candidate_power_product_terminal + S (bpv_result_kmcvznd_valuation_candidate) = S ((S (bpv_candidate_kmcvznd_valuation)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_terminal. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_kmcvznd_valuation)) * ff_v_kmcvznd_valuation_candidate_power_product) + (bpv_result_kmcvznd_valuation_candidate))) /\\ forall ff_i_kmcvznd_valuation_candidate_power_product. (exists ff_lt_kmcvznd_valuation_candidate_power_product_bound. ff_lt_kmcvznd_valuation_candidate_power_product_bound + S ff_i_kmcvznd_valuation_candidate_power_product = bpv_candidate_kmcvznd_valuation) -> exists ff_p_kmcvznd_valuation_candidate_power_product ff_r_kmcvznd_valuation_candidate_power_product ff_s_kmcvznd_valuation_candidate_power_product. ((((exists ff_h_kmcvznd_valuation_candidate_power_product_factor. ff_h_kmcvznd_valuation_candidate_power_product_factor + S (ff_p_kmcvznd_valuation_candidate_power_product) = S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_c_kmcvznd_valuation_candidate_power)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_factor. ff_b_kmcvznd_valuation_candidate_power = ff_q_kmcvznd_valuation_candidate_power_product_factor * S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_c_kmcvznd_valuation_candidate_power) + (ff_p_kmcvznd_valuation_candidate_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_candidate_power_product_partial. ff_h_kmcvznd_valuation_candidate_power_product_partial + S (ff_r_kmcvznd_valuation_candidate_power_product) = S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_partial. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_partial * S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product) + (ff_r_kmcvznd_valuation_candidate_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_candidate_power_product_successor. ff_h_kmcvznd_valuation_candidate_power_product_successor + S (ff_s_kmcvznd_valuation_candidate_power_product) = S ((S (S ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_successor. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_successor * S ((S (S ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product) + (ff_s_kmcvznd_valuation_candidate_power_product))) /\\ ff_s_kmcvznd_valuation_candidate_power_product = ff_r_kmcvznd_valuation_candidate_power_product * ff_p_kmcvznd_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_kmcvznd_valuation_candidate_divides. c = bpv_result_kmcvznd_valuation_candidate * bpv_factor_kmcvznd_valuation_candidate_divides))) -> (exists bpv_gap_kmcvznd_valuation_maximal. bpv_gap_kmcvznd_valuation_maximal + bpv_candidate_kmcvznd_valuation = e)) -> ((e = 0 -> ~(exists bpv_factor_kmcvznd_divides. c = p * bpv_factor_kmcvznd_divides)) /\\ (~(exists bpv_factor_kmcvznd_divides. c = p * bpv_factor_kmcvznd_divides) -> e = 0))", "statement_sha256": "292dc2c7648b6c57ffa192b9901964a6b3f4e3368cd922d70b9010c983d98977", "summary": "At a prime base and nonzero value, valuation zero is equivalent to nondivisibility.", "summary_sha256": "f353e04f402df2597afa900a3dfd2adf973d724fc429aa0336a6ed64f8a37a43" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "prime_divisor_power_valuation_nonzero", "power_valuation_nonzero_exponent_divides_base", "eq_decidable" ], 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((~(p = 1) /\\ forall frm_prime_left_kmcvznd_prime frm_prime_right_kmcvznd_prime. p = frm_prime_left_kmcvznd_prime * frm_prime_right_kmcvznd_prime -> frm_prime_left_kmcvznd_prime = 1 \\/ frm_prime_right_kmcvznd_prime = 1)) -> ~(c = 0) -> (((exists bpv_gap_kmcvznd_valuation_exponent_bound. bpv_gap_kmcvznd_valuation_exponent_bound + e = c) /\\ (exists bpv_result_kmcvznd_valuation_selected. ((exists ff_b_kmcvznd_valuation_selected_power ff_c_kmcvznd_valuation_selected_power. ((forall ff_i_kmcvznd_valuation_selected_power_repeat. (exists ff_lt_kmcvznd_valuation_selected_power_repeat_bound. ff_lt_kmcvznd_valuation_selected_power_repeat_bound + S ff_i_kmcvznd_valuation_selected_power_repeat = e) -> (((exists ff_h_kmcvznd_valuation_selected_power_repeat_decoded. ff_h_kmcvznd_valuation_selected_power_repeat_decoded + S (p) = S ((S (ff_i_kmcvznd_valuation_selected_power_repeat)) * ff_c_kmcvznd_valuation_selected_power)) /\\ exists ff_q_kmcvznd_valuation_selected_power_repeat_decoded. ff_b_kmcvznd_valuation_selected_power = ff_q_kmcvznd_valuation_selected_power_repeat_decoded * S ((S (ff_i_kmcvznd_valuation_selected_power_repeat)) * ff_c_kmcvznd_valuation_selected_power) + (p)))) /\\ (exists ff_u_kmcvznd_valuation_selected_power_product ff_v_kmcvznd_valuation_selected_power_product. ((((exists ff_h_kmcvznd_valuation_selected_power_product_start. ff_h_kmcvznd_valuation_selected_power_product_start + S (1) = S ((S (0)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_start. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_start * S ((S (0)) * ff_v_kmcvznd_valuation_selected_power_product) + (1))) /\\ ((((exists ff_h_kmcvznd_valuation_selected_power_product_terminal. ff_h_kmcvznd_valuation_selected_power_product_terminal + S (bpv_result_kmcvznd_valuation_selected) = S ((S (e)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_terminal. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_terminal * S ((S (e)) * ff_v_kmcvznd_valuation_selected_power_product) + (bpv_result_kmcvznd_valuation_selected))) /\\ forall ff_i_kmcvznd_valuation_selected_power_product. (exists ff_lt_kmcvznd_valuation_selected_power_product_bound. ff_lt_kmcvznd_valuation_selected_power_product_bound + S ff_i_kmcvznd_valuation_selected_power_product = e) -> exists ff_p_kmcvznd_valuation_selected_power_product ff_r_kmcvznd_valuation_selected_power_product ff_s_kmcvznd_valuation_selected_power_product. ((((exists ff_h_kmcvznd_valuation_selected_power_product_factor. ff_h_kmcvznd_valuation_selected_power_product_factor + S (ff_p_kmcvznd_valuation_selected_power_product) = S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_c_kmcvznd_valuation_selected_power)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_factor. ff_b_kmcvznd_valuation_selected_power = ff_q_kmcvznd_valuation_selected_power_product_factor * S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_c_kmcvznd_valuation_selected_power) + (ff_p_kmcvznd_valuation_selected_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_selected_power_product_partial. ff_h_kmcvznd_valuation_selected_power_product_partial + S (ff_r_kmcvznd_valuation_selected_power_product) = S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_partial. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_partial * S ((S (ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product) + (ff_r_kmcvznd_valuation_selected_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_selected_power_product_successor. ff_h_kmcvznd_valuation_selected_power_product_successor + S (ff_s_kmcvznd_valuation_selected_power_product) = S ((S (S ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product)) /\\ exists ff_q_kmcvznd_valuation_selected_power_product_successor. ff_u_kmcvznd_valuation_selected_power_product = ff_q_kmcvznd_valuation_selected_power_product_successor * S ((S (S ff_i_kmcvznd_valuation_selected_power_product)) * ff_v_kmcvznd_valuation_selected_power_product) + (ff_s_kmcvznd_valuation_selected_power_product))) /\\ ff_s_kmcvznd_valuation_selected_power_product = ff_r_kmcvznd_valuation_selected_power_product * ff_p_kmcvznd_valuation_selected_power_product)))))))) /\\ (exists bpv_factor_kmcvznd_valuation_selected_divides. c = bpv_result_kmcvznd_valuation_selected * bpv_factor_kmcvznd_valuation_selected_divides)))) /\\ forall bpv_candidate_kmcvznd_valuation. (exists bpv_gap_kmcvznd_valuation_candidate_bound. bpv_gap_kmcvznd_valuation_candidate_bound + bpv_candidate_kmcvznd_valuation = c) -> (exists bpv_result_kmcvznd_valuation_candidate. ((exists ff_b_kmcvznd_valuation_candidate_power ff_c_kmcvznd_valuation_candidate_power. ((forall ff_i_kmcvznd_valuation_candidate_power_repeat. (exists ff_lt_kmcvznd_valuation_candidate_power_repeat_bound. ff_lt_kmcvznd_valuation_candidate_power_repeat_bound + S ff_i_kmcvznd_valuation_candidate_power_repeat = bpv_candidate_kmcvznd_valuation) -> (((exists ff_h_kmcvznd_valuation_candidate_power_repeat_decoded. ff_h_kmcvznd_valuation_candidate_power_repeat_decoded + S (p) = S ((S (ff_i_kmcvznd_valuation_candidate_power_repeat)) * ff_c_kmcvznd_valuation_candidate_power)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_repeat_decoded. ff_b_kmcvznd_valuation_candidate_power = ff_q_kmcvznd_valuation_candidate_power_repeat_decoded * S ((S (ff_i_kmcvznd_valuation_candidate_power_repeat)) * ff_c_kmcvznd_valuation_candidate_power) + (p)))) /\\ (exists ff_u_kmcvznd_valuation_candidate_power_product ff_v_kmcvznd_valuation_candidate_power_product. ((((exists ff_h_kmcvznd_valuation_candidate_power_product_start. ff_h_kmcvznd_valuation_candidate_power_product_start + S (1) = S ((S (0)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_start. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_start * S ((S (0)) * ff_v_kmcvznd_valuation_candidate_power_product) + (1))) /\\ ((((exists ff_h_kmcvznd_valuation_candidate_power_product_terminal. ff_h_kmcvznd_valuation_candidate_power_product_terminal + S (bpv_result_kmcvznd_valuation_candidate) = S ((S (bpv_candidate_kmcvznd_valuation)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_terminal. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_terminal * S ((S (bpv_candidate_kmcvznd_valuation)) * ff_v_kmcvznd_valuation_candidate_power_product) + (bpv_result_kmcvznd_valuation_candidate))) /\\ forall ff_i_kmcvznd_valuation_candidate_power_product. (exists ff_lt_kmcvznd_valuation_candidate_power_product_bound. ff_lt_kmcvznd_valuation_candidate_power_product_bound + S ff_i_kmcvznd_valuation_candidate_power_product = bpv_candidate_kmcvznd_valuation) -> exists ff_p_kmcvznd_valuation_candidate_power_product ff_r_kmcvznd_valuation_candidate_power_product ff_s_kmcvznd_valuation_candidate_power_product. ((((exists ff_h_kmcvznd_valuation_candidate_power_product_factor. ff_h_kmcvznd_valuation_candidate_power_product_factor + S (ff_p_kmcvznd_valuation_candidate_power_product) = S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_c_kmcvznd_valuation_candidate_power)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_factor. ff_b_kmcvznd_valuation_candidate_power = ff_q_kmcvznd_valuation_candidate_power_product_factor * S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_c_kmcvznd_valuation_candidate_power) + (ff_p_kmcvznd_valuation_candidate_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_candidate_power_product_partial. ff_h_kmcvznd_valuation_candidate_power_product_partial + S (ff_r_kmcvznd_valuation_candidate_power_product) = S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_partial. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_partial * S ((S (ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product) + (ff_r_kmcvznd_valuation_candidate_power_product))) /\\ ((((exists ff_h_kmcvznd_valuation_candidate_power_product_successor. ff_h_kmcvznd_valuation_candidate_power_product_successor + S (ff_s_kmcvznd_valuation_candidate_power_product) = S ((S (S ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product)) /\\ exists ff_q_kmcvznd_valuation_candidate_power_product_successor. ff_u_kmcvznd_valuation_candidate_power_product = ff_q_kmcvznd_valuation_candidate_power_product_successor * S ((S (S ff_i_kmcvznd_valuation_candidate_power_product)) * ff_v_kmcvznd_valuation_candidate_power_product) + (ff_s_kmcvznd_valuation_candidate_power_product))) /\\ ff_s_kmcvznd_valuation_candidate_power_product = ff_r_kmcvznd_valuation_candidate_power_product * ff_p_kmcvznd_valuation_candidate_power_product)))))))) /\\ (exists bpv_factor_kmcvznd_valuation_candidate_divides. c = bpv_result_kmcvznd_valuation_candidate * bpv_factor_kmcvznd_valuation_candidate_divides))) -> (exists bpv_gap_kmcvznd_valuation_maximal. bpv_gap_kmcvznd_valuation_maximal + bpv_candidate_kmcvznd_valuation = e)) -> ((e = 0 -> ~(exists bpv_factor_kmcvznd_divides. c = p * bpv_factor_kmcvznd_divides)) /\\ (~(exists bpv_factor_kmcvznd_divides. c = p * bpv_factor_kmcvznd_divides) -> e = 0))", "statement_sha256": "292dc2c7648b6c57ffa192b9901964a6b3f4e3368cd922d70b9010c983d98977" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "linear_congruence_zero_residue_divides", "canonical_catalog_record": { "alpha_v19_frontier_enrollment": { "body_receipt_sha256": "596870a4517c6c09d6fe71d75b73391d422504f41c3799129d0f703f3343d7fe", "bundle_campaign": "frontier", "bundle_node_id": 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"cases hmod_witness", "specialize factor_difference d", "specialize factor_difference x1", "specialize factor_difference x", "specialize factor_difference n", "apply factor_difference", "trans 0 + d * x1", "symm", "apply zero_add", "trans n + d * x", "symm", "exact hmod_witness_witness", "apply add_comm" ], "script_sha256": "111fd7e65447015c45b36cc95f4d8b2b7c39ef9de316e673f7325b83a5c9552c", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_complete_candidate.py", "sha256": "133cab6b63fa7b2341a475ebb4a78ca22882d8176224c3fe4d91c13d7df2589f" }, "statement": "forall d n. 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"empty_context_closure": { "body_proof_depth": 47, "body_proof_nodes": 155, "bundle_campaign": "breakthrough_layer", "bundle_dependency_edge_count": 820, "bundle_node_count": 302, "bundle_node_id": 293, "bundle_path": "research/arithmetic-library/artifacts/alpha-v25-breakthrough-layer-proof-bundle-v1.json", "bundle_root_id": 301, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "d4532076049be869e4e397d0fcee81b668bd3fd5c7d9173028bb1bdb80b9793a", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "e3b28cbcdf65cdad1e51c834812bf2efb8a45cb534bb8a5daa1e4245b4d0a347", "status": "checked" }, "enrollment_index": 2072, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "1f88ba8af0b1169072387419c4bd9732cae20b94008ab476d3bfda3acaa00859", "kind": "alpha_v25_frontier_dependency_curried_body", "path": 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"independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=293]" }, { "document_sha256": "b8bc43dd4a4aa78077855ae27924def378bda190db977894e069c5e4c7bfb27d", "kind": "alpha_v25_breakthrough_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v25-breakthrough-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "94ac4d193cbfe8c2ec04e54024221bc2c3a534c0ae014d381663b86174b3dcc1", "kind": "sealed_alpha_v24_parent", "path": "artifacts/peano-library/alpha/catalog-v24.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "generalized_crt_compatibility", "logical_spec_sha256": "82069840a6f7541aa8e51298e51eb9c05e6187f0289212faac8c64a57082e14a", "membership": "alpha_only", "name": "crt_is_gcd_coprime_product", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro a", "intro b", "intro n", "intro ga", "intro gb", "intro P", "intro T", "intro hnzero", "intro hT", "intro hP", "intro hab", "intro hga", "intro hgb", "have haquot : exists q. a = ga * q", "specialize is_gcd_dvd_left ga", "specialize is_gcd_dvd_left a", "specialize is_gcd_dvd_left n", "apply is_gcd_dvd_left", "exact hga", "cases haquot", "have hnquot : exists q. n = ga * q", "specialize is_gcd_dvd_right ga", "specialize is_gcd_dvd_right a", "specialize is_gcd_dvd_right n", "apply is_gcd_dvd_right", "exact hga", "cases hnquot", "have hganzero : ~(ga = 0)", "specialize factor_nonzero_left n", "specialize factor_nonzero_left ga", "specialize factor_nonzero_left x1", "intro hzero", "apply factor_nonzero_left", "exact hnzero", "exact hnquot_witness", "exact hzero", "have hquotcop : forall frp_divisor_gcomp_product_quotients. (exists frp_left_factor_gcomp_product_quotients. x = frp_divisor_gcomp_product_quotients * frp_left_factor_gcomp_product_quotients) -> (exists frp_right_factor_gcomp_product_quotients. x1 = frp_divisor_gcomp_product_quotients * frp_right_factor_gcomp_product_quotients) -> frp_divisor_gcomp_product_quotients = 1", "specialize is_gcd_quotients_coprime_nonzero ga", "specialize is_gcd_quotients_coprime_nonzero a", "specialize is_gcd_quotients_coprime_nonzero n", "specialize is_gcd_quotients_coprime_nonzero x", "specialize is_gcd_quotients_coprime_nonzero x1", "apply is_gcd_quotients_coprime_nonzero", "exact hga", "exact hganzero", "exact haquot_witness", "exact hnquot_witness", "have hgbcop : forall frp_divisor_gcomp_product_scale_coprime. (exists frp_left_factor_gcomp_product_scale_coprime. ga = frp_divisor_gcomp_product_scale_coprime * frp_left_factor_gcomp_product_scale_coprime) -> (exists frp_right_factor_gcomp_product_scale_coprime. b = frp_divisor_gcomp_product_scale_coprime * frp_right_factor_gcomp_product_scale_coprime) -> frp_divisor_gcomp_product_scale_coprime = 1", "specialize crt_coprime_divisor_pair a", "specialize crt_coprime_divisor_pair b", "specialize crt_coprime_divisor_pair ga", "specialize crt_coprime_divisor_pair b", "apply crt_coprime_divisor_pair", "exact hab", "exists x", "exact haquot_witness", "specialize multiple_refl b", "exact multiple_refl", "specialize canonical_gcd_exists b", "specialize canonical_gcd_exists x1", "cases canonical_gcd_exists", "have hswap : (((exists hag_left_factor_gcomp_product_intermediate_swap. x1 = x2 * hag_left_factor_gcomp_product_intermediate_swap) /\\ (exists hag_right_factor_gcomp_product_intermediate_swap. b = x2 * hag_right_factor_gcomp_product_intermediate_swap)) /\\ forall hag_divisor_gcomp_product_intermediate_swap. (exists hag_common_left_gcomp_product_intermediate_swap. x1 = hag_divisor_gcomp_product_intermediate_swap * hag_common_left_gcomp_product_intermediate_swap) -> (exists hag_common_right_gcomp_product_intermediate_swap. b = hag_divisor_gcomp_product_intermediate_swap * hag_common_right_gcomp_product_intermediate_swap) -> exists hag_greatest_factor_gcomp_product_intermediate_swap. x2 = hag_divisor_gcomp_product_intermediate_swap * hag_greatest_factor_gcomp_product_intermediate_swap)", "specialize is_gcd_symm x2", "specialize is_gcd_symm b", "specialize is_gcd_symm x1", "apply is_gcd_symm", "exact canonical_gcd_exists_witness", "have hrestore : (((exists hag_left_factor_gcomp_product_restore_scaled. n = x2 * hag_left_factor_gcomp_product_restore_scaled) /\\ (exists hag_right_factor_gcomp_product_restore_scaled. b = x2 * hag_right_factor_gcomp_product_restore_scaled)) /\\ forall hag_divisor_gcomp_product_restore_scaled. (exists hag_common_left_gcomp_product_restore_scaled. n = hag_divisor_gcomp_product_restore_scaled * hag_common_left_gcomp_product_restore_scaled) -> (exists hag_common_right_gcomp_product_restore_scaled. b = hag_divisor_gcomp_product_restore_scaled * hag_common_right_gcomp_product_restore_scaled) -> exists hag_greatest_factor_gcomp_product_restore_scaled. x2 = hag_divisor_gcomp_product_restore_scaled * hag_greatest_factor_gcomp_product_restore_scaled)", "specialize crt_is_gcd_coprime_factor_remove ga", "specialize crt_is_gcd_coprime_factor_remove x1", "specialize crt_is_gcd_coprime_factor_remove b", "specialize crt_is_gcd_coprime_factor_remove x2", "specialize crt_is_gcd_coprime_factor_remove n", "apply crt_is_gcd_coprime_factor_remove", "exact hnquot_witness", "exact hgbcop", "exact hswap", "have hback : (((exists hag_left_factor_gcomp_product_restore_back. b = x2 * hag_left_factor_gcomp_product_restore_back) /\\ (exists hag_right_factor_gcomp_product_restore_back. n = x2 * hag_right_factor_gcomp_product_restore_back)) /\\ forall hag_divisor_gcomp_product_restore_back. (exists hag_common_left_gcomp_product_restore_back. b = hag_divisor_gcomp_product_restore_back * hag_common_left_gcomp_product_restore_back) -> (exists hag_common_right_gcomp_product_restore_back. n = hag_divisor_gcomp_product_restore_back * hag_common_right_gcomp_product_restore_back) -> exists hag_greatest_factor_gcomp_product_restore_back. x2 = hag_divisor_gcomp_product_restore_back * hag_greatest_factor_gcomp_product_restore_back)", "specialize is_gcd_symm x2", "specialize is_gcd_symm n", "specialize is_gcd_symm b", "apply is_gcd_symm", "exact hrestore", "have heq : x2 = gb", "specialize is_gcd_unique x2", "specialize is_gcd_unique gb", "specialize is_gcd_unique b", "specialize is_gcd_unique n", "apply is_gcd_unique", "exact hback", "exact hgb", "specialize crt_product_witness x", "specialize crt_product_witness b", "cases crt_product_witness", "have hbase : (((exists hag_left_factor_gcomp_product_base_temporary. x3 = x2 * hag_left_factor_gcomp_product_base_temporary) /\\ (exists hag_right_factor_gcomp_product_base_temporary. x1 = x2 * hag_right_factor_gcomp_product_base_temporary)) /\\ forall hag_divisor_gcomp_product_base_temporary. (exists hag_common_left_gcomp_product_base_temporary. x3 = hag_divisor_gcomp_product_base_temporary * hag_common_left_gcomp_product_base_temporary) -> (exists hag_common_right_gcomp_product_base_temporary. x1 = hag_divisor_gcomp_product_base_temporary * hag_common_right_gcomp_product_base_temporary) -> exists hag_greatest_factor_gcomp_product_base_temporary. x2 = hag_divisor_gcomp_product_base_temporary * hag_greatest_factor_gcomp_product_base_temporary)", "specialize crt_is_gcd_coprime_factor_remove x", "specialize crt_is_gcd_coprime_factor_remove b", "specialize crt_is_gcd_coprime_factor_remove x1", "specialize crt_is_gcd_coprime_factor_remove x2", "specialize crt_is_gcd_coprime_factor_remove x3", "apply crt_is_gcd_coprime_factor_remove", "exact crt_product_witness_witness", "exact hquotcop", "exact canonical_gcd_exists_witness", "specialize crt_is_gcd_scale ga", "specialize crt_is_gcd_scale x3", "specialize crt_is_gcd_scale x1", "specialize crt_is_gcd_scale x2", "specialize crt_is_gcd_scale T", "specialize crt_is_gcd_scale n", "specialize crt_is_gcd_scale P", "apply crt_is_gcd_scale", "trans a * b", "exact hT", "rewrite haquot_witness", "trans ga * (x * b)", "apply mul_assoc", "rewrite crt_product_witness_witness", "refl", "exact hnquot_witness", "rewrite heq", "exact hP", "exact hbase" ], "script_sha256": "f30553dd54b32e1622271459be908ae8bca745d7e54e6a454b2c888a92f3c6de", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/generalized_crt_compatibility_candidate.py", "sha256": "1f88ba8af0b1169072387419c4bd9732cae20b94008ab476d3bfda3acaa00859" }, "statement": "forall a b n ga gb P T. ~(n = 0) -> T = a * b -> P = ga * gb -> (forall frp_divisor_gcomp_product_coprime. (exists frp_left_factor_gcomp_product_coprime. a = frp_divisor_gcomp_product_coprime * frp_left_factor_gcomp_product_coprime) -> (exists frp_right_factor_gcomp_product_coprime. b = frp_divisor_gcomp_product_coprime * frp_right_factor_gcomp_product_coprime) -> frp_divisor_gcomp_product_coprime = 1) -> ((((exists hag_left_factor_gcomp_product_left. a = ga * hag_left_factor_gcomp_product_left) /\\ (exists hag_right_factor_gcomp_product_left. n = ga * hag_right_factor_gcomp_product_left)) /\\ forall hag_divisor_gcomp_product_left. (exists hag_common_left_gcomp_product_left. a = hag_divisor_gcomp_product_left * hag_common_left_gcomp_product_left) -> (exists hag_common_right_gcomp_product_left. n = hag_divisor_gcomp_product_left * hag_common_right_gcomp_product_left) -> exists hag_greatest_factor_gcomp_product_left. ga = hag_divisor_gcomp_product_left * hag_greatest_factor_gcomp_product_left)) -> ((((exists hag_left_factor_gcomp_product_right. b = gb * hag_left_factor_gcomp_product_right) /\\ (exists hag_right_factor_gcomp_product_right. n = gb * hag_right_factor_gcomp_product_right)) /\\ forall hag_divisor_gcomp_product_right. (exists hag_common_left_gcomp_product_right. b = hag_divisor_gcomp_product_right * hag_common_left_gcomp_product_right) -> (exists hag_common_right_gcomp_product_right. n = hag_divisor_gcomp_product_right * hag_common_right_gcomp_product_right) -> exists hag_greatest_factor_gcomp_product_right. gb = hag_divisor_gcomp_product_right * hag_greatest_factor_gcomp_product_right)) -> ((((exists hag_left_factor_gcomp_product_result. T = P * hag_left_factor_gcomp_product_result) /\\ (exists hag_right_factor_gcomp_product_result. n = P * hag_right_factor_gcomp_product_result)) /\\ forall hag_divisor_gcomp_product_result. (exists hag_common_left_gcomp_product_result. T = hag_divisor_gcomp_product_result * hag_common_left_gcomp_product_result) -> (exists hag_common_right_gcomp_product_result. n = hag_divisor_gcomp_product_result * hag_common_right_gcomp_product_result) -> exists hag_greatest_factor_gcomp_product_result. P = hag_divisor_gcomp_product_result * hag_greatest_factor_gcomp_product_result))", "statement_sha256": "e3b28cbcdf65cdad1e51c834812bf2efb8a45cb534bb8a5daa1e4245b4d0a347", "summary": "For coprime natural factors and any nonzero comparison input, the gcd of their product is exactly the product of their individual relational gcd values.", "summary_sha256": "60c2631ab8562a182f9b2436de68470307d0a4545c37e1e1f80d80ebece922c2" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "is_gcd_dvd_left", "is_gcd_dvd_right", "factor_nonzero_left", "is_gcd_quotients_coprime_nonzero", "multiple_refl", "crt_coprime_divisor_pair", "canonical_gcd_exists", "is_gcd_symm", "crt_is_gcd_coprime_factor_remove", "is_gcd_unique", "crt_product_witness", "mul_assoc", "crt_is_gcd_scale" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "1f88ba8af0b1169072387419c4bd9732cae20b94008ab476d3bfda3acaa00859", "kind": "alpha_v25_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/generalized_crt_compatibility_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "b1ba7735ed3cad65300229246b62e52ef2b5a2b04f0b8b8ad35c6f3a1198891f", "kind": "alpha_v25_frontier_executable_audit", "path": "peano-lab/py/tests/test_generalized_crt_compatibility_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "3edcd2f35ee632ca5010856144e8c952acae370e3e20cb58fbcd92046dac9dc5", "kind": "alpha_v25_frontier_campaign_rfc", "path": "research/arithmetic-library/generalized-crt-compatibility-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "d4532076049be869e4e397d0fcee81b668bd3fd5c7d9173028bb1bdb80b9793a", "kind": "alpha_v25_breakthrough_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v25-breakthrough-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=293]" }, { "document_sha256": "b8bc43dd4a4aa78077855ae27924def378bda190db977894e069c5e4c7bfb27d", "kind": "alpha_v25_breakthrough_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v25-breakthrough-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "94ac4d193cbfe8c2ec04e54024221bc2c3a534c0ae014d381663b86174b3dcc1", "kind": "sealed_alpha_v24_parent", "path": "artifacts/peano-library/alpha/catalog-v24.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "crt_is_gcd_coprime_product", "parent_alpha_version": "v34", "proof_bundle_node_id": 233, "reference_route": "jordan-totient/checkpoint.html#theorem-crt_is_gcd_coprime_product", "script": [ "intro a", "intro b", "intro n", "intro ga", "intro gb", "intro P", "intro T", "intro hnzero", "intro hT", "intro hP", "intro hab", "intro hga", "intro hgb", "have haquot : exists q. a = ga * q", "specialize is_gcd_dvd_left ga", "specialize is_gcd_dvd_left a", "specialize is_gcd_dvd_left n", "apply is_gcd_dvd_left", "exact hga", "cases haquot", "have hnquot : exists q. n = ga * q", "specialize is_gcd_dvd_right ga", "specialize is_gcd_dvd_right a", "specialize is_gcd_dvd_right n", "apply is_gcd_dvd_right", "exact hga", "cases hnquot", "have hganzero : ~(ga = 0)", "specialize factor_nonzero_left n", "specialize factor_nonzero_left ga", "specialize factor_nonzero_left x1", "intro hzero", "apply factor_nonzero_left", "exact hnzero", "exact hnquot_witness", "exact hzero", "have hquotcop : forall frp_divisor_gcomp_product_quotients. (exists frp_left_factor_gcomp_product_quotients. x = frp_divisor_gcomp_product_quotients * frp_left_factor_gcomp_product_quotients) -> (exists frp_right_factor_gcomp_product_quotients. x1 = frp_divisor_gcomp_product_quotients * frp_right_factor_gcomp_product_quotients) -> frp_divisor_gcomp_product_quotients = 1", "specialize is_gcd_quotients_coprime_nonzero ga", "specialize is_gcd_quotients_coprime_nonzero a", "specialize is_gcd_quotients_coprime_nonzero n", "specialize is_gcd_quotients_coprime_nonzero x", "specialize is_gcd_quotients_coprime_nonzero x1", "apply is_gcd_quotients_coprime_nonzero", "exact hga", "exact hganzero", "exact haquot_witness", "exact hnquot_witness", "have hgbcop : forall frp_divisor_gcomp_product_scale_coprime. (exists frp_left_factor_gcomp_product_scale_coprime. ga = frp_divisor_gcomp_product_scale_coprime * frp_left_factor_gcomp_product_scale_coprime) -> (exists frp_right_factor_gcomp_product_scale_coprime. b = frp_divisor_gcomp_product_scale_coprime * frp_right_factor_gcomp_product_scale_coprime) -> frp_divisor_gcomp_product_scale_coprime = 1", "specialize crt_coprime_divisor_pair a", "specialize crt_coprime_divisor_pair b", "specialize crt_coprime_divisor_pair ga", "specialize crt_coprime_divisor_pair b", "apply crt_coprime_divisor_pair", "exact hab", "exists x", "exact haquot_witness", "specialize multiple_refl b", "exact multiple_refl", "specialize canonical_gcd_exists b", "specialize canonical_gcd_exists x1", "cases canonical_gcd_exists", "have hswap : (((exists hag_left_factor_gcomp_product_intermediate_swap. x1 = x2 * hag_left_factor_gcomp_product_intermediate_swap) /\\ (exists hag_right_factor_gcomp_product_intermediate_swap. b = x2 * hag_right_factor_gcomp_product_intermediate_swap)) /\\ forall hag_divisor_gcomp_product_intermediate_swap. (exists hag_common_left_gcomp_product_intermediate_swap. x1 = hag_divisor_gcomp_product_intermediate_swap * hag_common_left_gcomp_product_intermediate_swap) -> (exists hag_common_right_gcomp_product_intermediate_swap. b = hag_divisor_gcomp_product_intermediate_swap * hag_common_right_gcomp_product_intermediate_swap) -> exists hag_greatest_factor_gcomp_product_intermediate_swap. x2 = hag_divisor_gcomp_product_intermediate_swap * hag_greatest_factor_gcomp_product_intermediate_swap)", "specialize is_gcd_symm x2", "specialize is_gcd_symm b", "specialize is_gcd_symm x1", "apply is_gcd_symm", "exact canonical_gcd_exists_witness", "have hrestore : (((exists hag_left_factor_gcomp_product_restore_scaled. n = x2 * hag_left_factor_gcomp_product_restore_scaled) /\\ (exists hag_right_factor_gcomp_product_restore_scaled. b = x2 * hag_right_factor_gcomp_product_restore_scaled)) /\\ forall hag_divisor_gcomp_product_restore_scaled. (exists hag_common_left_gcomp_product_restore_scaled. n = hag_divisor_gcomp_product_restore_scaled * hag_common_left_gcomp_product_restore_scaled) -> (exists hag_common_right_gcomp_product_restore_scaled. b = hag_divisor_gcomp_product_restore_scaled * hag_common_right_gcomp_product_restore_scaled) -> exists hag_greatest_factor_gcomp_product_restore_scaled. x2 = hag_divisor_gcomp_product_restore_scaled * hag_greatest_factor_gcomp_product_restore_scaled)", "specialize crt_is_gcd_coprime_factor_remove ga", "specialize crt_is_gcd_coprime_factor_remove x1", "specialize crt_is_gcd_coprime_factor_remove b", "specialize crt_is_gcd_coprime_factor_remove x2", "specialize crt_is_gcd_coprime_factor_remove n", "apply crt_is_gcd_coprime_factor_remove", "exact hnquot_witness", "exact hgbcop", "exact hswap", "have hback : (((exists hag_left_factor_gcomp_product_restore_back. b = x2 * hag_left_factor_gcomp_product_restore_back) /\\ (exists hag_right_factor_gcomp_product_restore_back. n = x2 * hag_right_factor_gcomp_product_restore_back)) /\\ forall hag_divisor_gcomp_product_restore_back. (exists hag_common_left_gcomp_product_restore_back. b = hag_divisor_gcomp_product_restore_back * hag_common_left_gcomp_product_restore_back) -> (exists hag_common_right_gcomp_product_restore_back. n = hag_divisor_gcomp_product_restore_back * hag_common_right_gcomp_product_restore_back) -> exists hag_greatest_factor_gcomp_product_restore_back. x2 = hag_divisor_gcomp_product_restore_back * hag_greatest_factor_gcomp_product_restore_back)", "specialize is_gcd_symm x2", "specialize is_gcd_symm n", "specialize is_gcd_symm b", "apply is_gcd_symm", "exact hrestore", "have heq : x2 = gb", "specialize is_gcd_unique x2", "specialize is_gcd_unique gb", "specialize is_gcd_unique b", "specialize is_gcd_unique n", "apply is_gcd_unique", "exact hback", "exact hgb", "specialize crt_product_witness x", "specialize crt_product_witness b", "cases crt_product_witness", "have hbase : (((exists hag_left_factor_gcomp_product_base_temporary. x3 = x2 * hag_left_factor_gcomp_product_base_temporary) /\\ (exists hag_right_factor_gcomp_product_base_temporary. x1 = x2 * hag_right_factor_gcomp_product_base_temporary)) /\\ forall hag_divisor_gcomp_product_base_temporary. (exists hag_common_left_gcomp_product_base_temporary. x3 = hag_divisor_gcomp_product_base_temporary * hag_common_left_gcomp_product_base_temporary) -> (exists hag_common_right_gcomp_product_base_temporary. x1 = hag_divisor_gcomp_product_base_temporary * hag_common_right_gcomp_product_base_temporary) -> exists hag_greatest_factor_gcomp_product_base_temporary. x2 = hag_divisor_gcomp_product_base_temporary * hag_greatest_factor_gcomp_product_base_temporary)", "specialize crt_is_gcd_coprime_factor_remove x", "specialize crt_is_gcd_coprime_factor_remove b", "specialize crt_is_gcd_coprime_factor_remove x1", "specialize crt_is_gcd_coprime_factor_remove x2", "specialize crt_is_gcd_coprime_factor_remove x3", "apply crt_is_gcd_coprime_factor_remove", "exact crt_product_witness_witness", "exact hquotcop", "exact canonical_gcd_exists_witness", "specialize crt_is_gcd_scale ga", "specialize crt_is_gcd_scale x3", "specialize crt_is_gcd_scale x1", "specialize crt_is_gcd_scale x2", "specialize crt_is_gcd_scale T", "specialize crt_is_gcd_scale n", "specialize crt_is_gcd_scale P", "apply crt_is_gcd_scale", "trans a * b", "exact hT", "rewrite haquot_witness", "trans ga * (x * b)", "apply mul_assoc", "rewrite crt_product_witness_witness", "refl", "exact hnquot_witness", "rewrite heq", "exact hP", "exact hbase" ], "script_sha256": "f30553dd54b32e1622271459be908ae8bca745d7e54e6a454b2c888a92f3c6de", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/generalized_crt_compatibility_candidate.py", "sha256": "1f88ba8af0b1169072387419c4bd9732cae20b94008ab476d3bfda3acaa00859" }, "stable_member": false, "statement": "forall a b n ga gb P T. ~(n = 0) -> T = a * b -> P = ga * gb -> (forall frp_divisor_gcomp_product_coprime. (exists frp_left_factor_gcomp_product_coprime. a = frp_divisor_gcomp_product_coprime * frp_left_factor_gcomp_product_coprime) -> (exists frp_right_factor_gcomp_product_coprime. b = frp_divisor_gcomp_product_coprime * frp_right_factor_gcomp_product_coprime) -> frp_divisor_gcomp_product_coprime = 1) -> ((((exists hag_left_factor_gcomp_product_left. a = ga * hag_left_factor_gcomp_product_left) /\\ (exists hag_right_factor_gcomp_product_left. n = ga * hag_right_factor_gcomp_product_left)) /\\ forall hag_divisor_gcomp_product_left. (exists hag_common_left_gcomp_product_left. a = hag_divisor_gcomp_product_left * hag_common_left_gcomp_product_left) -> (exists hag_common_right_gcomp_product_left. n = hag_divisor_gcomp_product_left * hag_common_right_gcomp_product_left) -> exists hag_greatest_factor_gcomp_product_left. ga = hag_divisor_gcomp_product_left * hag_greatest_factor_gcomp_product_left)) -> ((((exists hag_left_factor_gcomp_product_right. b = gb * hag_left_factor_gcomp_product_right) /\\ (exists hag_right_factor_gcomp_product_right. n = gb * hag_right_factor_gcomp_product_right)) /\\ forall hag_divisor_gcomp_product_right. (exists hag_common_left_gcomp_product_right. b = hag_divisor_gcomp_product_right * hag_common_left_gcomp_product_right) -> (exists hag_common_right_gcomp_product_right. n = hag_divisor_gcomp_product_right * hag_common_right_gcomp_product_right) -> exists hag_greatest_factor_gcomp_product_right. gb = hag_divisor_gcomp_product_right * hag_greatest_factor_gcomp_product_right)) -> ((((exists hag_left_factor_gcomp_product_result. T = P * hag_left_factor_gcomp_product_result) /\\ (exists hag_right_factor_gcomp_product_result. n = P * hag_right_factor_gcomp_product_result)) /\\ forall hag_divisor_gcomp_product_result. (exists hag_common_left_gcomp_product_result. T = hag_divisor_gcomp_product_result * hag_common_left_gcomp_product_result) -> (exists hag_common_right_gcomp_product_result. n = hag_divisor_gcomp_product_result * hag_common_right_gcomp_product_result) -> exists hag_greatest_factor_gcomp_product_result. P = hag_divisor_gcomp_product_result * hag_greatest_factor_gcomp_product_result))", "statement_sha256": "e3b28cbcdf65cdad1e51c834812bf2efb8a45cb534bb8a5daa1e4245b4d0a347" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "matrix_rank_bounded_prefix_value", "canonical_catalog_record": { "alpha_v27_frontier_enrollment": { "body_receipt_sha256": "f757132250d30b61d1754c952a39d182c22784981727ad4ad566fe16f6d24775", "bundle_campaign": "second_wave", "bundle_node_id": 845, "bundle_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "campaign": "matrix_determinants", "parent_catalog_sha256": "969c261f924060552dda393427b4fbc51515b9d4e69daa17f5e9f1691b5ab534", "rfc_sha256": "ec0ad8a949e318786c01b827ca5da63e8f06b2340af99dad134dc03be0239d96", "source_sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d", "test_sha256": "8a89ca5d61d9a2752c11ed012073b272aa30dcf0b8f794a28251f94b3a5b7d44" }, "body_checked": true, "body_receipt": { "command_count": 26, "dependency_count": 1, "dne_command_count": 0, "name": "matrix_rank_bounded_prefix_value", "proof_depth": 21, "proof_edges": 29, "proof_nodes": 30, "proof_objects": 30, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "beta_at_unique" ], "dependencies_sha256": "aee880e8fa2a31079a779e18aae39ebc46230ee14bf423dd743f20f3c6cfaad0", "empty_context_closure": { "body_proof_depth": 21, "body_proof_nodes": 30, "bundle_campaign": "second_wave", "bundle_dependency_edge_count": 3999, "bundle_node_count": 1224, "bundle_node_id": 845, "bundle_path": "research/arithmetic-library/artifacts/alpha-v27-second-wave-proof-bundle-v1.json", "bundle_root_id": 1223, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "6087cc007f5164665c5bf8cc8232594a35c74d0229c5249183a6b433d43f4f54", "status": "checked" }, "enrollment_index": 2182, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d", "kind": "alpha_v27_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "8a89ca5d61d9a2752c11ed012073b272aa30dcf0b8f794a28251f94b3a5b7d44", "kind": "alpha_v27_frontier_executable_audit", "path": "peano-lab/py/tests/test_matrix_rank_finite_coding_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "ec0ad8a949e318786c01b827ca5da63e8f06b2340af99dad134dc03be0239d96", "kind": "alpha_v27_frontier_campaign_rfc", "path": "research/arithmetic-library/matrix-rank-certificate-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "kind": "alpha_v27_second_wave_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v27-second-wave-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=845]" }, { "document_sha256": "e08edd23e4ba00dd4f91aa0445679261ea68027eeb4fbd8fa14e956eec79b29b", "kind": "alpha_v27_second_wave_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v27-second-wave-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "969c261f924060552dda393427b4fbc51515b9d4e69daa17f5e9f1691b5ab534", "kind": "sealed_alpha_v26_parent", "path": "artifacts/peano-library/alpha/catalog-v26.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "matrix_determinants", "logical_spec_sha256": "9b6124001bb585db8043fc3f0126528747351282f5e20115d71935e79516abe1", "membership": "alpha_only", "name": "matrix_rank_bounded_prefix_value", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro b", "intro c", "intro l", "intro B", "intro i", "intro a", "intro hbounded", "intro hi", "intro ha", "have hentry : exists v. 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(exists mdr_gap_invariant_bound. mdr_gap_invariant_bound + (n) = (k)) -> exists P z. (((~(P = 0)) /\\ ((forall mdr_i_all_invariantdiv. (exists mdr_gap_all_invariantdivi. mdr_gap_all_invariantdivi + S (mdr_i_all_invariantdiv) = (n)) -> (exists mdr_q_all_invariantdivd. P = (S ((S (mdr_i_all_invariantdiv)) * (c))) * mdr_q_all_invariantdivd)) /\\ ((forall mdr_i_all_invariantcong mdr_a_all_invariantcong. (exists mdr_gap_all_invariantcongi. mdr_gap_all_invariantcongi + S (mdr_i_all_invariantcong) = (n)) -> (((exists ff_h_mdr_all_invariantconga. ff_h_mdr_all_invariantconga + S (mdr_a_all_invariantcong) = S ((S (mdr_i_all_invariantcong)) * e)) /\\ exists ff_q_mdr_all_invariantconga. b = ff_q_mdr_all_invariantconga * S ((S (mdr_i_all_invariantcong)) * e) + (mdr_a_all_invariantcong))) -> (exists mdr_u_all_invariantcongm mdr_v_all_invariantcongm. 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(((~(P = 0)) /\\ ((forall mdr_i_terminal_invariantdiv. (exists mdr_gap_terminal_invariantdivi. mdr_gap_terminal_invariantdivi + S (mdr_i_terminal_invariantdiv) = (k)) -> (exists mdr_q_terminal_invariantdivd. P = (S ((S (mdr_i_terminal_invariantdiv)) * (c))) * mdr_q_terminal_invariantdivd)) /\\ ((forall mdr_i_terminal_invariantcong mdr_a_terminal_invariantcong. (exists mdr_gap_terminal_invariantcongi. mdr_gap_terminal_invariantcongi + S (mdr_i_terminal_invariantcong) = (k)) -> (((exists ff_h_mdr_terminal_invariantconga. ff_h_mdr_terminal_invariantconga + S (mdr_a_terminal_invariantcong) = S ((S (mdr_i_terminal_invariantcong)) * e)) /\\ exists ff_q_mdr_terminal_invariantconga. b = ff_q_mdr_terminal_invariantconga * S ((S (mdr_i_terminal_invariantcong)) * e) + (mdr_a_terminal_invariantcong))) -> (exists mdr_u_terminal_invariantcongm mdr_v_terminal_invariantcongm. (z) + (S ((S (mdr_i_terminal_invariantcong)) * (c))) * mdr_u_terminal_invariantcongm = (mdr_a_terminal_invariantcong) + (S ((S (mdr_i_terminal_invariantcong)) * (c))) * mdr_v_terminal_invariantcongm)) /\\ (forall mdr_j_terminal_invariant. (exists mdr_gap_terminal_invariantlow. mdr_gap_terminal_invariantlow + (k) = (mdr_j_terminal_invariant)) -> (exists mdr_gap_terminal_invarianthigh. mdr_gap_terminal_invarianthigh + (mdr_j_terminal_invariant) = (k)) -> forall mdr_d_terminal_invariant. (exists mdr_q_terminal_invariantfactor. P = (mdr_d_terminal_invariant) * mdr_q_terminal_invariantfactor) -> (exists mdr_q_terminal_invariantmod. S ((S (mdr_j_terminal_invariant)) * (c)) = (mdr_d_terminal_invariant) * mdr_q_terminal_invariantmod) -> mdr_d_terminal_invariant = 1)))))", "specialize hall (k)", "apply hall", "specialize le_refl (k)", "apply le_refl", "cases hinv", "cases hinv_witness", "cases hinv_witness_witness", "cases hinv_witness_witness_right", "cases hinv_witness_witness_right_right", "exists x1", "exact hinv_witness_witness_right_right_left" ], "script_sha256": "1bf18a487fb4ee6728315ea7c70c3561bf84dbe6230b3fe466422540b62aaa96", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d" }, "stable_member": false, "statement": "forall k c b e. (forall mdr_t_recode_common. (exists mdr_h_recode_common. S mdr_t_recode_common + S mdr_h_recode_common = S (k)) -> exists mdr_q_recode_common. c = S mdr_t_recode_common * mdr_q_recode_common) -> exists z. (forall mdr_i_recode_result mdr_a_recode_result. (exists mdr_gap_recode_resulti. mdr_gap_recode_resulti + S (mdr_i_recode_result) = (k)) -> (((exists ff_h_mdr_recode_resulta. ff_h_mdr_recode_resulta + S (mdr_a_recode_result) = S ((S (mdr_i_recode_result)) * e)) /\\ exists ff_q_mdr_recode_resulta. b = ff_q_mdr_recode_resulta * S ((S (mdr_i_recode_result)) * e) + (mdr_a_recode_result))) -> (exists mdr_u_recode_resultm mdr_v_recode_resultm. (z) + (S ((S (mdr_i_recode_result)) * (c))) * mdr_u_recode_resultm = (mdr_a_recode_result) + (S ((S (mdr_i_recode_result)) * (c))) * mdr_v_recode_resultm))", "statement_sha256": "f14c01c5209b149353e6f945532d0cf0634d1dd40e0229e3dd1841a211de5a64" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "matrix_rank_bounded_recode_in_fixed_box", "canonical_catalog_record": { "alpha_v27_frontier_enrollment": { "body_receipt_sha256": "32ea702c8d0c212ef00de86d7650ce4c867ac036f2882ac0983bdf343b305540", "bundle_campaign": "second_wave", "bundle_node_id": 849, "bundle_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "campaign": "matrix_determinants", "parent_catalog_sha256": "969c261f924060552dda393427b4fbc51515b9d4e69daa17f5e9f1691b5ab534", "rfc_sha256": "ec0ad8a949e318786c01b827ca5da63e8f06b2340af99dad134dc03be0239d96", "source_sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d", "test_sha256": "8a89ca5d61d9a2752c11ed012073b272aa30dcf0b8f794a28251f94b3a5b7d44" }, "body_checked": true, "body_receipt": { "command_count": 98, "dependency_count": 12, "dne_command_count": 0, "name": "matrix_rank_bounded_recode_in_fixed_box", "proof_depth": 46, "proof_edges": 124, "proof_nodes": 125, "proof_objects": 125, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "matrix_rank_recode_congruences_exists", "division_remainder_exists", "mul_comm", "remainder_decomposition_to_mod_eq", "matrix_rank_bounded_prefix_value", "base_le_beta_modulus", "le_trans", "lt_of_lt_of_le", "mod_eq_of_mod_eq_multiple", "mod_eq_symm", "mod_eq_trans", "beta_at_of_mod_eq_bound" ], "dependencies_sha256": "da6784eedf66e78c6b92379d8af69433db006b7a9df986de2ab68c95bc670257", "empty_context_closure": { "body_proof_depth": 46, "body_proof_nodes": 125, "bundle_campaign": "second_wave", "bundle_dependency_edge_count": 3999, "bundle_node_count": 1224, "bundle_node_id": 849, "bundle_path": "research/arithmetic-library/artifacts/alpha-v27-second-wave-proof-bundle-v1.json", "bundle_root_id": 1223, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "665f3cf0e6776ee7838f79b39c76b92099af819603fb5515dc07fc15bb833cab", "status": "checked" }, "enrollment_index": 2186, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d", "kind": "alpha_v27_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "8a89ca5d61d9a2752c11ed012073b272aa30dcf0b8f794a28251f94b3a5b7d44", "kind": "alpha_v27_frontier_executable_audit", "path": "peano-lab/py/tests/test_matrix_rank_finite_coding_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "ec0ad8a949e318786c01b827ca5da63e8f06b2340af99dad134dc03be0239d96", "kind": "alpha_v27_frontier_campaign_rfc", "path": "research/arithmetic-library/matrix-rank-certificate-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "kind": "alpha_v27_second_wave_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v27-second-wave-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=849]" }, { "document_sha256": "e08edd23e4ba00dd4f91aa0445679261ea68027eeb4fbd8fa14e956eec79b29b", "kind": "alpha_v27_second_wave_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v27-second-wave-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "969c261f924060552dda393427b4fbc51515b9d4e69daa17f5e9f1691b5ab534", "kind": "sealed_alpha_v26_parent", "path": "artifacts/peano-library/alpha/catalog-v26.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "matrix_determinants", "logical_spec_sha256": "a8104453e2f69e5e7c439481c407cee7f20c5f4464cf2768f00bd5151fdac7a7", "membership": "alpha_only", "name": "matrix_rank_bounded_recode_in_fixed_box", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro k", "intro B", "intro c", "intro T", "intro b", "intro e", "intro hscale", "intro hcommon", "intro hT", "intro hmoduli", "intro hbounded", "have hcodes : exists z. (forall mdr_i_fixed_crt mdr_a_fixed_crt. (exists mdr_gap_fixed_crti. mdr_gap_fixed_crti + S (mdr_i_fixed_crt) = (k)) -> (((exists ff_h_mdr_fixed_crta. ff_h_mdr_fixed_crta + S (mdr_a_fixed_crt) = S ((S (mdr_i_fixed_crt)) * e)) /\\ exists ff_q_mdr_fixed_crta. b = ff_q_mdr_fixed_crta * S ((S (mdr_i_fixed_crt)) * e) + (mdr_a_fixed_crt))) -> (exists mdr_u_fixed_crtm mdr_v_fixed_crtm. (z) + (S ((S (mdr_i_fixed_crt)) * (c))) * mdr_u_fixed_crtm = (mdr_a_fixed_crt) + (S ((S (mdr_i_fixed_crt)) * (c))) * mdr_v_fixed_crtm))", "specialize matrix_rank_recode_congruences_exists (k)", "specialize matrix_rank_recode_congruences_exists (c)", "specialize matrix_rank_recode_congruences_exists (b)", "specialize matrix_rank_recode_congruences_exists (e)", "apply matrix_rank_recode_congruences_exists", "exact hcommon", "cases hcodes", "have hdivision : exists q r. x = T * q + r /\\ (exists mdr_gap_division_bound. mdr_gap_division_bound + S (r) = (T))", "specialize division_remainder_exists (T)", "specialize division_remainder_exists (x)", "apply division_remainder_exists", "exact hT", "cases hdivision", "cases hdivision_witness", "cases hdivision_witness_witness", "have hcommute : T * x1 = x1 * T", "apply mul_comm", "rewrite hcommute at hdivision_witness_witness_left", "have hremainder : exists mdr_u_fixed_remainder mdr_v_fixed_remainder. 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((((exists fom_beta_height_mrf_fixed_source_entry. fom_beta_height_mrf_fixed_source_entry + S (fom_value_mrf_fixed_source) = S ((S (fom_index_mrf_fixed_source)) * e)) /\\ exists fom_beta_quotient_mrf_fixed_source_entry. b = fom_beta_quotient_mrf_fixed_source_entry * S ((S (fom_index_mrf_fixed_source)) * e) + (fom_value_mrf_fixed_source))) /\\ (exists fom_gap_mrf_fixed_source_value_bound. fom_gap_mrf_fixed_source_value_bound + S (fom_value_mrf_fixed_source) = B))) -> exists z. (((exists mdr_gap_fixed_result_bound. mdr_gap_fixed_result_bound + S (z) = (T)) /\\ (forall mdr_i_fixed_result_prefix mdr_a_fixed_result_prefix. (exists mdr_gap_fixed_result_prefixb. mdr_gap_fixed_result_prefixb + S (mdr_i_fixed_result_prefix) = (k)) -> (((exists ff_h_mdr_fixed_result_prefixo. ff_h_mdr_fixed_result_prefixo + S (mdr_a_fixed_result_prefix) = S ((S (mdr_i_fixed_result_prefix)) * e)) /\\ exists ff_q_mdr_fixed_result_prefixo. b = ff_q_mdr_fixed_result_prefixo * S ((S (mdr_i_fixed_result_prefix)) * e) + (mdr_a_fixed_result_prefix))) -> (((exists ff_h_mdr_fixed_result_prefixn. ff_h_mdr_fixed_result_prefixn + S (mdr_a_fixed_result_prefix) = S ((S (mdr_i_fixed_result_prefix)) * c)) /\\ exists ff_q_mdr_fixed_result_prefixn. z = ff_q_mdr_fixed_result_prefixn * S ((S (mdr_i_fixed_result_prefix)) * c) + (mdr_a_fixed_result_prefix))))))", "statement_sha256": "665f3cf0e6776ee7838f79b39c76b92099af819603fb5515dc07fc15bb833cab", "summary": "Reducing a genuine CRT recoding modulo a fixed common multiple gives a strictly bounded code with every finite source value preserved.", "summary_sha256": "0c4c5229e8b0cbc125ff405861ce1cca699bfabdc5ade5b2ddab3803805bcfae" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "matrix_rank_recode_congruences_exists", "division_remainder_exists", "mul_comm", "remainder_decomposition_to_mod_eq", "matrix_rank_bounded_prefix_value", "base_le_beta_modulus", "le_trans", "lt_of_lt_of_le", "mod_eq_of_mod_eq_multiple", "mod_eq_symm", "mod_eq_trans", "beta_at_of_mod_eq_bound" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d", "kind": "alpha_v27_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "8a89ca5d61d9a2752c11ed012073b272aa30dcf0b8f794a28251f94b3a5b7d44", "kind": "alpha_v27_frontier_executable_audit", "path": "peano-lab/py/tests/test_matrix_rank_finite_coding_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "ec0ad8a949e318786c01b827ca5da63e8f06b2340af99dad134dc03be0239d96", "kind": "alpha_v27_frontier_campaign_rfc", "path": "research/arithmetic-library/matrix-rank-certificate-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "kind": "alpha_v27_second_wave_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v27-second-wave-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=849]" }, { "document_sha256": "e08edd23e4ba00dd4f91aa0445679261ea68027eeb4fbd8fa14e956eec79b29b", "kind": "alpha_v27_second_wave_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v27-second-wave-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "969c261f924060552dda393427b4fbc51515b9d4e69daa17f5e9f1691b5ab534", "kind": "sealed_alpha_v26_parent", "path": "artifacts/peano-library/alpha/catalog-v26.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "matrix_rank_bounded_recode_in_fixed_box", "parent_alpha_version": "v34", "proof_bundle_node_id": 238, "reference_route": "jordan-totient/checkpoint.html#theorem-matrix_rank_bounded_recode_in_fixed_box", "script": [ "intro k", "intro B", "intro c", "intro T", "intro b", "intro e", "intro hscale", "intro hcommon", "intro hT", "intro hmoduli", "intro hbounded", "have hcodes : exists z. (forall mdr_i_fixed_crt mdr_a_fixed_crt. (exists mdr_gap_fixed_crti. mdr_gap_fixed_crti + S (mdr_i_fixed_crt) = (k)) -> (((exists ff_h_mdr_fixed_crta. ff_h_mdr_fixed_crta + S (mdr_a_fixed_crt) = S ((S (mdr_i_fixed_crt)) * e)) /\\ exists ff_q_mdr_fixed_crta. b = ff_q_mdr_fixed_crta * S ((S (mdr_i_fixed_crt)) * e) + (mdr_a_fixed_crt))) -> (exists mdr_u_fixed_crtm mdr_v_fixed_crtm. (z) + (S ((S (mdr_i_fixed_crt)) * (c))) * mdr_u_fixed_crtm = (mdr_a_fixed_crt) + (S ((S (mdr_i_fixed_crt)) * (c))) * mdr_v_fixed_crtm))", "specialize matrix_rank_recode_congruences_exists (k)", "specialize matrix_rank_recode_congruences_exists (c)", "specialize matrix_rank_recode_congruences_exists (b)", "specialize matrix_rank_recode_congruences_exists (e)", "apply matrix_rank_recode_congruences_exists", "exact hcommon", "cases hcodes", "have hdivision : exists q r. x = T * q + r /\\ (exists mdr_gap_division_bound. mdr_gap_division_bound + S (r) = (T))", "specialize division_remainder_exists (T)", "specialize division_remainder_exists (x)", "apply division_remainder_exists", "exact hT", "cases hdivision", "cases hdivision_witness", "cases hdivision_witness_witness", "have hcommute : T * x1 = x1 * T", "apply mul_comm", "rewrite hcommute at hdivision_witness_witness_left", "have hremainder : exists mdr_u_fixed_remainder mdr_v_fixed_remainder. (x) + (T) * mdr_u_fixed_remainder = (x2) + (T) * mdr_v_fixed_remainder", "specialize remainder_decomposition_to_mod_eq (T)", "specialize remainder_decomposition_to_mod_eq (x)", "specialize remainder_decomposition_to_mod_eq (x1)", "specialize remainder_decomposition_to_mod_eq (x2)", "apply remainder_decomposition_to_mod_eq", "exact hdivision_witness_witness_left", "exists x2", "split", "exact hdivision_witness_witness_right", "intro i", "intro a", "intro hi", "intro ha", "have hvalue : exists mdr_gap_fixed_value_bound. mdr_gap_fixed_value_bound + S (a) = (B)", "specialize matrix_rank_bounded_prefix_value (b)", "specialize matrix_rank_bounded_prefix_value (e)", "specialize matrix_rank_bounded_prefix_value (k)", "specialize matrix_rank_bounded_prefix_value (B)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (a)", "apply matrix_rank_bounded_prefix_value", "exact hbounded", "exact hi", "exact ha", "have hmodbound : exists mdr_gap_fixed_mod_bound. mdr_gap_fixed_mod_bound + (B) = (S ((S i) * c))", "specialize le_trans (B)", "specialize le_trans (c)", "specialize le_trans (S ((S i) * c))", "apply le_trans", "exact hscale", "specialize base_le_beta_modulus (c)", "specialize base_le_beta_modulus (i)", "apply base_le_beta_modulus", "specialize beta_at_of_mod_eq_bound (x2)", "specialize beta_at_of_mod_eq_bound (c)", "specialize beta_at_of_mod_eq_bound (i)", "specialize beta_at_of_mod_eq_bound (a)", "apply beta_at_of_mod_eq_bound", "specialize lt_of_lt_of_le (a)", "specialize lt_of_lt_of_le (B)", "specialize lt_of_lt_of_le (S ((S i) * c))", "apply lt_of_lt_of_le", "exact hvalue", "exact hmodbound", "specialize mod_eq_trans (S ((S i) * c))", "specialize mod_eq_trans (x2)", "specialize mod_eq_trans (x)", "specialize mod_eq_trans (a)", "apply mod_eq_trans", "specialize mod_eq_symm (S ((S i) * c))", "specialize mod_eq_symm (x)", "specialize mod_eq_symm (x2)", "apply mod_eq_symm", "specialize mod_eq_of_mod_eq_multiple (S ((S i) * c))", "specialize mod_eq_of_mod_eq_multiple (T)", "specialize mod_eq_of_mod_eq_multiple (x)", "specialize mod_eq_of_mod_eq_multiple (x2)", "apply mod_eq_of_mod_eq_multiple", "specialize hmoduli (i)", "apply hmoduli", "exact hi", "exact hremainder", "specialize hcodes_witness (i)", "specialize hcodes_witness (a)", "apply hcodes_witness", "exact hi", "exact ha" ], "script_sha256": "2a185ec7d8aa035f8e71cd84a90fceb64c74af2e732ae5c31fdd5a6001fbe437", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d" }, "stable_member": false, "statement": "forall k B c T b e. (exists mdr_gap_fixed_scale. mdr_gap_fixed_scale + (B) = (c)) -> (forall mdr_t_fixed_common. (exists mdr_h_fixed_common. S mdr_t_fixed_common + S mdr_h_fixed_common = S (k)) -> exists mdr_q_fixed_common. c = S mdr_t_fixed_common * mdr_q_fixed_common) -> ~(T = 0) -> (forall mdr_i_fixed_divides. (exists mdr_gap_fixed_dividesi. mdr_gap_fixed_dividesi + S (mdr_i_fixed_divides) = (k)) -> (exists mdr_q_fixed_dividesd. T = (S ((S (mdr_i_fixed_divides)) * (c))) * mdr_q_fixed_dividesd)) -> (forall fom_index_mrf_fixed_source. (exists fom_gap_mrf_fixed_source_index_bound. fom_gap_mrf_fixed_source_index_bound + S (fom_index_mrf_fixed_source) = k) -> exists fom_value_mrf_fixed_source. ((((exists fom_beta_height_mrf_fixed_source_entry. fom_beta_height_mrf_fixed_source_entry + S (fom_value_mrf_fixed_source) = S ((S (fom_index_mrf_fixed_source)) * e)) /\\ exists fom_beta_quotient_mrf_fixed_source_entry. b = fom_beta_quotient_mrf_fixed_source_entry * S ((S (fom_index_mrf_fixed_source)) * e) + (fom_value_mrf_fixed_source))) /\\ (exists fom_gap_mrf_fixed_source_value_bound. fom_gap_mrf_fixed_source_value_bound + S (fom_value_mrf_fixed_source) = B))) -> exists z. (((exists mdr_gap_fixed_result_bound. mdr_gap_fixed_result_bound + S (z) = (T)) /\\ (forall mdr_i_fixed_result_prefix mdr_a_fixed_result_prefix. (exists mdr_gap_fixed_result_prefixb. mdr_gap_fixed_result_prefixb + S (mdr_i_fixed_result_prefix) = (k)) -> (((exists ff_h_mdr_fixed_result_prefixo. ff_h_mdr_fixed_result_prefixo + S (mdr_a_fixed_result_prefix) = S ((S (mdr_i_fixed_result_prefix)) * e)) /\\ exists ff_q_mdr_fixed_result_prefixo. b = ff_q_mdr_fixed_result_prefixo * S ((S (mdr_i_fixed_result_prefix)) * e) + (mdr_a_fixed_result_prefix))) -> (((exists ff_h_mdr_fixed_result_prefixn. ff_h_mdr_fixed_result_prefixn + S (mdr_a_fixed_result_prefix) = S ((S (mdr_i_fixed_result_prefix)) * c)) /\\ exists ff_q_mdr_fixed_result_prefixn. z = ff_q_mdr_fixed_result_prefixn * S ((S (mdr_i_fixed_result_prefix)) * c) + (mdr_a_fixed_result_prefix))))))", "statement_sha256": "665f3cf0e6776ee7838f79b39c76b92099af819603fb5515dc07fc15bb833cab" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "matrix_rank_uniform_beta_prefix_box_exists", 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"logical_spec_sha256": "71e069023f93da2210b3c964576bfdb45a4e1d1d742943b3afd056f594458ded", "membership": "alpha_only", "name": "matrix_rank_uniform_beta_prefix_box_exists", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro k", "intro B", "have hC : exists C. ((~(C = 0)) /\\ (forall mdr_t_uniform_common. (exists mdr_h_uniform_common. S mdr_t_uniform_common + S mdr_h_uniform_common = S (k)) -> exists mdr_q_uniform_common. C = S mdr_t_uniform_common * mdr_q_uniform_common))", "specialize bounded_common_multiple_exists (k)", "apply bounded_common_multiple_exists", "cases hC", "cases hC_witness", "have hscalecommon : forall mdr_t_scaled_common. (exists mdr_h_scaled_common. S mdr_t_scaled_common + S mdr_h_scaled_common = S (k)) -> exists mdr_q_scaled_common. x * B = S mdr_t_scaled_common * mdr_q_scaled_common", "specialize scaled_bounded_common_multiple (k)", "specialize scaled_bounded_common_multiple (x)", "specialize scaled_bounded_common_multiple (B)", "apply scaled_bounded_common_multiple", "exact hC_witness_right", "have hscale : exists mdr_gap_uniform_scale. mdr_gap_uniform_scale + (B) = (x * B)", "specialize le_scaled_nonzero (x)", "specialize le_scaled_nonzero (B)", "apply le_scaled_nonzero", "exact hC_witness_left", "have hT : exists T. ((~(T = 0)) /\\ (forall mdr_t_uniform_moduli. (exists mdr_h_uniform_moduli. S mdr_t_uniform_moduli + S mdr_h_uniform_moduli = S (S (k * (x * B)))) -> exists mdr_q_uniform_moduli. T = S mdr_t_uniform_moduli * mdr_q_uniform_moduli))", "specialize bounded_common_multiple_exists (S (k * (x * B)))", "apply bounded_common_multiple_exists", "cases hT", "cases hT_witness", "exists x * B", "exists x1", "split", "exact hT_witness_left", "intro b", "intro e", "intro hbounded", "specialize matrix_rank_bounded_recode_in_fixed_box (k)", "specialize matrix_rank_bounded_recode_in_fixed_box (B)", "specialize matrix_rank_bounded_recode_in_fixed_box (x * B)", "specialize matrix_rank_bounded_recode_in_fixed_box (x1)", "specialize matrix_rank_bounded_recode_in_fixed_box (b)", "specialize matrix_rank_bounded_recode_in_fixed_box (e)", "apply matrix_rank_bounded_recode_in_fixed_box", "exact hscale", "exact hscalecommon", "exact hT_witness_left", "specialize matrix_rank_beta_moduli_common_multiple (k)", "specialize matrix_rank_beta_moduli_common_multiple (x * B)", "specialize matrix_rank_beta_moduli_common_multiple (x1)", "apply matrix_rank_beta_moduli_common_multiple", "exact hT_witness_right", "exact hbounded" ], "script_sha256": "b90668ef5f7151100fcd67aa681cfec6be35c0a279e3adddc2104b824a1ce067", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d" }, "statement": "forall k B. exists c T. (((~(T = 0)) /\\ (forall mdr_b_uniform_box mdr_e_uniform_box. (forall fom_index_mrf_uniform_boxsource. (exists fom_gap_mrf_uniform_boxsource_index_bound. fom_gap_mrf_uniform_boxsource_index_bound + S (fom_index_mrf_uniform_boxsource) = k) -> exists fom_value_mrf_uniform_boxsource. ((((exists fom_beta_height_mrf_uniform_boxsource_entry. fom_beta_height_mrf_uniform_boxsource_entry + S (fom_value_mrf_uniform_boxsource) = S ((S (fom_index_mrf_uniform_boxsource)) * mdr_e_uniform_box)) /\\ exists fom_beta_quotient_mrf_uniform_boxsource_entry. mdr_b_uniform_box = fom_beta_quotient_mrf_uniform_boxsource_entry * S ((S (fom_index_mrf_uniform_boxsource)) * mdr_e_uniform_box) + (fom_value_mrf_uniform_boxsource))) /\\ (exists fom_gap_mrf_uniform_boxsource_value_bound. fom_gap_mrf_uniform_boxsource_value_bound + S (fom_value_mrf_uniform_boxsource) = B))) -> exists mdr_z_uniform_box. (((exists mdr_gap_uniform_boxbound. mdr_gap_uniform_boxbound + S (mdr_z_uniform_box) = (T)) /\\ (forall mdr_i_uniform_boxprefix mdr_a_uniform_boxprefix. (exists mdr_gap_uniform_boxprefixb. mdr_gap_uniform_boxprefixb + S (mdr_i_uniform_boxprefix) = (k)) -> (((exists ff_h_mdr_uniform_boxprefixo. ff_h_mdr_uniform_boxprefixo + S (mdr_a_uniform_boxprefix) = S ((S (mdr_i_uniform_boxprefix)) * mdr_e_uniform_box)) /\\ exists ff_q_mdr_uniform_boxprefixo. mdr_b_uniform_box = ff_q_mdr_uniform_boxprefixo * S ((S (mdr_i_uniform_boxprefix)) * mdr_e_uniform_box) + (mdr_a_uniform_boxprefix))) -> (((exists ff_h_mdr_uniform_boxprefixn. ff_h_mdr_uniform_boxprefixn + S (mdr_a_uniform_boxprefix) = S ((S (mdr_i_uniform_boxprefix)) * c)) /\\ exists ff_q_mdr_uniform_boxprefixn. mdr_z_uniform_box = ff_q_mdr_uniform_boxprefixn * S ((S (mdr_i_uniform_boxprefix)) * c) + (mdr_a_uniform_boxprefix)))))))))", "statement_sha256": "15c6b9386a3c36f27f5f5a76d419c121b626d7c50820bf597df47f419e21b10d", "summary": "Unconditionally construct one fixed scale and one positive finite code bound representing every bounded prefix of the requested length.", "summary_sha256": "c5e4f2d2904e548856721da8e408f2a81f776f85704f92dafc5c3e792a6c89c1" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "bounded_common_multiple_exists", "scaled_bounded_common_multiple", "le_scaled_nonzero", "matrix_rank_beta_moduli_common_multiple", "matrix_rank_bounded_recode_in_fixed_box" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "9a72aed5aa215816b5e26868c04453e0a3042486580e79a13234431b5f45952d", "kind": "alpha_v27_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/matrix_rank_finite_coding_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "8a89ca5d61d9a2752c11ed012073b272aa30dcf0b8f794a28251f94b3a5b7d44", "kind": "alpha_v27_frontier_executable_audit", "path": "peano-lab/py/tests/test_matrix_rank_finite_coding_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "ec0ad8a949e318786c01b827ca5da63e8f06b2340af99dad134dc03be0239d96", "kind": "alpha_v27_frontier_campaign_rfc", "path": "research/arithmetic-library/matrix-rank-certificate-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "c4711433c92b67d2ebeb30131669c60563c70e0464dafa851d417fb88fb21a6d", "kind": "alpha_v27_second_wave_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v27-second-wave-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=850]" }, { "document_sha256": "e08edd23e4ba00dd4f91aa0445679261ea68027eeb4fbd8fa14e956eec79b29b", "kind": "alpha_v27_second_wave_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v27-second-wave-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "969c261f924060552dda393427b4fbc51515b9d4e69daa17f5e9f1691b5ab534", "kind": "sealed_alpha_v26_parent", "path": "artifacts/peano-library/alpha/catalog-v26.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "matrix_rank_uniform_beta_prefix_box_exists", "parent_alpha_version": "v34", "proof_bundle_node_id": 239, "reference_route": "jordan-totient/checkpoint.html#theorem-matrix_rank_uniform_beta_prefix_box_exists", "script": [ "intro k", "intro B", "have hC : exists C. ((~(C = 0)) /\\ (forall mdr_t_uniform_common. (exists mdr_h_uniform_common. S mdr_t_uniform_common + S mdr_h_uniform_common = S (k)) -> exists mdr_q_uniform_common. C = S mdr_t_uniform_common * mdr_q_uniform_common))", "specialize bounded_common_multiple_exists (k)", "apply bounded_common_multiple_exists", "cases hC", "cases hC_witness", "have hscalecommon : forall mdr_t_scaled_common. (exists mdr_h_scaled_common. 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((((exists bpvi_h_pvs_transport_source_selected_power_start. bpvi_h_pvs_transport_source_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_start. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_start * S ((S (0)) * bpvi_v_pvs_transport_source_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_source_selected_power_terminal. bpvi_h_pvs_transport_source_selected_power_terminal + S (bpvi_result_pvs_transport_source_selected) = S ((S (e)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_terminal. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_transport_source_selected_power) + (bpvi_result_pvs_transport_source_selected))) /\\ forall bpvi_j_pvs_transport_source_selected_power. (exists bpvi_product_gap_pvs_transport_source_selected_power. bpvi_product_gap_pvs_transport_source_selected_power + S bpvi_j_pvs_transport_source_selected_power = e) -> exists bpvi_factor_pvs_transport_source_selected_power bpvi_partial_pvs_transport_source_selected_power bpvi_successor_pvs_transport_source_selected_power. ((((exists bpvi_h_pvs_transport_source_selected_power_factor. bpvi_h_pvs_transport_source_selected_power_factor + S (bpvi_factor_pvs_transport_source_selected_power) = S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_c_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_factor. bpvi_b_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_factor * S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_c_pvs_transport_source_selected_power) + (bpvi_factor_pvs_transport_source_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_source_selected_power_partial. bpvi_h_pvs_transport_source_selected_power_partial + S (bpvi_partial_pvs_transport_source_selected_power) = S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_partial. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_partial * S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power) + (bpvi_partial_pvs_transport_source_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_source_selected_power_successor. bpvi_h_pvs_transport_source_selected_power_successor + S (bpvi_successor_pvs_transport_source_selected_power) = S ((S (S bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_successor. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_successor * S ((S (S bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power) + (bpvi_successor_pvs_transport_source_selected_power))) /\\ bpvi_successor_pvs_transport_source_selected_power = bpvi_partial_pvs_transport_source_selected_power * bpvi_factor_pvs_transport_source_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_source_selected. n = bpvi_result_pvs_transport_source_selected * bpvi_divisor_factor_pvs_transport_source_selected))) /\\ forall bpd_candidate_pvs_transport_source. (exists bpd_gap_pvs_transport_source_candidate_bound. bpd_gap_pvs_transport_source_candidate_bound + (bpd_candidate_pvs_transport_source) = (n)) -> (exists bpvi_result_pvs_transport_source_candidate. ((exists bpvi_b_pvs_transport_source_candidate_power bpvi_c_pvs_transport_source_candidate_power. ((forall bpvi_i_pvs_transport_source_candidate_power. (exists bpvi_repeat_gap_pvs_transport_source_candidate_power. bpvi_repeat_gap_pvs_transport_source_candidate_power + S bpvi_i_pvs_transport_source_candidate_power = bpd_candidate_pvs_transport_source) -> (((exists bpvi_h_pvs_transport_source_candidate_power_repeat. bpvi_h_pvs_transport_source_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_repeat. bpvi_b_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_repeat * S ((S (bpvi_i_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_source_candidate_power bpvi_v_pvs_transport_source_candidate_power. ((((exists bpvi_h_pvs_transport_source_candidate_power_start. bpvi_h_pvs_transport_source_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_start. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_start * S ((S (0)) * bpvi_v_pvs_transport_source_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_source_candidate_power_terminal. bpvi_h_pvs_transport_source_candidate_power_terminal + S (bpvi_result_pvs_transport_source_candidate) = S ((S (bpd_candidate_pvs_transport_source)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_terminal. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_terminal * S ((S (bpd_candidate_pvs_transport_source)) * bpvi_v_pvs_transport_source_candidate_power) + (bpvi_result_pvs_transport_source_candidate))) /\\ forall bpvi_j_pvs_transport_source_candidate_power. (exists bpvi_product_gap_pvs_transport_source_candidate_power. bpvi_product_gap_pvs_transport_source_candidate_power + S bpvi_j_pvs_transport_source_candidate_power = bpd_candidate_pvs_transport_source) -> exists bpvi_factor_pvs_transport_source_candidate_power bpvi_partial_pvs_transport_source_candidate_power bpvi_successor_pvs_transport_source_candidate_power. ((((exists bpvi_h_pvs_transport_source_candidate_power_factor. bpvi_h_pvs_transport_source_candidate_power_factor + S (bpvi_factor_pvs_transport_source_candidate_power) = S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_factor. bpvi_b_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_factor * S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power) + (bpvi_factor_pvs_transport_source_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_source_candidate_power_partial. bpvi_h_pvs_transport_source_candidate_power_partial + S (bpvi_partial_pvs_transport_source_candidate_power) = S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_partial. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_partial * S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power) + (bpvi_partial_pvs_transport_source_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_source_candidate_power_successor. bpvi_h_pvs_transport_source_candidate_power_successor + S (bpvi_successor_pvs_transport_source_candidate_power) = S ((S (S bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_successor. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_successor * S ((S (S bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power) + (bpvi_successor_pvs_transport_source_candidate_power))) /\\ bpvi_successor_pvs_transport_source_candidate_power = bpvi_partial_pvs_transport_source_candidate_power * bpvi_factor_pvs_transport_source_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_source_candidate. n = bpvi_result_pvs_transport_source_candidate * bpvi_divisor_factor_pvs_transport_source_candidate)) -> (exists bpd_gap_pvs_transport_source_maximal. bpd_gap_pvs_transport_source_maximal + (bpd_candidate_pvs_transport_source) = (e))) -> (((exists bpd_gap_pvs_transport_target_selected_bound. bpd_gap_pvs_transport_target_selected_bound + (f) = (n)) /\\ (exists bpvi_result_pvs_transport_target_selected. ((exists bpvi_b_pvs_transport_target_selected_power bpvi_c_pvs_transport_target_selected_power. ((forall bpvi_i_pvs_transport_target_selected_power. (exists bpvi_repeat_gap_pvs_transport_target_selected_power. bpvi_repeat_gap_pvs_transport_target_selected_power + S bpvi_i_pvs_transport_target_selected_power = f) -> (((exists bpvi_h_pvs_transport_target_selected_power_repeat. bpvi_h_pvs_transport_target_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_repeat. bpvi_b_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_repeat * S ((S (bpvi_i_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_target_selected_power bpvi_v_pvs_transport_target_selected_power. ((((exists bpvi_h_pvs_transport_target_selected_power_start. bpvi_h_pvs_transport_target_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_start. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_start * S ((S (0)) * bpvi_v_pvs_transport_target_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_target_selected_power_terminal. bpvi_h_pvs_transport_target_selected_power_terminal + S (bpvi_result_pvs_transport_target_selected) = S ((S (f)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_terminal. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_transport_target_selected_power) + (bpvi_result_pvs_transport_target_selected))) /\\ forall bpvi_j_pvs_transport_target_selected_power. (exists bpvi_product_gap_pvs_transport_target_selected_power. bpvi_product_gap_pvs_transport_target_selected_power + S bpvi_j_pvs_transport_target_selected_power = f) -> exists bpvi_factor_pvs_transport_target_selected_power bpvi_partial_pvs_transport_target_selected_power bpvi_successor_pvs_transport_target_selected_power. ((((exists bpvi_h_pvs_transport_target_selected_power_factor. bpvi_h_pvs_transport_target_selected_power_factor + S (bpvi_factor_pvs_transport_target_selected_power) = S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_factor. bpvi_b_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_factor * S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power) + (bpvi_factor_pvs_transport_target_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_target_selected_power_partial. bpvi_h_pvs_transport_target_selected_power_partial + S (bpvi_partial_pvs_transport_target_selected_power) = S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_partial. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_partial * S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power) + (bpvi_partial_pvs_transport_target_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_target_selected_power_successor. bpvi_h_pvs_transport_target_selected_power_successor + S (bpvi_successor_pvs_transport_target_selected_power) = S ((S (S bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_successor. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_successor * S ((S (S bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power) + (bpvi_successor_pvs_transport_target_selected_power))) /\\ bpvi_successor_pvs_transport_target_selected_power = bpvi_partial_pvs_transport_target_selected_power * bpvi_factor_pvs_transport_target_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_target_selected. n = bpvi_result_pvs_transport_target_selected * bpvi_divisor_factor_pvs_transport_target_selected))) /\\ forall bpd_candidate_pvs_transport_target. (exists bpd_gap_pvs_transport_target_candidate_bound. bpd_gap_pvs_transport_target_candidate_bound + (bpd_candidate_pvs_transport_target) = (n)) -> (exists bpvi_result_pvs_transport_target_candidate. ((exists bpvi_b_pvs_transport_target_candidate_power bpvi_c_pvs_transport_target_candidate_power. ((forall bpvi_i_pvs_transport_target_candidate_power. (exists bpvi_repeat_gap_pvs_transport_target_candidate_power. bpvi_repeat_gap_pvs_transport_target_candidate_power + S bpvi_i_pvs_transport_target_candidate_power = bpd_candidate_pvs_transport_target) -> (((exists bpvi_h_pvs_transport_target_candidate_power_repeat. bpvi_h_pvs_transport_target_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_repeat. bpvi_b_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_repeat * S ((S (bpvi_i_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_target_candidate_power bpvi_v_pvs_transport_target_candidate_power. ((((exists bpvi_h_pvs_transport_target_candidate_power_start. bpvi_h_pvs_transport_target_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_start. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_start * S ((S (0)) * bpvi_v_pvs_transport_target_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_target_candidate_power_terminal. bpvi_h_pvs_transport_target_candidate_power_terminal + S (bpvi_result_pvs_transport_target_candidate) = S ((S (bpd_candidate_pvs_transport_target)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_terminal. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_terminal * S ((S (bpd_candidate_pvs_transport_target)) * bpvi_v_pvs_transport_target_candidate_power) + (bpvi_result_pvs_transport_target_candidate))) /\\ forall bpvi_j_pvs_transport_target_candidate_power. (exists bpvi_product_gap_pvs_transport_target_candidate_power. bpvi_product_gap_pvs_transport_target_candidate_power + S bpvi_j_pvs_transport_target_candidate_power = bpd_candidate_pvs_transport_target) -> exists bpvi_factor_pvs_transport_target_candidate_power bpvi_partial_pvs_transport_target_candidate_power bpvi_successor_pvs_transport_target_candidate_power. ((((exists bpvi_h_pvs_transport_target_candidate_power_factor. bpvi_h_pvs_transport_target_candidate_power_factor + S (bpvi_factor_pvs_transport_target_candidate_power) = S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_factor. bpvi_b_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_factor * S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power) + (bpvi_factor_pvs_transport_target_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_target_candidate_power_partial. bpvi_h_pvs_transport_target_candidate_power_partial + S (bpvi_partial_pvs_transport_target_candidate_power) = S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_partial. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_partial * S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power) + (bpvi_partial_pvs_transport_target_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_target_candidate_power_successor. bpvi_h_pvs_transport_target_candidate_power_successor + S (bpvi_successor_pvs_transport_target_candidate_power) = S ((S (S bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_successor. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_successor * S ((S (S bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power) + (bpvi_successor_pvs_transport_target_candidate_power))) /\\ bpvi_successor_pvs_transport_target_candidate_power = bpvi_partial_pvs_transport_target_candidate_power * bpvi_factor_pvs_transport_target_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_target_candidate. n = bpvi_result_pvs_transport_target_candidate * bpvi_divisor_factor_pvs_transport_target_candidate)) -> (exists bpd_gap_pvs_transport_target_maximal. bpd_gap_pvs_transport_target_maximal + (bpd_candidate_pvs_transport_target) = (f)))", "statement_sha256": "869aedd8826d4fd0ef3cf25eaaa3364b6f764e36488bc8d25f53dba784683683", "summary": "Equality transports an actual bounded valuation exponent without changing its prime or value.", "summary_sha256": "7cd887b9494bec5b0129857e0dc5a15e08ca4628e6f20df2a517ad474061d23b" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": 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at hval", "rewrite heq at hval", "rewrite heq at hval", "rewrite heq at hval", "rewrite heq at hval", "exact hval" ], "script_sha256": "762f1612b8a4d9a98d5884dd477ec16b1d6fc58d9a71853544ecd0f35cffcdc2", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall p n e f. e = f -> (((exists bpd_gap_pvs_transport_source_selected_bound. bpd_gap_pvs_transport_source_selected_bound + (e) = (n)) /\\ (exists bpvi_result_pvs_transport_source_selected. ((exists bpvi_b_pvs_transport_source_selected_power bpvi_c_pvs_transport_source_selected_power. ((forall bpvi_i_pvs_transport_source_selected_power. (exists bpvi_repeat_gap_pvs_transport_source_selected_power. bpvi_repeat_gap_pvs_transport_source_selected_power + S bpvi_i_pvs_transport_source_selected_power = e) -> (((exists bpvi_h_pvs_transport_source_selected_power_repeat. bpvi_h_pvs_transport_source_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_source_selected_power)) * bpvi_c_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_repeat. bpvi_b_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_repeat * S ((S (bpvi_i_pvs_transport_source_selected_power)) * bpvi_c_pvs_transport_source_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_source_selected_power bpvi_v_pvs_transport_source_selected_power. ((((exists bpvi_h_pvs_transport_source_selected_power_start. bpvi_h_pvs_transport_source_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_start. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_start * S ((S (0)) * bpvi_v_pvs_transport_source_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_source_selected_power_terminal. bpvi_h_pvs_transport_source_selected_power_terminal + S (bpvi_result_pvs_transport_source_selected) = S ((S (e)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_terminal. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_transport_source_selected_power) + (bpvi_result_pvs_transport_source_selected))) /\\ forall bpvi_j_pvs_transport_source_selected_power. (exists bpvi_product_gap_pvs_transport_source_selected_power. bpvi_product_gap_pvs_transport_source_selected_power + S bpvi_j_pvs_transport_source_selected_power = e) -> exists bpvi_factor_pvs_transport_source_selected_power bpvi_partial_pvs_transport_source_selected_power bpvi_successor_pvs_transport_source_selected_power. ((((exists bpvi_h_pvs_transport_source_selected_power_factor. bpvi_h_pvs_transport_source_selected_power_factor + S (bpvi_factor_pvs_transport_source_selected_power) = S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_c_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_factor. bpvi_b_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_factor * S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_c_pvs_transport_source_selected_power) + (bpvi_factor_pvs_transport_source_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_source_selected_power_partial. bpvi_h_pvs_transport_source_selected_power_partial + S (bpvi_partial_pvs_transport_source_selected_power) = S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_partial. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_partial * S ((S (bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power) + (bpvi_partial_pvs_transport_source_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_source_selected_power_successor. bpvi_h_pvs_transport_source_selected_power_successor + S (bpvi_successor_pvs_transport_source_selected_power) = S ((S (S bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power)) /\\ exists bpvi_q_pvs_transport_source_selected_power_successor. bpvi_u_pvs_transport_source_selected_power = bpvi_q_pvs_transport_source_selected_power_successor * S ((S (S bpvi_j_pvs_transport_source_selected_power)) * bpvi_v_pvs_transport_source_selected_power) + (bpvi_successor_pvs_transport_source_selected_power))) /\\ bpvi_successor_pvs_transport_source_selected_power = bpvi_partial_pvs_transport_source_selected_power * bpvi_factor_pvs_transport_source_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_source_selected. n = bpvi_result_pvs_transport_source_selected * bpvi_divisor_factor_pvs_transport_source_selected))) /\\ forall bpd_candidate_pvs_transport_source. (exists bpd_gap_pvs_transport_source_candidate_bound. bpd_gap_pvs_transport_source_candidate_bound + (bpd_candidate_pvs_transport_source) = (n)) -> (exists bpvi_result_pvs_transport_source_candidate. ((exists bpvi_b_pvs_transport_source_candidate_power bpvi_c_pvs_transport_source_candidate_power. ((forall bpvi_i_pvs_transport_source_candidate_power. (exists bpvi_repeat_gap_pvs_transport_source_candidate_power. bpvi_repeat_gap_pvs_transport_source_candidate_power + S bpvi_i_pvs_transport_source_candidate_power = bpd_candidate_pvs_transport_source) -> (((exists bpvi_h_pvs_transport_source_candidate_power_repeat. bpvi_h_pvs_transport_source_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_repeat. bpvi_b_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_repeat * S ((S (bpvi_i_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_source_candidate_power bpvi_v_pvs_transport_source_candidate_power. ((((exists bpvi_h_pvs_transport_source_candidate_power_start. bpvi_h_pvs_transport_source_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_start. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_start * S ((S (0)) * bpvi_v_pvs_transport_source_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_source_candidate_power_terminal. bpvi_h_pvs_transport_source_candidate_power_terminal + S (bpvi_result_pvs_transport_source_candidate) = S ((S (bpd_candidate_pvs_transport_source)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_terminal. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_terminal * S ((S (bpd_candidate_pvs_transport_source)) * bpvi_v_pvs_transport_source_candidate_power) + (bpvi_result_pvs_transport_source_candidate))) /\\ forall bpvi_j_pvs_transport_source_candidate_power. (exists bpvi_product_gap_pvs_transport_source_candidate_power. bpvi_product_gap_pvs_transport_source_candidate_power + S bpvi_j_pvs_transport_source_candidate_power = bpd_candidate_pvs_transport_source) -> exists bpvi_factor_pvs_transport_source_candidate_power bpvi_partial_pvs_transport_source_candidate_power bpvi_successor_pvs_transport_source_candidate_power. ((((exists bpvi_h_pvs_transport_source_candidate_power_factor. bpvi_h_pvs_transport_source_candidate_power_factor + S (bpvi_factor_pvs_transport_source_candidate_power) = S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_factor. bpvi_b_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_factor * S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_c_pvs_transport_source_candidate_power) + (bpvi_factor_pvs_transport_source_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_source_candidate_power_partial. bpvi_h_pvs_transport_source_candidate_power_partial + S (bpvi_partial_pvs_transport_source_candidate_power) = S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_partial. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_partial * S ((S (bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power) + (bpvi_partial_pvs_transport_source_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_source_candidate_power_successor. bpvi_h_pvs_transport_source_candidate_power_successor + S (bpvi_successor_pvs_transport_source_candidate_power) = S ((S (S bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power)) /\\ exists bpvi_q_pvs_transport_source_candidate_power_successor. bpvi_u_pvs_transport_source_candidate_power = bpvi_q_pvs_transport_source_candidate_power_successor * S ((S (S bpvi_j_pvs_transport_source_candidate_power)) * bpvi_v_pvs_transport_source_candidate_power) + (bpvi_successor_pvs_transport_source_candidate_power))) /\\ bpvi_successor_pvs_transport_source_candidate_power = bpvi_partial_pvs_transport_source_candidate_power * bpvi_factor_pvs_transport_source_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_source_candidate. n = bpvi_result_pvs_transport_source_candidate * bpvi_divisor_factor_pvs_transport_source_candidate)) -> (exists bpd_gap_pvs_transport_source_maximal. bpd_gap_pvs_transport_source_maximal + (bpd_candidate_pvs_transport_source) = (e))) -> (((exists bpd_gap_pvs_transport_target_selected_bound. bpd_gap_pvs_transport_target_selected_bound + (f) = (n)) /\\ (exists bpvi_result_pvs_transport_target_selected. ((exists bpvi_b_pvs_transport_target_selected_power bpvi_c_pvs_transport_target_selected_power. ((forall bpvi_i_pvs_transport_target_selected_power. (exists bpvi_repeat_gap_pvs_transport_target_selected_power. bpvi_repeat_gap_pvs_transport_target_selected_power + S bpvi_i_pvs_transport_target_selected_power = f) -> (((exists bpvi_h_pvs_transport_target_selected_power_repeat. bpvi_h_pvs_transport_target_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_repeat. bpvi_b_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_repeat * S ((S (bpvi_i_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_target_selected_power bpvi_v_pvs_transport_target_selected_power. ((((exists bpvi_h_pvs_transport_target_selected_power_start. bpvi_h_pvs_transport_target_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_start. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_start * S ((S (0)) * bpvi_v_pvs_transport_target_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_target_selected_power_terminal. bpvi_h_pvs_transport_target_selected_power_terminal + S (bpvi_result_pvs_transport_target_selected) = S ((S (f)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_terminal. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_transport_target_selected_power) + (bpvi_result_pvs_transport_target_selected))) /\\ forall bpvi_j_pvs_transport_target_selected_power. (exists bpvi_product_gap_pvs_transport_target_selected_power. bpvi_product_gap_pvs_transport_target_selected_power + S bpvi_j_pvs_transport_target_selected_power = f) -> exists bpvi_factor_pvs_transport_target_selected_power bpvi_partial_pvs_transport_target_selected_power bpvi_successor_pvs_transport_target_selected_power. ((((exists bpvi_h_pvs_transport_target_selected_power_factor. bpvi_h_pvs_transport_target_selected_power_factor + S (bpvi_factor_pvs_transport_target_selected_power) = S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_factor. bpvi_b_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_factor * S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_c_pvs_transport_target_selected_power) + (bpvi_factor_pvs_transport_target_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_target_selected_power_partial. bpvi_h_pvs_transport_target_selected_power_partial + S (bpvi_partial_pvs_transport_target_selected_power) = S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_partial. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_partial * S ((S (bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power) + (bpvi_partial_pvs_transport_target_selected_power))) /\\ ((((exists bpvi_h_pvs_transport_target_selected_power_successor. bpvi_h_pvs_transport_target_selected_power_successor + S (bpvi_successor_pvs_transport_target_selected_power) = S ((S (S bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power)) /\\ exists bpvi_q_pvs_transport_target_selected_power_successor. bpvi_u_pvs_transport_target_selected_power = bpvi_q_pvs_transport_target_selected_power_successor * S ((S (S bpvi_j_pvs_transport_target_selected_power)) * bpvi_v_pvs_transport_target_selected_power) + (bpvi_successor_pvs_transport_target_selected_power))) /\\ bpvi_successor_pvs_transport_target_selected_power = bpvi_partial_pvs_transport_target_selected_power * bpvi_factor_pvs_transport_target_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_transport_target_selected. n = bpvi_result_pvs_transport_target_selected * bpvi_divisor_factor_pvs_transport_target_selected))) /\\ forall bpd_candidate_pvs_transport_target. (exists bpd_gap_pvs_transport_target_candidate_bound. bpd_gap_pvs_transport_target_candidate_bound + (bpd_candidate_pvs_transport_target) = (n)) -> (exists bpvi_result_pvs_transport_target_candidate. ((exists bpvi_b_pvs_transport_target_candidate_power bpvi_c_pvs_transport_target_candidate_power. ((forall bpvi_i_pvs_transport_target_candidate_power. (exists bpvi_repeat_gap_pvs_transport_target_candidate_power. bpvi_repeat_gap_pvs_transport_target_candidate_power + S bpvi_i_pvs_transport_target_candidate_power = bpd_candidate_pvs_transport_target) -> (((exists bpvi_h_pvs_transport_target_candidate_power_repeat. bpvi_h_pvs_transport_target_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_repeat. bpvi_b_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_repeat * S ((S (bpvi_i_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_transport_target_candidate_power bpvi_v_pvs_transport_target_candidate_power. ((((exists bpvi_h_pvs_transport_target_candidate_power_start. bpvi_h_pvs_transport_target_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_start. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_start * S ((S (0)) * bpvi_v_pvs_transport_target_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_transport_target_candidate_power_terminal. bpvi_h_pvs_transport_target_candidate_power_terminal + S (bpvi_result_pvs_transport_target_candidate) = S ((S (bpd_candidate_pvs_transport_target)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_terminal. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_terminal * S ((S (bpd_candidate_pvs_transport_target)) * bpvi_v_pvs_transport_target_candidate_power) + (bpvi_result_pvs_transport_target_candidate))) /\\ forall bpvi_j_pvs_transport_target_candidate_power. (exists bpvi_product_gap_pvs_transport_target_candidate_power. bpvi_product_gap_pvs_transport_target_candidate_power + S bpvi_j_pvs_transport_target_candidate_power = bpd_candidate_pvs_transport_target) -> exists bpvi_factor_pvs_transport_target_candidate_power bpvi_partial_pvs_transport_target_candidate_power bpvi_successor_pvs_transport_target_candidate_power. ((((exists bpvi_h_pvs_transport_target_candidate_power_factor. bpvi_h_pvs_transport_target_candidate_power_factor + S (bpvi_factor_pvs_transport_target_candidate_power) = S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_factor. bpvi_b_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_factor * S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_c_pvs_transport_target_candidate_power) + (bpvi_factor_pvs_transport_target_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_target_candidate_power_partial. bpvi_h_pvs_transport_target_candidate_power_partial + S (bpvi_partial_pvs_transport_target_candidate_power) = S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_partial. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_partial * S ((S (bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power) + (bpvi_partial_pvs_transport_target_candidate_power))) /\\ ((((exists bpvi_h_pvs_transport_target_candidate_power_successor. bpvi_h_pvs_transport_target_candidate_power_successor + S (bpvi_successor_pvs_transport_target_candidate_power) = S ((S (S bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power)) /\\ exists bpvi_q_pvs_transport_target_candidate_power_successor. bpvi_u_pvs_transport_target_candidate_power = bpvi_q_pvs_transport_target_candidate_power_successor * S ((S (S bpvi_j_pvs_transport_target_candidate_power)) * bpvi_v_pvs_transport_target_candidate_power) + (bpvi_successor_pvs_transport_target_candidate_power))) /\\ bpvi_successor_pvs_transport_target_candidate_power = bpvi_partial_pvs_transport_target_candidate_power * bpvi_factor_pvs_transport_target_candidate_power)))))))) /\\ 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(((exists bpd_gap_pvs_zero_exists_selected_bound. bpd_gap_pvs_zero_exists_selected_bound + (e) = (n)) /\\ (exists bpvi_result_pvs_zero_exists_selected. ((exists bpvi_b_pvs_zero_exists_selected_power bpvi_c_pvs_zero_exists_selected_power. ((forall bpvi_i_pvs_zero_exists_selected_power. (exists bpvi_repeat_gap_pvs_zero_exists_selected_power. bpvi_repeat_gap_pvs_zero_exists_selected_power + S bpvi_i_pvs_zero_exists_selected_power = e) -> (((exists bpvi_h_pvs_zero_exists_selected_power_repeat. bpvi_h_pvs_zero_exists_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_repeat. bpvi_b_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_repeat * S ((S (bpvi_i_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_exists_selected_power bpvi_v_pvs_zero_exists_selected_power. ((((exists bpvi_h_pvs_zero_exists_selected_power_start. bpvi_h_pvs_zero_exists_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_start. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_start * S ((S (0)) * bpvi_v_pvs_zero_exists_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_exists_selected_power_terminal. bpvi_h_pvs_zero_exists_selected_power_terminal + S (bpvi_result_pvs_zero_exists_selected) = S ((S (e)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_terminal. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_zero_exists_selected_power) + (bpvi_result_pvs_zero_exists_selected))) /\\ forall bpvi_j_pvs_zero_exists_selected_power. (exists bpvi_product_gap_pvs_zero_exists_selected_power. bpvi_product_gap_pvs_zero_exists_selected_power + S bpvi_j_pvs_zero_exists_selected_power = e) -> exists bpvi_factor_pvs_zero_exists_selected_power bpvi_partial_pvs_zero_exists_selected_power bpvi_successor_pvs_zero_exists_selected_power. ((((exists bpvi_h_pvs_zero_exists_selected_power_factor. bpvi_h_pvs_zero_exists_selected_power_factor + S (bpvi_factor_pvs_zero_exists_selected_power) = S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_factor. bpvi_b_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_factor * S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power) + (bpvi_factor_pvs_zero_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_selected_power_partial. bpvi_h_pvs_zero_exists_selected_power_partial + S (bpvi_partial_pvs_zero_exists_selected_power) = S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_partial. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_partial * S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power) + (bpvi_partial_pvs_zero_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_selected_power_successor. bpvi_h_pvs_zero_exists_selected_power_successor + S (bpvi_successor_pvs_zero_exists_selected_power) = S ((S (S bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_successor. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_successor * S ((S (S bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power) + (bpvi_successor_pvs_zero_exists_selected_power))) /\\ bpvi_successor_pvs_zero_exists_selected_power = bpvi_partial_pvs_zero_exists_selected_power * bpvi_factor_pvs_zero_exists_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_exists_selected. n = bpvi_result_pvs_zero_exists_selected * bpvi_divisor_factor_pvs_zero_exists_selected))) /\\ forall bpd_candidate_pvs_zero_exists. (exists bpd_gap_pvs_zero_exists_candidate_bound. bpd_gap_pvs_zero_exists_candidate_bound + (bpd_candidate_pvs_zero_exists) = (n)) -> (exists bpvi_result_pvs_zero_exists_candidate. ((exists bpvi_b_pvs_zero_exists_candidate_power bpvi_c_pvs_zero_exists_candidate_power. ((forall bpvi_i_pvs_zero_exists_candidate_power. (exists bpvi_repeat_gap_pvs_zero_exists_candidate_power. bpvi_repeat_gap_pvs_zero_exists_candidate_power + S bpvi_i_pvs_zero_exists_candidate_power = bpd_candidate_pvs_zero_exists) -> (((exists bpvi_h_pvs_zero_exists_candidate_power_repeat. bpvi_h_pvs_zero_exists_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_repeat. bpvi_b_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_repeat * S ((S (bpvi_i_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_exists_candidate_power bpvi_v_pvs_zero_exists_candidate_power. ((((exists bpvi_h_pvs_zero_exists_candidate_power_start. bpvi_h_pvs_zero_exists_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_start. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_start * S ((S (0)) * bpvi_v_pvs_zero_exists_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_exists_candidate_power_terminal. bpvi_h_pvs_zero_exists_candidate_power_terminal + S (bpvi_result_pvs_zero_exists_candidate) = S ((S (bpd_candidate_pvs_zero_exists)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_terminal. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_terminal * S ((S (bpd_candidate_pvs_zero_exists)) * bpvi_v_pvs_zero_exists_candidate_power) + (bpvi_result_pvs_zero_exists_candidate))) /\\ forall bpvi_j_pvs_zero_exists_candidate_power. (exists bpvi_product_gap_pvs_zero_exists_candidate_power. bpvi_product_gap_pvs_zero_exists_candidate_power + S bpvi_j_pvs_zero_exists_candidate_power = bpd_candidate_pvs_zero_exists) -> exists bpvi_factor_pvs_zero_exists_candidate_power bpvi_partial_pvs_zero_exists_candidate_power bpvi_successor_pvs_zero_exists_candidate_power. ((((exists bpvi_h_pvs_zero_exists_candidate_power_factor. bpvi_h_pvs_zero_exists_candidate_power_factor + S (bpvi_factor_pvs_zero_exists_candidate_power) = S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_factor. bpvi_b_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_factor * S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power) + (bpvi_factor_pvs_zero_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_candidate_power_partial. bpvi_h_pvs_zero_exists_candidate_power_partial + S (bpvi_partial_pvs_zero_exists_candidate_power) = S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_partial. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_partial * S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power) + (bpvi_partial_pvs_zero_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_candidate_power_successor. bpvi_h_pvs_zero_exists_candidate_power_successor + S (bpvi_successor_pvs_zero_exists_candidate_power) = S ((S (S bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_successor. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_successor * S ((S (S bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power) + (bpvi_successor_pvs_zero_exists_candidate_power))) /\\ bpvi_successor_pvs_zero_exists_candidate_power = bpvi_partial_pvs_zero_exists_candidate_power * bpvi_factor_pvs_zero_exists_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_exists_candidate. n = bpvi_result_pvs_zero_exists_candidate * bpvi_divisor_factor_pvs_zero_exists_candidate)) -> (exists bpd_gap_pvs_zero_exists_maximal. bpd_gap_pvs_zero_exists_maximal + (bpd_candidate_pvs_zero_exists) = (e)))", "specialize power_valuation_exists (p)", "specialize power_valuation_exists (n)", "apply power_valuation_exists", "cases hex", "have hiff : (x = 0 -> ~(exists pvs_factor_zero_forward. (n) = (p) * pvs_factor_zero_forward)) /\\ (~(exists pvs_factor_zero_reverse. (n) = (p) * pvs_factor_zero_reverse) -> x = 0)", "specialize prime_power_valuation_zero_iff_not_divides (p)", "specialize prime_power_valuation_zero_iff_not_divides (n)", "specialize prime_power_valuation_zero_iff_not_divides (x)", "apply prime_power_valuation_zero_iff_not_divides", "exact hp", "exact hn", "exact hex_witness", "cases hiff", "specialize prime_valuation_exponent_eq_transport (p)", "specialize prime_valuation_exponent_eq_transport (n)", "specialize prime_valuation_exponent_eq_transport (x)", "specialize prime_valuation_exponent_eq_transport (0)", "apply prime_valuation_exponent_eq_transport", "apply hiff_right", "exact hnot", "exact hex_witness" ], "script_sha256": "591018cc22d076b21dc4c979b72cb39aca8539162c9fff031eb0639b87524956", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall p n. (~((p) = 1) /\\ forall pvs_left_zero_domain pvs_right_zero_domain. (p) = pvs_left_zero_domain * pvs_right_zero_domain -> pvs_left_zero_domain = 1 \\/ pvs_right_zero_domain = 1) -> ~(n = 0) -> ~(exists pvs_factor_zero_nondivisor. (n) = (p) * pvs_factor_zero_nondivisor) -> (((exists bpd_gap_pvs_zero_value_selected_bound. bpd_gap_pvs_zero_value_selected_bound + (0) = (n)) /\\ (exists bpvi_result_pvs_zero_value_selected. ((exists bpvi_b_pvs_zero_value_selected_power bpvi_c_pvs_zero_value_selected_power. ((forall bpvi_i_pvs_zero_value_selected_power. (exists bpvi_repeat_gap_pvs_zero_value_selected_power. bpvi_repeat_gap_pvs_zero_value_selected_power + S bpvi_i_pvs_zero_value_selected_power = 0) -> (((exists bpvi_h_pvs_zero_value_selected_power_repeat. bpvi_h_pvs_zero_value_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_repeat. bpvi_b_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_repeat * S ((S (bpvi_i_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_value_selected_power bpvi_v_pvs_zero_value_selected_power. ((((exists bpvi_h_pvs_zero_value_selected_power_start. bpvi_h_pvs_zero_value_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_start. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_start * S ((S (0)) * bpvi_v_pvs_zero_value_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_value_selected_power_terminal. bpvi_h_pvs_zero_value_selected_power_terminal + S (bpvi_result_pvs_zero_value_selected) = S ((S (0)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_terminal. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_zero_value_selected_power) + (bpvi_result_pvs_zero_value_selected))) /\\ forall bpvi_j_pvs_zero_value_selected_power. (exists bpvi_product_gap_pvs_zero_value_selected_power. bpvi_product_gap_pvs_zero_value_selected_power + S bpvi_j_pvs_zero_value_selected_power = 0) -> exists bpvi_factor_pvs_zero_value_selected_power bpvi_partial_pvs_zero_value_selected_power bpvi_successor_pvs_zero_value_selected_power. ((((exists bpvi_h_pvs_zero_value_selected_power_factor. bpvi_h_pvs_zero_value_selected_power_factor + S (bpvi_factor_pvs_zero_value_selected_power) = S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_factor. bpvi_b_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_factor * S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power) + (bpvi_factor_pvs_zero_value_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_value_selected_power_partial. bpvi_h_pvs_zero_value_selected_power_partial + S (bpvi_partial_pvs_zero_value_selected_power) = S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_partial. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_partial * S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power) + (bpvi_partial_pvs_zero_value_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_value_selected_power_successor. bpvi_h_pvs_zero_value_selected_power_successor + S (bpvi_successor_pvs_zero_value_selected_power) = S ((S (S bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_successor. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_successor * S ((S (S bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power) + (bpvi_successor_pvs_zero_value_selected_power))) /\\ bpvi_successor_pvs_zero_value_selected_power = bpvi_partial_pvs_zero_value_selected_power * bpvi_factor_pvs_zero_value_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_value_selected. n = bpvi_result_pvs_zero_value_selected * bpvi_divisor_factor_pvs_zero_value_selected))) /\\ forall bpd_candidate_pvs_zero_value. (exists bpd_gap_pvs_zero_value_candidate_bound. bpd_gap_pvs_zero_value_candidate_bound + (bpd_candidate_pvs_zero_value) = (n)) -> (exists bpvi_result_pvs_zero_value_candidate. ((exists bpvi_b_pvs_zero_value_candidate_power bpvi_c_pvs_zero_value_candidate_power. ((forall bpvi_i_pvs_zero_value_candidate_power. (exists bpvi_repeat_gap_pvs_zero_value_candidate_power. bpvi_repeat_gap_pvs_zero_value_candidate_power + S bpvi_i_pvs_zero_value_candidate_power = bpd_candidate_pvs_zero_value) -> (((exists bpvi_h_pvs_zero_value_candidate_power_repeat. bpvi_h_pvs_zero_value_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_repeat. bpvi_b_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_repeat * S ((S (bpvi_i_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_value_candidate_power bpvi_v_pvs_zero_value_candidate_power. ((((exists bpvi_h_pvs_zero_value_candidate_power_start. bpvi_h_pvs_zero_value_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_start. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_start * S ((S (0)) * bpvi_v_pvs_zero_value_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_value_candidate_power_terminal. bpvi_h_pvs_zero_value_candidate_power_terminal + S (bpvi_result_pvs_zero_value_candidate) = S ((S (bpd_candidate_pvs_zero_value)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_terminal. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_terminal * S ((S (bpd_candidate_pvs_zero_value)) * bpvi_v_pvs_zero_value_candidate_power) + (bpvi_result_pvs_zero_value_candidate))) /\\ forall bpvi_j_pvs_zero_value_candidate_power. (exists bpvi_product_gap_pvs_zero_value_candidate_power. bpvi_product_gap_pvs_zero_value_candidate_power + S bpvi_j_pvs_zero_value_candidate_power = bpd_candidate_pvs_zero_value) -> exists bpvi_factor_pvs_zero_value_candidate_power bpvi_partial_pvs_zero_value_candidate_power bpvi_successor_pvs_zero_value_candidate_power. ((((exists bpvi_h_pvs_zero_value_candidate_power_factor. bpvi_h_pvs_zero_value_candidate_power_factor + S (bpvi_factor_pvs_zero_value_candidate_power) = S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_factor. bpvi_b_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_factor * S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power) + (bpvi_factor_pvs_zero_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_value_candidate_power_partial. bpvi_h_pvs_zero_value_candidate_power_partial + S (bpvi_partial_pvs_zero_value_candidate_power) = S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_partial. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_partial * S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power) + (bpvi_partial_pvs_zero_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_value_candidate_power_successor. bpvi_h_pvs_zero_value_candidate_power_successor + S (bpvi_successor_pvs_zero_value_candidate_power) = S ((S (S bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_successor. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_successor * S ((S (S bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power) + (bpvi_successor_pvs_zero_value_candidate_power))) /\\ bpvi_successor_pvs_zero_value_candidate_power = bpvi_partial_pvs_zero_value_candidate_power * bpvi_factor_pvs_zero_value_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_value_candidate. n = bpvi_result_pvs_zero_value_candidate * bpvi_divisor_factor_pvs_zero_value_candidate)) -> (exists bpd_gap_pvs_zero_value_maximal. bpd_gap_pvs_zero_value_maximal + (bpd_candidate_pvs_zero_value) = (0)))", "statement_sha256": "b3d6af3f399a1d07c39e6a1f66b60d9774f7c576fddc3646304b21886ce84813", "summary": "Construct valuation zero for a positive value not divisible by the actual prime.", "summary_sha256": "e923467e9fb198eee2d22b1a285877bf6ba93955a1fca3e422791f6d47e1b00b" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_exists", "prime_power_valuation_zero_iff_not_divides", "prime_valuation_exponent_eq_transport" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=288]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_valuation_zero_of_nondivisor", "parent_alpha_version": "v34", "proof_bundle_node_id": 249, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_valuation_zero_of_nondivisor", "script": [ "intro p", "intro n", "intro hp", "intro hn", "intro hnot", "have hex : exists e. (((exists bpd_gap_pvs_zero_exists_selected_bound. bpd_gap_pvs_zero_exists_selected_bound + (e) = (n)) /\\ (exists bpvi_result_pvs_zero_exists_selected. ((exists bpvi_b_pvs_zero_exists_selected_power bpvi_c_pvs_zero_exists_selected_power. ((forall bpvi_i_pvs_zero_exists_selected_power. (exists bpvi_repeat_gap_pvs_zero_exists_selected_power. bpvi_repeat_gap_pvs_zero_exists_selected_power + S bpvi_i_pvs_zero_exists_selected_power = e) -> (((exists bpvi_h_pvs_zero_exists_selected_power_repeat. bpvi_h_pvs_zero_exists_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_repeat. bpvi_b_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_repeat * S ((S (bpvi_i_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_exists_selected_power bpvi_v_pvs_zero_exists_selected_power. ((((exists bpvi_h_pvs_zero_exists_selected_power_start. bpvi_h_pvs_zero_exists_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_start. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_start * S ((S (0)) * bpvi_v_pvs_zero_exists_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_exists_selected_power_terminal. bpvi_h_pvs_zero_exists_selected_power_terminal + S (bpvi_result_pvs_zero_exists_selected) = S ((S (e)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_terminal. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_zero_exists_selected_power) + (bpvi_result_pvs_zero_exists_selected))) /\\ forall bpvi_j_pvs_zero_exists_selected_power. (exists bpvi_product_gap_pvs_zero_exists_selected_power. bpvi_product_gap_pvs_zero_exists_selected_power + S bpvi_j_pvs_zero_exists_selected_power = e) -> exists bpvi_factor_pvs_zero_exists_selected_power bpvi_partial_pvs_zero_exists_selected_power bpvi_successor_pvs_zero_exists_selected_power. ((((exists bpvi_h_pvs_zero_exists_selected_power_factor. bpvi_h_pvs_zero_exists_selected_power_factor + S (bpvi_factor_pvs_zero_exists_selected_power) = S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_factor. bpvi_b_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_factor * S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_c_pvs_zero_exists_selected_power) + (bpvi_factor_pvs_zero_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_selected_power_partial. bpvi_h_pvs_zero_exists_selected_power_partial + S (bpvi_partial_pvs_zero_exists_selected_power) = S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_partial. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_partial * S ((S (bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power) + (bpvi_partial_pvs_zero_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_selected_power_successor. bpvi_h_pvs_zero_exists_selected_power_successor + S (bpvi_successor_pvs_zero_exists_selected_power) = S ((S (S bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power)) /\\ exists bpvi_q_pvs_zero_exists_selected_power_successor. bpvi_u_pvs_zero_exists_selected_power = bpvi_q_pvs_zero_exists_selected_power_successor * S ((S (S bpvi_j_pvs_zero_exists_selected_power)) * bpvi_v_pvs_zero_exists_selected_power) + (bpvi_successor_pvs_zero_exists_selected_power))) /\\ bpvi_successor_pvs_zero_exists_selected_power = bpvi_partial_pvs_zero_exists_selected_power * bpvi_factor_pvs_zero_exists_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_exists_selected. n = bpvi_result_pvs_zero_exists_selected * bpvi_divisor_factor_pvs_zero_exists_selected))) /\\ forall bpd_candidate_pvs_zero_exists. (exists bpd_gap_pvs_zero_exists_candidate_bound. bpd_gap_pvs_zero_exists_candidate_bound + (bpd_candidate_pvs_zero_exists) = (n)) -> (exists bpvi_result_pvs_zero_exists_candidate. ((exists bpvi_b_pvs_zero_exists_candidate_power bpvi_c_pvs_zero_exists_candidate_power. ((forall bpvi_i_pvs_zero_exists_candidate_power. (exists bpvi_repeat_gap_pvs_zero_exists_candidate_power. bpvi_repeat_gap_pvs_zero_exists_candidate_power + S bpvi_i_pvs_zero_exists_candidate_power = bpd_candidate_pvs_zero_exists) -> (((exists bpvi_h_pvs_zero_exists_candidate_power_repeat. bpvi_h_pvs_zero_exists_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_repeat. bpvi_b_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_repeat * S ((S (bpvi_i_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_exists_candidate_power bpvi_v_pvs_zero_exists_candidate_power. ((((exists bpvi_h_pvs_zero_exists_candidate_power_start. bpvi_h_pvs_zero_exists_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_start. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_start * S ((S (0)) * bpvi_v_pvs_zero_exists_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_exists_candidate_power_terminal. bpvi_h_pvs_zero_exists_candidate_power_terminal + S (bpvi_result_pvs_zero_exists_candidate) = S ((S (bpd_candidate_pvs_zero_exists)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_terminal. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_terminal * S ((S (bpd_candidate_pvs_zero_exists)) * bpvi_v_pvs_zero_exists_candidate_power) + (bpvi_result_pvs_zero_exists_candidate))) /\\ forall bpvi_j_pvs_zero_exists_candidate_power. (exists bpvi_product_gap_pvs_zero_exists_candidate_power. bpvi_product_gap_pvs_zero_exists_candidate_power + S bpvi_j_pvs_zero_exists_candidate_power = bpd_candidate_pvs_zero_exists) -> exists bpvi_factor_pvs_zero_exists_candidate_power bpvi_partial_pvs_zero_exists_candidate_power bpvi_successor_pvs_zero_exists_candidate_power. ((((exists bpvi_h_pvs_zero_exists_candidate_power_factor. bpvi_h_pvs_zero_exists_candidate_power_factor + S (bpvi_factor_pvs_zero_exists_candidate_power) = S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_factor. bpvi_b_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_factor * S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_c_pvs_zero_exists_candidate_power) + (bpvi_factor_pvs_zero_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_candidate_power_partial. bpvi_h_pvs_zero_exists_candidate_power_partial + S (bpvi_partial_pvs_zero_exists_candidate_power) = S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_partial. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_partial * S ((S (bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power) + (bpvi_partial_pvs_zero_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_exists_candidate_power_successor. bpvi_h_pvs_zero_exists_candidate_power_successor + S (bpvi_successor_pvs_zero_exists_candidate_power) = S ((S (S bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power)) /\\ exists bpvi_q_pvs_zero_exists_candidate_power_successor. bpvi_u_pvs_zero_exists_candidate_power = bpvi_q_pvs_zero_exists_candidate_power_successor * S ((S (S bpvi_j_pvs_zero_exists_candidate_power)) * bpvi_v_pvs_zero_exists_candidate_power) + (bpvi_successor_pvs_zero_exists_candidate_power))) /\\ bpvi_successor_pvs_zero_exists_candidate_power = bpvi_partial_pvs_zero_exists_candidate_power * bpvi_factor_pvs_zero_exists_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_exists_candidate. n = bpvi_result_pvs_zero_exists_candidate * bpvi_divisor_factor_pvs_zero_exists_candidate)) -> (exists bpd_gap_pvs_zero_exists_maximal. bpd_gap_pvs_zero_exists_maximal + (bpd_candidate_pvs_zero_exists) = (e)))", "specialize power_valuation_exists (p)", "specialize power_valuation_exists (n)", "apply power_valuation_exists", "cases hex", "have hiff : (x = 0 -> ~(exists pvs_factor_zero_forward. (n) = (p) * pvs_factor_zero_forward)) /\\ (~(exists pvs_factor_zero_reverse. (n) = (p) * pvs_factor_zero_reverse) -> x = 0)", "specialize prime_power_valuation_zero_iff_not_divides (p)", "specialize prime_power_valuation_zero_iff_not_divides (n)", "specialize prime_power_valuation_zero_iff_not_divides (x)", "apply prime_power_valuation_zero_iff_not_divides", "exact hp", "exact hn", "exact hex_witness", "cases hiff", "specialize prime_valuation_exponent_eq_transport (p)", "specialize prime_valuation_exponent_eq_transport (n)", "specialize prime_valuation_exponent_eq_transport (x)", "specialize prime_valuation_exponent_eq_transport (0)", "apply prime_valuation_exponent_eq_transport", "apply hiff_right", "exact hnot", "exact hex_witness" ], "script_sha256": "591018cc22d076b21dc4c979b72cb39aca8539162c9fff031eb0639b87524956", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall p n. (~((p) = 1) /\\ forall pvs_left_zero_domain pvs_right_zero_domain. (p) = pvs_left_zero_domain * pvs_right_zero_domain -> pvs_left_zero_domain = 1 \\/ pvs_right_zero_domain = 1) -> ~(n = 0) -> ~(exists pvs_factor_zero_nondivisor. (n) = (p) * pvs_factor_zero_nondivisor) -> (((exists bpd_gap_pvs_zero_value_selected_bound. bpd_gap_pvs_zero_value_selected_bound + (0) = (n)) /\\ (exists bpvi_result_pvs_zero_value_selected. ((exists bpvi_b_pvs_zero_value_selected_power bpvi_c_pvs_zero_value_selected_power. ((forall bpvi_i_pvs_zero_value_selected_power. (exists bpvi_repeat_gap_pvs_zero_value_selected_power. bpvi_repeat_gap_pvs_zero_value_selected_power + S bpvi_i_pvs_zero_value_selected_power = 0) -> (((exists bpvi_h_pvs_zero_value_selected_power_repeat. bpvi_h_pvs_zero_value_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_repeat. bpvi_b_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_repeat * S ((S (bpvi_i_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_value_selected_power bpvi_v_pvs_zero_value_selected_power. ((((exists bpvi_h_pvs_zero_value_selected_power_start. bpvi_h_pvs_zero_value_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_start. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_start * S ((S (0)) * bpvi_v_pvs_zero_value_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_value_selected_power_terminal. bpvi_h_pvs_zero_value_selected_power_terminal + S (bpvi_result_pvs_zero_value_selected) = S ((S (0)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_terminal. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_zero_value_selected_power) + (bpvi_result_pvs_zero_value_selected))) /\\ forall bpvi_j_pvs_zero_value_selected_power. (exists bpvi_product_gap_pvs_zero_value_selected_power. bpvi_product_gap_pvs_zero_value_selected_power + S bpvi_j_pvs_zero_value_selected_power = 0) -> exists bpvi_factor_pvs_zero_value_selected_power bpvi_partial_pvs_zero_value_selected_power bpvi_successor_pvs_zero_value_selected_power. ((((exists bpvi_h_pvs_zero_value_selected_power_factor. bpvi_h_pvs_zero_value_selected_power_factor + S (bpvi_factor_pvs_zero_value_selected_power) = S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_factor. bpvi_b_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_factor * S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_c_pvs_zero_value_selected_power) + (bpvi_factor_pvs_zero_value_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_value_selected_power_partial. bpvi_h_pvs_zero_value_selected_power_partial + S (bpvi_partial_pvs_zero_value_selected_power) = S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_partial. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_partial * S ((S (bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power) + (bpvi_partial_pvs_zero_value_selected_power))) /\\ ((((exists bpvi_h_pvs_zero_value_selected_power_successor. bpvi_h_pvs_zero_value_selected_power_successor + S (bpvi_successor_pvs_zero_value_selected_power) = S ((S (S bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power)) /\\ exists bpvi_q_pvs_zero_value_selected_power_successor. bpvi_u_pvs_zero_value_selected_power = bpvi_q_pvs_zero_value_selected_power_successor * S ((S (S bpvi_j_pvs_zero_value_selected_power)) * bpvi_v_pvs_zero_value_selected_power) + (bpvi_successor_pvs_zero_value_selected_power))) /\\ bpvi_successor_pvs_zero_value_selected_power = bpvi_partial_pvs_zero_value_selected_power * bpvi_factor_pvs_zero_value_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_value_selected. n = bpvi_result_pvs_zero_value_selected * bpvi_divisor_factor_pvs_zero_value_selected))) /\\ forall bpd_candidate_pvs_zero_value. (exists bpd_gap_pvs_zero_value_candidate_bound. bpd_gap_pvs_zero_value_candidate_bound + (bpd_candidate_pvs_zero_value) = (n)) -> (exists bpvi_result_pvs_zero_value_candidate. ((exists bpvi_b_pvs_zero_value_candidate_power bpvi_c_pvs_zero_value_candidate_power. ((forall bpvi_i_pvs_zero_value_candidate_power. (exists bpvi_repeat_gap_pvs_zero_value_candidate_power. bpvi_repeat_gap_pvs_zero_value_candidate_power + S bpvi_i_pvs_zero_value_candidate_power = bpd_candidate_pvs_zero_value) -> (((exists bpvi_h_pvs_zero_value_candidate_power_repeat. bpvi_h_pvs_zero_value_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_repeat. bpvi_b_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_repeat * S ((S (bpvi_i_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_zero_value_candidate_power bpvi_v_pvs_zero_value_candidate_power. ((((exists bpvi_h_pvs_zero_value_candidate_power_start. bpvi_h_pvs_zero_value_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_start. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_start * S ((S (0)) * bpvi_v_pvs_zero_value_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_zero_value_candidate_power_terminal. bpvi_h_pvs_zero_value_candidate_power_terminal + S (bpvi_result_pvs_zero_value_candidate) = S ((S (bpd_candidate_pvs_zero_value)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_terminal. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_terminal * S ((S (bpd_candidate_pvs_zero_value)) * bpvi_v_pvs_zero_value_candidate_power) + (bpvi_result_pvs_zero_value_candidate))) /\\ forall bpvi_j_pvs_zero_value_candidate_power. (exists bpvi_product_gap_pvs_zero_value_candidate_power. bpvi_product_gap_pvs_zero_value_candidate_power + S bpvi_j_pvs_zero_value_candidate_power = bpd_candidate_pvs_zero_value) -> exists bpvi_factor_pvs_zero_value_candidate_power bpvi_partial_pvs_zero_value_candidate_power bpvi_successor_pvs_zero_value_candidate_power. ((((exists bpvi_h_pvs_zero_value_candidate_power_factor. bpvi_h_pvs_zero_value_candidate_power_factor + S (bpvi_factor_pvs_zero_value_candidate_power) = S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_factor. bpvi_b_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_factor * S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_c_pvs_zero_value_candidate_power) + (bpvi_factor_pvs_zero_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_value_candidate_power_partial. bpvi_h_pvs_zero_value_candidate_power_partial + S (bpvi_partial_pvs_zero_value_candidate_power) = S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_partial. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_partial * S ((S (bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power) + (bpvi_partial_pvs_zero_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_zero_value_candidate_power_successor. bpvi_h_pvs_zero_value_candidate_power_successor + S (bpvi_successor_pvs_zero_value_candidate_power) = S ((S (S bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power)) /\\ exists bpvi_q_pvs_zero_value_candidate_power_successor. bpvi_u_pvs_zero_value_candidate_power = bpvi_q_pvs_zero_value_candidate_power_successor * S ((S (S bpvi_j_pvs_zero_value_candidate_power)) * bpvi_v_pvs_zero_value_candidate_power) + (bpvi_successor_pvs_zero_value_candidate_power))) /\\ bpvi_successor_pvs_zero_value_candidate_power = bpvi_partial_pvs_zero_value_candidate_power * bpvi_factor_pvs_zero_value_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_zero_value_candidate. n = bpvi_result_pvs_zero_value_candidate * bpvi_divisor_factor_pvs_zero_value_candidate)) -> (exists bpd_gap_pvs_zero_value_maximal. 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(n) = (p) * pvs_factor_nondivisor_forward)) /\\ (~(exists pvs_factor_nondivisor_reverse. (n) = (p) * pvs_factor_nondivisor_reverse) -> 0 = 0)", "specialize prime_power_valuation_zero_iff_not_divides (p)", "specialize prime_power_valuation_zero_iff_not_divides (n)", "specialize prime_power_valuation_zero_iff_not_divides (0)", "apply prime_power_valuation_zero_iff_not_divides", "exact hp", "exact hn", "exact hval", "cases hiff", "intro hdiv", "apply hiff_left", "refl", "exact hdiv" ], "script_sha256": "cebc8e433d9352bbe3e5d82b6da86f7fb791ae4294899fb97fbe8266afab0e83", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall p n. (~((p) = 1) /\\ forall pvs_left_nondivisor_domain pvs_right_nondivisor_domain. (p) = pvs_left_nondivisor_domain * pvs_right_nondivisor_domain -> pvs_left_nondivisor_domain = 1 \\/ pvs_right_nondivisor_domain = 1) -> ~(n = 0) -> (((exists bpd_gap_pvs_nondivisor_value_selected_bound. bpd_gap_pvs_nondivisor_value_selected_bound + (0) = (n)) /\\ (exists bpvi_result_pvs_nondivisor_value_selected. ((exists bpvi_b_pvs_nondivisor_value_selected_power bpvi_c_pvs_nondivisor_value_selected_power. ((forall bpvi_i_pvs_nondivisor_value_selected_power. (exists bpvi_repeat_gap_pvs_nondivisor_value_selected_power. bpvi_repeat_gap_pvs_nondivisor_value_selected_power + S bpvi_i_pvs_nondivisor_value_selected_power = 0) -> (((exists bpvi_h_pvs_nondivisor_value_selected_power_repeat. bpvi_h_pvs_nondivisor_value_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_repeat. bpvi_b_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_repeat * S ((S (bpvi_i_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_nondivisor_value_selected_power bpvi_v_pvs_nondivisor_value_selected_power. ((((exists bpvi_h_pvs_nondivisor_value_selected_power_start. bpvi_h_pvs_nondivisor_value_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_start. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_start * S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_terminal. bpvi_h_pvs_nondivisor_value_selected_power_terminal + S (bpvi_result_pvs_nondivisor_value_selected) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_terminal. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_result_pvs_nondivisor_value_selected))) /\\ forall bpvi_j_pvs_nondivisor_value_selected_power. (exists bpvi_product_gap_pvs_nondivisor_value_selected_power. bpvi_product_gap_pvs_nondivisor_value_selected_power + S bpvi_j_pvs_nondivisor_value_selected_power = 0) -> exists bpvi_factor_pvs_nondivisor_value_selected_power bpvi_partial_pvs_nondivisor_value_selected_power bpvi_successor_pvs_nondivisor_value_selected_power. ((((exists bpvi_h_pvs_nondivisor_value_selected_power_factor. bpvi_h_pvs_nondivisor_value_selected_power_factor + S (bpvi_factor_pvs_nondivisor_value_selected_power) = S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_factor. bpvi_b_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_factor * S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power) + (bpvi_factor_pvs_nondivisor_value_selected_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_partial. bpvi_h_pvs_nondivisor_value_selected_power_partial + S (bpvi_partial_pvs_nondivisor_value_selected_power) = S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_partial. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_partial * S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_partial_pvs_nondivisor_value_selected_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_successor. bpvi_h_pvs_nondivisor_value_selected_power_successor + S (bpvi_successor_pvs_nondivisor_value_selected_power) = S ((S (S bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_successor. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_successor * S ((S (S bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_successor_pvs_nondivisor_value_selected_power))) /\\ bpvi_successor_pvs_nondivisor_value_selected_power = bpvi_partial_pvs_nondivisor_value_selected_power * bpvi_factor_pvs_nondivisor_value_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_nondivisor_value_selected. n = bpvi_result_pvs_nondivisor_value_selected * bpvi_divisor_factor_pvs_nondivisor_value_selected))) /\\ forall bpd_candidate_pvs_nondivisor_value. (exists bpd_gap_pvs_nondivisor_value_candidate_bound. bpd_gap_pvs_nondivisor_value_candidate_bound + (bpd_candidate_pvs_nondivisor_value) = (n)) -> (exists bpvi_result_pvs_nondivisor_value_candidate. ((exists bpvi_b_pvs_nondivisor_value_candidate_power bpvi_c_pvs_nondivisor_value_candidate_power. ((forall bpvi_i_pvs_nondivisor_value_candidate_power. (exists bpvi_repeat_gap_pvs_nondivisor_value_candidate_power. bpvi_repeat_gap_pvs_nondivisor_value_candidate_power + S bpvi_i_pvs_nondivisor_value_candidate_power = bpd_candidate_pvs_nondivisor_value) -> (((exists bpvi_h_pvs_nondivisor_value_candidate_power_repeat. bpvi_h_pvs_nondivisor_value_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_repeat. bpvi_b_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_repeat * S ((S (bpvi_i_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_nondivisor_value_candidate_power bpvi_v_pvs_nondivisor_value_candidate_power. ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_start. bpvi_h_pvs_nondivisor_value_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_start. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_start * S ((S (0)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_terminal. bpvi_h_pvs_nondivisor_value_candidate_power_terminal + S (bpvi_result_pvs_nondivisor_value_candidate) = S ((S (bpd_candidate_pvs_nondivisor_value)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_terminal. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_terminal * S ((S (bpd_candidate_pvs_nondivisor_value)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_result_pvs_nondivisor_value_candidate))) /\\ forall bpvi_j_pvs_nondivisor_value_candidate_power. (exists bpvi_product_gap_pvs_nondivisor_value_candidate_power. bpvi_product_gap_pvs_nondivisor_value_candidate_power + S bpvi_j_pvs_nondivisor_value_candidate_power = bpd_candidate_pvs_nondivisor_value) -> exists bpvi_factor_pvs_nondivisor_value_candidate_power bpvi_partial_pvs_nondivisor_value_candidate_power bpvi_successor_pvs_nondivisor_value_candidate_power. ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_factor. bpvi_h_pvs_nondivisor_value_candidate_power_factor + S (bpvi_factor_pvs_nondivisor_value_candidate_power) = S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_factor. bpvi_b_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_factor * S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power) + (bpvi_factor_pvs_nondivisor_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_partial. bpvi_h_pvs_nondivisor_value_candidate_power_partial + S (bpvi_partial_pvs_nondivisor_value_candidate_power) = S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_partial. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_partial * S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_partial_pvs_nondivisor_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_successor. bpvi_h_pvs_nondivisor_value_candidate_power_successor + S (bpvi_successor_pvs_nondivisor_value_candidate_power) = S ((S (S bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_successor. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_successor * S ((S (S bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_successor_pvs_nondivisor_value_candidate_power))) /\\ bpvi_successor_pvs_nondivisor_value_candidate_power = bpvi_partial_pvs_nondivisor_value_candidate_power * bpvi_factor_pvs_nondivisor_value_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_nondivisor_value_candidate. n = bpvi_result_pvs_nondivisor_value_candidate * bpvi_divisor_factor_pvs_nondivisor_value_candidate)) -> (exists bpd_gap_pvs_nondivisor_value_maximal. bpd_gap_pvs_nondivisor_value_maximal + (bpd_candidate_pvs_nondivisor_value) = (0))) -> ~(exists pvs_factor_nondivisor_result. (n) = (p) * pvs_factor_nondivisor_result)", "statement_sha256": "b750abe102bb8f37365a0a69813b38974ff9a0d79eab9b60c9871b6f02a7ae0a", "summary": "Valuation zero excludes divisibility, with both intended domain guards explicit.", "summary_sha256": "31a2f040571cade1983044be892afed2545858a0b31199f70ed066759e3d0230" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "prime_power_valuation_zero_iff_not_divides" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=289]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_valuation_nondivisor_of_zero", "parent_alpha_version": "v34", "proof_bundle_node_id": 250, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_valuation_nondivisor_of_zero", "script": [ "intro p", "intro n", "intro hp", "intro hn", "intro hval", "have hiff : (0 = 0 -> ~(exists pvs_factor_nondivisor_forward. (n) = (p) * pvs_factor_nondivisor_forward)) /\\ (~(exists pvs_factor_nondivisor_reverse. (n) = (p) * pvs_factor_nondivisor_reverse) -> 0 = 0)", "specialize prime_power_valuation_zero_iff_not_divides (p)", "specialize prime_power_valuation_zero_iff_not_divides (n)", "specialize prime_power_valuation_zero_iff_not_divides (0)", "apply prime_power_valuation_zero_iff_not_divides", "exact hp", "exact hn", "exact hval", "cases hiff", "intro hdiv", "apply hiff_left", "refl", "exact hdiv" ], "script_sha256": "cebc8e433d9352bbe3e5d82b6da86f7fb791ae4294899fb97fbe8266afab0e83", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall p n. (~((p) = 1) /\\ forall pvs_left_nondivisor_domain pvs_right_nondivisor_domain. (p) = pvs_left_nondivisor_domain * pvs_right_nondivisor_domain -> pvs_left_nondivisor_domain = 1 \\/ pvs_right_nondivisor_domain = 1) -> ~(n = 0) -> (((exists bpd_gap_pvs_nondivisor_value_selected_bound. bpd_gap_pvs_nondivisor_value_selected_bound + (0) = (n)) /\\ (exists bpvi_result_pvs_nondivisor_value_selected. ((exists bpvi_b_pvs_nondivisor_value_selected_power bpvi_c_pvs_nondivisor_value_selected_power. ((forall bpvi_i_pvs_nondivisor_value_selected_power. (exists bpvi_repeat_gap_pvs_nondivisor_value_selected_power. bpvi_repeat_gap_pvs_nondivisor_value_selected_power + S bpvi_i_pvs_nondivisor_value_selected_power = 0) -> (((exists bpvi_h_pvs_nondivisor_value_selected_power_repeat. bpvi_h_pvs_nondivisor_value_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_repeat. bpvi_b_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_repeat * S ((S (bpvi_i_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_nondivisor_value_selected_power bpvi_v_pvs_nondivisor_value_selected_power. ((((exists bpvi_h_pvs_nondivisor_value_selected_power_start. bpvi_h_pvs_nondivisor_value_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_start. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_start * S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_terminal. bpvi_h_pvs_nondivisor_value_selected_power_terminal + S (bpvi_result_pvs_nondivisor_value_selected) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_terminal. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_result_pvs_nondivisor_value_selected))) /\\ forall bpvi_j_pvs_nondivisor_value_selected_power. (exists bpvi_product_gap_pvs_nondivisor_value_selected_power. bpvi_product_gap_pvs_nondivisor_value_selected_power + S bpvi_j_pvs_nondivisor_value_selected_power = 0) -> exists bpvi_factor_pvs_nondivisor_value_selected_power bpvi_partial_pvs_nondivisor_value_selected_power bpvi_successor_pvs_nondivisor_value_selected_power. ((((exists bpvi_h_pvs_nondivisor_value_selected_power_factor. bpvi_h_pvs_nondivisor_value_selected_power_factor + S (bpvi_factor_pvs_nondivisor_value_selected_power) = S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_factor. bpvi_b_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_factor * S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_c_pvs_nondivisor_value_selected_power) + (bpvi_factor_pvs_nondivisor_value_selected_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_partial. bpvi_h_pvs_nondivisor_value_selected_power_partial + S (bpvi_partial_pvs_nondivisor_value_selected_power) = S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_partial. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_partial * S ((S (bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_partial_pvs_nondivisor_value_selected_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_selected_power_successor. bpvi_h_pvs_nondivisor_value_selected_power_successor + S (bpvi_successor_pvs_nondivisor_value_selected_power) = S ((S (S bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power)) /\\ exists bpvi_q_pvs_nondivisor_value_selected_power_successor. bpvi_u_pvs_nondivisor_value_selected_power = bpvi_q_pvs_nondivisor_value_selected_power_successor * S ((S (S bpvi_j_pvs_nondivisor_value_selected_power)) * bpvi_v_pvs_nondivisor_value_selected_power) + (bpvi_successor_pvs_nondivisor_value_selected_power))) /\\ bpvi_successor_pvs_nondivisor_value_selected_power = bpvi_partial_pvs_nondivisor_value_selected_power * bpvi_factor_pvs_nondivisor_value_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_nondivisor_value_selected. n = bpvi_result_pvs_nondivisor_value_selected * bpvi_divisor_factor_pvs_nondivisor_value_selected))) /\\ forall bpd_candidate_pvs_nondivisor_value. (exists bpd_gap_pvs_nondivisor_value_candidate_bound. bpd_gap_pvs_nondivisor_value_candidate_bound + (bpd_candidate_pvs_nondivisor_value) = (n)) -> (exists bpvi_result_pvs_nondivisor_value_candidate. ((exists bpvi_b_pvs_nondivisor_value_candidate_power bpvi_c_pvs_nondivisor_value_candidate_power. ((forall bpvi_i_pvs_nondivisor_value_candidate_power. (exists bpvi_repeat_gap_pvs_nondivisor_value_candidate_power. bpvi_repeat_gap_pvs_nondivisor_value_candidate_power + S bpvi_i_pvs_nondivisor_value_candidate_power = bpd_candidate_pvs_nondivisor_value) -> (((exists bpvi_h_pvs_nondivisor_value_candidate_power_repeat. bpvi_h_pvs_nondivisor_value_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_repeat. bpvi_b_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_repeat * S ((S (bpvi_i_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_nondivisor_value_candidate_power bpvi_v_pvs_nondivisor_value_candidate_power. ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_start. bpvi_h_pvs_nondivisor_value_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_start. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_start * S ((S (0)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_terminal. bpvi_h_pvs_nondivisor_value_candidate_power_terminal + S (bpvi_result_pvs_nondivisor_value_candidate) = S ((S (bpd_candidate_pvs_nondivisor_value)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_terminal. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_terminal * S ((S (bpd_candidate_pvs_nondivisor_value)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_result_pvs_nondivisor_value_candidate))) /\\ forall bpvi_j_pvs_nondivisor_value_candidate_power. (exists bpvi_product_gap_pvs_nondivisor_value_candidate_power. bpvi_product_gap_pvs_nondivisor_value_candidate_power + S bpvi_j_pvs_nondivisor_value_candidate_power = bpd_candidate_pvs_nondivisor_value) -> exists bpvi_factor_pvs_nondivisor_value_candidate_power bpvi_partial_pvs_nondivisor_value_candidate_power bpvi_successor_pvs_nondivisor_value_candidate_power. ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_factor. bpvi_h_pvs_nondivisor_value_candidate_power_factor + S (bpvi_factor_pvs_nondivisor_value_candidate_power) = S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_factor. bpvi_b_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_factor * S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_c_pvs_nondivisor_value_candidate_power) + (bpvi_factor_pvs_nondivisor_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_partial. bpvi_h_pvs_nondivisor_value_candidate_power_partial + S (bpvi_partial_pvs_nondivisor_value_candidate_power) = S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_partial. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_partial * S ((S (bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_partial_pvs_nondivisor_value_candidate_power))) /\\ ((((exists bpvi_h_pvs_nondivisor_value_candidate_power_successor. bpvi_h_pvs_nondivisor_value_candidate_power_successor + S (bpvi_successor_pvs_nondivisor_value_candidate_power) = S ((S (S bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power)) /\\ exists bpvi_q_pvs_nondivisor_value_candidate_power_successor. bpvi_u_pvs_nondivisor_value_candidate_power = bpvi_q_pvs_nondivisor_value_candidate_power_successor * S ((S (S bpvi_j_pvs_nondivisor_value_candidate_power)) * bpvi_v_pvs_nondivisor_value_candidate_power) + (bpvi_successor_pvs_nondivisor_value_candidate_power))) /\\ bpvi_successor_pvs_nondivisor_value_candidate_power = bpvi_partial_pvs_nondivisor_value_candidate_power * bpvi_factor_pvs_nondivisor_value_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_nondivisor_value_candidate. n = bpvi_result_pvs_nondivisor_value_candidate * bpvi_divisor_factor_pvs_nondivisor_value_candidate)) -> (exists bpd_gap_pvs_nondivisor_value_maximal. bpd_gap_pvs_nondivisor_value_maximal + (bpd_candidate_pvs_nondivisor_value) = (0))) -> ~(exists pvs_factor_nondivisor_result. 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"exact hpow", "trans 0", "specialize prime_power_valuation_one_zero (p)", "specialize prime_power_valuation_one_zero (z)", "specialize prime_power_valuation_one_zero (f)", "apply prime_power_valuation_one_zero", "exact hz", "exact hp", "exact hval", "symm", "apply mul_zero_left", "intro e", "intro z", "intro f", "intro hp", "intro ha", "intro hbase", "intro hpow", "intro hval", "have hprev : exists r. (exists pa_b_pvs_pow_predecessor pa_c_pvs_pow_predecessor. ((forall pa_i_pvs_pow_predecessor_repeat. (exists pa_lt_pvs_pow_predecessor_repeat_bound. pa_lt_pvs_pow_predecessor_repeat_bound + S pa_i_pvs_pow_predecessor_repeat = k) -> (((exists pa_h_pvs_pow_predecessor_repeat_decoded. pa_h_pvs_pow_predecessor_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_predecessor_repeat)) * pa_c_pvs_pow_predecessor)) /\\ exists pa_q_pvs_pow_predecessor_repeat_decoded. pa_b_pvs_pow_predecessor = pa_q_pvs_pow_predecessor_repeat_decoded * S ((S (pa_i_pvs_pow_predecessor_repeat)) * pa_c_pvs_pow_predecessor) + (a)))) /\\ (exists pa_u_pvs_pow_predecessor_product pa_v_pvs_pow_predecessor_product. ((((exists pa_h_pvs_pow_predecessor_product_start. pa_h_pvs_pow_predecessor_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_start. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_start * S ((S (0)) * pa_v_pvs_pow_predecessor_product) + (1))) /\\ ((((exists pa_h_pvs_pow_predecessor_product_terminal. pa_h_pvs_pow_predecessor_product_terminal + S (r) = S ((S (k)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_terminal. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_terminal * S ((S (k)) * pa_v_pvs_pow_predecessor_product) + (r))) /\\ forall pa_i_pvs_pow_predecessor_product. (exists pa_lt_pvs_pow_predecessor_product_bound. pa_lt_pvs_pow_predecessor_product_bound + S pa_i_pvs_pow_predecessor_product = k) -> exists pa_p_pvs_pow_predecessor_product pa_r_pvs_pow_predecessor_product pa_s_pvs_pow_predecessor_product. ((((exists pa_h_pvs_pow_predecessor_product_factor. pa_h_pvs_pow_predecessor_product_factor + S (pa_p_pvs_pow_predecessor_product) = S ((S (pa_i_pvs_pow_predecessor_product)) * pa_c_pvs_pow_predecessor)) /\\ exists pa_q_pvs_pow_predecessor_product_factor. pa_b_pvs_pow_predecessor = pa_q_pvs_pow_predecessor_product_factor * S ((S (pa_i_pvs_pow_predecessor_product)) * pa_c_pvs_pow_predecessor) + (pa_p_pvs_pow_predecessor_product))) /\\ ((((exists pa_h_pvs_pow_predecessor_product_partial. pa_h_pvs_pow_predecessor_product_partial + S (pa_r_pvs_pow_predecessor_product) = S ((S (pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_partial. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_partial * S ((S (pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product) + (pa_r_pvs_pow_predecessor_product))) /\\ ((((exists pa_h_pvs_pow_predecessor_product_successor. pa_h_pvs_pow_predecessor_product_successor + S (pa_s_pvs_pow_predecessor_product) = S ((S (S pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_successor. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_successor * S ((S (S pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product) + (pa_s_pvs_pow_predecessor_product))) /\\ pa_s_pvs_pow_predecessor_product = pa_r_pvs_pow_predecessor_product * pa_p_pvs_pow_predecessor_product)))))))) /\\ z = r * a", "specialize pow_successor_decompose (a)", "specialize pow_successor_decompose (k)", "specialize pow_successor_decompose (S k)", "specialize pow_successor_decompose (z)", "apply pow_successor_decompose", "refl", "exact hpow", "cases hprev", "cases hprev_witness", "have hv : exists j. (((exists bpd_gap_pvs_pow_predecessor_val_selected_bound. bpd_gap_pvs_pow_predecessor_val_selected_bound + (j) = (x)) /\\ (exists bpvi_result_pvs_pow_predecessor_val_selected. ((exists bpvi_b_pvs_pow_predecessor_val_selected_power bpvi_c_pvs_pow_predecessor_val_selected_power. ((forall bpvi_i_pvs_pow_predecessor_val_selected_power. (exists bpvi_repeat_gap_pvs_pow_predecessor_val_selected_power. bpvi_repeat_gap_pvs_pow_predecessor_val_selected_power + S bpvi_i_pvs_pow_predecessor_val_selected_power = j) -> (((exists bpvi_h_pvs_pow_predecessor_val_selected_power_repeat. bpvi_h_pvs_pow_predecessor_val_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_repeat. bpvi_b_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_repeat * S ((S (bpvi_i_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_predecessor_val_selected_power bpvi_v_pvs_pow_predecessor_val_selected_power. ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_start. bpvi_h_pvs_pow_predecessor_val_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_start. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_terminal. bpvi_h_pvs_pow_predecessor_val_selected_power_terminal + S (bpvi_result_pvs_pow_predecessor_val_selected) = S ((S (j)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_terminal. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_terminal * S ((S (j)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (bpvi_result_pvs_pow_predecessor_val_selected))) /\\ forall bpvi_j_pvs_pow_predecessor_val_selected_power. (exists bpvi_product_gap_pvs_pow_predecessor_val_selected_power. bpvi_product_gap_pvs_pow_predecessor_val_selected_power + S bpvi_j_pvs_pow_predecessor_val_selected_power = j) -> exists bpvi_factor_pvs_pow_predecessor_val_selected_power bpvi_partial_pvs_pow_predecessor_val_selected_power bpvi_successor_pvs_pow_predecessor_val_selected_power. ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_factor. bpvi_h_pvs_pow_predecessor_val_selected_power_factor + S (bpvi_factor_pvs_pow_predecessor_val_selected_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_factor. bpvi_b_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_factor * S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power) + (bpvi_factor_pvs_pow_predecessor_val_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_partial. bpvi_h_pvs_pow_predecessor_val_selected_power_partial + S (bpvi_partial_pvs_pow_predecessor_val_selected_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_partial. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_partial * S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (bpvi_partial_pvs_pow_predecessor_val_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_successor. bpvi_h_pvs_pow_predecessor_val_selected_power_successor + S (bpvi_successor_pvs_pow_predecessor_val_selected_power) = S ((S (S bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_successor. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_successor * S ((S (S bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (bpvi_successor_pvs_pow_predecessor_val_selected_power))) /\\ bpvi_successor_pvs_pow_predecessor_val_selected_power = bpvi_partial_pvs_pow_predecessor_val_selected_power * bpvi_factor_pvs_pow_predecessor_val_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_predecessor_val_selected. x = bpvi_result_pvs_pow_predecessor_val_selected * bpvi_divisor_factor_pvs_pow_predecessor_val_selected))) /\\ forall bpd_candidate_pvs_pow_predecessor_val. (exists bpd_gap_pvs_pow_predecessor_val_candidate_bound. bpd_gap_pvs_pow_predecessor_val_candidate_bound + (bpd_candidate_pvs_pow_predecessor_val) = (x)) -> (exists bpvi_result_pvs_pow_predecessor_val_candidate. ((exists bpvi_b_pvs_pow_predecessor_val_candidate_power bpvi_c_pvs_pow_predecessor_val_candidate_power. ((forall bpvi_i_pvs_pow_predecessor_val_candidate_power. (exists bpvi_repeat_gap_pvs_pow_predecessor_val_candidate_power. bpvi_repeat_gap_pvs_pow_predecessor_val_candidate_power + S bpvi_i_pvs_pow_predecessor_val_candidate_power = bpd_candidate_pvs_pow_predecessor_val) -> (((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_repeat. bpvi_h_pvs_pow_predecessor_val_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_repeat. bpvi_b_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_predecessor_val_candidate_power bpvi_v_pvs_pow_predecessor_val_candidate_power. ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_start. bpvi_h_pvs_pow_predecessor_val_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_start. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_terminal. bpvi_h_pvs_pow_predecessor_val_candidate_power_terminal + S (bpvi_result_pvs_pow_predecessor_val_candidate) = S ((S (bpd_candidate_pvs_pow_predecessor_val)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_terminal. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_predecessor_val)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (bpvi_result_pvs_pow_predecessor_val_candidate))) /\\ forall bpvi_j_pvs_pow_predecessor_val_candidate_power. (exists bpvi_product_gap_pvs_pow_predecessor_val_candidate_power. bpvi_product_gap_pvs_pow_predecessor_val_candidate_power + S bpvi_j_pvs_pow_predecessor_val_candidate_power = bpd_candidate_pvs_pow_predecessor_val) -> exists bpvi_factor_pvs_pow_predecessor_val_candidate_power bpvi_partial_pvs_pow_predecessor_val_candidate_power bpvi_successor_pvs_pow_predecessor_val_candidate_power. ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_factor. bpvi_h_pvs_pow_predecessor_val_candidate_power_factor + S (bpvi_factor_pvs_pow_predecessor_val_candidate_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_factor. bpvi_b_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_factor * S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power) + (bpvi_factor_pvs_pow_predecessor_val_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_partial. bpvi_h_pvs_pow_predecessor_val_candidate_power_partial + S (bpvi_partial_pvs_pow_predecessor_val_candidate_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_partial. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_partial * S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (bpvi_partial_pvs_pow_predecessor_val_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_successor. bpvi_h_pvs_pow_predecessor_val_candidate_power_successor + S (bpvi_successor_pvs_pow_predecessor_val_candidate_power) = S ((S (S bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_successor. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (bpvi_successor_pvs_pow_predecessor_val_candidate_power))) /\\ bpvi_successor_pvs_pow_predecessor_val_candidate_power = bpvi_partial_pvs_pow_predecessor_val_candidate_power * bpvi_factor_pvs_pow_predecessor_val_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_predecessor_val_candidate. x = bpvi_result_pvs_pow_predecessor_val_candidate * bpvi_divisor_factor_pvs_pow_predecessor_val_candidate)) -> (exists bpd_gap_pvs_pow_predecessor_val_maximal. bpd_gap_pvs_pow_predecessor_val_maximal + (bpd_candidate_pvs_pow_predecessor_val) = (j)))", "specialize power_valuation_exists (p)", "specialize power_valuation_exists (x)", "apply power_valuation_exists", "cases hv", "have hindex : x1 = k * e", "specialize IH (e)", "specialize IH (x)", "specialize IH (x1)", "apply IH", "exact hp", "exact ha", "exact hbase", "exact hprev_witness_left", "exact hv_witness", "have hx : ~(x = 0)", "intro hxzero", "specialize pow_nonzero_of_one_le (a)", "specialize pow_nonzero_of_one_le (k)", "specialize pow_nonzero_of_one_le (x)", "apply pow_nonzero_of_one_le", "specialize one_le_of_ne_zero (a)", "apply one_le_of_ne_zero", "exact ha", "exact hprev_witness_left", "exact hxzero", "have hproduct : ((exists bpd_gap_pvs_pow_product_selected_bound. bpd_gap_pvs_pow_product_selected_bound + (f) = (x * a)) /\\ (exists bpvi_result_pvs_pow_product_selected. ((exists bpvi_b_pvs_pow_product_selected_power bpvi_c_pvs_pow_product_selected_power. ((forall bpvi_i_pvs_pow_product_selected_power. (exists bpvi_repeat_gap_pvs_pow_product_selected_power. bpvi_repeat_gap_pvs_pow_product_selected_power + S bpvi_i_pvs_pow_product_selected_power = f) -> (((exists bpvi_h_pvs_pow_product_selected_power_repeat. bpvi_h_pvs_pow_product_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_repeat. bpvi_b_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_repeat * S ((S (bpvi_i_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_product_selected_power bpvi_v_pvs_pow_product_selected_power. ((((exists bpvi_h_pvs_pow_product_selected_power_start. bpvi_h_pvs_pow_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_start. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_product_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_product_selected_power_terminal. bpvi_h_pvs_pow_product_selected_power_terminal + S (bpvi_result_pvs_pow_product_selected) = S ((S (f)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_terminal. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_product_selected_power) + (bpvi_result_pvs_pow_product_selected))) /\\ forall bpvi_j_pvs_pow_product_selected_power. (exists bpvi_product_gap_pvs_pow_product_selected_power. bpvi_product_gap_pvs_pow_product_selected_power + S bpvi_j_pvs_pow_product_selected_power = f) -> exists bpvi_factor_pvs_pow_product_selected_power bpvi_partial_pvs_pow_product_selected_power bpvi_successor_pvs_pow_product_selected_power. ((((exists bpvi_h_pvs_pow_product_selected_power_factor. bpvi_h_pvs_pow_product_selected_power_factor + S (bpvi_factor_pvs_pow_product_selected_power) = S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_factor. bpvi_b_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_factor * S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power) + (bpvi_factor_pvs_pow_product_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_product_selected_power_partial. bpvi_h_pvs_pow_product_selected_power_partial + S (bpvi_partial_pvs_pow_product_selected_power) = S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_partial. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_partial * S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power) + (bpvi_partial_pvs_pow_product_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_product_selected_power_successor. bpvi_h_pvs_pow_product_selected_power_successor + S (bpvi_successor_pvs_pow_product_selected_power) = S ((S (S bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_successor. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_successor * S ((S (S bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power) + (bpvi_successor_pvs_pow_product_selected_power))) /\\ bpvi_successor_pvs_pow_product_selected_power = bpvi_partial_pvs_pow_product_selected_power * bpvi_factor_pvs_pow_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_product_selected. x * a = bpvi_result_pvs_pow_product_selected * bpvi_divisor_factor_pvs_pow_product_selected))) /\\ forall bpd_candidate_pvs_pow_product. (exists bpd_gap_pvs_pow_product_candidate_bound. bpd_gap_pvs_pow_product_candidate_bound + (bpd_candidate_pvs_pow_product) = (x * a)) -> (exists bpvi_result_pvs_pow_product_candidate. ((exists bpvi_b_pvs_pow_product_candidate_power bpvi_c_pvs_pow_product_candidate_power. ((forall bpvi_i_pvs_pow_product_candidate_power. (exists bpvi_repeat_gap_pvs_pow_product_candidate_power. bpvi_repeat_gap_pvs_pow_product_candidate_power + S bpvi_i_pvs_pow_product_candidate_power = bpd_candidate_pvs_pow_product) -> (((exists bpvi_h_pvs_pow_product_candidate_power_repeat. bpvi_h_pvs_pow_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_repeat. bpvi_b_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_product_candidate_power bpvi_v_pvs_pow_product_candidate_power. ((((exists bpvi_h_pvs_pow_product_candidate_power_start. bpvi_h_pvs_pow_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_start. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_product_candidate_power_terminal. bpvi_h_pvs_pow_product_candidate_power_terminal + S (bpvi_result_pvs_pow_product_candidate) = S ((S (bpd_candidate_pvs_pow_product)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_terminal. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_product)) * bpvi_v_pvs_pow_product_candidate_power) + (bpvi_result_pvs_pow_product_candidate))) /\\ forall bpvi_j_pvs_pow_product_candidate_power. (exists bpvi_product_gap_pvs_pow_product_candidate_power. bpvi_product_gap_pvs_pow_product_candidate_power + S bpvi_j_pvs_pow_product_candidate_power = bpd_candidate_pvs_pow_product) -> exists bpvi_factor_pvs_pow_product_candidate_power bpvi_partial_pvs_pow_product_candidate_power bpvi_successor_pvs_pow_product_candidate_power. ((((exists bpvi_h_pvs_pow_product_candidate_power_factor. bpvi_h_pvs_pow_product_candidate_power_factor + S (bpvi_factor_pvs_pow_product_candidate_power) = S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_factor. bpvi_b_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_factor * S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power) + (bpvi_factor_pvs_pow_product_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_product_candidate_power_partial. bpvi_h_pvs_pow_product_candidate_power_partial + S (bpvi_partial_pvs_pow_product_candidate_power) = S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_partial. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_partial * S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power) + (bpvi_partial_pvs_pow_product_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_product_candidate_power_successor. bpvi_h_pvs_pow_product_candidate_power_successor + S (bpvi_successor_pvs_pow_product_candidate_power) = S ((S (S bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_successor. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power) + (bpvi_successor_pvs_pow_product_candidate_power))) /\\ bpvi_successor_pvs_pow_product_candidate_power = bpvi_partial_pvs_pow_product_candidate_power * bpvi_factor_pvs_pow_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_product_candidate. x * a = bpvi_result_pvs_pow_product_candidate * bpvi_divisor_factor_pvs_pow_product_candidate)) -> (exists bpd_gap_pvs_pow_product_maximal. bpd_gap_pvs_pow_product_maximal + (bpd_candidate_pvs_pow_product) = (f))", "specialize power_valuation_value_eq_transport (p)", "specialize power_valuation_value_eq_transport (z)", "specialize power_valuation_value_eq_transport (x * a)", "specialize power_valuation_value_eq_transport (f)", "apply power_valuation_value_eq_transport", "exact hprev_witness_right", "exact hval", "trans x1 + e", "specialize prime_power_valuation_mul (p)", "specialize prime_power_valuation_mul (x)", "specialize prime_power_valuation_mul (a)", "specialize prime_power_valuation_mul (x1)", "specialize prime_power_valuation_mul (e)", "specialize prime_power_valuation_mul (f)", "apply prime_power_valuation_mul", "exact hp", "exact hx", "exact ha", "exact hv_witness", "exact hbase", "exact hproduct", "rewrite hindex", "symm", "apply mul_succ_left" ], "script_sha256": "cd3ac0f35fd4ff21a7ec025fe6da02245b535c227351d9bf38ee0f6ae9a7b31b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall p a k e z f. (~((p) = 1) /\\ forall pvs_left_pow_domain pvs_right_pow_domain. (p) = pvs_left_pow_domain * pvs_right_pow_domain -> pvs_left_pow_domain = 1 \\/ pvs_right_pow_domain = 1) -> ~(a = 0) -> (((exists bpd_gap_pvs_pow_base_selected_bound. bpd_gap_pvs_pow_base_selected_bound + (e) = (a)) /\\ (exists bpvi_result_pvs_pow_base_selected. ((exists bpvi_b_pvs_pow_base_selected_power bpvi_c_pvs_pow_base_selected_power. ((forall bpvi_i_pvs_pow_base_selected_power. (exists bpvi_repeat_gap_pvs_pow_base_selected_power. bpvi_repeat_gap_pvs_pow_base_selected_power + S bpvi_i_pvs_pow_base_selected_power = e) -> (((exists bpvi_h_pvs_pow_base_selected_power_repeat. bpvi_h_pvs_pow_base_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_repeat. bpvi_b_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_repeat * S ((S (bpvi_i_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_base_selected_power bpvi_v_pvs_pow_base_selected_power. ((((exists bpvi_h_pvs_pow_base_selected_power_start. bpvi_h_pvs_pow_base_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_start. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_base_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_base_selected_power_terminal. bpvi_h_pvs_pow_base_selected_power_terminal + S (bpvi_result_pvs_pow_base_selected) = S ((S (e)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_terminal. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_pow_base_selected_power) + (bpvi_result_pvs_pow_base_selected))) /\\ forall bpvi_j_pvs_pow_base_selected_power. (exists bpvi_product_gap_pvs_pow_base_selected_power. bpvi_product_gap_pvs_pow_base_selected_power + S bpvi_j_pvs_pow_base_selected_power = e) -> exists bpvi_factor_pvs_pow_base_selected_power bpvi_partial_pvs_pow_base_selected_power bpvi_successor_pvs_pow_base_selected_power. ((((exists bpvi_h_pvs_pow_base_selected_power_factor. bpvi_h_pvs_pow_base_selected_power_factor + S (bpvi_factor_pvs_pow_base_selected_power) = S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_factor. bpvi_b_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_factor * S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power) + (bpvi_factor_pvs_pow_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_base_selected_power_partial. bpvi_h_pvs_pow_base_selected_power_partial + S (bpvi_partial_pvs_pow_base_selected_power) = S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_partial. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_partial * S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power) + (bpvi_partial_pvs_pow_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_base_selected_power_successor. bpvi_h_pvs_pow_base_selected_power_successor + S (bpvi_successor_pvs_pow_base_selected_power) = S ((S (S bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_successor. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_successor * S ((S (S bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power) + (bpvi_successor_pvs_pow_base_selected_power))) /\\ bpvi_successor_pvs_pow_base_selected_power = bpvi_partial_pvs_pow_base_selected_power * bpvi_factor_pvs_pow_base_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_base_selected. a = bpvi_result_pvs_pow_base_selected * bpvi_divisor_factor_pvs_pow_base_selected))) /\\ forall bpd_candidate_pvs_pow_base. (exists bpd_gap_pvs_pow_base_candidate_bound. bpd_gap_pvs_pow_base_candidate_bound + (bpd_candidate_pvs_pow_base) = (a)) -> (exists bpvi_result_pvs_pow_base_candidate. ((exists bpvi_b_pvs_pow_base_candidate_power bpvi_c_pvs_pow_base_candidate_power. ((forall bpvi_i_pvs_pow_base_candidate_power. (exists bpvi_repeat_gap_pvs_pow_base_candidate_power. bpvi_repeat_gap_pvs_pow_base_candidate_power + S bpvi_i_pvs_pow_base_candidate_power = bpd_candidate_pvs_pow_base) -> (((exists bpvi_h_pvs_pow_base_candidate_power_repeat. bpvi_h_pvs_pow_base_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_repeat. bpvi_b_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_base_candidate_power bpvi_v_pvs_pow_base_candidate_power. ((((exists bpvi_h_pvs_pow_base_candidate_power_start. bpvi_h_pvs_pow_base_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_start. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_base_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_base_candidate_power_terminal. bpvi_h_pvs_pow_base_candidate_power_terminal + S (bpvi_result_pvs_pow_base_candidate) = S ((S (bpd_candidate_pvs_pow_base)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_terminal. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_base)) * bpvi_v_pvs_pow_base_candidate_power) + (bpvi_result_pvs_pow_base_candidate))) /\\ forall bpvi_j_pvs_pow_base_candidate_power. (exists bpvi_product_gap_pvs_pow_base_candidate_power. bpvi_product_gap_pvs_pow_base_candidate_power + S bpvi_j_pvs_pow_base_candidate_power = bpd_candidate_pvs_pow_base) -> exists bpvi_factor_pvs_pow_base_candidate_power bpvi_partial_pvs_pow_base_candidate_power bpvi_successor_pvs_pow_base_candidate_power. ((((exists bpvi_h_pvs_pow_base_candidate_power_factor. bpvi_h_pvs_pow_base_candidate_power_factor + S (bpvi_factor_pvs_pow_base_candidate_power) = S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_factor. bpvi_b_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_factor * S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power) + (bpvi_factor_pvs_pow_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_base_candidate_power_partial. bpvi_h_pvs_pow_base_candidate_power_partial + S (bpvi_partial_pvs_pow_base_candidate_power) = S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_partial. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_partial * S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power) + (bpvi_partial_pvs_pow_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_base_candidate_power_successor. bpvi_h_pvs_pow_base_candidate_power_successor + S (bpvi_successor_pvs_pow_base_candidate_power) = S ((S (S bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_successor. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power) + (bpvi_successor_pvs_pow_base_candidate_power))) /\\ bpvi_successor_pvs_pow_base_candidate_power = bpvi_partial_pvs_pow_base_candidate_power * bpvi_factor_pvs_pow_base_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_base_candidate. a = bpvi_result_pvs_pow_base_candidate * bpvi_divisor_factor_pvs_pow_base_candidate)) -> (exists bpd_gap_pvs_pow_base_maximal. bpd_gap_pvs_pow_base_maximal + (bpd_candidate_pvs_pow_base) = (e))) -> (exists pa_b_pvs_pow_source pa_c_pvs_pow_source. ((forall pa_i_pvs_pow_source_repeat. (exists pa_lt_pvs_pow_source_repeat_bound. pa_lt_pvs_pow_source_repeat_bound + S pa_i_pvs_pow_source_repeat = k) -> (((exists pa_h_pvs_pow_source_repeat_decoded. pa_h_pvs_pow_source_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_source_repeat)) * pa_c_pvs_pow_source)) /\\ exists pa_q_pvs_pow_source_repeat_decoded. pa_b_pvs_pow_source = pa_q_pvs_pow_source_repeat_decoded * S ((S (pa_i_pvs_pow_source_repeat)) * pa_c_pvs_pow_source) + (a)))) /\\ (exists pa_u_pvs_pow_source_product pa_v_pvs_pow_source_product. ((((exists pa_h_pvs_pow_source_product_start. pa_h_pvs_pow_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_start. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_start * S ((S (0)) * pa_v_pvs_pow_source_product) + (1))) /\\ ((((exists pa_h_pvs_pow_source_product_terminal. pa_h_pvs_pow_source_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_terminal. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_terminal * S ((S (k)) * pa_v_pvs_pow_source_product) + (z))) /\\ forall pa_i_pvs_pow_source_product. (exists pa_lt_pvs_pow_source_product_bound. pa_lt_pvs_pow_source_product_bound + S pa_i_pvs_pow_source_product = k) -> exists pa_p_pvs_pow_source_product pa_r_pvs_pow_source_product pa_s_pvs_pow_source_product. ((((exists pa_h_pvs_pow_source_product_factor. pa_h_pvs_pow_source_product_factor + S (pa_p_pvs_pow_source_product) = S ((S (pa_i_pvs_pow_source_product)) * pa_c_pvs_pow_source)) /\\ exists pa_q_pvs_pow_source_product_factor. pa_b_pvs_pow_source = pa_q_pvs_pow_source_product_factor * S ((S (pa_i_pvs_pow_source_product)) * pa_c_pvs_pow_source) + (pa_p_pvs_pow_source_product))) /\\ ((((exists pa_h_pvs_pow_source_product_partial. pa_h_pvs_pow_source_product_partial + S (pa_r_pvs_pow_source_product) = S ((S (pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_partial. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_partial * S ((S (pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product) + (pa_r_pvs_pow_source_product))) /\\ ((((exists pa_h_pvs_pow_source_product_successor. pa_h_pvs_pow_source_product_successor + S (pa_s_pvs_pow_source_product) = S ((S (S pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_successor. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_successor * S ((S (S pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product) + (pa_s_pvs_pow_source_product))) /\\ pa_s_pvs_pow_source_product = pa_r_pvs_pow_source_product * pa_p_pvs_pow_source_product)))))))) -> (((exists bpd_gap_pvs_pow_output_selected_bound. bpd_gap_pvs_pow_output_selected_bound + (f) = (z)) /\\ (exists bpvi_result_pvs_pow_output_selected. ((exists bpvi_b_pvs_pow_output_selected_power bpvi_c_pvs_pow_output_selected_power. ((forall bpvi_i_pvs_pow_output_selected_power. (exists bpvi_repeat_gap_pvs_pow_output_selected_power. bpvi_repeat_gap_pvs_pow_output_selected_power + S bpvi_i_pvs_pow_output_selected_power = f) -> (((exists bpvi_h_pvs_pow_output_selected_power_repeat. bpvi_h_pvs_pow_output_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_repeat. bpvi_b_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_repeat * S ((S (bpvi_i_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_output_selected_power bpvi_v_pvs_pow_output_selected_power. ((((exists bpvi_h_pvs_pow_output_selected_power_start. bpvi_h_pvs_pow_output_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_start. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_output_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_output_selected_power_terminal. bpvi_h_pvs_pow_output_selected_power_terminal + S (bpvi_result_pvs_pow_output_selected) = S ((S (f)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_terminal. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_output_selected_power) + (bpvi_result_pvs_pow_output_selected))) /\\ forall bpvi_j_pvs_pow_output_selected_power. (exists bpvi_product_gap_pvs_pow_output_selected_power. bpvi_product_gap_pvs_pow_output_selected_power + S bpvi_j_pvs_pow_output_selected_power = f) -> exists bpvi_factor_pvs_pow_output_selected_power bpvi_partial_pvs_pow_output_selected_power bpvi_successor_pvs_pow_output_selected_power. ((((exists bpvi_h_pvs_pow_output_selected_power_factor. bpvi_h_pvs_pow_output_selected_power_factor + S (bpvi_factor_pvs_pow_output_selected_power) = S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_factor. bpvi_b_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_factor * S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power) + (bpvi_factor_pvs_pow_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_output_selected_power_partial. bpvi_h_pvs_pow_output_selected_power_partial + S (bpvi_partial_pvs_pow_output_selected_power) = S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_partial. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_partial * S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power) + (bpvi_partial_pvs_pow_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_output_selected_power_successor. bpvi_h_pvs_pow_output_selected_power_successor + S (bpvi_successor_pvs_pow_output_selected_power) = S ((S (S bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_successor. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_successor * S ((S (S bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power) + (bpvi_successor_pvs_pow_output_selected_power))) /\\ bpvi_successor_pvs_pow_output_selected_power = bpvi_partial_pvs_pow_output_selected_power * bpvi_factor_pvs_pow_output_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_output_selected. z = bpvi_result_pvs_pow_output_selected * bpvi_divisor_factor_pvs_pow_output_selected))) /\\ forall bpd_candidate_pvs_pow_output. (exists bpd_gap_pvs_pow_output_candidate_bound. bpd_gap_pvs_pow_output_candidate_bound + (bpd_candidate_pvs_pow_output) = (z)) -> (exists bpvi_result_pvs_pow_output_candidate. ((exists bpvi_b_pvs_pow_output_candidate_power bpvi_c_pvs_pow_output_candidate_power. ((forall bpvi_i_pvs_pow_output_candidate_power. (exists bpvi_repeat_gap_pvs_pow_output_candidate_power. bpvi_repeat_gap_pvs_pow_output_candidate_power + S bpvi_i_pvs_pow_output_candidate_power = bpd_candidate_pvs_pow_output) -> (((exists bpvi_h_pvs_pow_output_candidate_power_repeat. bpvi_h_pvs_pow_output_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_repeat. bpvi_b_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_output_candidate_power bpvi_v_pvs_pow_output_candidate_power. ((((exists bpvi_h_pvs_pow_output_candidate_power_start. bpvi_h_pvs_pow_output_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_start. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_output_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_output_candidate_power_terminal. bpvi_h_pvs_pow_output_candidate_power_terminal + S (bpvi_result_pvs_pow_output_candidate) = S ((S (bpd_candidate_pvs_pow_output)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_terminal. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_output)) * bpvi_v_pvs_pow_output_candidate_power) + (bpvi_result_pvs_pow_output_candidate))) /\\ forall bpvi_j_pvs_pow_output_candidate_power. (exists bpvi_product_gap_pvs_pow_output_candidate_power. bpvi_product_gap_pvs_pow_output_candidate_power + S bpvi_j_pvs_pow_output_candidate_power = bpd_candidate_pvs_pow_output) -> exists bpvi_factor_pvs_pow_output_candidate_power bpvi_partial_pvs_pow_output_candidate_power bpvi_successor_pvs_pow_output_candidate_power. ((((exists bpvi_h_pvs_pow_output_candidate_power_factor. bpvi_h_pvs_pow_output_candidate_power_factor + S (bpvi_factor_pvs_pow_output_candidate_power) = S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_factor. bpvi_b_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_factor * S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power) + (bpvi_factor_pvs_pow_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_output_candidate_power_partial. bpvi_h_pvs_pow_output_candidate_power_partial + S (bpvi_partial_pvs_pow_output_candidate_power) = S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_partial. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_partial * S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power) + (bpvi_partial_pvs_pow_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_output_candidate_power_successor. bpvi_h_pvs_pow_output_candidate_power_successor + S (bpvi_successor_pvs_pow_output_candidate_power) = S ((S (S bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_successor. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power) + (bpvi_successor_pvs_pow_output_candidate_power))) /\\ bpvi_successor_pvs_pow_output_candidate_power = bpvi_partial_pvs_pow_output_candidate_power * bpvi_factor_pvs_pow_output_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_output_candidate. z = bpvi_result_pvs_pow_output_candidate * bpvi_divisor_factor_pvs_pow_output_candidate)) -> (exists bpd_gap_pvs_pow_output_maximal. bpd_gap_pvs_pow_output_maximal + (bpd_candidate_pvs_pow_output) = (f))) -> f = k * e", "statement_sha256": "e3d8fa56e8d8d7d6a4bcabba8834b1a96127c075e907484a4d95c2fcf31478b9", "summary": "The exact valuation of any witnessed nonnegative power is its exponent times the base valuation; zero powers are included.", "summary_sha256": "775749b5afc8fabfae2c941680519fab45214d7725eb3c6c40f37104b69544dc" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "pow_zero", "prime_power_valuation_one_zero", "mul_zero_left", "pow_successor_decompose", "power_valuation_exists", "one_le_of_ne_zero", "pow_nonzero_of_one_le", "power_valuation_value_eq_transport", "prime_power_valuation_mul", "mul_succ_left" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=290]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_power_valuation_pow_value", "parent_alpha_version": "v34", "proof_bundle_node_id": 251, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_power_valuation_pow_value", "script": [ "intro p", "intro a", "intro k", "induction k", "intro e", "intro z", "intro f", "intro hp", "intro ha", "intro hbase", "intro hpow", "intro hval", "have hz : z = 1", "specialize pow_zero (a)", "specialize pow_zero (0)", "specialize pow_zero (z)", "apply pow_zero", "refl", "exact hpow", "trans 0", "specialize prime_power_valuation_one_zero (p)", "specialize prime_power_valuation_one_zero (z)", "specialize prime_power_valuation_one_zero (f)", "apply prime_power_valuation_one_zero", "exact hz", "exact hp", "exact hval", "symm", "apply mul_zero_left", "intro e", "intro z", "intro f", "intro hp", "intro ha", "intro hbase", "intro hpow", "intro hval", "have hprev : exists r. (exists pa_b_pvs_pow_predecessor pa_c_pvs_pow_predecessor. ((forall pa_i_pvs_pow_predecessor_repeat. (exists pa_lt_pvs_pow_predecessor_repeat_bound. pa_lt_pvs_pow_predecessor_repeat_bound + S pa_i_pvs_pow_predecessor_repeat = k) -> (((exists pa_h_pvs_pow_predecessor_repeat_decoded. pa_h_pvs_pow_predecessor_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_predecessor_repeat)) * pa_c_pvs_pow_predecessor)) /\\ exists pa_q_pvs_pow_predecessor_repeat_decoded. pa_b_pvs_pow_predecessor = pa_q_pvs_pow_predecessor_repeat_decoded * S ((S (pa_i_pvs_pow_predecessor_repeat)) * pa_c_pvs_pow_predecessor) + (a)))) /\\ (exists pa_u_pvs_pow_predecessor_product pa_v_pvs_pow_predecessor_product. ((((exists pa_h_pvs_pow_predecessor_product_start. pa_h_pvs_pow_predecessor_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_start. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_start * S ((S (0)) * pa_v_pvs_pow_predecessor_product) + (1))) /\\ ((((exists pa_h_pvs_pow_predecessor_product_terminal. pa_h_pvs_pow_predecessor_product_terminal + S (r) = S ((S (k)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_terminal. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_terminal * S ((S (k)) * pa_v_pvs_pow_predecessor_product) + (r))) /\\ forall pa_i_pvs_pow_predecessor_product. (exists pa_lt_pvs_pow_predecessor_product_bound. pa_lt_pvs_pow_predecessor_product_bound + S pa_i_pvs_pow_predecessor_product = k) -> exists pa_p_pvs_pow_predecessor_product pa_r_pvs_pow_predecessor_product pa_s_pvs_pow_predecessor_product. ((((exists pa_h_pvs_pow_predecessor_product_factor. pa_h_pvs_pow_predecessor_product_factor + S (pa_p_pvs_pow_predecessor_product) = S ((S (pa_i_pvs_pow_predecessor_product)) * pa_c_pvs_pow_predecessor)) /\\ exists pa_q_pvs_pow_predecessor_product_factor. pa_b_pvs_pow_predecessor = pa_q_pvs_pow_predecessor_product_factor * S ((S (pa_i_pvs_pow_predecessor_product)) * pa_c_pvs_pow_predecessor) + (pa_p_pvs_pow_predecessor_product))) /\\ ((((exists pa_h_pvs_pow_predecessor_product_partial. pa_h_pvs_pow_predecessor_product_partial + S (pa_r_pvs_pow_predecessor_product) = S ((S (pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_partial. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_partial * S ((S (pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product) + (pa_r_pvs_pow_predecessor_product))) /\\ ((((exists pa_h_pvs_pow_predecessor_product_successor. pa_h_pvs_pow_predecessor_product_successor + S (pa_s_pvs_pow_predecessor_product) = S ((S (S pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product)) /\\ exists pa_q_pvs_pow_predecessor_product_successor. pa_u_pvs_pow_predecessor_product = pa_q_pvs_pow_predecessor_product_successor * S ((S (S pa_i_pvs_pow_predecessor_product)) * pa_v_pvs_pow_predecessor_product) + (pa_s_pvs_pow_predecessor_product))) /\\ pa_s_pvs_pow_predecessor_product = pa_r_pvs_pow_predecessor_product * pa_p_pvs_pow_predecessor_product)))))))) /\\ z = r * a", "specialize pow_successor_decompose (a)", "specialize pow_successor_decompose (k)", "specialize pow_successor_decompose (S k)", "specialize pow_successor_decompose (z)", "apply pow_successor_decompose", "refl", "exact hpow", "cases hprev", "cases hprev_witness", "have hv : exists j. (((exists bpd_gap_pvs_pow_predecessor_val_selected_bound. bpd_gap_pvs_pow_predecessor_val_selected_bound + (j) = (x)) /\\ (exists bpvi_result_pvs_pow_predecessor_val_selected. ((exists bpvi_b_pvs_pow_predecessor_val_selected_power bpvi_c_pvs_pow_predecessor_val_selected_power. ((forall bpvi_i_pvs_pow_predecessor_val_selected_power. (exists bpvi_repeat_gap_pvs_pow_predecessor_val_selected_power. bpvi_repeat_gap_pvs_pow_predecessor_val_selected_power + S bpvi_i_pvs_pow_predecessor_val_selected_power = j) -> (((exists bpvi_h_pvs_pow_predecessor_val_selected_power_repeat. bpvi_h_pvs_pow_predecessor_val_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_repeat. bpvi_b_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_repeat * S ((S (bpvi_i_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_predecessor_val_selected_power bpvi_v_pvs_pow_predecessor_val_selected_power. ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_start. bpvi_h_pvs_pow_predecessor_val_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_start. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_terminal. bpvi_h_pvs_pow_predecessor_val_selected_power_terminal + S (bpvi_result_pvs_pow_predecessor_val_selected) = S ((S (j)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_terminal. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_terminal * S ((S (j)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (bpvi_result_pvs_pow_predecessor_val_selected))) /\\ forall bpvi_j_pvs_pow_predecessor_val_selected_power. (exists bpvi_product_gap_pvs_pow_predecessor_val_selected_power. bpvi_product_gap_pvs_pow_predecessor_val_selected_power + S bpvi_j_pvs_pow_predecessor_val_selected_power = j) -> exists bpvi_factor_pvs_pow_predecessor_val_selected_power bpvi_partial_pvs_pow_predecessor_val_selected_power bpvi_successor_pvs_pow_predecessor_val_selected_power. ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_factor. bpvi_h_pvs_pow_predecessor_val_selected_power_factor + S (bpvi_factor_pvs_pow_predecessor_val_selected_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_factor. bpvi_b_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_factor * S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_c_pvs_pow_predecessor_val_selected_power) + (bpvi_factor_pvs_pow_predecessor_val_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_partial. bpvi_h_pvs_pow_predecessor_val_selected_power_partial + S (bpvi_partial_pvs_pow_predecessor_val_selected_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_partial. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_partial * S ((S (bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (bpvi_partial_pvs_pow_predecessor_val_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_selected_power_successor. bpvi_h_pvs_pow_predecessor_val_selected_power_successor + S (bpvi_successor_pvs_pow_predecessor_val_selected_power) = S ((S (S bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_selected_power_successor. bpvi_u_pvs_pow_predecessor_val_selected_power = bpvi_q_pvs_pow_predecessor_val_selected_power_successor * S ((S (S bpvi_j_pvs_pow_predecessor_val_selected_power)) * bpvi_v_pvs_pow_predecessor_val_selected_power) + (bpvi_successor_pvs_pow_predecessor_val_selected_power))) /\\ bpvi_successor_pvs_pow_predecessor_val_selected_power = bpvi_partial_pvs_pow_predecessor_val_selected_power * bpvi_factor_pvs_pow_predecessor_val_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_predecessor_val_selected. x = bpvi_result_pvs_pow_predecessor_val_selected * bpvi_divisor_factor_pvs_pow_predecessor_val_selected))) /\\ forall bpd_candidate_pvs_pow_predecessor_val. (exists bpd_gap_pvs_pow_predecessor_val_candidate_bound. bpd_gap_pvs_pow_predecessor_val_candidate_bound + (bpd_candidate_pvs_pow_predecessor_val) = (x)) -> (exists bpvi_result_pvs_pow_predecessor_val_candidate. ((exists bpvi_b_pvs_pow_predecessor_val_candidate_power bpvi_c_pvs_pow_predecessor_val_candidate_power. ((forall bpvi_i_pvs_pow_predecessor_val_candidate_power. (exists bpvi_repeat_gap_pvs_pow_predecessor_val_candidate_power. bpvi_repeat_gap_pvs_pow_predecessor_val_candidate_power + S bpvi_i_pvs_pow_predecessor_val_candidate_power = bpd_candidate_pvs_pow_predecessor_val) -> (((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_repeat. bpvi_h_pvs_pow_predecessor_val_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_repeat. bpvi_b_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_predecessor_val_candidate_power bpvi_v_pvs_pow_predecessor_val_candidate_power. ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_start. bpvi_h_pvs_pow_predecessor_val_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_start. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_terminal. bpvi_h_pvs_pow_predecessor_val_candidate_power_terminal + S (bpvi_result_pvs_pow_predecessor_val_candidate) = S ((S (bpd_candidate_pvs_pow_predecessor_val)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_terminal. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_predecessor_val)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (bpvi_result_pvs_pow_predecessor_val_candidate))) /\\ forall bpvi_j_pvs_pow_predecessor_val_candidate_power. (exists bpvi_product_gap_pvs_pow_predecessor_val_candidate_power. bpvi_product_gap_pvs_pow_predecessor_val_candidate_power + S bpvi_j_pvs_pow_predecessor_val_candidate_power = bpd_candidate_pvs_pow_predecessor_val) -> exists bpvi_factor_pvs_pow_predecessor_val_candidate_power bpvi_partial_pvs_pow_predecessor_val_candidate_power bpvi_successor_pvs_pow_predecessor_val_candidate_power. ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_factor. bpvi_h_pvs_pow_predecessor_val_candidate_power_factor + S (bpvi_factor_pvs_pow_predecessor_val_candidate_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_factor. bpvi_b_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_factor * S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_c_pvs_pow_predecessor_val_candidate_power) + (bpvi_factor_pvs_pow_predecessor_val_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_partial. bpvi_h_pvs_pow_predecessor_val_candidate_power_partial + S (bpvi_partial_pvs_pow_predecessor_val_candidate_power) = S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_partial. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_partial * S ((S (bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (bpvi_partial_pvs_pow_predecessor_val_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_predecessor_val_candidate_power_successor. bpvi_h_pvs_pow_predecessor_val_candidate_power_successor + S (bpvi_successor_pvs_pow_predecessor_val_candidate_power) = S ((S (S bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power)) /\\ exists bpvi_q_pvs_pow_predecessor_val_candidate_power_successor. bpvi_u_pvs_pow_predecessor_val_candidate_power = bpvi_q_pvs_pow_predecessor_val_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_predecessor_val_candidate_power)) * bpvi_v_pvs_pow_predecessor_val_candidate_power) + (bpvi_successor_pvs_pow_predecessor_val_candidate_power))) /\\ bpvi_successor_pvs_pow_predecessor_val_candidate_power = bpvi_partial_pvs_pow_predecessor_val_candidate_power * bpvi_factor_pvs_pow_predecessor_val_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_predecessor_val_candidate. x = bpvi_result_pvs_pow_predecessor_val_candidate * bpvi_divisor_factor_pvs_pow_predecessor_val_candidate)) -> (exists bpd_gap_pvs_pow_predecessor_val_maximal. bpd_gap_pvs_pow_predecessor_val_maximal + (bpd_candidate_pvs_pow_predecessor_val) = (j)))", "specialize power_valuation_exists (p)", "specialize power_valuation_exists (x)", "apply power_valuation_exists", "cases hv", "have hindex : x1 = k * e", "specialize IH (e)", "specialize IH (x)", "specialize IH (x1)", "apply IH", "exact hp", "exact ha", "exact hbase", "exact hprev_witness_left", "exact hv_witness", "have hx : ~(x = 0)", "intro hxzero", "specialize pow_nonzero_of_one_le (a)", "specialize pow_nonzero_of_one_le (k)", "specialize pow_nonzero_of_one_le (x)", "apply pow_nonzero_of_one_le", "specialize one_le_of_ne_zero (a)", "apply one_le_of_ne_zero", "exact ha", "exact hprev_witness_left", "exact hxzero", "have hproduct : ((exists bpd_gap_pvs_pow_product_selected_bound. bpd_gap_pvs_pow_product_selected_bound + (f) = (x * a)) /\\ (exists bpvi_result_pvs_pow_product_selected. ((exists bpvi_b_pvs_pow_product_selected_power bpvi_c_pvs_pow_product_selected_power. ((forall bpvi_i_pvs_pow_product_selected_power. (exists bpvi_repeat_gap_pvs_pow_product_selected_power. bpvi_repeat_gap_pvs_pow_product_selected_power + S bpvi_i_pvs_pow_product_selected_power = f) -> (((exists bpvi_h_pvs_pow_product_selected_power_repeat. bpvi_h_pvs_pow_product_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_repeat. bpvi_b_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_repeat * S ((S (bpvi_i_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_product_selected_power bpvi_v_pvs_pow_product_selected_power. ((((exists bpvi_h_pvs_pow_product_selected_power_start. bpvi_h_pvs_pow_product_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_start. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_product_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_product_selected_power_terminal. bpvi_h_pvs_pow_product_selected_power_terminal + S (bpvi_result_pvs_pow_product_selected) = S ((S (f)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_terminal. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_product_selected_power) + (bpvi_result_pvs_pow_product_selected))) /\\ forall bpvi_j_pvs_pow_product_selected_power. (exists bpvi_product_gap_pvs_pow_product_selected_power. bpvi_product_gap_pvs_pow_product_selected_power + S bpvi_j_pvs_pow_product_selected_power = f) -> exists bpvi_factor_pvs_pow_product_selected_power bpvi_partial_pvs_pow_product_selected_power bpvi_successor_pvs_pow_product_selected_power. ((((exists bpvi_h_pvs_pow_product_selected_power_factor. bpvi_h_pvs_pow_product_selected_power_factor + S (bpvi_factor_pvs_pow_product_selected_power) = S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_factor. bpvi_b_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_factor * S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_c_pvs_pow_product_selected_power) + (bpvi_factor_pvs_pow_product_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_product_selected_power_partial. bpvi_h_pvs_pow_product_selected_power_partial + S (bpvi_partial_pvs_pow_product_selected_power) = S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_partial. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_partial * S ((S (bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power) + (bpvi_partial_pvs_pow_product_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_product_selected_power_successor. bpvi_h_pvs_pow_product_selected_power_successor + S (bpvi_successor_pvs_pow_product_selected_power) = S ((S (S bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power)) /\\ exists bpvi_q_pvs_pow_product_selected_power_successor. bpvi_u_pvs_pow_product_selected_power = bpvi_q_pvs_pow_product_selected_power_successor * S ((S (S bpvi_j_pvs_pow_product_selected_power)) * bpvi_v_pvs_pow_product_selected_power) + (bpvi_successor_pvs_pow_product_selected_power))) /\\ bpvi_successor_pvs_pow_product_selected_power = bpvi_partial_pvs_pow_product_selected_power * bpvi_factor_pvs_pow_product_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_product_selected. x * a = bpvi_result_pvs_pow_product_selected * bpvi_divisor_factor_pvs_pow_product_selected))) /\\ forall bpd_candidate_pvs_pow_product. (exists bpd_gap_pvs_pow_product_candidate_bound. bpd_gap_pvs_pow_product_candidate_bound + (bpd_candidate_pvs_pow_product) = (x * a)) -> (exists bpvi_result_pvs_pow_product_candidate. ((exists bpvi_b_pvs_pow_product_candidate_power bpvi_c_pvs_pow_product_candidate_power. ((forall bpvi_i_pvs_pow_product_candidate_power. (exists bpvi_repeat_gap_pvs_pow_product_candidate_power. bpvi_repeat_gap_pvs_pow_product_candidate_power + S bpvi_i_pvs_pow_product_candidate_power = bpd_candidate_pvs_pow_product) -> (((exists bpvi_h_pvs_pow_product_candidate_power_repeat. bpvi_h_pvs_pow_product_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_repeat. bpvi_b_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_product_candidate_power bpvi_v_pvs_pow_product_candidate_power. ((((exists bpvi_h_pvs_pow_product_candidate_power_start. bpvi_h_pvs_pow_product_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_start. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_product_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_product_candidate_power_terminal. bpvi_h_pvs_pow_product_candidate_power_terminal + S (bpvi_result_pvs_pow_product_candidate) = S ((S (bpd_candidate_pvs_pow_product)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_terminal. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_product)) * bpvi_v_pvs_pow_product_candidate_power) + (bpvi_result_pvs_pow_product_candidate))) /\\ forall bpvi_j_pvs_pow_product_candidate_power. (exists bpvi_product_gap_pvs_pow_product_candidate_power. bpvi_product_gap_pvs_pow_product_candidate_power + S bpvi_j_pvs_pow_product_candidate_power = bpd_candidate_pvs_pow_product) -> exists bpvi_factor_pvs_pow_product_candidate_power bpvi_partial_pvs_pow_product_candidate_power bpvi_successor_pvs_pow_product_candidate_power. ((((exists bpvi_h_pvs_pow_product_candidate_power_factor. bpvi_h_pvs_pow_product_candidate_power_factor + S (bpvi_factor_pvs_pow_product_candidate_power) = S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_factor. bpvi_b_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_factor * S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_c_pvs_pow_product_candidate_power) + (bpvi_factor_pvs_pow_product_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_product_candidate_power_partial. bpvi_h_pvs_pow_product_candidate_power_partial + S (bpvi_partial_pvs_pow_product_candidate_power) = S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_partial. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_partial * S ((S (bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power) + (bpvi_partial_pvs_pow_product_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_product_candidate_power_successor. bpvi_h_pvs_pow_product_candidate_power_successor + S (bpvi_successor_pvs_pow_product_candidate_power) = S ((S (S bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power)) /\\ exists bpvi_q_pvs_pow_product_candidate_power_successor. bpvi_u_pvs_pow_product_candidate_power = bpvi_q_pvs_pow_product_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_product_candidate_power)) * bpvi_v_pvs_pow_product_candidate_power) + (bpvi_successor_pvs_pow_product_candidate_power))) /\\ bpvi_successor_pvs_pow_product_candidate_power = bpvi_partial_pvs_pow_product_candidate_power * bpvi_factor_pvs_pow_product_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_product_candidate. x * a = bpvi_result_pvs_pow_product_candidate * bpvi_divisor_factor_pvs_pow_product_candidate)) -> (exists bpd_gap_pvs_pow_product_maximal. bpd_gap_pvs_pow_product_maximal + (bpd_candidate_pvs_pow_product) = (f))", "specialize power_valuation_value_eq_transport (p)", "specialize power_valuation_value_eq_transport (z)", "specialize power_valuation_value_eq_transport (x * a)", "specialize power_valuation_value_eq_transport (f)", "apply power_valuation_value_eq_transport", "exact hprev_witness_right", "exact hval", "trans x1 + e", "specialize prime_power_valuation_mul (p)", "specialize prime_power_valuation_mul (x)", "specialize prime_power_valuation_mul (a)", "specialize prime_power_valuation_mul (x1)", "specialize prime_power_valuation_mul (e)", "specialize prime_power_valuation_mul (f)", "apply prime_power_valuation_mul", "exact hp", "exact hx", "exact ha", "exact hv_witness", "exact hbase", "exact hproduct", "rewrite hindex", "symm", "apply mul_succ_left" ], "script_sha256": "cd3ac0f35fd4ff21a7ec025fe6da02245b535c227351d9bf38ee0f6ae9a7b31b", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall p a k e z f. (~((p) = 1) /\\ forall pvs_left_pow_domain pvs_right_pow_domain. (p) = pvs_left_pow_domain * pvs_right_pow_domain -> pvs_left_pow_domain = 1 \\/ pvs_right_pow_domain = 1) -> ~(a = 0) -> (((exists bpd_gap_pvs_pow_base_selected_bound. bpd_gap_pvs_pow_base_selected_bound + (e) = (a)) /\\ (exists bpvi_result_pvs_pow_base_selected. ((exists bpvi_b_pvs_pow_base_selected_power bpvi_c_pvs_pow_base_selected_power. ((forall bpvi_i_pvs_pow_base_selected_power. (exists bpvi_repeat_gap_pvs_pow_base_selected_power. bpvi_repeat_gap_pvs_pow_base_selected_power + S bpvi_i_pvs_pow_base_selected_power = e) -> (((exists bpvi_h_pvs_pow_base_selected_power_repeat. bpvi_h_pvs_pow_base_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_repeat. bpvi_b_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_repeat * S ((S (bpvi_i_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_base_selected_power bpvi_v_pvs_pow_base_selected_power. ((((exists bpvi_h_pvs_pow_base_selected_power_start. bpvi_h_pvs_pow_base_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_start. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_base_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_base_selected_power_terminal. bpvi_h_pvs_pow_base_selected_power_terminal + S (bpvi_result_pvs_pow_base_selected) = S ((S (e)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_terminal. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_pow_base_selected_power) + (bpvi_result_pvs_pow_base_selected))) /\\ forall bpvi_j_pvs_pow_base_selected_power. (exists bpvi_product_gap_pvs_pow_base_selected_power. bpvi_product_gap_pvs_pow_base_selected_power + S bpvi_j_pvs_pow_base_selected_power = e) -> exists bpvi_factor_pvs_pow_base_selected_power bpvi_partial_pvs_pow_base_selected_power bpvi_successor_pvs_pow_base_selected_power. ((((exists bpvi_h_pvs_pow_base_selected_power_factor. bpvi_h_pvs_pow_base_selected_power_factor + S (bpvi_factor_pvs_pow_base_selected_power) = S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_factor. bpvi_b_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_factor * S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_c_pvs_pow_base_selected_power) + (bpvi_factor_pvs_pow_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_base_selected_power_partial. bpvi_h_pvs_pow_base_selected_power_partial + S (bpvi_partial_pvs_pow_base_selected_power) = S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_partial. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_partial * S ((S (bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power) + (bpvi_partial_pvs_pow_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_base_selected_power_successor. bpvi_h_pvs_pow_base_selected_power_successor + S (bpvi_successor_pvs_pow_base_selected_power) = S ((S (S bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power)) /\\ exists bpvi_q_pvs_pow_base_selected_power_successor. bpvi_u_pvs_pow_base_selected_power = bpvi_q_pvs_pow_base_selected_power_successor * S ((S (S bpvi_j_pvs_pow_base_selected_power)) * bpvi_v_pvs_pow_base_selected_power) + (bpvi_successor_pvs_pow_base_selected_power))) /\\ bpvi_successor_pvs_pow_base_selected_power = bpvi_partial_pvs_pow_base_selected_power * bpvi_factor_pvs_pow_base_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_base_selected. a = bpvi_result_pvs_pow_base_selected * bpvi_divisor_factor_pvs_pow_base_selected))) /\\ forall bpd_candidate_pvs_pow_base. (exists bpd_gap_pvs_pow_base_candidate_bound. bpd_gap_pvs_pow_base_candidate_bound + (bpd_candidate_pvs_pow_base) = (a)) -> (exists bpvi_result_pvs_pow_base_candidate. ((exists bpvi_b_pvs_pow_base_candidate_power bpvi_c_pvs_pow_base_candidate_power. ((forall bpvi_i_pvs_pow_base_candidate_power. (exists bpvi_repeat_gap_pvs_pow_base_candidate_power. bpvi_repeat_gap_pvs_pow_base_candidate_power + S bpvi_i_pvs_pow_base_candidate_power = bpd_candidate_pvs_pow_base) -> (((exists bpvi_h_pvs_pow_base_candidate_power_repeat. bpvi_h_pvs_pow_base_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_repeat. bpvi_b_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_base_candidate_power bpvi_v_pvs_pow_base_candidate_power. ((((exists bpvi_h_pvs_pow_base_candidate_power_start. bpvi_h_pvs_pow_base_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_start. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_base_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_base_candidate_power_terminal. bpvi_h_pvs_pow_base_candidate_power_terminal + S (bpvi_result_pvs_pow_base_candidate) = S ((S (bpd_candidate_pvs_pow_base)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_terminal. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_base)) * bpvi_v_pvs_pow_base_candidate_power) + (bpvi_result_pvs_pow_base_candidate))) /\\ forall bpvi_j_pvs_pow_base_candidate_power. (exists bpvi_product_gap_pvs_pow_base_candidate_power. bpvi_product_gap_pvs_pow_base_candidate_power + S bpvi_j_pvs_pow_base_candidate_power = bpd_candidate_pvs_pow_base) -> exists bpvi_factor_pvs_pow_base_candidate_power bpvi_partial_pvs_pow_base_candidate_power bpvi_successor_pvs_pow_base_candidate_power. ((((exists bpvi_h_pvs_pow_base_candidate_power_factor. bpvi_h_pvs_pow_base_candidate_power_factor + S (bpvi_factor_pvs_pow_base_candidate_power) = S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_factor. bpvi_b_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_factor * S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_c_pvs_pow_base_candidate_power) + (bpvi_factor_pvs_pow_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_base_candidate_power_partial. bpvi_h_pvs_pow_base_candidate_power_partial + S (bpvi_partial_pvs_pow_base_candidate_power) = S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_partial. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_partial * S ((S (bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power) + (bpvi_partial_pvs_pow_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_base_candidate_power_successor. bpvi_h_pvs_pow_base_candidate_power_successor + S (bpvi_successor_pvs_pow_base_candidate_power) = S ((S (S bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_base_candidate_power_successor. bpvi_u_pvs_pow_base_candidate_power = bpvi_q_pvs_pow_base_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_base_candidate_power)) * bpvi_v_pvs_pow_base_candidate_power) + (bpvi_successor_pvs_pow_base_candidate_power))) /\\ bpvi_successor_pvs_pow_base_candidate_power = bpvi_partial_pvs_pow_base_candidate_power * bpvi_factor_pvs_pow_base_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_base_candidate. a = bpvi_result_pvs_pow_base_candidate * bpvi_divisor_factor_pvs_pow_base_candidate)) -> (exists bpd_gap_pvs_pow_base_maximal. bpd_gap_pvs_pow_base_maximal + (bpd_candidate_pvs_pow_base) = (e))) -> (exists pa_b_pvs_pow_source pa_c_pvs_pow_source. ((forall pa_i_pvs_pow_source_repeat. (exists pa_lt_pvs_pow_source_repeat_bound. pa_lt_pvs_pow_source_repeat_bound + S pa_i_pvs_pow_source_repeat = k) -> (((exists pa_h_pvs_pow_source_repeat_decoded. pa_h_pvs_pow_source_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_source_repeat)) * pa_c_pvs_pow_source)) /\\ exists pa_q_pvs_pow_source_repeat_decoded. pa_b_pvs_pow_source = pa_q_pvs_pow_source_repeat_decoded * S ((S (pa_i_pvs_pow_source_repeat)) * pa_c_pvs_pow_source) + (a)))) /\\ (exists pa_u_pvs_pow_source_product pa_v_pvs_pow_source_product. ((((exists pa_h_pvs_pow_source_product_start. pa_h_pvs_pow_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_start. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_start * S ((S (0)) * pa_v_pvs_pow_source_product) + (1))) /\\ ((((exists pa_h_pvs_pow_source_product_terminal. pa_h_pvs_pow_source_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_terminal. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_terminal * S ((S (k)) * pa_v_pvs_pow_source_product) + (z))) /\\ forall pa_i_pvs_pow_source_product. (exists pa_lt_pvs_pow_source_product_bound. pa_lt_pvs_pow_source_product_bound + S pa_i_pvs_pow_source_product = k) -> exists pa_p_pvs_pow_source_product pa_r_pvs_pow_source_product pa_s_pvs_pow_source_product. ((((exists pa_h_pvs_pow_source_product_factor. pa_h_pvs_pow_source_product_factor + S (pa_p_pvs_pow_source_product) = S ((S (pa_i_pvs_pow_source_product)) * pa_c_pvs_pow_source)) /\\ exists pa_q_pvs_pow_source_product_factor. pa_b_pvs_pow_source = pa_q_pvs_pow_source_product_factor * S ((S (pa_i_pvs_pow_source_product)) * pa_c_pvs_pow_source) + (pa_p_pvs_pow_source_product))) /\\ ((((exists pa_h_pvs_pow_source_product_partial. pa_h_pvs_pow_source_product_partial + S (pa_r_pvs_pow_source_product) = S ((S (pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_partial. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_partial * S ((S (pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product) + (pa_r_pvs_pow_source_product))) /\\ ((((exists pa_h_pvs_pow_source_product_successor. pa_h_pvs_pow_source_product_successor + S (pa_s_pvs_pow_source_product) = S ((S (S pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product)) /\\ exists pa_q_pvs_pow_source_product_successor. pa_u_pvs_pow_source_product = pa_q_pvs_pow_source_product_successor * S ((S (S pa_i_pvs_pow_source_product)) * pa_v_pvs_pow_source_product) + (pa_s_pvs_pow_source_product))) /\\ pa_s_pvs_pow_source_product = pa_r_pvs_pow_source_product * pa_p_pvs_pow_source_product)))))))) -> (((exists bpd_gap_pvs_pow_output_selected_bound. bpd_gap_pvs_pow_output_selected_bound + (f) = (z)) /\\ (exists bpvi_result_pvs_pow_output_selected. ((exists bpvi_b_pvs_pow_output_selected_power bpvi_c_pvs_pow_output_selected_power. ((forall bpvi_i_pvs_pow_output_selected_power. (exists bpvi_repeat_gap_pvs_pow_output_selected_power. bpvi_repeat_gap_pvs_pow_output_selected_power + S bpvi_i_pvs_pow_output_selected_power = f) -> (((exists bpvi_h_pvs_pow_output_selected_power_repeat. bpvi_h_pvs_pow_output_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_repeat. bpvi_b_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_repeat * S ((S (bpvi_i_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_output_selected_power bpvi_v_pvs_pow_output_selected_power. ((((exists bpvi_h_pvs_pow_output_selected_power_start. bpvi_h_pvs_pow_output_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_start. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_output_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_output_selected_power_terminal. bpvi_h_pvs_pow_output_selected_power_terminal + S (bpvi_result_pvs_pow_output_selected) = S ((S (f)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_terminal. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_output_selected_power) + (bpvi_result_pvs_pow_output_selected))) /\\ forall bpvi_j_pvs_pow_output_selected_power. (exists bpvi_product_gap_pvs_pow_output_selected_power. bpvi_product_gap_pvs_pow_output_selected_power + S bpvi_j_pvs_pow_output_selected_power = f) -> exists bpvi_factor_pvs_pow_output_selected_power bpvi_partial_pvs_pow_output_selected_power bpvi_successor_pvs_pow_output_selected_power. ((((exists bpvi_h_pvs_pow_output_selected_power_factor. bpvi_h_pvs_pow_output_selected_power_factor + S (bpvi_factor_pvs_pow_output_selected_power) = S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_factor. bpvi_b_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_factor * S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_c_pvs_pow_output_selected_power) + (bpvi_factor_pvs_pow_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_output_selected_power_partial. bpvi_h_pvs_pow_output_selected_power_partial + S (bpvi_partial_pvs_pow_output_selected_power) = S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_partial. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_partial * S ((S (bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power) + (bpvi_partial_pvs_pow_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_output_selected_power_successor. bpvi_h_pvs_pow_output_selected_power_successor + S (bpvi_successor_pvs_pow_output_selected_power) = S ((S (S bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power)) /\\ exists bpvi_q_pvs_pow_output_selected_power_successor. bpvi_u_pvs_pow_output_selected_power = bpvi_q_pvs_pow_output_selected_power_successor * S ((S (S bpvi_j_pvs_pow_output_selected_power)) * bpvi_v_pvs_pow_output_selected_power) + (bpvi_successor_pvs_pow_output_selected_power))) /\\ bpvi_successor_pvs_pow_output_selected_power = bpvi_partial_pvs_pow_output_selected_power * bpvi_factor_pvs_pow_output_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_output_selected. z = bpvi_result_pvs_pow_output_selected * bpvi_divisor_factor_pvs_pow_output_selected))) /\\ forall bpd_candidate_pvs_pow_output. (exists bpd_gap_pvs_pow_output_candidate_bound. bpd_gap_pvs_pow_output_candidate_bound + (bpd_candidate_pvs_pow_output) = (z)) -> (exists bpvi_result_pvs_pow_output_candidate. ((exists bpvi_b_pvs_pow_output_candidate_power bpvi_c_pvs_pow_output_candidate_power. ((forall bpvi_i_pvs_pow_output_candidate_power. (exists bpvi_repeat_gap_pvs_pow_output_candidate_power. bpvi_repeat_gap_pvs_pow_output_candidate_power + S bpvi_i_pvs_pow_output_candidate_power = bpd_candidate_pvs_pow_output) -> (((exists bpvi_h_pvs_pow_output_candidate_power_repeat. bpvi_h_pvs_pow_output_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_repeat. bpvi_b_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_output_candidate_power bpvi_v_pvs_pow_output_candidate_power. ((((exists bpvi_h_pvs_pow_output_candidate_power_start. bpvi_h_pvs_pow_output_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_start. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_output_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_output_candidate_power_terminal. bpvi_h_pvs_pow_output_candidate_power_terminal + S (bpvi_result_pvs_pow_output_candidate) = S ((S (bpd_candidate_pvs_pow_output)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_terminal. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_output)) * bpvi_v_pvs_pow_output_candidate_power) + (bpvi_result_pvs_pow_output_candidate))) /\\ forall bpvi_j_pvs_pow_output_candidate_power. (exists bpvi_product_gap_pvs_pow_output_candidate_power. bpvi_product_gap_pvs_pow_output_candidate_power + S bpvi_j_pvs_pow_output_candidate_power = bpd_candidate_pvs_pow_output) -> exists bpvi_factor_pvs_pow_output_candidate_power bpvi_partial_pvs_pow_output_candidate_power bpvi_successor_pvs_pow_output_candidate_power. ((((exists bpvi_h_pvs_pow_output_candidate_power_factor. bpvi_h_pvs_pow_output_candidate_power_factor + S (bpvi_factor_pvs_pow_output_candidate_power) = S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_factor. bpvi_b_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_factor * S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_c_pvs_pow_output_candidate_power) + (bpvi_factor_pvs_pow_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_output_candidate_power_partial. bpvi_h_pvs_pow_output_candidate_power_partial + S (bpvi_partial_pvs_pow_output_candidate_power) = S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_partial. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_partial * S ((S (bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power) + (bpvi_partial_pvs_pow_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_output_candidate_power_successor. bpvi_h_pvs_pow_output_candidate_power_successor + S (bpvi_successor_pvs_pow_output_candidate_power) = S ((S (S bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_output_candidate_power_successor. bpvi_u_pvs_pow_output_candidate_power = bpvi_q_pvs_pow_output_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_output_candidate_power)) * bpvi_v_pvs_pow_output_candidate_power) + (bpvi_successor_pvs_pow_output_candidate_power))) /\\ bpvi_successor_pvs_pow_output_candidate_power = bpvi_partial_pvs_pow_output_candidate_power * bpvi_factor_pvs_pow_output_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_output_candidate. z = bpvi_result_pvs_pow_output_candidate * bpvi_divisor_factor_pvs_pow_output_candidate)) -> (exists bpd_gap_pvs_pow_output_maximal. bpd_gap_pvs_pow_output_maximal + (bpd_candidate_pvs_pow_output) = (f))) -> f = k * e", "statement_sha256": "e3d8fa56e8d8d7d6a4bcabba8834b1a96127c075e907484a4d95c2fcf31478b9" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_power_valuation_pow", "canonical_catalog_record": { "alpha_v29_frontier_enrollment": { "body_receipt_sha256": "c9f62dc97b1814b596d1bc78bb12681ba7215c3e076678062e267fad85cf33b9", "bundle_campaign": "priority_layer", "bundle_node_id": 291, "bundle_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "campaign": "prime_valuation_support", "parent_catalog_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "rfc_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "source_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "test_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c" }, "body_checked": true, "body_receipt": { "command_count": 32, "dependency_count": 3, "dne_command_count": 0, "name": "prime_power_valuation_pow", "proof_depth": 27, "proof_edges": 40, "proof_nodes": 41, "proof_objects": 41, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "power_valuation_exists", "prime_power_valuation_pow_value", "prime_valuation_exponent_eq_transport" ], "dependencies_sha256": "f99f1ac34dceed580c8cb2705a8f4613b0ae7b14be35d995726cc7694fdbbeba", "empty_context_closure": { "body_proof_depth": 27, "body_proof_nodes": 41, "bundle_campaign": "priority_layer", "bundle_dependency_edge_count": 1690, "bundle_node_count": 566, "bundle_node_id": 291, "bundle_path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "bundle_root_id": 565, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": 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"897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "prime_valuation_support", "logical_spec_sha256": "5ed19b8d9076ff42bda6aa0c973acb2c136868e03fad8be93509aa0300d02949", "membership": "alpha_only", "name": "prime_power_valuation_pow", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro a", "intro k", "intro e", "intro z", "intro hp", "intro ha", "intro hbase", "intro hpow", "have hv : exists f. (((exists bpd_gap_pvs_pow_construct_exists_selected_bound. bpd_gap_pvs_pow_construct_exists_selected_bound + (f) = (z)) /\\ (exists bpvi_result_pvs_pow_construct_exists_selected. ((exists bpvi_b_pvs_pow_construct_exists_selected_power bpvi_c_pvs_pow_construct_exists_selected_power. ((forall bpvi_i_pvs_pow_construct_exists_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_exists_selected_power. bpvi_repeat_gap_pvs_pow_construct_exists_selected_power + S bpvi_i_pvs_pow_construct_exists_selected_power = f) -> (((exists bpvi_h_pvs_pow_construct_exists_selected_power_repeat. bpvi_h_pvs_pow_construct_exists_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_repeat. bpvi_b_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_exists_selected_power bpvi_v_pvs_pow_construct_exists_selected_power. ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_start. bpvi_h_pvs_pow_construct_exists_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_start. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_terminal. bpvi_h_pvs_pow_construct_exists_selected_power_terminal + S (bpvi_result_pvs_pow_construct_exists_selected) = S ((S (f)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_terminal. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_result_pvs_pow_construct_exists_selected))) /\\ forall bpvi_j_pvs_pow_construct_exists_selected_power. (exists bpvi_product_gap_pvs_pow_construct_exists_selected_power. bpvi_product_gap_pvs_pow_construct_exists_selected_power + S bpvi_j_pvs_pow_construct_exists_selected_power = f) -> exists bpvi_factor_pvs_pow_construct_exists_selected_power bpvi_partial_pvs_pow_construct_exists_selected_power bpvi_successor_pvs_pow_construct_exists_selected_power. ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_factor. bpvi_h_pvs_pow_construct_exists_selected_power_factor + S (bpvi_factor_pvs_pow_construct_exists_selected_power) = S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_factor. bpvi_b_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power) + (bpvi_factor_pvs_pow_construct_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_partial. bpvi_h_pvs_pow_construct_exists_selected_power_partial + S (bpvi_partial_pvs_pow_construct_exists_selected_power) = S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_partial. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_partial_pvs_pow_construct_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_successor. bpvi_h_pvs_pow_construct_exists_selected_power_successor + S (bpvi_successor_pvs_pow_construct_exists_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_successor. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_successor_pvs_pow_construct_exists_selected_power))) /\\ bpvi_successor_pvs_pow_construct_exists_selected_power = bpvi_partial_pvs_pow_construct_exists_selected_power * bpvi_factor_pvs_pow_construct_exists_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_exists_selected. z = bpvi_result_pvs_pow_construct_exists_selected * bpvi_divisor_factor_pvs_pow_construct_exists_selected))) /\\ forall bpd_candidate_pvs_pow_construct_exists. (exists bpd_gap_pvs_pow_construct_exists_candidate_bound. bpd_gap_pvs_pow_construct_exists_candidate_bound + (bpd_candidate_pvs_pow_construct_exists) = (z)) -> (exists bpvi_result_pvs_pow_construct_exists_candidate. ((exists bpvi_b_pvs_pow_construct_exists_candidate_power bpvi_c_pvs_pow_construct_exists_candidate_power. ((forall bpvi_i_pvs_pow_construct_exists_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_exists_candidate_power. bpvi_repeat_gap_pvs_pow_construct_exists_candidate_power + S bpvi_i_pvs_pow_construct_exists_candidate_power = bpd_candidate_pvs_pow_construct_exists) -> (((exists bpvi_h_pvs_pow_construct_exists_candidate_power_repeat. bpvi_h_pvs_pow_construct_exists_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_repeat. bpvi_b_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_exists_candidate_power bpvi_v_pvs_pow_construct_exists_candidate_power. ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_start. bpvi_h_pvs_pow_construct_exists_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_start. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_terminal. bpvi_h_pvs_pow_construct_exists_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_exists_candidate) = S ((S (bpd_candidate_pvs_pow_construct_exists)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_terminal. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_exists)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_result_pvs_pow_construct_exists_candidate))) /\\ forall bpvi_j_pvs_pow_construct_exists_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_exists_candidate_power. bpvi_product_gap_pvs_pow_construct_exists_candidate_power + S bpvi_j_pvs_pow_construct_exists_candidate_power = bpd_candidate_pvs_pow_construct_exists) -> exists bpvi_factor_pvs_pow_construct_exists_candidate_power bpvi_partial_pvs_pow_construct_exists_candidate_power bpvi_successor_pvs_pow_construct_exists_candidate_power. ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_factor. bpvi_h_pvs_pow_construct_exists_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_exists_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_factor. bpvi_b_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power) + (bpvi_factor_pvs_pow_construct_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_partial. bpvi_h_pvs_pow_construct_exists_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_exists_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_partial. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_partial_pvs_pow_construct_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_successor. bpvi_h_pvs_pow_construct_exists_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_exists_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_successor. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_successor_pvs_pow_construct_exists_candidate_power))) /\\ bpvi_successor_pvs_pow_construct_exists_candidate_power = bpvi_partial_pvs_pow_construct_exists_candidate_power * bpvi_factor_pvs_pow_construct_exists_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_exists_candidate. z = bpvi_result_pvs_pow_construct_exists_candidate * bpvi_divisor_factor_pvs_pow_construct_exists_candidate)) -> (exists bpd_gap_pvs_pow_construct_exists_maximal. bpd_gap_pvs_pow_construct_exists_maximal + (bpd_candidate_pvs_pow_construct_exists) = (f)))", "specialize power_valuation_exists (p)", "specialize power_valuation_exists (z)", "apply power_valuation_exists", "cases hv", "specialize prime_valuation_exponent_eq_transport (p)", "specialize prime_valuation_exponent_eq_transport (z)", "specialize prime_valuation_exponent_eq_transport (x)", "specialize prime_valuation_exponent_eq_transport (k * e)", "apply prime_valuation_exponent_eq_transport", "specialize prime_power_valuation_pow_value (p)", "specialize prime_power_valuation_pow_value (a)", "specialize prime_power_valuation_pow_value (k)", "specialize prime_power_valuation_pow_value (e)", "specialize prime_power_valuation_pow_value (z)", "specialize prime_power_valuation_pow_value (x)", "apply prime_power_valuation_pow_value", "exact hp", "exact ha", "exact hbase", "exact hpow", "exact hv_witness", "exact hv_witness" ], "script_sha256": "fdb2ce4886216d49605c9e82e100fb888f926efe953f54b55a6c68e58ca59c0a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall p a k e z. (~((p) = 1) /\\ forall pvs_left_pow_construct_domain pvs_right_pow_construct_domain. (p) = pvs_left_pow_construct_domain * pvs_right_pow_construct_domain -> pvs_left_pow_construct_domain = 1 \\/ pvs_right_pow_construct_domain = 1) -> ~(a = 0) -> (((exists bpd_gap_pvs_pow_construct_base_selected_bound. bpd_gap_pvs_pow_construct_base_selected_bound + (e) = (a)) /\\ (exists bpvi_result_pvs_pow_construct_base_selected. ((exists bpvi_b_pvs_pow_construct_base_selected_power bpvi_c_pvs_pow_construct_base_selected_power. ((forall bpvi_i_pvs_pow_construct_base_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_base_selected_power. bpvi_repeat_gap_pvs_pow_construct_base_selected_power + S bpvi_i_pvs_pow_construct_base_selected_power = e) -> (((exists bpvi_h_pvs_pow_construct_base_selected_power_repeat. bpvi_h_pvs_pow_construct_base_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_repeat. bpvi_b_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_base_selected_power bpvi_v_pvs_pow_construct_base_selected_power. ((((exists bpvi_h_pvs_pow_construct_base_selected_power_start. bpvi_h_pvs_pow_construct_base_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_start. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_base_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_terminal. bpvi_h_pvs_pow_construct_base_selected_power_terminal + S (bpvi_result_pvs_pow_construct_base_selected) = S ((S (e)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_terminal. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_result_pvs_pow_construct_base_selected))) /\\ forall bpvi_j_pvs_pow_construct_base_selected_power. (exists bpvi_product_gap_pvs_pow_construct_base_selected_power. bpvi_product_gap_pvs_pow_construct_base_selected_power + S bpvi_j_pvs_pow_construct_base_selected_power = e) -> exists bpvi_factor_pvs_pow_construct_base_selected_power bpvi_partial_pvs_pow_construct_base_selected_power bpvi_successor_pvs_pow_construct_base_selected_power. ((((exists bpvi_h_pvs_pow_construct_base_selected_power_factor. bpvi_h_pvs_pow_construct_base_selected_power_factor + S (bpvi_factor_pvs_pow_construct_base_selected_power) = S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_factor. bpvi_b_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power) + (bpvi_factor_pvs_pow_construct_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_partial. bpvi_h_pvs_pow_construct_base_selected_power_partial + S (bpvi_partial_pvs_pow_construct_base_selected_power) = S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_partial. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_partial_pvs_pow_construct_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_successor. bpvi_h_pvs_pow_construct_base_selected_power_successor + S (bpvi_successor_pvs_pow_construct_base_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_successor. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_successor_pvs_pow_construct_base_selected_power))) /\\ bpvi_successor_pvs_pow_construct_base_selected_power = bpvi_partial_pvs_pow_construct_base_selected_power * bpvi_factor_pvs_pow_construct_base_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_base_selected. a = bpvi_result_pvs_pow_construct_base_selected * bpvi_divisor_factor_pvs_pow_construct_base_selected))) /\\ forall bpd_candidate_pvs_pow_construct_base. (exists bpd_gap_pvs_pow_construct_base_candidate_bound. bpd_gap_pvs_pow_construct_base_candidate_bound + (bpd_candidate_pvs_pow_construct_base) = (a)) -> (exists bpvi_result_pvs_pow_construct_base_candidate. ((exists bpvi_b_pvs_pow_construct_base_candidate_power bpvi_c_pvs_pow_construct_base_candidate_power. ((forall bpvi_i_pvs_pow_construct_base_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_base_candidate_power. bpvi_repeat_gap_pvs_pow_construct_base_candidate_power + S bpvi_i_pvs_pow_construct_base_candidate_power = bpd_candidate_pvs_pow_construct_base) -> (((exists bpvi_h_pvs_pow_construct_base_candidate_power_repeat. bpvi_h_pvs_pow_construct_base_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_repeat. bpvi_b_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_base_candidate_power bpvi_v_pvs_pow_construct_base_candidate_power. ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_start. bpvi_h_pvs_pow_construct_base_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_start. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_terminal. bpvi_h_pvs_pow_construct_base_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_base_candidate) = S ((S (bpd_candidate_pvs_pow_construct_base)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_terminal. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_base)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_result_pvs_pow_construct_base_candidate))) /\\ forall bpvi_j_pvs_pow_construct_base_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_base_candidate_power. bpvi_product_gap_pvs_pow_construct_base_candidate_power + S bpvi_j_pvs_pow_construct_base_candidate_power = bpd_candidate_pvs_pow_construct_base) -> exists bpvi_factor_pvs_pow_construct_base_candidate_power bpvi_partial_pvs_pow_construct_base_candidate_power bpvi_successor_pvs_pow_construct_base_candidate_power. ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_factor. bpvi_h_pvs_pow_construct_base_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_base_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_factor. bpvi_b_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power) + (bpvi_factor_pvs_pow_construct_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_partial. bpvi_h_pvs_pow_construct_base_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_base_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_partial. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_partial_pvs_pow_construct_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_successor. bpvi_h_pvs_pow_construct_base_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_base_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_successor. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_successor_pvs_pow_construct_base_candidate_power))) /\\ bpvi_successor_pvs_pow_construct_base_candidate_power = bpvi_partial_pvs_pow_construct_base_candidate_power * bpvi_factor_pvs_pow_construct_base_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_base_candidate. a = bpvi_result_pvs_pow_construct_base_candidate * bpvi_divisor_factor_pvs_pow_construct_base_candidate)) -> (exists bpd_gap_pvs_pow_construct_base_maximal. bpd_gap_pvs_pow_construct_base_maximal + (bpd_candidate_pvs_pow_construct_base) = (e))) -> (exists pa_b_pvs_pow_construct_source pa_c_pvs_pow_construct_source. ((forall pa_i_pvs_pow_construct_source_repeat. (exists pa_lt_pvs_pow_construct_source_repeat_bound. pa_lt_pvs_pow_construct_source_repeat_bound + S pa_i_pvs_pow_construct_source_repeat = k) -> (((exists pa_h_pvs_pow_construct_source_repeat_decoded. pa_h_pvs_pow_construct_source_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_construct_source_repeat)) * pa_c_pvs_pow_construct_source)) /\\ exists pa_q_pvs_pow_construct_source_repeat_decoded. pa_b_pvs_pow_construct_source = pa_q_pvs_pow_construct_source_repeat_decoded * S ((S (pa_i_pvs_pow_construct_source_repeat)) * pa_c_pvs_pow_construct_source) + (a)))) /\\ (exists pa_u_pvs_pow_construct_source_product pa_v_pvs_pow_construct_source_product. ((((exists pa_h_pvs_pow_construct_source_product_start. pa_h_pvs_pow_construct_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_start. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_start * S ((S (0)) * pa_v_pvs_pow_construct_source_product) + (1))) /\\ ((((exists pa_h_pvs_pow_construct_source_product_terminal. pa_h_pvs_pow_construct_source_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_terminal. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_terminal * S ((S (k)) * pa_v_pvs_pow_construct_source_product) + (z))) /\\ forall pa_i_pvs_pow_construct_source_product. (exists pa_lt_pvs_pow_construct_source_product_bound. pa_lt_pvs_pow_construct_source_product_bound + S pa_i_pvs_pow_construct_source_product = k) -> exists pa_p_pvs_pow_construct_source_product pa_r_pvs_pow_construct_source_product pa_s_pvs_pow_construct_source_product. ((((exists pa_h_pvs_pow_construct_source_product_factor. pa_h_pvs_pow_construct_source_product_factor + S (pa_p_pvs_pow_construct_source_product) = S ((S (pa_i_pvs_pow_construct_source_product)) * pa_c_pvs_pow_construct_source)) /\\ exists pa_q_pvs_pow_construct_source_product_factor. pa_b_pvs_pow_construct_source = pa_q_pvs_pow_construct_source_product_factor * S ((S (pa_i_pvs_pow_construct_source_product)) * pa_c_pvs_pow_construct_source) + (pa_p_pvs_pow_construct_source_product))) /\\ ((((exists pa_h_pvs_pow_construct_source_product_partial. pa_h_pvs_pow_construct_source_product_partial + S (pa_r_pvs_pow_construct_source_product) = S ((S (pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_partial. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_partial * S ((S (pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product) + (pa_r_pvs_pow_construct_source_product))) /\\ ((((exists pa_h_pvs_pow_construct_source_product_successor. pa_h_pvs_pow_construct_source_product_successor + S (pa_s_pvs_pow_construct_source_product) = S ((S (S pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_successor. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_successor * S ((S (S pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product) + (pa_s_pvs_pow_construct_source_product))) /\\ pa_s_pvs_pow_construct_source_product = pa_r_pvs_pow_construct_source_product * pa_p_pvs_pow_construct_source_product)))))))) -> (((exists bpd_gap_pvs_pow_construct_output_selected_bound. bpd_gap_pvs_pow_construct_output_selected_bound + (k * e) = (z)) /\\ (exists bpvi_result_pvs_pow_construct_output_selected. ((exists bpvi_b_pvs_pow_construct_output_selected_power bpvi_c_pvs_pow_construct_output_selected_power. ((forall bpvi_i_pvs_pow_construct_output_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_output_selected_power. bpvi_repeat_gap_pvs_pow_construct_output_selected_power + S bpvi_i_pvs_pow_construct_output_selected_power = k * e) -> (((exists bpvi_h_pvs_pow_construct_output_selected_power_repeat. bpvi_h_pvs_pow_construct_output_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_repeat. bpvi_b_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_output_selected_power bpvi_v_pvs_pow_construct_output_selected_power. ((((exists bpvi_h_pvs_pow_construct_output_selected_power_start. bpvi_h_pvs_pow_construct_output_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_start. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_output_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_terminal. bpvi_h_pvs_pow_construct_output_selected_power_terminal + S (bpvi_result_pvs_pow_construct_output_selected) = S ((S (k * e)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_terminal. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_terminal * S ((S (k * e)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_result_pvs_pow_construct_output_selected))) /\\ forall bpvi_j_pvs_pow_construct_output_selected_power. (exists bpvi_product_gap_pvs_pow_construct_output_selected_power. bpvi_product_gap_pvs_pow_construct_output_selected_power + S bpvi_j_pvs_pow_construct_output_selected_power = k * e) -> exists bpvi_factor_pvs_pow_construct_output_selected_power bpvi_partial_pvs_pow_construct_output_selected_power bpvi_successor_pvs_pow_construct_output_selected_power. ((((exists bpvi_h_pvs_pow_construct_output_selected_power_factor. bpvi_h_pvs_pow_construct_output_selected_power_factor + S (bpvi_factor_pvs_pow_construct_output_selected_power) = S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_factor. bpvi_b_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power) + (bpvi_factor_pvs_pow_construct_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_partial. bpvi_h_pvs_pow_construct_output_selected_power_partial + S (bpvi_partial_pvs_pow_construct_output_selected_power) = S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_partial. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_partial_pvs_pow_construct_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_successor. bpvi_h_pvs_pow_construct_output_selected_power_successor + S (bpvi_successor_pvs_pow_construct_output_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_successor. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_successor_pvs_pow_construct_output_selected_power))) /\\ bpvi_successor_pvs_pow_construct_output_selected_power = bpvi_partial_pvs_pow_construct_output_selected_power * bpvi_factor_pvs_pow_construct_output_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_output_selected. z = bpvi_result_pvs_pow_construct_output_selected * bpvi_divisor_factor_pvs_pow_construct_output_selected))) /\\ forall bpd_candidate_pvs_pow_construct_output. (exists bpd_gap_pvs_pow_construct_output_candidate_bound. bpd_gap_pvs_pow_construct_output_candidate_bound + (bpd_candidate_pvs_pow_construct_output) = (z)) -> (exists bpvi_result_pvs_pow_construct_output_candidate. ((exists bpvi_b_pvs_pow_construct_output_candidate_power bpvi_c_pvs_pow_construct_output_candidate_power. ((forall bpvi_i_pvs_pow_construct_output_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_output_candidate_power. bpvi_repeat_gap_pvs_pow_construct_output_candidate_power + S bpvi_i_pvs_pow_construct_output_candidate_power = bpd_candidate_pvs_pow_construct_output) -> (((exists bpvi_h_pvs_pow_construct_output_candidate_power_repeat. bpvi_h_pvs_pow_construct_output_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_repeat. bpvi_b_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_output_candidate_power bpvi_v_pvs_pow_construct_output_candidate_power. ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_start. bpvi_h_pvs_pow_construct_output_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_start. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_terminal. bpvi_h_pvs_pow_construct_output_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_output_candidate) = S ((S (bpd_candidate_pvs_pow_construct_output)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_terminal. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_output)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_result_pvs_pow_construct_output_candidate))) /\\ forall bpvi_j_pvs_pow_construct_output_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_output_candidate_power. bpvi_product_gap_pvs_pow_construct_output_candidate_power + S bpvi_j_pvs_pow_construct_output_candidate_power = bpd_candidate_pvs_pow_construct_output) -> exists bpvi_factor_pvs_pow_construct_output_candidate_power bpvi_partial_pvs_pow_construct_output_candidate_power bpvi_successor_pvs_pow_construct_output_candidate_power. ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_factor. bpvi_h_pvs_pow_construct_output_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_output_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_factor. bpvi_b_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power) + (bpvi_factor_pvs_pow_construct_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_partial. bpvi_h_pvs_pow_construct_output_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_output_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_partial. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_partial_pvs_pow_construct_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_successor. bpvi_h_pvs_pow_construct_output_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_output_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_successor. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_successor_pvs_pow_construct_output_candidate_power))) /\\ bpvi_successor_pvs_pow_construct_output_candidate_power = bpvi_partial_pvs_pow_construct_output_candidate_power * bpvi_factor_pvs_pow_construct_output_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_output_candidate. z = bpvi_result_pvs_pow_construct_output_candidate * bpvi_divisor_factor_pvs_pow_construct_output_candidate)) -> (exists bpd_gap_pvs_pow_construct_output_maximal. bpd_gap_pvs_pow_construct_output_maximal + (bpd_candidate_pvs_pow_construct_output) = (k * e)))", "statement_sha256": "30e2347cbb82399420bcfec8fc56d030a7843825de85f761da4bf6152d897782", "summary": "Construct the actual maximal valuation graph of a witnessed power, not merely an equation between supplied output valuations.", "summary_sha256": "d50ea1b4165be3d17b0d0c3da6524a87c7576e97bb77006e461580ef04530f3a" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "power_valuation_exists", "prime_power_valuation_pow_value", "prime_valuation_exponent_eq_transport" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=291]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "prime_power_valuation_pow", "parent_alpha_version": "v34", "proof_bundle_node_id": 252, "reference_route": "jordan-totient/checkpoint.html#theorem-prime_power_valuation_pow", "script": [ "intro p", "intro a", "intro k", "intro e", "intro z", "intro hp", "intro ha", "intro hbase", "intro hpow", "have hv : exists f. (((exists bpd_gap_pvs_pow_construct_exists_selected_bound. bpd_gap_pvs_pow_construct_exists_selected_bound + (f) = (z)) /\\ (exists bpvi_result_pvs_pow_construct_exists_selected. ((exists bpvi_b_pvs_pow_construct_exists_selected_power bpvi_c_pvs_pow_construct_exists_selected_power. ((forall bpvi_i_pvs_pow_construct_exists_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_exists_selected_power. bpvi_repeat_gap_pvs_pow_construct_exists_selected_power + S bpvi_i_pvs_pow_construct_exists_selected_power = f) -> (((exists bpvi_h_pvs_pow_construct_exists_selected_power_repeat. bpvi_h_pvs_pow_construct_exists_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_repeat. bpvi_b_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_exists_selected_power bpvi_v_pvs_pow_construct_exists_selected_power. ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_start. bpvi_h_pvs_pow_construct_exists_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_start. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_terminal. bpvi_h_pvs_pow_construct_exists_selected_power_terminal + S (bpvi_result_pvs_pow_construct_exists_selected) = S ((S (f)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_terminal. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_terminal * S ((S (f)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_result_pvs_pow_construct_exists_selected))) /\\ forall bpvi_j_pvs_pow_construct_exists_selected_power. (exists bpvi_product_gap_pvs_pow_construct_exists_selected_power. bpvi_product_gap_pvs_pow_construct_exists_selected_power + S bpvi_j_pvs_pow_construct_exists_selected_power = f) -> exists bpvi_factor_pvs_pow_construct_exists_selected_power bpvi_partial_pvs_pow_construct_exists_selected_power bpvi_successor_pvs_pow_construct_exists_selected_power. ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_factor. bpvi_h_pvs_pow_construct_exists_selected_power_factor + S (bpvi_factor_pvs_pow_construct_exists_selected_power) = S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_factor. bpvi_b_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_c_pvs_pow_construct_exists_selected_power) + (bpvi_factor_pvs_pow_construct_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_partial. bpvi_h_pvs_pow_construct_exists_selected_power_partial + S (bpvi_partial_pvs_pow_construct_exists_selected_power) = S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_partial. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_partial_pvs_pow_construct_exists_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_selected_power_successor. bpvi_h_pvs_pow_construct_exists_selected_power_successor + S (bpvi_successor_pvs_pow_construct_exists_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_selected_power_successor. bpvi_u_pvs_pow_construct_exists_selected_power = bpvi_q_pvs_pow_construct_exists_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_exists_selected_power)) * bpvi_v_pvs_pow_construct_exists_selected_power) + (bpvi_successor_pvs_pow_construct_exists_selected_power))) /\\ bpvi_successor_pvs_pow_construct_exists_selected_power = bpvi_partial_pvs_pow_construct_exists_selected_power * bpvi_factor_pvs_pow_construct_exists_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_exists_selected. z = bpvi_result_pvs_pow_construct_exists_selected * bpvi_divisor_factor_pvs_pow_construct_exists_selected))) /\\ forall bpd_candidate_pvs_pow_construct_exists. (exists bpd_gap_pvs_pow_construct_exists_candidate_bound. bpd_gap_pvs_pow_construct_exists_candidate_bound + (bpd_candidate_pvs_pow_construct_exists) = (z)) -> (exists bpvi_result_pvs_pow_construct_exists_candidate. ((exists bpvi_b_pvs_pow_construct_exists_candidate_power bpvi_c_pvs_pow_construct_exists_candidate_power. ((forall bpvi_i_pvs_pow_construct_exists_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_exists_candidate_power. bpvi_repeat_gap_pvs_pow_construct_exists_candidate_power + S bpvi_i_pvs_pow_construct_exists_candidate_power = bpd_candidate_pvs_pow_construct_exists) -> (((exists bpvi_h_pvs_pow_construct_exists_candidate_power_repeat. bpvi_h_pvs_pow_construct_exists_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_repeat. bpvi_b_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_exists_candidate_power bpvi_v_pvs_pow_construct_exists_candidate_power. ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_start. bpvi_h_pvs_pow_construct_exists_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_start. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_terminal. bpvi_h_pvs_pow_construct_exists_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_exists_candidate) = S ((S (bpd_candidate_pvs_pow_construct_exists)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_terminal. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_exists)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_result_pvs_pow_construct_exists_candidate))) /\\ forall bpvi_j_pvs_pow_construct_exists_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_exists_candidate_power. bpvi_product_gap_pvs_pow_construct_exists_candidate_power + S bpvi_j_pvs_pow_construct_exists_candidate_power = bpd_candidate_pvs_pow_construct_exists) -> exists bpvi_factor_pvs_pow_construct_exists_candidate_power bpvi_partial_pvs_pow_construct_exists_candidate_power bpvi_successor_pvs_pow_construct_exists_candidate_power. ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_factor. bpvi_h_pvs_pow_construct_exists_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_exists_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_factor. bpvi_b_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_c_pvs_pow_construct_exists_candidate_power) + (bpvi_factor_pvs_pow_construct_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_partial. bpvi_h_pvs_pow_construct_exists_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_exists_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_partial. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_partial_pvs_pow_construct_exists_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_exists_candidate_power_successor. bpvi_h_pvs_pow_construct_exists_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_exists_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_exists_candidate_power_successor. bpvi_u_pvs_pow_construct_exists_candidate_power = bpvi_q_pvs_pow_construct_exists_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_exists_candidate_power)) * bpvi_v_pvs_pow_construct_exists_candidate_power) + (bpvi_successor_pvs_pow_construct_exists_candidate_power))) /\\ bpvi_successor_pvs_pow_construct_exists_candidate_power = bpvi_partial_pvs_pow_construct_exists_candidate_power * bpvi_factor_pvs_pow_construct_exists_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_exists_candidate. z = bpvi_result_pvs_pow_construct_exists_candidate * bpvi_divisor_factor_pvs_pow_construct_exists_candidate)) -> (exists bpd_gap_pvs_pow_construct_exists_maximal. bpd_gap_pvs_pow_construct_exists_maximal + (bpd_candidate_pvs_pow_construct_exists) = (f)))", "specialize power_valuation_exists (p)", "specialize power_valuation_exists (z)", "apply power_valuation_exists", "cases hv", "specialize prime_valuation_exponent_eq_transport (p)", "specialize prime_valuation_exponent_eq_transport (z)", "specialize prime_valuation_exponent_eq_transport (x)", "specialize prime_valuation_exponent_eq_transport (k * e)", "apply prime_valuation_exponent_eq_transport", "specialize prime_power_valuation_pow_value (p)", "specialize prime_power_valuation_pow_value (a)", "specialize prime_power_valuation_pow_value (k)", "specialize prime_power_valuation_pow_value (e)", "specialize prime_power_valuation_pow_value (z)", "specialize prime_power_valuation_pow_value (x)", "apply prime_power_valuation_pow_value", "exact hp", "exact ha", "exact hbase", "exact hpow", "exact hv_witness", "exact hv_witness" ], "script_sha256": "fdb2ce4886216d49605c9e82e100fb888f926efe953f54b55a6c68e58ca59c0a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall p a k e z. (~((p) = 1) /\\ forall pvs_left_pow_construct_domain pvs_right_pow_construct_domain. (p) = pvs_left_pow_construct_domain * pvs_right_pow_construct_domain -> pvs_left_pow_construct_domain = 1 \\/ pvs_right_pow_construct_domain = 1) -> ~(a = 0) -> (((exists bpd_gap_pvs_pow_construct_base_selected_bound. bpd_gap_pvs_pow_construct_base_selected_bound + (e) = (a)) /\\ (exists bpvi_result_pvs_pow_construct_base_selected. ((exists bpvi_b_pvs_pow_construct_base_selected_power bpvi_c_pvs_pow_construct_base_selected_power. ((forall bpvi_i_pvs_pow_construct_base_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_base_selected_power. bpvi_repeat_gap_pvs_pow_construct_base_selected_power + S bpvi_i_pvs_pow_construct_base_selected_power = e) -> (((exists bpvi_h_pvs_pow_construct_base_selected_power_repeat. bpvi_h_pvs_pow_construct_base_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_repeat. bpvi_b_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_base_selected_power bpvi_v_pvs_pow_construct_base_selected_power. ((((exists bpvi_h_pvs_pow_construct_base_selected_power_start. bpvi_h_pvs_pow_construct_base_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_start. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_base_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_terminal. bpvi_h_pvs_pow_construct_base_selected_power_terminal + S (bpvi_result_pvs_pow_construct_base_selected) = S ((S (e)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_terminal. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_terminal * S ((S (e)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_result_pvs_pow_construct_base_selected))) /\\ forall bpvi_j_pvs_pow_construct_base_selected_power. (exists bpvi_product_gap_pvs_pow_construct_base_selected_power. bpvi_product_gap_pvs_pow_construct_base_selected_power + S bpvi_j_pvs_pow_construct_base_selected_power = e) -> exists bpvi_factor_pvs_pow_construct_base_selected_power bpvi_partial_pvs_pow_construct_base_selected_power bpvi_successor_pvs_pow_construct_base_selected_power. ((((exists bpvi_h_pvs_pow_construct_base_selected_power_factor. bpvi_h_pvs_pow_construct_base_selected_power_factor + S (bpvi_factor_pvs_pow_construct_base_selected_power) = S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_factor. bpvi_b_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_c_pvs_pow_construct_base_selected_power) + (bpvi_factor_pvs_pow_construct_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_partial. bpvi_h_pvs_pow_construct_base_selected_power_partial + S (bpvi_partial_pvs_pow_construct_base_selected_power) = S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_partial. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_partial_pvs_pow_construct_base_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_selected_power_successor. bpvi_h_pvs_pow_construct_base_selected_power_successor + S (bpvi_successor_pvs_pow_construct_base_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_base_selected_power_successor. bpvi_u_pvs_pow_construct_base_selected_power = bpvi_q_pvs_pow_construct_base_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_base_selected_power)) * bpvi_v_pvs_pow_construct_base_selected_power) + (bpvi_successor_pvs_pow_construct_base_selected_power))) /\\ bpvi_successor_pvs_pow_construct_base_selected_power = bpvi_partial_pvs_pow_construct_base_selected_power * bpvi_factor_pvs_pow_construct_base_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_base_selected. a = bpvi_result_pvs_pow_construct_base_selected * bpvi_divisor_factor_pvs_pow_construct_base_selected))) /\\ forall bpd_candidate_pvs_pow_construct_base. (exists bpd_gap_pvs_pow_construct_base_candidate_bound. bpd_gap_pvs_pow_construct_base_candidate_bound + (bpd_candidate_pvs_pow_construct_base) = (a)) -> (exists bpvi_result_pvs_pow_construct_base_candidate. ((exists bpvi_b_pvs_pow_construct_base_candidate_power bpvi_c_pvs_pow_construct_base_candidate_power. ((forall bpvi_i_pvs_pow_construct_base_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_base_candidate_power. bpvi_repeat_gap_pvs_pow_construct_base_candidate_power + S bpvi_i_pvs_pow_construct_base_candidate_power = bpd_candidate_pvs_pow_construct_base) -> (((exists bpvi_h_pvs_pow_construct_base_candidate_power_repeat. bpvi_h_pvs_pow_construct_base_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_repeat. bpvi_b_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_base_candidate_power bpvi_v_pvs_pow_construct_base_candidate_power. ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_start. bpvi_h_pvs_pow_construct_base_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_start. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_terminal. bpvi_h_pvs_pow_construct_base_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_base_candidate) = S ((S (bpd_candidate_pvs_pow_construct_base)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_terminal. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_base)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_result_pvs_pow_construct_base_candidate))) /\\ forall bpvi_j_pvs_pow_construct_base_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_base_candidate_power. bpvi_product_gap_pvs_pow_construct_base_candidate_power + S bpvi_j_pvs_pow_construct_base_candidate_power = bpd_candidate_pvs_pow_construct_base) -> exists bpvi_factor_pvs_pow_construct_base_candidate_power bpvi_partial_pvs_pow_construct_base_candidate_power bpvi_successor_pvs_pow_construct_base_candidate_power. ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_factor. bpvi_h_pvs_pow_construct_base_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_base_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_factor. bpvi_b_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_c_pvs_pow_construct_base_candidate_power) + (bpvi_factor_pvs_pow_construct_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_partial. bpvi_h_pvs_pow_construct_base_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_base_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_partial. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_partial_pvs_pow_construct_base_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_base_candidate_power_successor. bpvi_h_pvs_pow_construct_base_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_base_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_base_candidate_power_successor. bpvi_u_pvs_pow_construct_base_candidate_power = bpvi_q_pvs_pow_construct_base_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_base_candidate_power)) * bpvi_v_pvs_pow_construct_base_candidate_power) + (bpvi_successor_pvs_pow_construct_base_candidate_power))) /\\ bpvi_successor_pvs_pow_construct_base_candidate_power = bpvi_partial_pvs_pow_construct_base_candidate_power * bpvi_factor_pvs_pow_construct_base_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_base_candidate. a = bpvi_result_pvs_pow_construct_base_candidate * bpvi_divisor_factor_pvs_pow_construct_base_candidate)) -> (exists bpd_gap_pvs_pow_construct_base_maximal. bpd_gap_pvs_pow_construct_base_maximal + (bpd_candidate_pvs_pow_construct_base) = (e))) -> (exists pa_b_pvs_pow_construct_source pa_c_pvs_pow_construct_source. ((forall pa_i_pvs_pow_construct_source_repeat. (exists pa_lt_pvs_pow_construct_source_repeat_bound. pa_lt_pvs_pow_construct_source_repeat_bound + S pa_i_pvs_pow_construct_source_repeat = k) -> (((exists pa_h_pvs_pow_construct_source_repeat_decoded. pa_h_pvs_pow_construct_source_repeat_decoded + S (a) = S ((S (pa_i_pvs_pow_construct_source_repeat)) * pa_c_pvs_pow_construct_source)) /\\ exists pa_q_pvs_pow_construct_source_repeat_decoded. pa_b_pvs_pow_construct_source = pa_q_pvs_pow_construct_source_repeat_decoded * S ((S (pa_i_pvs_pow_construct_source_repeat)) * pa_c_pvs_pow_construct_source) + (a)))) /\\ (exists pa_u_pvs_pow_construct_source_product pa_v_pvs_pow_construct_source_product. ((((exists pa_h_pvs_pow_construct_source_product_start. pa_h_pvs_pow_construct_source_product_start + S (1) = S ((S (0)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_start. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_start * S ((S (0)) * pa_v_pvs_pow_construct_source_product) + (1))) /\\ ((((exists pa_h_pvs_pow_construct_source_product_terminal. pa_h_pvs_pow_construct_source_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_terminal. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_terminal * S ((S (k)) * pa_v_pvs_pow_construct_source_product) + (z))) /\\ forall pa_i_pvs_pow_construct_source_product. (exists pa_lt_pvs_pow_construct_source_product_bound. pa_lt_pvs_pow_construct_source_product_bound + S pa_i_pvs_pow_construct_source_product = k) -> exists pa_p_pvs_pow_construct_source_product pa_r_pvs_pow_construct_source_product pa_s_pvs_pow_construct_source_product. ((((exists pa_h_pvs_pow_construct_source_product_factor. pa_h_pvs_pow_construct_source_product_factor + S (pa_p_pvs_pow_construct_source_product) = S ((S (pa_i_pvs_pow_construct_source_product)) * pa_c_pvs_pow_construct_source)) /\\ exists pa_q_pvs_pow_construct_source_product_factor. pa_b_pvs_pow_construct_source = pa_q_pvs_pow_construct_source_product_factor * S ((S (pa_i_pvs_pow_construct_source_product)) * pa_c_pvs_pow_construct_source) + (pa_p_pvs_pow_construct_source_product))) /\\ ((((exists pa_h_pvs_pow_construct_source_product_partial. pa_h_pvs_pow_construct_source_product_partial + S (pa_r_pvs_pow_construct_source_product) = S ((S (pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_partial. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_partial * S ((S (pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product) + (pa_r_pvs_pow_construct_source_product))) /\\ ((((exists pa_h_pvs_pow_construct_source_product_successor. pa_h_pvs_pow_construct_source_product_successor + S (pa_s_pvs_pow_construct_source_product) = S ((S (S pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product)) /\\ exists pa_q_pvs_pow_construct_source_product_successor. pa_u_pvs_pow_construct_source_product = pa_q_pvs_pow_construct_source_product_successor * S ((S (S pa_i_pvs_pow_construct_source_product)) * pa_v_pvs_pow_construct_source_product) + (pa_s_pvs_pow_construct_source_product))) /\\ pa_s_pvs_pow_construct_source_product = pa_r_pvs_pow_construct_source_product * pa_p_pvs_pow_construct_source_product)))))))) -> (((exists bpd_gap_pvs_pow_construct_output_selected_bound. bpd_gap_pvs_pow_construct_output_selected_bound + (k * e) = (z)) /\\ (exists bpvi_result_pvs_pow_construct_output_selected. ((exists bpvi_b_pvs_pow_construct_output_selected_power bpvi_c_pvs_pow_construct_output_selected_power. ((forall bpvi_i_pvs_pow_construct_output_selected_power. (exists bpvi_repeat_gap_pvs_pow_construct_output_selected_power. bpvi_repeat_gap_pvs_pow_construct_output_selected_power + S bpvi_i_pvs_pow_construct_output_selected_power = k * e) -> (((exists bpvi_h_pvs_pow_construct_output_selected_power_repeat. bpvi_h_pvs_pow_construct_output_selected_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_repeat. bpvi_b_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_repeat * S ((S (bpvi_i_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_output_selected_power bpvi_v_pvs_pow_construct_output_selected_power. ((((exists bpvi_h_pvs_pow_construct_output_selected_power_start. bpvi_h_pvs_pow_construct_output_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_start. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_output_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_terminal. bpvi_h_pvs_pow_construct_output_selected_power_terminal + S (bpvi_result_pvs_pow_construct_output_selected) = S ((S (k * e)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_terminal. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_terminal * S ((S (k * e)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_result_pvs_pow_construct_output_selected))) /\\ forall bpvi_j_pvs_pow_construct_output_selected_power. (exists bpvi_product_gap_pvs_pow_construct_output_selected_power. bpvi_product_gap_pvs_pow_construct_output_selected_power + S bpvi_j_pvs_pow_construct_output_selected_power = k * e) -> exists bpvi_factor_pvs_pow_construct_output_selected_power bpvi_partial_pvs_pow_construct_output_selected_power bpvi_successor_pvs_pow_construct_output_selected_power. ((((exists bpvi_h_pvs_pow_construct_output_selected_power_factor. bpvi_h_pvs_pow_construct_output_selected_power_factor + S (bpvi_factor_pvs_pow_construct_output_selected_power) = S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_factor. bpvi_b_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_factor * S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_c_pvs_pow_construct_output_selected_power) + (bpvi_factor_pvs_pow_construct_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_partial. bpvi_h_pvs_pow_construct_output_selected_power_partial + S (bpvi_partial_pvs_pow_construct_output_selected_power) = S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_partial. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_partial * S ((S (bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_partial_pvs_pow_construct_output_selected_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_selected_power_successor. bpvi_h_pvs_pow_construct_output_selected_power_successor + S (bpvi_successor_pvs_pow_construct_output_selected_power) = S ((S (S bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power)) /\\ exists bpvi_q_pvs_pow_construct_output_selected_power_successor. bpvi_u_pvs_pow_construct_output_selected_power = bpvi_q_pvs_pow_construct_output_selected_power_successor * S ((S (S bpvi_j_pvs_pow_construct_output_selected_power)) * bpvi_v_pvs_pow_construct_output_selected_power) + (bpvi_successor_pvs_pow_construct_output_selected_power))) /\\ bpvi_successor_pvs_pow_construct_output_selected_power = bpvi_partial_pvs_pow_construct_output_selected_power * bpvi_factor_pvs_pow_construct_output_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_output_selected. z = bpvi_result_pvs_pow_construct_output_selected * bpvi_divisor_factor_pvs_pow_construct_output_selected))) /\\ forall bpd_candidate_pvs_pow_construct_output. (exists bpd_gap_pvs_pow_construct_output_candidate_bound. bpd_gap_pvs_pow_construct_output_candidate_bound + (bpd_candidate_pvs_pow_construct_output) = (z)) -> (exists bpvi_result_pvs_pow_construct_output_candidate. ((exists bpvi_b_pvs_pow_construct_output_candidate_power bpvi_c_pvs_pow_construct_output_candidate_power. ((forall bpvi_i_pvs_pow_construct_output_candidate_power. (exists bpvi_repeat_gap_pvs_pow_construct_output_candidate_power. bpvi_repeat_gap_pvs_pow_construct_output_candidate_power + S bpvi_i_pvs_pow_construct_output_candidate_power = bpd_candidate_pvs_pow_construct_output) -> (((exists bpvi_h_pvs_pow_construct_output_candidate_power_repeat. bpvi_h_pvs_pow_construct_output_candidate_power_repeat + S (p) = S ((S (bpvi_i_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_repeat. bpvi_b_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_repeat * S ((S (bpvi_i_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power) + (p)))) /\\ (exists bpvi_u_pvs_pow_construct_output_candidate_power bpvi_v_pvs_pow_construct_output_candidate_power. ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_start. bpvi_h_pvs_pow_construct_output_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_start. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_start * S ((S (0)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_terminal. bpvi_h_pvs_pow_construct_output_candidate_power_terminal + S (bpvi_result_pvs_pow_construct_output_candidate) = S ((S (bpd_candidate_pvs_pow_construct_output)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_terminal. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_terminal * S ((S (bpd_candidate_pvs_pow_construct_output)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_result_pvs_pow_construct_output_candidate))) /\\ forall bpvi_j_pvs_pow_construct_output_candidate_power. (exists bpvi_product_gap_pvs_pow_construct_output_candidate_power. bpvi_product_gap_pvs_pow_construct_output_candidate_power + S bpvi_j_pvs_pow_construct_output_candidate_power = bpd_candidate_pvs_pow_construct_output) -> exists bpvi_factor_pvs_pow_construct_output_candidate_power bpvi_partial_pvs_pow_construct_output_candidate_power bpvi_successor_pvs_pow_construct_output_candidate_power. ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_factor. bpvi_h_pvs_pow_construct_output_candidate_power_factor + S (bpvi_factor_pvs_pow_construct_output_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_factor. bpvi_b_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_factor * S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_c_pvs_pow_construct_output_candidate_power) + (bpvi_factor_pvs_pow_construct_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_partial. bpvi_h_pvs_pow_construct_output_candidate_power_partial + S (bpvi_partial_pvs_pow_construct_output_candidate_power) = S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_partial. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_partial * S ((S (bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_partial_pvs_pow_construct_output_candidate_power))) /\\ ((((exists bpvi_h_pvs_pow_construct_output_candidate_power_successor. bpvi_h_pvs_pow_construct_output_candidate_power_successor + S (bpvi_successor_pvs_pow_construct_output_candidate_power) = S ((S (S bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power)) /\\ exists bpvi_q_pvs_pow_construct_output_candidate_power_successor. bpvi_u_pvs_pow_construct_output_candidate_power = bpvi_q_pvs_pow_construct_output_candidate_power_successor * S ((S (S bpvi_j_pvs_pow_construct_output_candidate_power)) * bpvi_v_pvs_pow_construct_output_candidate_power) + (bpvi_successor_pvs_pow_construct_output_candidate_power))) /\\ bpvi_successor_pvs_pow_construct_output_candidate_power = bpvi_partial_pvs_pow_construct_output_candidate_power * bpvi_factor_pvs_pow_construct_output_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_pow_construct_output_candidate. z = bpvi_result_pvs_pow_construct_output_candidate * bpvi_divisor_factor_pvs_pow_construct_output_candidate)) -> (exists bpd_gap_pvs_pow_construct_output_maximal. bpd_gap_pvs_pow_construct_output_maximal + (bpd_candidate_pvs_pow_construct_output) = (k * e)))", "statement_sha256": "30e2347cbb82399420bcfec8fc56d030a7843825de85f761da4bf6152d897782" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "pow_positive_exponent_base_divides", "canonical_catalog_record": { "alpha_v29_frontier_enrollment": { "body_receipt_sha256": "c4bcd0ffdf1cee5b1a985f42b5e84ede243fbdd2f7c1a0bb638c3f301018b5a9", "bundle_campaign": "priority_layer", "bundle_node_id": 292, "bundle_sha256": 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nonzero_is_succ", "exact hk", "cases hs", "have hprev : exists r. (exists pa_b_pvs_positive_previous pa_c_pvs_positive_previous. ((forall pa_i_pvs_positive_previous_repeat. (exists pa_lt_pvs_positive_previous_repeat_bound. pa_lt_pvs_positive_previous_repeat_bound + S pa_i_pvs_positive_previous_repeat = x) -> (((exists pa_h_pvs_positive_previous_repeat_decoded. pa_h_pvs_positive_previous_repeat_decoded + S (a) = S ((S (pa_i_pvs_positive_previous_repeat)) * pa_c_pvs_positive_previous)) /\\ exists pa_q_pvs_positive_previous_repeat_decoded. pa_b_pvs_positive_previous = pa_q_pvs_positive_previous_repeat_decoded * S ((S (pa_i_pvs_positive_previous_repeat)) * pa_c_pvs_positive_previous) + (a)))) /\\ (exists pa_u_pvs_positive_previous_product pa_v_pvs_positive_previous_product. ((((exists pa_h_pvs_positive_previous_product_start. pa_h_pvs_positive_previous_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_start. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_start * S ((S (0)) * pa_v_pvs_positive_previous_product) + (1))) /\\ ((((exists pa_h_pvs_positive_previous_product_terminal. pa_h_pvs_positive_previous_product_terminal + S (r) = S ((S (x)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_terminal. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_terminal * S ((S (x)) * pa_v_pvs_positive_previous_product) + (r))) /\\ forall pa_i_pvs_positive_previous_product. (exists pa_lt_pvs_positive_previous_product_bound. pa_lt_pvs_positive_previous_product_bound + S pa_i_pvs_positive_previous_product = x) -> exists pa_p_pvs_positive_previous_product pa_r_pvs_positive_previous_product pa_s_pvs_positive_previous_product. ((((exists pa_h_pvs_positive_previous_product_factor. pa_h_pvs_positive_previous_product_factor + S (pa_p_pvs_positive_previous_product) = S ((S (pa_i_pvs_positive_previous_product)) * pa_c_pvs_positive_previous)) /\\ exists pa_q_pvs_positive_previous_product_factor. pa_b_pvs_positive_previous = pa_q_pvs_positive_previous_product_factor * S ((S (pa_i_pvs_positive_previous_product)) * pa_c_pvs_positive_previous) + (pa_p_pvs_positive_previous_product))) /\\ ((((exists pa_h_pvs_positive_previous_product_partial. pa_h_pvs_positive_previous_product_partial + S (pa_r_pvs_positive_previous_product) = S ((S (pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_partial. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_partial * S ((S (pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product) + (pa_r_pvs_positive_previous_product))) /\\ ((((exists pa_h_pvs_positive_previous_product_successor. pa_h_pvs_positive_previous_product_successor + S (pa_s_pvs_positive_previous_product) = S ((S (S pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_successor. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_successor * S ((S (S pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product) + (pa_s_pvs_positive_previous_product))) /\\ pa_s_pvs_positive_previous_product = pa_r_pvs_positive_previous_product * pa_p_pvs_positive_previous_product)))))))) /\\ z = r * a", "specialize pow_successor_decompose (a)", "specialize pow_successor_decompose (x)", "specialize pow_successor_decompose (k)", "specialize pow_successor_decompose (z)", "apply pow_successor_decompose", "exact hs_witness", "exact hpow", "cases hprev", "cases hprev_witness", "exists x1", "trans x1 * a", "exact hprev_witness_right", "apply mul_comm" ], "script_sha256": "1c68878b3515f04811dd3f23ca31c8fe9a3e01c714e52cd10d102f2fe4189d31", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall a k z. ~(k = 0) -> (exists pa_b_pvs_positive_power pa_c_pvs_positive_power. ((forall pa_i_pvs_positive_power_repeat. (exists pa_lt_pvs_positive_power_repeat_bound. pa_lt_pvs_positive_power_repeat_bound + S pa_i_pvs_positive_power_repeat = k) -> (((exists pa_h_pvs_positive_power_repeat_decoded. pa_h_pvs_positive_power_repeat_decoded + S (a) = S ((S (pa_i_pvs_positive_power_repeat)) * pa_c_pvs_positive_power)) /\\ exists pa_q_pvs_positive_power_repeat_decoded. pa_b_pvs_positive_power = pa_q_pvs_positive_power_repeat_decoded * S ((S (pa_i_pvs_positive_power_repeat)) * pa_c_pvs_positive_power) + (a)))) /\\ (exists pa_u_pvs_positive_power_product pa_v_pvs_positive_power_product. ((((exists pa_h_pvs_positive_power_product_start. pa_h_pvs_positive_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_start. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_start * S ((S (0)) * pa_v_pvs_positive_power_product) + (1))) /\\ ((((exists pa_h_pvs_positive_power_product_terminal. pa_h_pvs_positive_power_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_terminal. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_terminal * S ((S (k)) * pa_v_pvs_positive_power_product) + (z))) /\\ forall pa_i_pvs_positive_power_product. (exists pa_lt_pvs_positive_power_product_bound. pa_lt_pvs_positive_power_product_bound + S pa_i_pvs_positive_power_product = k) -> exists pa_p_pvs_positive_power_product pa_r_pvs_positive_power_product pa_s_pvs_positive_power_product. ((((exists pa_h_pvs_positive_power_product_factor. pa_h_pvs_positive_power_product_factor + S (pa_p_pvs_positive_power_product) = S ((S (pa_i_pvs_positive_power_product)) * pa_c_pvs_positive_power)) /\\ exists pa_q_pvs_positive_power_product_factor. pa_b_pvs_positive_power = pa_q_pvs_positive_power_product_factor * S ((S (pa_i_pvs_positive_power_product)) * pa_c_pvs_positive_power) + (pa_p_pvs_positive_power_product))) /\\ ((((exists pa_h_pvs_positive_power_product_partial. pa_h_pvs_positive_power_product_partial + S (pa_r_pvs_positive_power_product) = S ((S (pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_partial. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_partial * S ((S (pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product) + (pa_r_pvs_positive_power_product))) /\\ ((((exists pa_h_pvs_positive_power_product_successor. pa_h_pvs_positive_power_product_successor + S (pa_s_pvs_positive_power_product) = S ((S (S pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_successor. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_successor * S ((S (S pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product) + (pa_s_pvs_positive_power_product))) /\\ pa_s_pvs_positive_power_product = pa_r_pvs_positive_power_product * pa_p_pvs_positive_power_product)))))))) -> (exists pvs_factor_positive_divides. (z) = (a) * pvs_factor_positive_divides)", "statement_sha256": "04eeef7afd546e26ef34eb2ed414bbcb8cc5e1875fd41a8a79be7d0e06a1b189", "summary": "Every positive power has its base as an actual divisor; no prime or positivity oracle is needed.", "summary_sha256": "968dbef23393e9d83a72d1bf24e96a7d141b67044d78bdf20ac8e0bf16330b73" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "nonzero_is_succ", "pow_successor_decompose", "mul_comm" ], "direct_prerequisite_of_owned_theorem": true, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=292]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "pow_positive_exponent_base_divides", "parent_alpha_version": "v34", "proof_bundle_node_id": 253, "reference_route": "jordan-totient/checkpoint.html#theorem-pow_positive_exponent_base_divides", "script": [ "intro a", "intro k", "intro z", "intro hk", "intro hpow", "have hs : exists j. k = S j", "specialize nonzero_is_succ (k)", "apply nonzero_is_succ", "exact hk", "cases hs", "have hprev : exists r. (exists pa_b_pvs_positive_previous pa_c_pvs_positive_previous. ((forall pa_i_pvs_positive_previous_repeat. (exists pa_lt_pvs_positive_previous_repeat_bound. pa_lt_pvs_positive_previous_repeat_bound + S pa_i_pvs_positive_previous_repeat = x) -> (((exists pa_h_pvs_positive_previous_repeat_decoded. pa_h_pvs_positive_previous_repeat_decoded + S (a) = S ((S (pa_i_pvs_positive_previous_repeat)) * pa_c_pvs_positive_previous)) /\\ exists pa_q_pvs_positive_previous_repeat_decoded. pa_b_pvs_positive_previous = pa_q_pvs_positive_previous_repeat_decoded * S ((S (pa_i_pvs_positive_previous_repeat)) * pa_c_pvs_positive_previous) + (a)))) /\\ (exists pa_u_pvs_positive_previous_product pa_v_pvs_positive_previous_product. ((((exists pa_h_pvs_positive_previous_product_start. pa_h_pvs_positive_previous_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_start. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_start * S ((S (0)) * pa_v_pvs_positive_previous_product) + (1))) /\\ ((((exists pa_h_pvs_positive_previous_product_terminal. pa_h_pvs_positive_previous_product_terminal + S (r) = S ((S (x)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_terminal. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_terminal * S ((S (x)) * pa_v_pvs_positive_previous_product) + (r))) /\\ forall pa_i_pvs_positive_previous_product. (exists pa_lt_pvs_positive_previous_product_bound. pa_lt_pvs_positive_previous_product_bound + S pa_i_pvs_positive_previous_product = x) -> exists pa_p_pvs_positive_previous_product pa_r_pvs_positive_previous_product pa_s_pvs_positive_previous_product. ((((exists pa_h_pvs_positive_previous_product_factor. pa_h_pvs_positive_previous_product_factor + S (pa_p_pvs_positive_previous_product) = S ((S (pa_i_pvs_positive_previous_product)) * pa_c_pvs_positive_previous)) /\\ exists pa_q_pvs_positive_previous_product_factor. pa_b_pvs_positive_previous = pa_q_pvs_positive_previous_product_factor * S ((S (pa_i_pvs_positive_previous_product)) * pa_c_pvs_positive_previous) + (pa_p_pvs_positive_previous_product))) /\\ ((((exists pa_h_pvs_positive_previous_product_partial. pa_h_pvs_positive_previous_product_partial + S (pa_r_pvs_positive_previous_product) = S ((S (pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_partial. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_partial * S ((S (pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product) + (pa_r_pvs_positive_previous_product))) /\\ ((((exists pa_h_pvs_positive_previous_product_successor. pa_h_pvs_positive_previous_product_successor + S (pa_s_pvs_positive_previous_product) = S ((S (S pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product)) /\\ exists pa_q_pvs_positive_previous_product_successor. pa_u_pvs_positive_previous_product = pa_q_pvs_positive_previous_product_successor * S ((S (S pa_i_pvs_positive_previous_product)) * pa_v_pvs_positive_previous_product) + (pa_s_pvs_positive_previous_product))) /\\ pa_s_pvs_positive_previous_product = pa_r_pvs_positive_previous_product * pa_p_pvs_positive_previous_product)))))))) /\\ z = r * a", "specialize pow_successor_decompose (a)", "specialize pow_successor_decompose (x)", "specialize pow_successor_decompose (k)", "specialize pow_successor_decompose (z)", "apply pow_successor_decompose", "exact hs_witness", "exact hpow", "cases hprev", "cases hprev_witness", "exists x1", "trans x1 * a", "exact hprev_witness_right", "apply mul_comm" ], "script_sha256": "1c68878b3515f04811dd3f23ca31c8fe9a3e01c714e52cd10d102f2fe4189d31", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall a k z. ~(k = 0) -> (exists pa_b_pvs_positive_power pa_c_pvs_positive_power. ((forall pa_i_pvs_positive_power_repeat. (exists pa_lt_pvs_positive_power_repeat_bound. pa_lt_pvs_positive_power_repeat_bound + S pa_i_pvs_positive_power_repeat = k) -> (((exists pa_h_pvs_positive_power_repeat_decoded. pa_h_pvs_positive_power_repeat_decoded + S (a) = S ((S (pa_i_pvs_positive_power_repeat)) * pa_c_pvs_positive_power)) /\\ exists pa_q_pvs_positive_power_repeat_decoded. pa_b_pvs_positive_power = pa_q_pvs_positive_power_repeat_decoded * S ((S (pa_i_pvs_positive_power_repeat)) * pa_c_pvs_positive_power) + (a)))) /\\ (exists pa_u_pvs_positive_power_product pa_v_pvs_positive_power_product. ((((exists pa_h_pvs_positive_power_product_start. pa_h_pvs_positive_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_start. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_start * S ((S (0)) * pa_v_pvs_positive_power_product) + (1))) /\\ ((((exists pa_h_pvs_positive_power_product_terminal. pa_h_pvs_positive_power_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_terminal. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_terminal * S ((S (k)) * pa_v_pvs_positive_power_product) + (z))) /\\ forall pa_i_pvs_positive_power_product. (exists pa_lt_pvs_positive_power_product_bound. pa_lt_pvs_positive_power_product_bound + S pa_i_pvs_positive_power_product = k) -> exists pa_p_pvs_positive_power_product pa_r_pvs_positive_power_product pa_s_pvs_positive_power_product. ((((exists pa_h_pvs_positive_power_product_factor. pa_h_pvs_positive_power_product_factor + S (pa_p_pvs_positive_power_product) = S ((S (pa_i_pvs_positive_power_product)) * pa_c_pvs_positive_power)) /\\ exists pa_q_pvs_positive_power_product_factor. pa_b_pvs_positive_power = pa_q_pvs_positive_power_product_factor * S ((S (pa_i_pvs_positive_power_product)) * pa_c_pvs_positive_power) + (pa_p_pvs_positive_power_product))) /\\ ((((exists pa_h_pvs_positive_power_product_partial. pa_h_pvs_positive_power_product_partial + S (pa_r_pvs_positive_power_product) = S ((S (pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_partial. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_partial * S ((S (pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product) + (pa_r_pvs_positive_power_product))) /\\ ((((exists pa_h_pvs_positive_power_product_successor. pa_h_pvs_positive_power_product_successor + S (pa_s_pvs_positive_power_product) = S ((S (S pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product)) /\\ exists pa_q_pvs_positive_power_product_successor. pa_u_pvs_positive_power_product = pa_q_pvs_positive_power_product_successor * S ((S (S pa_i_pvs_positive_power_product)) * pa_v_pvs_positive_power_product) + (pa_s_pvs_positive_power_product))) /\\ pa_s_pvs_positive_power_product = pa_r_pvs_positive_power_product * pa_p_pvs_positive_power_product)))))))) -> (exists pvs_factor_positive_divides. (z) = (a) * pvs_factor_positive_divides)", "statement_sha256": "04eeef7afd546e26ef34eb2ed414bbcb8cc5e1875fd41a8a79be7d0e06a1b189" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_valuation_distinct_prime_power_zero", "canonical_catalog_record": { "alpha_v29_frontier_enrollment": { "body_receipt_sha256": "c76ef85e3e80dec5e8c400cef7a36afa12bb807e1a9db1ff5597533d892db563", "bundle_campaign": "priority_layer", "bundle_node_id": 293, "bundle_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "campaign": "prime_valuation_support", "parent_catalog_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "rfc_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "source_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "test_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c" }, "body_checked": true, "body_receipt": { "command_count": 44, "dependency_count": 5, "dne_command_count": 0, "name": "prime_valuation_distinct_prime_power_zero", "proof_depth": 25, "proof_edges": 66, "proof_nodes": 67, "proof_objects": 67, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "prime_nonzero", "distinct_primes_left_not_divide_right", "prime_valuation_zero_of_nondivisor", "prime_power_valuation_pow", "prime_valuation_exponent_eq_transport" ], "dependencies_sha256": "2153fba4aac54db29dd0e382cdbb55357c3c22b874f27083cee486db3c12c20e", "empty_context_closure": { "body_proof_depth": 25, "body_proof_nodes": 67, "bundle_campaign": "priority_layer", "bundle_dependency_edge_count": 1690, "bundle_node_count": 566, "bundle_node_id": 293, "bundle_path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "bundle_root_id": 565, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "af998c853267c251d0d8d205ce8149f49cac250218a5c7fef550f772332d53af", "status": "checked" }, "enrollment_index": 2770, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=293]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "prime_valuation_support", "logical_spec_sha256": "3771d7707c8829aeaa346a9917843e65cdda1b98e154bd4ed5c03c1b939ade47", "membership": "alpha_only", "name": "prime_valuation_distinct_prime_power_zero", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro q", "intro k", "intro z", "intro hp", "intro hq", "intro hne", "intro hpow", "have hpzero : ~(p = 0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hbase : ((exists bpd_gap_pvs_distinct_base_zero_selected_bound. bpd_gap_pvs_distinct_base_zero_selected_bound + (0) = (p)) /\\ (exists bpvi_result_pvs_distinct_base_zero_selected. ((exists bpvi_b_pvs_distinct_base_zero_selected_power bpvi_c_pvs_distinct_base_zero_selected_power. ((forall bpvi_i_pvs_distinct_base_zero_selected_power. (exists bpvi_repeat_gap_pvs_distinct_base_zero_selected_power. bpvi_repeat_gap_pvs_distinct_base_zero_selected_power + S bpvi_i_pvs_distinct_base_zero_selected_power = 0) -> (((exists bpvi_h_pvs_distinct_base_zero_selected_power_repeat. bpvi_h_pvs_distinct_base_zero_selected_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_repeat. bpvi_b_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_repeat * S ((S (bpvi_i_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_base_zero_selected_power bpvi_v_pvs_distinct_base_zero_selected_power. ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_start. bpvi_h_pvs_distinct_base_zero_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_start. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_start * S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_terminal. bpvi_h_pvs_distinct_base_zero_selected_power_terminal + S (bpvi_result_pvs_distinct_base_zero_selected) = S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_terminal. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (bpvi_result_pvs_distinct_base_zero_selected))) /\\ forall bpvi_j_pvs_distinct_base_zero_selected_power. (exists bpvi_product_gap_pvs_distinct_base_zero_selected_power. bpvi_product_gap_pvs_distinct_base_zero_selected_power + S bpvi_j_pvs_distinct_base_zero_selected_power = 0) -> exists bpvi_factor_pvs_distinct_base_zero_selected_power bpvi_partial_pvs_distinct_base_zero_selected_power bpvi_successor_pvs_distinct_base_zero_selected_power. ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_factor. bpvi_h_pvs_distinct_base_zero_selected_power_factor + S (bpvi_factor_pvs_distinct_base_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_factor. bpvi_b_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_factor * S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power) + (bpvi_factor_pvs_distinct_base_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_partial. bpvi_h_pvs_distinct_base_zero_selected_power_partial + S (bpvi_partial_pvs_distinct_base_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_partial. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_partial * S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (bpvi_partial_pvs_distinct_base_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_successor. bpvi_h_pvs_distinct_base_zero_selected_power_successor + S (bpvi_successor_pvs_distinct_base_zero_selected_power) = S ((S (S bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_successor. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_successor * S ((S (S bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (bpvi_successor_pvs_distinct_base_zero_selected_power))) /\\ bpvi_successor_pvs_distinct_base_zero_selected_power = bpvi_partial_pvs_distinct_base_zero_selected_power * bpvi_factor_pvs_distinct_base_zero_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_base_zero_selected. p = bpvi_result_pvs_distinct_base_zero_selected * bpvi_divisor_factor_pvs_distinct_base_zero_selected))) /\\ forall bpd_candidate_pvs_distinct_base_zero. (exists bpd_gap_pvs_distinct_base_zero_candidate_bound. bpd_gap_pvs_distinct_base_zero_candidate_bound + (bpd_candidate_pvs_distinct_base_zero) = (p)) -> (exists bpvi_result_pvs_distinct_base_zero_candidate. ((exists bpvi_b_pvs_distinct_base_zero_candidate_power bpvi_c_pvs_distinct_base_zero_candidate_power. ((forall bpvi_i_pvs_distinct_base_zero_candidate_power. (exists bpvi_repeat_gap_pvs_distinct_base_zero_candidate_power. bpvi_repeat_gap_pvs_distinct_base_zero_candidate_power + S bpvi_i_pvs_distinct_base_zero_candidate_power = bpd_candidate_pvs_distinct_base_zero) -> (((exists bpvi_h_pvs_distinct_base_zero_candidate_power_repeat. bpvi_h_pvs_distinct_base_zero_candidate_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_repeat. bpvi_b_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_repeat * S ((S (bpvi_i_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_base_zero_candidate_power bpvi_v_pvs_distinct_base_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_start. bpvi_h_pvs_distinct_base_zero_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_start. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_start * S ((S (0)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_terminal. bpvi_h_pvs_distinct_base_zero_candidate_power_terminal + S (bpvi_result_pvs_distinct_base_zero_candidate) = S ((S (bpd_candidate_pvs_distinct_base_zero)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_terminal. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_terminal * S ((S (bpd_candidate_pvs_distinct_base_zero)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (bpvi_result_pvs_distinct_base_zero_candidate))) /\\ forall bpvi_j_pvs_distinct_base_zero_candidate_power. (exists bpvi_product_gap_pvs_distinct_base_zero_candidate_power. bpvi_product_gap_pvs_distinct_base_zero_candidate_power + S bpvi_j_pvs_distinct_base_zero_candidate_power = bpd_candidate_pvs_distinct_base_zero) -> exists bpvi_factor_pvs_distinct_base_zero_candidate_power bpvi_partial_pvs_distinct_base_zero_candidate_power bpvi_successor_pvs_distinct_base_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_factor. bpvi_h_pvs_distinct_base_zero_candidate_power_factor + S (bpvi_factor_pvs_distinct_base_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_factor. bpvi_b_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_factor * S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power) + (bpvi_factor_pvs_distinct_base_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_partial. bpvi_h_pvs_distinct_base_zero_candidate_power_partial + S (bpvi_partial_pvs_distinct_base_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_partial. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_partial * S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (bpvi_partial_pvs_distinct_base_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_successor. bpvi_h_pvs_distinct_base_zero_candidate_power_successor + S (bpvi_successor_pvs_distinct_base_zero_candidate_power) = S ((S (S bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_successor. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_successor * S ((S (S bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (bpvi_successor_pvs_distinct_base_zero_candidate_power))) /\\ bpvi_successor_pvs_distinct_base_zero_candidate_power = bpvi_partial_pvs_distinct_base_zero_candidate_power * bpvi_factor_pvs_distinct_base_zero_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_base_zero_candidate. p = bpvi_result_pvs_distinct_base_zero_candidate * bpvi_divisor_factor_pvs_distinct_base_zero_candidate)) -> (exists bpd_gap_pvs_distinct_base_zero_maximal. bpd_gap_pvs_distinct_base_zero_maximal + (bpd_candidate_pvs_distinct_base_zero) = (0))", "specialize prime_valuation_zero_of_nondivisor (q)", "specialize prime_valuation_zero_of_nondivisor (p)", "apply prime_valuation_zero_of_nondivisor", "exact hq", "exact hpzero", "intro hdiv", "specialize distinct_primes_left_not_divide_right (q)", "specialize distinct_primes_left_not_divide_right (p)", "apply distinct_primes_left_not_divide_right", "exact hq", "exact hp", "exact hne", "exact hdiv", "specialize prime_valuation_exponent_eq_transport (q)", "specialize prime_valuation_exponent_eq_transport (z)", "specialize prime_valuation_exponent_eq_transport (k * 0)", "specialize prime_valuation_exponent_eq_transport (0)", "apply prime_valuation_exponent_eq_transport", "apply PA5", "specialize prime_power_valuation_pow (q)", "specialize prime_power_valuation_pow (p)", "specialize prime_power_valuation_pow (k)", "specialize prime_power_valuation_pow (0)", "specialize prime_power_valuation_pow (z)", "apply prime_power_valuation_pow", "exact hq", "exact hpzero", "exact hbase", "exact hpow" ], "script_sha256": "c523997e526be44ed099369b01354603ce18c36c3422734422fc01d6205d7589", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall p q k z. (~((p) = 1) /\\ forall pvs_left_distinct_base pvs_right_distinct_base. (p) = pvs_left_distinct_base * pvs_right_distinct_base -> pvs_left_distinct_base = 1 \\/ pvs_right_distinct_base = 1) -> (~((q) = 1) /\\ forall pvs_left_distinct_valuation pvs_right_distinct_valuation. (q) = pvs_left_distinct_valuation * pvs_right_distinct_valuation -> pvs_left_distinct_valuation = 1 \\/ pvs_right_distinct_valuation = 1) -> ~(q = p) -> (exists pa_b_pvs_distinct_power pa_c_pvs_distinct_power. ((forall pa_i_pvs_distinct_power_repeat. (exists pa_lt_pvs_distinct_power_repeat_bound. pa_lt_pvs_distinct_power_repeat_bound + S pa_i_pvs_distinct_power_repeat = k) -> (((exists pa_h_pvs_distinct_power_repeat_decoded. pa_h_pvs_distinct_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_distinct_power_repeat)) * pa_c_pvs_distinct_power)) /\\ exists pa_q_pvs_distinct_power_repeat_decoded. pa_b_pvs_distinct_power = pa_q_pvs_distinct_power_repeat_decoded * S ((S (pa_i_pvs_distinct_power_repeat)) * pa_c_pvs_distinct_power) + (p)))) /\\ (exists pa_u_pvs_distinct_power_product pa_v_pvs_distinct_power_product. ((((exists pa_h_pvs_distinct_power_product_start. pa_h_pvs_distinct_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_start. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_start * S ((S (0)) * pa_v_pvs_distinct_power_product) + (1))) /\\ ((((exists pa_h_pvs_distinct_power_product_terminal. pa_h_pvs_distinct_power_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_terminal. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_terminal * S ((S (k)) * pa_v_pvs_distinct_power_product) + (z))) /\\ forall pa_i_pvs_distinct_power_product. (exists pa_lt_pvs_distinct_power_product_bound. pa_lt_pvs_distinct_power_product_bound + S pa_i_pvs_distinct_power_product = k) -> exists pa_p_pvs_distinct_power_product pa_r_pvs_distinct_power_product pa_s_pvs_distinct_power_product. ((((exists pa_h_pvs_distinct_power_product_factor. pa_h_pvs_distinct_power_product_factor + S (pa_p_pvs_distinct_power_product) = S ((S (pa_i_pvs_distinct_power_product)) * pa_c_pvs_distinct_power)) /\\ exists pa_q_pvs_distinct_power_product_factor. pa_b_pvs_distinct_power = pa_q_pvs_distinct_power_product_factor * S ((S (pa_i_pvs_distinct_power_product)) * pa_c_pvs_distinct_power) + (pa_p_pvs_distinct_power_product))) /\\ ((((exists pa_h_pvs_distinct_power_product_partial. pa_h_pvs_distinct_power_product_partial + S (pa_r_pvs_distinct_power_product) = S ((S (pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_partial. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_partial * S ((S (pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product) + (pa_r_pvs_distinct_power_product))) /\\ ((((exists pa_h_pvs_distinct_power_product_successor. pa_h_pvs_distinct_power_product_successor + S (pa_s_pvs_distinct_power_product) = S ((S (S pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_successor. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_successor * S ((S (S pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product) + (pa_s_pvs_distinct_power_product))) /\\ pa_s_pvs_distinct_power_product = pa_r_pvs_distinct_power_product * pa_p_pvs_distinct_power_product)))))))) -> (((exists bpd_gap_pvs_distinct_zero_selected_bound. bpd_gap_pvs_distinct_zero_selected_bound + (0) = (z)) /\\ (exists bpvi_result_pvs_distinct_zero_selected. ((exists bpvi_b_pvs_distinct_zero_selected_power bpvi_c_pvs_distinct_zero_selected_power. ((forall bpvi_i_pvs_distinct_zero_selected_power. (exists bpvi_repeat_gap_pvs_distinct_zero_selected_power. bpvi_repeat_gap_pvs_distinct_zero_selected_power + S bpvi_i_pvs_distinct_zero_selected_power = 0) -> (((exists bpvi_h_pvs_distinct_zero_selected_power_repeat. bpvi_h_pvs_distinct_zero_selected_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_repeat. bpvi_b_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_repeat * S ((S (bpvi_i_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_zero_selected_power bpvi_v_pvs_distinct_zero_selected_power. ((((exists bpvi_h_pvs_distinct_zero_selected_power_start. bpvi_h_pvs_distinct_zero_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_start. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_start * S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_terminal. bpvi_h_pvs_distinct_zero_selected_power_terminal + S (bpvi_result_pvs_distinct_zero_selected) = S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_terminal. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_result_pvs_distinct_zero_selected))) /\\ forall bpvi_j_pvs_distinct_zero_selected_power. (exists bpvi_product_gap_pvs_distinct_zero_selected_power. bpvi_product_gap_pvs_distinct_zero_selected_power + S bpvi_j_pvs_distinct_zero_selected_power = 0) -> exists bpvi_factor_pvs_distinct_zero_selected_power bpvi_partial_pvs_distinct_zero_selected_power bpvi_successor_pvs_distinct_zero_selected_power. ((((exists bpvi_h_pvs_distinct_zero_selected_power_factor. bpvi_h_pvs_distinct_zero_selected_power_factor + S (bpvi_factor_pvs_distinct_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_factor. bpvi_b_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_factor * S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power) + (bpvi_factor_pvs_distinct_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_partial. bpvi_h_pvs_distinct_zero_selected_power_partial + S (bpvi_partial_pvs_distinct_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_partial. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_partial * S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_partial_pvs_distinct_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_successor. bpvi_h_pvs_distinct_zero_selected_power_successor + S (bpvi_successor_pvs_distinct_zero_selected_power) = S ((S (S bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_successor. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_successor * S ((S (S bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_successor_pvs_distinct_zero_selected_power))) /\\ bpvi_successor_pvs_distinct_zero_selected_power = bpvi_partial_pvs_distinct_zero_selected_power * bpvi_factor_pvs_distinct_zero_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_zero_selected. z = bpvi_result_pvs_distinct_zero_selected * bpvi_divisor_factor_pvs_distinct_zero_selected))) /\\ forall bpd_candidate_pvs_distinct_zero. (exists bpd_gap_pvs_distinct_zero_candidate_bound. bpd_gap_pvs_distinct_zero_candidate_bound + (bpd_candidate_pvs_distinct_zero) = (z)) -> (exists bpvi_result_pvs_distinct_zero_candidate. ((exists bpvi_b_pvs_distinct_zero_candidate_power bpvi_c_pvs_distinct_zero_candidate_power. ((forall bpvi_i_pvs_distinct_zero_candidate_power. (exists bpvi_repeat_gap_pvs_distinct_zero_candidate_power. bpvi_repeat_gap_pvs_distinct_zero_candidate_power + S bpvi_i_pvs_distinct_zero_candidate_power = bpd_candidate_pvs_distinct_zero) -> (((exists bpvi_h_pvs_distinct_zero_candidate_power_repeat. bpvi_h_pvs_distinct_zero_candidate_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_repeat. bpvi_b_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_repeat * S ((S (bpvi_i_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_zero_candidate_power bpvi_v_pvs_distinct_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_zero_candidate_power_start. bpvi_h_pvs_distinct_zero_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_start. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_start * S ((S (0)) * bpvi_v_pvs_distinct_zero_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_terminal. bpvi_h_pvs_distinct_zero_candidate_power_terminal + S (bpvi_result_pvs_distinct_zero_candidate) = S ((S (bpd_candidate_pvs_distinct_zero)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_terminal. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_terminal * S ((S (bpd_candidate_pvs_distinct_zero)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_result_pvs_distinct_zero_candidate))) /\\ forall bpvi_j_pvs_distinct_zero_candidate_power. (exists bpvi_product_gap_pvs_distinct_zero_candidate_power. bpvi_product_gap_pvs_distinct_zero_candidate_power + S bpvi_j_pvs_distinct_zero_candidate_power = bpd_candidate_pvs_distinct_zero) -> exists bpvi_factor_pvs_distinct_zero_candidate_power bpvi_partial_pvs_distinct_zero_candidate_power bpvi_successor_pvs_distinct_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_zero_candidate_power_factor. bpvi_h_pvs_distinct_zero_candidate_power_factor + S (bpvi_factor_pvs_distinct_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_factor. bpvi_b_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_factor * S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power) + (bpvi_factor_pvs_distinct_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_partial. bpvi_h_pvs_distinct_zero_candidate_power_partial + S (bpvi_partial_pvs_distinct_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_partial. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_partial * S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_partial_pvs_distinct_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_successor. bpvi_h_pvs_distinct_zero_candidate_power_successor + S (bpvi_successor_pvs_distinct_zero_candidate_power) = S ((S (S bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_successor. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_successor * S ((S (S bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_successor_pvs_distinct_zero_candidate_power))) /\\ bpvi_successor_pvs_distinct_zero_candidate_power = bpvi_partial_pvs_distinct_zero_candidate_power * bpvi_factor_pvs_distinct_zero_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_zero_candidate. z = bpvi_result_pvs_distinct_zero_candidate * bpvi_divisor_factor_pvs_distinct_zero_candidate)) -> (exists bpd_gap_pvs_distinct_zero_maximal. bpd_gap_pvs_distinct_zero_maximal + (bpd_candidate_pvs_distinct_zero) = (0)))", "statement_sha256": "af998c853267c251d0d8d205ce8149f49cac250218a5c7fef550f772332d53af", "summary": "A power of a prime has zero valuation at every genuinely distinct prime, including exponent zero.", "summary_sha256": "b46a24182b1a858ecfd1b93f8ba4a23f3b0b503a3e2a657b3084cfa5e23cb91b" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "prime_nonzero", "distinct_primes_left_not_divide_right", "prime_valuation_zero_of_nondivisor", "prime_power_valuation_pow", "prime_valuation_exponent_eq_transport" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": 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"script": [ "intro p", "intro q", "intro k", "intro z", "intro hp", "intro hq", "intro hne", "intro hpow", "have hpzero : ~(p = 0)", "intro hz", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hz", "have hbase : ((exists bpd_gap_pvs_distinct_base_zero_selected_bound. bpd_gap_pvs_distinct_base_zero_selected_bound + (0) = (p)) /\\ (exists bpvi_result_pvs_distinct_base_zero_selected. ((exists bpvi_b_pvs_distinct_base_zero_selected_power bpvi_c_pvs_distinct_base_zero_selected_power. ((forall bpvi_i_pvs_distinct_base_zero_selected_power. (exists bpvi_repeat_gap_pvs_distinct_base_zero_selected_power. bpvi_repeat_gap_pvs_distinct_base_zero_selected_power + S bpvi_i_pvs_distinct_base_zero_selected_power = 0) -> (((exists bpvi_h_pvs_distinct_base_zero_selected_power_repeat. bpvi_h_pvs_distinct_base_zero_selected_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_repeat. bpvi_b_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_repeat * S ((S (bpvi_i_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_base_zero_selected_power bpvi_v_pvs_distinct_base_zero_selected_power. ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_start. bpvi_h_pvs_distinct_base_zero_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_start. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_start * S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_terminal. bpvi_h_pvs_distinct_base_zero_selected_power_terminal + S (bpvi_result_pvs_distinct_base_zero_selected) = S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_terminal. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (bpvi_result_pvs_distinct_base_zero_selected))) /\\ forall bpvi_j_pvs_distinct_base_zero_selected_power. (exists bpvi_product_gap_pvs_distinct_base_zero_selected_power. bpvi_product_gap_pvs_distinct_base_zero_selected_power + S bpvi_j_pvs_distinct_base_zero_selected_power = 0) -> exists bpvi_factor_pvs_distinct_base_zero_selected_power bpvi_partial_pvs_distinct_base_zero_selected_power bpvi_successor_pvs_distinct_base_zero_selected_power. ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_factor. bpvi_h_pvs_distinct_base_zero_selected_power_factor + S (bpvi_factor_pvs_distinct_base_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_factor. bpvi_b_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_factor * S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_c_pvs_distinct_base_zero_selected_power) + (bpvi_factor_pvs_distinct_base_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_partial. bpvi_h_pvs_distinct_base_zero_selected_power_partial + S (bpvi_partial_pvs_distinct_base_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_partial. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_partial * S ((S (bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (bpvi_partial_pvs_distinct_base_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_selected_power_successor. bpvi_h_pvs_distinct_base_zero_selected_power_successor + S (bpvi_successor_pvs_distinct_base_zero_selected_power) = S ((S (S bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_selected_power_successor. bpvi_u_pvs_distinct_base_zero_selected_power = bpvi_q_pvs_distinct_base_zero_selected_power_successor * S ((S (S bpvi_j_pvs_distinct_base_zero_selected_power)) * bpvi_v_pvs_distinct_base_zero_selected_power) + (bpvi_successor_pvs_distinct_base_zero_selected_power))) /\\ bpvi_successor_pvs_distinct_base_zero_selected_power = bpvi_partial_pvs_distinct_base_zero_selected_power * bpvi_factor_pvs_distinct_base_zero_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_base_zero_selected. p = bpvi_result_pvs_distinct_base_zero_selected * bpvi_divisor_factor_pvs_distinct_base_zero_selected))) /\\ forall bpd_candidate_pvs_distinct_base_zero. (exists bpd_gap_pvs_distinct_base_zero_candidate_bound. bpd_gap_pvs_distinct_base_zero_candidate_bound + (bpd_candidate_pvs_distinct_base_zero) = (p)) -> (exists bpvi_result_pvs_distinct_base_zero_candidate. ((exists bpvi_b_pvs_distinct_base_zero_candidate_power bpvi_c_pvs_distinct_base_zero_candidate_power. ((forall bpvi_i_pvs_distinct_base_zero_candidate_power. (exists bpvi_repeat_gap_pvs_distinct_base_zero_candidate_power. bpvi_repeat_gap_pvs_distinct_base_zero_candidate_power + S bpvi_i_pvs_distinct_base_zero_candidate_power = bpd_candidate_pvs_distinct_base_zero) -> (((exists bpvi_h_pvs_distinct_base_zero_candidate_power_repeat. bpvi_h_pvs_distinct_base_zero_candidate_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_repeat. bpvi_b_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_repeat * S ((S (bpvi_i_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_base_zero_candidate_power bpvi_v_pvs_distinct_base_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_start. bpvi_h_pvs_distinct_base_zero_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_start. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_start * S ((S (0)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_terminal. bpvi_h_pvs_distinct_base_zero_candidate_power_terminal + S (bpvi_result_pvs_distinct_base_zero_candidate) = S ((S (bpd_candidate_pvs_distinct_base_zero)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_terminal. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_terminal * S ((S (bpd_candidate_pvs_distinct_base_zero)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (bpvi_result_pvs_distinct_base_zero_candidate))) /\\ forall bpvi_j_pvs_distinct_base_zero_candidate_power. (exists bpvi_product_gap_pvs_distinct_base_zero_candidate_power. bpvi_product_gap_pvs_distinct_base_zero_candidate_power + S bpvi_j_pvs_distinct_base_zero_candidate_power = bpd_candidate_pvs_distinct_base_zero) -> exists bpvi_factor_pvs_distinct_base_zero_candidate_power bpvi_partial_pvs_distinct_base_zero_candidate_power bpvi_successor_pvs_distinct_base_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_factor. bpvi_h_pvs_distinct_base_zero_candidate_power_factor + S (bpvi_factor_pvs_distinct_base_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_factor. bpvi_b_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_factor * S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_c_pvs_distinct_base_zero_candidate_power) + (bpvi_factor_pvs_distinct_base_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_partial. bpvi_h_pvs_distinct_base_zero_candidate_power_partial + S (bpvi_partial_pvs_distinct_base_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_partial. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_partial * S ((S (bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (bpvi_partial_pvs_distinct_base_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_base_zero_candidate_power_successor. bpvi_h_pvs_distinct_base_zero_candidate_power_successor + S (bpvi_successor_pvs_distinct_base_zero_candidate_power) = S ((S (S bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_base_zero_candidate_power_successor. bpvi_u_pvs_distinct_base_zero_candidate_power = bpvi_q_pvs_distinct_base_zero_candidate_power_successor * S ((S (S bpvi_j_pvs_distinct_base_zero_candidate_power)) * bpvi_v_pvs_distinct_base_zero_candidate_power) + (bpvi_successor_pvs_distinct_base_zero_candidate_power))) /\\ bpvi_successor_pvs_distinct_base_zero_candidate_power = bpvi_partial_pvs_distinct_base_zero_candidate_power * bpvi_factor_pvs_distinct_base_zero_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_base_zero_candidate. p = bpvi_result_pvs_distinct_base_zero_candidate * bpvi_divisor_factor_pvs_distinct_base_zero_candidate)) -> (exists bpd_gap_pvs_distinct_base_zero_maximal. bpd_gap_pvs_distinct_base_zero_maximal + (bpd_candidate_pvs_distinct_base_zero) = (0))", "specialize prime_valuation_zero_of_nondivisor (q)", "specialize prime_valuation_zero_of_nondivisor (p)", "apply prime_valuation_zero_of_nondivisor", "exact hq", "exact hpzero", "intro hdiv", "specialize distinct_primes_left_not_divide_right (q)", "specialize distinct_primes_left_not_divide_right (p)", "apply distinct_primes_left_not_divide_right", "exact hq", "exact hp", "exact hne", "exact hdiv", "specialize prime_valuation_exponent_eq_transport (q)", "specialize prime_valuation_exponent_eq_transport (z)", "specialize prime_valuation_exponent_eq_transport (k * 0)", "specialize prime_valuation_exponent_eq_transport (0)", "apply prime_valuation_exponent_eq_transport", "apply PA5", "specialize prime_power_valuation_pow (q)", "specialize prime_power_valuation_pow (p)", "specialize prime_power_valuation_pow (k)", "specialize prime_power_valuation_pow (0)", "specialize prime_power_valuation_pow (z)", "apply prime_power_valuation_pow", "exact hq", "exact hpzero", "exact hbase", "exact hpow" ], "script_sha256": "c523997e526be44ed099369b01354603ce18c36c3422734422fc01d6205d7589", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "stable_member": false, "statement": "forall p q k z. (~((p) = 1) /\\ forall pvs_left_distinct_base pvs_right_distinct_base. (p) = pvs_left_distinct_base * pvs_right_distinct_base -> pvs_left_distinct_base = 1 \\/ pvs_right_distinct_base = 1) -> (~((q) = 1) /\\ forall pvs_left_distinct_valuation pvs_right_distinct_valuation. (q) = pvs_left_distinct_valuation * pvs_right_distinct_valuation -> pvs_left_distinct_valuation = 1 \\/ pvs_right_distinct_valuation = 1) -> ~(q = p) -> (exists pa_b_pvs_distinct_power pa_c_pvs_distinct_power. ((forall pa_i_pvs_distinct_power_repeat. (exists pa_lt_pvs_distinct_power_repeat_bound. pa_lt_pvs_distinct_power_repeat_bound + S pa_i_pvs_distinct_power_repeat = k) -> (((exists pa_h_pvs_distinct_power_repeat_decoded. pa_h_pvs_distinct_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_distinct_power_repeat)) * pa_c_pvs_distinct_power)) /\\ exists pa_q_pvs_distinct_power_repeat_decoded. pa_b_pvs_distinct_power = pa_q_pvs_distinct_power_repeat_decoded * S ((S (pa_i_pvs_distinct_power_repeat)) * pa_c_pvs_distinct_power) + (p)))) /\\ (exists pa_u_pvs_distinct_power_product pa_v_pvs_distinct_power_product. ((((exists pa_h_pvs_distinct_power_product_start. pa_h_pvs_distinct_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_start. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_start * S ((S (0)) * pa_v_pvs_distinct_power_product) + (1))) /\\ ((((exists pa_h_pvs_distinct_power_product_terminal. pa_h_pvs_distinct_power_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_terminal. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_terminal * S ((S (k)) * pa_v_pvs_distinct_power_product) + (z))) /\\ forall pa_i_pvs_distinct_power_product. (exists pa_lt_pvs_distinct_power_product_bound. pa_lt_pvs_distinct_power_product_bound + S pa_i_pvs_distinct_power_product = k) -> exists pa_p_pvs_distinct_power_product pa_r_pvs_distinct_power_product pa_s_pvs_distinct_power_product. ((((exists pa_h_pvs_distinct_power_product_factor. pa_h_pvs_distinct_power_product_factor + S (pa_p_pvs_distinct_power_product) = S ((S (pa_i_pvs_distinct_power_product)) * pa_c_pvs_distinct_power)) /\\ exists pa_q_pvs_distinct_power_product_factor. pa_b_pvs_distinct_power = pa_q_pvs_distinct_power_product_factor * S ((S (pa_i_pvs_distinct_power_product)) * pa_c_pvs_distinct_power) + (pa_p_pvs_distinct_power_product))) /\\ ((((exists pa_h_pvs_distinct_power_product_partial. pa_h_pvs_distinct_power_product_partial + S (pa_r_pvs_distinct_power_product) = S ((S (pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_partial. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_partial * S ((S (pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product) + (pa_r_pvs_distinct_power_product))) /\\ ((((exists pa_h_pvs_distinct_power_product_successor. pa_h_pvs_distinct_power_product_successor + S (pa_s_pvs_distinct_power_product) = S ((S (S pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product)) /\\ exists pa_q_pvs_distinct_power_product_successor. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_successor * S ((S (S pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product) + (pa_s_pvs_distinct_power_product))) /\\ pa_s_pvs_distinct_power_product = pa_r_pvs_distinct_power_product * pa_p_pvs_distinct_power_product)))))))) -> (((exists bpd_gap_pvs_distinct_zero_selected_bound. bpd_gap_pvs_distinct_zero_selected_bound + (0) = (z)) /\\ (exists bpvi_result_pvs_distinct_zero_selected. ((exists bpvi_b_pvs_distinct_zero_selected_power bpvi_c_pvs_distinct_zero_selected_power. ((forall bpvi_i_pvs_distinct_zero_selected_power. (exists bpvi_repeat_gap_pvs_distinct_zero_selected_power. bpvi_repeat_gap_pvs_distinct_zero_selected_power + S bpvi_i_pvs_distinct_zero_selected_power = 0) -> (((exists bpvi_h_pvs_distinct_zero_selected_power_repeat. bpvi_h_pvs_distinct_zero_selected_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_repeat. bpvi_b_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_repeat * S ((S (bpvi_i_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_zero_selected_power bpvi_v_pvs_distinct_zero_selected_power. ((((exists bpvi_h_pvs_distinct_zero_selected_power_start. bpvi_h_pvs_distinct_zero_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_start. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_start * S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_terminal. bpvi_h_pvs_distinct_zero_selected_power_terminal + S (bpvi_result_pvs_distinct_zero_selected) = S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_terminal. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_result_pvs_distinct_zero_selected))) /\\ forall bpvi_j_pvs_distinct_zero_selected_power. (exists bpvi_product_gap_pvs_distinct_zero_selected_power. bpvi_product_gap_pvs_distinct_zero_selected_power + S bpvi_j_pvs_distinct_zero_selected_power = 0) -> exists bpvi_factor_pvs_distinct_zero_selected_power bpvi_partial_pvs_distinct_zero_selected_power bpvi_successor_pvs_distinct_zero_selected_power. ((((exists bpvi_h_pvs_distinct_zero_selected_power_factor. bpvi_h_pvs_distinct_zero_selected_power_factor + S (bpvi_factor_pvs_distinct_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_factor. bpvi_b_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_factor * S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power) + (bpvi_factor_pvs_distinct_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_partial. bpvi_h_pvs_distinct_zero_selected_power_partial + S (bpvi_partial_pvs_distinct_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_partial. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_partial * S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_partial_pvs_distinct_zero_selected_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_successor. bpvi_h_pvs_distinct_zero_selected_power_successor + S (bpvi_successor_pvs_distinct_zero_selected_power) = S ((S (S bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power)) /\\ exists bpvi_q_pvs_distinct_zero_selected_power_successor. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_successor * S ((S (S bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_successor_pvs_distinct_zero_selected_power))) /\\ bpvi_successor_pvs_distinct_zero_selected_power = bpvi_partial_pvs_distinct_zero_selected_power * bpvi_factor_pvs_distinct_zero_selected_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_zero_selected. z = bpvi_result_pvs_distinct_zero_selected * bpvi_divisor_factor_pvs_distinct_zero_selected))) /\\ forall bpd_candidate_pvs_distinct_zero. (exists bpd_gap_pvs_distinct_zero_candidate_bound. bpd_gap_pvs_distinct_zero_candidate_bound + (bpd_candidate_pvs_distinct_zero) = (z)) -> (exists bpvi_result_pvs_distinct_zero_candidate. ((exists bpvi_b_pvs_distinct_zero_candidate_power bpvi_c_pvs_distinct_zero_candidate_power. ((forall bpvi_i_pvs_distinct_zero_candidate_power. (exists bpvi_repeat_gap_pvs_distinct_zero_candidate_power. bpvi_repeat_gap_pvs_distinct_zero_candidate_power + S bpvi_i_pvs_distinct_zero_candidate_power = bpd_candidate_pvs_distinct_zero) -> (((exists bpvi_h_pvs_distinct_zero_candidate_power_repeat. bpvi_h_pvs_distinct_zero_candidate_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_repeat. bpvi_b_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_repeat * S ((S (bpvi_i_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power) + (q)))) /\\ (exists bpvi_u_pvs_distinct_zero_candidate_power bpvi_v_pvs_distinct_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_zero_candidate_power_start. bpvi_h_pvs_distinct_zero_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_start. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_start * S ((S (0)) * bpvi_v_pvs_distinct_zero_candidate_power) + (1))) /\\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_terminal. bpvi_h_pvs_distinct_zero_candidate_power_terminal + S (bpvi_result_pvs_distinct_zero_candidate) = S ((S (bpd_candidate_pvs_distinct_zero)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_terminal. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_terminal * S ((S (bpd_candidate_pvs_distinct_zero)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_result_pvs_distinct_zero_candidate))) /\\ forall bpvi_j_pvs_distinct_zero_candidate_power. (exists bpvi_product_gap_pvs_distinct_zero_candidate_power. bpvi_product_gap_pvs_distinct_zero_candidate_power + S bpvi_j_pvs_distinct_zero_candidate_power = bpd_candidate_pvs_distinct_zero) -> exists bpvi_factor_pvs_distinct_zero_candidate_power bpvi_partial_pvs_distinct_zero_candidate_power bpvi_successor_pvs_distinct_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_zero_candidate_power_factor. bpvi_h_pvs_distinct_zero_candidate_power_factor + S (bpvi_factor_pvs_distinct_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_factor. bpvi_b_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_factor * S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power) + (bpvi_factor_pvs_distinct_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_partial. bpvi_h_pvs_distinct_zero_candidate_power_partial + S (bpvi_partial_pvs_distinct_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_partial. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_partial * S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_partial_pvs_distinct_zero_candidate_power))) /\\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_successor. bpvi_h_pvs_distinct_zero_candidate_power_successor + S (bpvi_successor_pvs_distinct_zero_candidate_power) = S ((S (S bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\\ exists bpvi_q_pvs_distinct_zero_candidate_power_successor. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_successor * S ((S (S bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_successor_pvs_distinct_zero_candidate_power))) /\\ bpvi_successor_pvs_distinct_zero_candidate_power = bpvi_partial_pvs_distinct_zero_candidate_power * bpvi_factor_pvs_distinct_zero_candidate_power)))))))) /\\ exists bpvi_divisor_factor_pvs_distinct_zero_candidate. z = bpvi_result_pvs_distinct_zero_candidate * bpvi_divisor_factor_pvs_distinct_zero_candidate)) -> (exists bpd_gap_pvs_distinct_zero_maximal. bpd_gap_pvs_distinct_zero_maximal + (bpd_candidate_pvs_distinct_zero) = (0)))", "statement_sha256": "af998c853267c251d0d8d205ce8149f49cac250218a5c7fef550f772332d53af" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "prime_divisor_of_prime_power", "canonical_catalog_record": { "alpha_v29_frontier_enrollment": { "body_receipt_sha256": "17cd049a36d87ef9a6d1662946fbef2a59d1a48c1baa0ced8b5d9b28df583009", "bundle_campaign": "priority_layer", "bundle_node_id": 294, "bundle_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "campaign": "prime_valuation_support", "parent_catalog_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "rfc_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "source_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "test_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c" }, "body_checked": true, "body_receipt": { "command_count": 41, "dependency_count": 6, "dne_command_count": 0, "name": "prime_divisor_of_prime_power", "proof_depth": 46, "proof_edges": 100, "proof_nodes": 101, "proof_objects": 101, "reused_objects": 0, "status": "kernel_checked_dependency_curried_body" }, "checked_use": true, "dependencies": [ "eq_decidable", "prime_nonzero", "one_le_of_ne_zero", "pow_nonzero_of_one_le", "prime_valuation_distinct_prime_power_zero", "prime_valuation_nondivisor_of_zero" ], "dependencies_sha256": "bbb18ebb623e665eb9eb880a4e6d9370b641aa66441a1731cb6e82b82092b85e", "empty_context_closure": { "body_proof_depth": 46, "body_proof_nodes": 101, "bundle_campaign": "priority_layer", "bundle_dependency_edge_count": 1690, "bundle_node_count": 566, "bundle_node_id": 294, "bundle_path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "bundle_root_id": 565, "certificate_representation": "peano-lab-bundle-v1", "certificate_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "closure_kind": "dependency_closed_bundle_node", "digest_kind": "self-contained-proof-bundle-sha256", "kernel_mode": "intuitionistic", "node_statement_sha256": "26dd0b5cb8863a7b7c8dff981648b47ee3d24e5bca35b8b61e591bedfd3844a6", "status": "checked" }, "enrollment_index": 2771, "enrollment_origin": "ha", "evidence_links": [ { "document_sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2", "kind": "alpha_v29_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "0bfd659bdb4d48177c87b34154afce11ebbe1319e59a30ecd5bea8e61425390c", "kind": "alpha_v29_frontier_executable_audit", "path": "peano-lab/py/tests/test_prime_valuation_support_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "12d6501d0fe9a24c821cf2a20e0ecf232f431f6093dacad70f9423b5fd729522", "kind": "alpha_v29_frontier_campaign_rfc", "path": "research/arithmetic-library/prime-valuation-support-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "4fcb3cd45e83448776abb9e33692496a7acfa98a051cae15761826a0b15fda44", "kind": "alpha_v29_priority_layer_self_contained_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/alpha-v29-priority-layer-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=294]" }, { "document_sha256": "b75fbe789b873e421bfd4a939f67ba477dc8ddb94730bebf6999748d0c480b92", "kind": "alpha_v29_priority_layer_original_kernel_receipt", "path": "research/arithmetic-library/alpha-v29-priority-layer-receipt.md", "role": "original_kernel_independent_dependency_closure_verification", "selector": "document" }, { "document_sha256": "897410581b66552c7f01f4b1266de887e52b3198b1ff2d2ac5135ab694d467e9", "kind": "sealed_alpha_v28_parent", "path": "artifacts/peano-library/alpha/catalog-v28.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "prime_valuation_support", "logical_spec_sha256": "667e9816f4e698f08f36713dbb8903a7e1b46d278c5cb5a369a0008963eb5bbb", "membership": "alpha_only", "name": "prime_divisor_of_prime_power", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro p", "intro q", "intro k", "intro z", "intro hp", "intro hq", "intro hpow", "intro hdiv", "specialize eq_decidable q", "specialize eq_decidable p", "cases eq_decidable", "exact eq_decidable_left", "exfalso", "specialize prime_valuation_nondivisor_of_zero (q)", "specialize prime_valuation_nondivisor_of_zero (z)", "apply prime_valuation_nondivisor_of_zero", "exact hq", "intro hz", "specialize pow_nonzero_of_one_le (p)", "specialize pow_nonzero_of_one_le (k)", "specialize pow_nonzero_of_one_le (z)", "apply pow_nonzero_of_one_le", "specialize one_le_of_ne_zero (p)", "apply one_le_of_ne_zero", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "exact hpow", "exact hz", "specialize prime_valuation_distinct_prime_power_zero (p)", "specialize prime_valuation_distinct_prime_power_zero (q)", "specialize prime_valuation_distinct_prime_power_zero (k)", "specialize prime_valuation_distinct_prime_power_zero (z)", "apply prime_valuation_distinct_prime_power_zero", "exact hp", "exact hq", "exact eq_decidable_right", "exact hpow", "exact hdiv" ], "script_sha256": "2d635d25e6c75a567c472f34a64b1aef5ed6369702ec52f72f030b8eca39f1a4", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/prime_valuation_support_candidate.py", "sha256": "bbd6e661a575f6a39f7a71424611da36a16d34cb6704cbae2b918387cc0f66d2" }, "statement": "forall p q k z. (~((p) = 1) /\\ forall pvs_left_prime_power_base pvs_right_prime_power_base. (p) = pvs_left_prime_power_base * pvs_right_prime_power_base -> pvs_left_prime_power_base = 1 \\/ pvs_right_prime_power_base = 1) -> (~((q) = 1) /\\ forall pvs_left_prime_power_divisor pvs_right_prime_power_divisor. (q) = pvs_left_prime_power_divisor * pvs_right_prime_power_divisor -> pvs_left_prime_power_divisor = 1 \\/ pvs_right_prime_power_divisor = 1) -> (exists pa_b_pvs_prime_power_value pa_c_pvs_prime_power_value. ((forall pa_i_pvs_prime_power_value_repeat. (exists pa_lt_pvs_prime_power_value_repeat_bound. pa_lt_pvs_prime_power_value_repeat_bound + S pa_i_pvs_prime_power_value_repeat = k) -> (((exists pa_h_pvs_prime_power_value_repeat_decoded. pa_h_pvs_prime_power_value_repeat_decoded + S (p) = S ((S (pa_i_pvs_prime_power_value_repeat)) * pa_c_pvs_prime_power_value)) /\\ exists pa_q_pvs_prime_power_value_repeat_decoded. pa_b_pvs_prime_power_value = pa_q_pvs_prime_power_value_repeat_decoded * S ((S (pa_i_pvs_prime_power_value_repeat)) * pa_c_pvs_prime_power_value) + (p)))) /\\ (exists pa_u_pvs_prime_power_value_product pa_v_pvs_prime_power_value_product. ((((exists pa_h_pvs_prime_power_value_product_start. pa_h_pvs_prime_power_value_product_start + S (1) = S ((S (0)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_start. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_start * S ((S (0)) * pa_v_pvs_prime_power_value_product) + (1))) /\\ ((((exists pa_h_pvs_prime_power_value_product_terminal. pa_h_pvs_prime_power_value_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_terminal. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_terminal * S ((S (k)) * pa_v_pvs_prime_power_value_product) + (z))) /\\ forall pa_i_pvs_prime_power_value_product. (exists pa_lt_pvs_prime_power_value_product_bound. pa_lt_pvs_prime_power_value_product_bound + S pa_i_pvs_prime_power_value_product = k) -> exists pa_p_pvs_prime_power_value_product pa_r_pvs_prime_power_value_product pa_s_pvs_prime_power_value_product. ((((exists pa_h_pvs_prime_power_value_product_factor. pa_h_pvs_prime_power_value_product_factor + S (pa_p_pvs_prime_power_value_product) = S ((S (pa_i_pvs_prime_power_value_product)) * pa_c_pvs_prime_power_value)) /\\ exists pa_q_pvs_prime_power_value_product_factor. pa_b_pvs_prime_power_value = pa_q_pvs_prime_power_value_product_factor * S ((S (pa_i_pvs_prime_power_value_product)) * pa_c_pvs_prime_power_value) + (pa_p_pvs_prime_power_value_product))) /\\ ((((exists pa_h_pvs_prime_power_value_product_partial. pa_h_pvs_prime_power_value_product_partial + S (pa_r_pvs_prime_power_value_product) = S ((S (pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_partial. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_partial * S ((S (pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product) + (pa_r_pvs_prime_power_value_product))) /\\ ((((exists pa_h_pvs_prime_power_value_product_successor. pa_h_pvs_prime_power_value_product_successor + S (pa_s_pvs_prime_power_value_product) = S ((S (S pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_successor. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_successor * S ((S (S pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product) + (pa_s_pvs_prime_power_value_product))) /\\ pa_s_pvs_prime_power_value_product = pa_r_pvs_prime_power_value_product * pa_p_pvs_prime_power_value_product)))))))) -> (exists pvs_factor_prime_power_divides. 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(~((p) = 1) /\\ forall pvs_left_prime_power_base pvs_right_prime_power_base. (p) = pvs_left_prime_power_base * pvs_right_prime_power_base -> pvs_left_prime_power_base = 1 \\/ pvs_right_prime_power_base = 1) -> (~((q) = 1) /\\ forall pvs_left_prime_power_divisor pvs_right_prime_power_divisor. (q) = pvs_left_prime_power_divisor * pvs_right_prime_power_divisor -> pvs_left_prime_power_divisor = 1 \\/ pvs_right_prime_power_divisor = 1) -> (exists pa_b_pvs_prime_power_value pa_c_pvs_prime_power_value. ((forall pa_i_pvs_prime_power_value_repeat. (exists pa_lt_pvs_prime_power_value_repeat_bound. pa_lt_pvs_prime_power_value_repeat_bound + S pa_i_pvs_prime_power_value_repeat = k) -> (((exists pa_h_pvs_prime_power_value_repeat_decoded. pa_h_pvs_prime_power_value_repeat_decoded + S (p) = S ((S (pa_i_pvs_prime_power_value_repeat)) * pa_c_pvs_prime_power_value)) /\\ exists pa_q_pvs_prime_power_value_repeat_decoded. pa_b_pvs_prime_power_value = pa_q_pvs_prime_power_value_repeat_decoded * S ((S (pa_i_pvs_prime_power_value_repeat)) * pa_c_pvs_prime_power_value) + (p)))) /\\ (exists pa_u_pvs_prime_power_value_product pa_v_pvs_prime_power_value_product. ((((exists pa_h_pvs_prime_power_value_product_start. pa_h_pvs_prime_power_value_product_start + S (1) = S ((S (0)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_start. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_start * S ((S (0)) * pa_v_pvs_prime_power_value_product) + (1))) /\\ ((((exists pa_h_pvs_prime_power_value_product_terminal. pa_h_pvs_prime_power_value_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_terminal. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_terminal * S ((S (k)) * pa_v_pvs_prime_power_value_product) + (z))) /\\ forall pa_i_pvs_prime_power_value_product. (exists pa_lt_pvs_prime_power_value_product_bound. pa_lt_pvs_prime_power_value_product_bound + S pa_i_pvs_prime_power_value_product = k) -> exists pa_p_pvs_prime_power_value_product pa_r_pvs_prime_power_value_product pa_s_pvs_prime_power_value_product. ((((exists pa_h_pvs_prime_power_value_product_factor. pa_h_pvs_prime_power_value_product_factor + S (pa_p_pvs_prime_power_value_product) = S ((S (pa_i_pvs_prime_power_value_product)) * pa_c_pvs_prime_power_value)) /\\ exists pa_q_pvs_prime_power_value_product_factor. pa_b_pvs_prime_power_value = pa_q_pvs_prime_power_value_product_factor * S ((S (pa_i_pvs_prime_power_value_product)) * pa_c_pvs_prime_power_value) + (pa_p_pvs_prime_power_value_product))) /\\ ((((exists pa_h_pvs_prime_power_value_product_partial. pa_h_pvs_prime_power_value_product_partial + S (pa_r_pvs_prime_power_value_product) = S ((S (pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_partial. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_partial * S ((S (pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product) + (pa_r_pvs_prime_power_value_product))) /\\ ((((exists pa_h_pvs_prime_power_value_product_successor. pa_h_pvs_prime_power_value_product_successor + S (pa_s_pvs_prime_power_value_product) = S ((S (S pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product)) /\\ exists pa_q_pvs_prime_power_value_product_successor. pa_u_pvs_prime_power_value_product = pa_q_pvs_prime_power_value_product_successor * S ((S (S pa_i_pvs_prime_power_value_product)) * pa_v_pvs_prime_power_value_product) + (pa_s_pvs_prime_power_value_product))) /\\ pa_s_pvs_prime_power_value_product = pa_r_pvs_prime_power_value_product * pa_p_pvs_prime_power_value_product)))))))) -> (exists pvs_factor_prime_power_divides. 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"coprime_divisor_gcd_product", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro m", "intro n", "intro d", "intro a", "intro b", "intro hd", "intro hc", "intro hdiv", "intro ha", "intro hb", "have hprod : (((exists ec_gcd_left_cor_cdp_product. (m * n) = (a * b) * ec_gcd_left_cor_cdp_product) /\\ (exists ec_gcd_right_cor_cdp_product. d = (a * b) * ec_gcd_right_cor_cdp_product)) /\\ forall ec_gcd_common_cor_cdp_product. (exists ec_gcd_common_left_cor_cdp_product. (m * n) = ec_gcd_common_cor_cdp_product * ec_gcd_common_left_cor_cdp_product) -> (exists ec_gcd_common_right_cor_cdp_product. d = ec_gcd_common_cor_cdp_product * ec_gcd_common_right_cor_cdp_product) -> exists ec_gcd_greatest_cor_cdp_product. (a * b) = ec_gcd_common_cor_cdp_product * ec_gcd_greatest_cor_cdp_product)", "specialize crt_is_gcd_coprime_product (m)", "specialize crt_is_gcd_coprime_product (n)", "specialize crt_is_gcd_coprime_product (d)", "specialize crt_is_gcd_coprime_product (a)", "specialize crt_is_gcd_coprime_product (b)", "specialize crt_is_gcd_coprime_product (a*b)", "specialize crt_is_gcd_coprime_product (m*n)", "apply crt_is_gcd_coprime_product", "exact hd", "refl", "refl", "exact hc", "exact ha", "exact hb", "have hself : (((exists ec_gcd_left_cor_cdp_self. d = d * ec_gcd_left_cor_cdp_self) /\\ (exists ec_gcd_right_cor_cdp_self. (m * n) = d * ec_gcd_right_cor_cdp_self)) /\\ forall ec_gcd_common_cor_cdp_self. (exists ec_gcd_common_left_cor_cdp_self. d = ec_gcd_common_cor_cdp_self * ec_gcd_common_left_cor_cdp_self) -> (exists ec_gcd_common_right_cor_cdp_self. (m * n) = ec_gcd_common_cor_cdp_self * ec_gcd_common_right_cor_cdp_self) -> exists ec_gcd_greatest_cor_cdp_self. d = ec_gcd_common_cor_cdp_self * ec_gcd_greatest_cor_cdp_self)", "specialize is_gcd_of_dvd (d)", "specialize is_gcd_of_dvd (m*n)", "apply is_gcd_of_dvd", "exact hdiv", "have hswap : (((exists ec_gcd_left_cor_cdp_swap. (m * n) = d * ec_gcd_left_cor_cdp_swap) /\\ (exists ec_gcd_right_cor_cdp_swap. d = d * ec_gcd_right_cor_cdp_swap)) /\\ forall ec_gcd_common_cor_cdp_swap. (exists ec_gcd_common_left_cor_cdp_swap. (m * n) = ec_gcd_common_cor_cdp_swap * ec_gcd_common_left_cor_cdp_swap) -> (exists ec_gcd_common_right_cor_cdp_swap. d = ec_gcd_common_cor_cdp_swap * ec_gcd_common_right_cor_cdp_swap) -> exists ec_gcd_greatest_cor_cdp_swap. d = ec_gcd_common_cor_cdp_swap * ec_gcd_greatest_cor_cdp_swap)", "specialize is_gcd_symm (d)", "specialize is_gcd_symm (d)", "specialize is_gcd_symm (m*n)", "apply is_gcd_symm", "exact hself", "specialize is_gcd_unique (d)", "specialize is_gcd_unique (a*b)", "specialize is_gcd_unique (m*n)", "specialize is_gcd_unique (d)", "apply is_gcd_unique", "exact hswap", "exact hprod" ], "script_sha256": "e65bfd95eee16115d2432c451a0cfc53ac0183fa3b5c9dfb2a95b18cb4e8b5cc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/coprime_divisor_decomposition_candidate.py", "sha256": "de19bb61543f5d7ab3a1d1b675c96ae4b31c7c96b58d6107904e7188973a2e1c" }, "statement": "forall m n d a b. ~(d=0) -> (forall sfd_common_divisor_cdp_product_coprime. (exists pvs_factor_cdp_product_coprimeleft. (m) = (sfd_common_divisor_cdp_product_coprime) * pvs_factor_cdp_product_coprimeleft) -> (exists pvs_factor_cdp_product_coprimeright. (n) = (sfd_common_divisor_cdp_product_coprime) * pvs_factor_cdp_product_coprimeright) -> sfd_common_divisor_cdp_product_coprime = 1) -> (exists pvs_factor_cdp_product_divisor. (m*n) = (d) * pvs_factor_cdp_product_divisor) -> ((((exists ec_gcd_left_cor_cdp_product_left. m = a * ec_gcd_left_cor_cdp_product_left) /\\ (exists ec_gcd_right_cor_cdp_product_left. d = a * ec_gcd_right_cor_cdp_product_left)) /\\ forall ec_gcd_common_cor_cdp_product_left. (exists ec_gcd_common_left_cor_cdp_product_left. m = ec_gcd_common_cor_cdp_product_left * ec_gcd_common_left_cor_cdp_product_left) -> (exists ec_gcd_common_right_cor_cdp_product_left. d = ec_gcd_common_cor_cdp_product_left * ec_gcd_common_right_cor_cdp_product_left) -> exists ec_gcd_greatest_cor_cdp_product_left. a = ec_gcd_common_cor_cdp_product_left * ec_gcd_greatest_cor_cdp_product_left)) -> ((((exists ec_gcd_left_cor_cdp_product_right. n = b * ec_gcd_left_cor_cdp_product_right) /\\ (exists ec_gcd_right_cor_cdp_product_right. d = b * ec_gcd_right_cor_cdp_product_right)) /\\ forall ec_gcd_common_cor_cdp_product_right. (exists ec_gcd_common_left_cor_cdp_product_right. n = ec_gcd_common_cor_cdp_product_right * ec_gcd_common_left_cor_cdp_product_right) -> (exists ec_gcd_common_right_cor_cdp_product_right. d = ec_gcd_common_cor_cdp_product_right * ec_gcd_common_right_cor_cdp_product_right) -> exists ec_gcd_greatest_cor_cdp_product_right. b = ec_gcd_common_cor_cdp_product_right * ec_gcd_greatest_cor_cdp_product_right)) -> d=a*b", "statement_sha256": "17f2a4129e61dfe3e3f7124a35c99131bc69425fd2a82838f4cf51cf57295f37", "summary": "The two genuine gcds multiply to the given positive divisor of a coprime product.", "summary_sha256": "0dd5b8f6649667dca7e615bca2e2768be9dbc8e924e72648ce4294da9089db95" }, "canonical_theorem_route": null, "counted_as_new_owned_theorem": false, "dependencies": [ "crt_is_gcd_coprime_product", "is_gcd_of_dvd", "is_gcd_symm", "is_gcd_unique" ], "direct_prerequisite_of_owned_theorem": false, "enrolled_in_alpha": true, "evidence_links": [ { "document_sha256": "de19bb61543f5d7ab3a1d1b675c96ae4b31c7c96b58d6107904e7188973a2e1c", "kind": "alpha_v32_frontier_dependency_curried_body", "path": "peano-lab/py/peano_lab/library/coprime_divisor_decomposition_candidate.py", "role": "dependency_curried_body", "selector": "document" }, { "document_sha256": "4af2835b5e81d1b971eb7be7cab5e5acb2cc2fac820ac7a6a5c531a0826b7160", "kind": "alpha_v32_frontier_executable_audit", "path": "peano-lab/py/tests/test_coprime_divisor_decomposition_candidate.py", "role": "statement_dependency_replay_mutation_audit", "selector": "document" }, { "document_sha256": "ef5c63dbad6899e1c54c2044bb296393d293ee4aa4f6ea197ee417f167e81513", "kind": "alpha_v32_frontier_campaign_rfc", "path": "research/arithmetic-library/g009-multiplicative-convolution-rfc-v1.md", "role": "reviewed_constructive_campaign_contract", "selector": "document" }, { "document_sha256": "953dc5ef340379b1e34883c2f9ab2181e91c872f5bbb7943c52b2fb70ce76959", "kind": "alpha_v32_complete_constructive_proof_bundle", "path": "research/arithmetic-library/artifacts/g009-multiplicative-convolution-proof-bundle-v1.json", "role": "independently_kernel_checked_dependency_closed_proof", "selector": "nodes[id=382]" }, { "document_sha256": "6f0a09144ed53e95c5bee7ca5c033c684e5d6531b830a5b92c0c6042535fcce4", "kind": "alpha_v32_live_original_kernel_and_lean_receipt", "path": "research/arithmetic-library/artifacts/alpha-v32-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=multiplicative-convolution]" }, { "document_sha256": "6c9ebfb3c37e42aefab200b710f78e7693dc5826c80f053544deea41caf44aab", "kind": "sealed_alpha_v31_parent", "path": "artifacts/peano-library/alpha/catalog-v31.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "first_admission_reclassified": false, "inventory_role": "inherited_alpha_v34", "is_inherited_source_alias": false, "name": "coprime_divisor_gcd_product", "parent_alpha_version": "v34", "proof_bundle_node_id": 266, "reference_route": "jordan-totient/checkpoint.html#theorem-coprime_divisor_gcd_product", "script": [ "intro m", "intro n", "intro d", "intro a", "intro b", "intro hd", "intro hc", "intro hdiv", "intro ha", "intro hb", "have hprod : (((exists ec_gcd_left_cor_cdp_product. (m * n) = (a * b) * ec_gcd_left_cor_cdp_product) /\\ (exists ec_gcd_right_cor_cdp_product. d = (a * b) * ec_gcd_right_cor_cdp_product)) /\\ forall ec_gcd_common_cor_cdp_product. (exists ec_gcd_common_left_cor_cdp_product. (m * n) = ec_gcd_common_cor_cdp_product * ec_gcd_common_left_cor_cdp_product) -> (exists ec_gcd_common_right_cor_cdp_product. d = ec_gcd_common_cor_cdp_product * ec_gcd_common_right_cor_cdp_product) -> exists ec_gcd_greatest_cor_cdp_product. (a * b) = ec_gcd_common_cor_cdp_product * ec_gcd_greatest_cor_cdp_product)", "specialize crt_is_gcd_coprime_product (m)", "specialize crt_is_gcd_coprime_product (n)", "specialize crt_is_gcd_coprime_product (d)", "specialize crt_is_gcd_coprime_product (a)", "specialize crt_is_gcd_coprime_product (b)", "specialize crt_is_gcd_coprime_product (a*b)", "specialize crt_is_gcd_coprime_product (m*n)", "apply crt_is_gcd_coprime_product", "exact hd", "refl", "refl", "exact hc", "exact ha", "exact hb", "have hself : (((exists ec_gcd_left_cor_cdp_self. d = d * ec_gcd_left_cor_cdp_self) /\\ (exists ec_gcd_right_cor_cdp_self. (m * n) = d * ec_gcd_right_cor_cdp_self)) /\\ forall ec_gcd_common_cor_cdp_self. (exists ec_gcd_common_left_cor_cdp_self. d = ec_gcd_common_cor_cdp_self * ec_gcd_common_left_cor_cdp_self) -> (exists ec_gcd_common_right_cor_cdp_self. (m * n) = ec_gcd_common_cor_cdp_self * ec_gcd_common_right_cor_cdp_self) -> exists ec_gcd_greatest_cor_cdp_self. d = ec_gcd_common_cor_cdp_self * ec_gcd_greatest_cor_cdp_self)", "specialize is_gcd_of_dvd (d)", "specialize is_gcd_of_dvd (m*n)", "apply is_gcd_of_dvd", "exact hdiv", "have hswap : (((exists ec_gcd_left_cor_cdp_swap. (m * n) = d * ec_gcd_left_cor_cdp_swap) /\\ (exists ec_gcd_right_cor_cdp_swap. d = d * ec_gcd_right_cor_cdp_swap)) /\\ forall ec_gcd_common_cor_cdp_swap. (exists ec_gcd_common_left_cor_cdp_swap. (m * n) = ec_gcd_common_cor_cdp_swap * ec_gcd_common_left_cor_cdp_swap) -> (exists ec_gcd_common_right_cor_cdp_swap. d = ec_gcd_common_cor_cdp_swap * ec_gcd_common_right_cor_cdp_swap) -> exists ec_gcd_greatest_cor_cdp_swap. d = ec_gcd_common_cor_cdp_swap * ec_gcd_greatest_cor_cdp_swap)", "specialize is_gcd_symm (d)", "specialize is_gcd_symm (d)", "specialize is_gcd_symm (m*n)", "apply is_gcd_symm", "exact hself", "specialize is_gcd_unique (d)", "specialize is_gcd_unique (a*b)", "specialize is_gcd_unique (m*n)", "specialize is_gcd_unique (d)", "apply is_gcd_unique", "exact hswap", "exact hprod" ], "script_sha256": "e65bfd95eee16115d2432c451a0cfc53ac0183fa3b5c9dfb2a95b18cb4e8b5cc", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/coprime_divisor_decomposition_candidate.py", "sha256": "de19bb61543f5d7ab3a1d1b675c96ae4b31c7c96b58d6107904e7188973a2e1c" }, "stable_member": false, "statement": "forall m n d a b. ~(d=0) -> (forall sfd_common_divisor_cdp_product_coprime. 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(exists ec_gcd_common_left_cor_cdp_product_left. m = ec_gcd_common_cor_cdp_product_left * ec_gcd_common_left_cor_cdp_product_left) -> (exists ec_gcd_common_right_cor_cdp_product_left. d = ec_gcd_common_cor_cdp_product_left * ec_gcd_common_right_cor_cdp_product_left) -> exists ec_gcd_greatest_cor_cdp_product_left. a = ec_gcd_common_cor_cdp_product_left * ec_gcd_greatest_cor_cdp_product_left)) -> ((((exists ec_gcd_left_cor_cdp_product_right. n = b * ec_gcd_left_cor_cdp_product_right) /\\ (exists ec_gcd_right_cor_cdp_product_right. d = b * ec_gcd_right_cor_cdp_product_right)) /\\ forall ec_gcd_common_cor_cdp_product_right. (exists ec_gcd_common_left_cor_cdp_product_right. n = ec_gcd_common_cor_cdp_product_right * ec_gcd_common_left_cor_cdp_product_right) -> (exists ec_gcd_common_right_cor_cdp_product_right. d = ec_gcd_common_cor_cdp_product_right * ec_gcd_common_right_cor_cdp_product_right) -> exists ec_gcd_greatest_cor_cdp_product_right. b = ec_gcd_common_cor_cdp_product_right * ec_gcd_greatest_cor_cdp_product_right)) -> d=a*b", "statement_sha256": "17f2a4129e61dfe3e3f7124a35c99131bc69425fd2a82838f4cf51cf57295f37" }, { "admitted_to_alpha": true, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_first_enrolled_version": null, "canonical_admission_name": "coprime_divisor_factor_pair_exists", "canonical_catalog_record": { "alpha_v32_frontier_enrollment": { "body_receipt_sha256": "7560d951c3678a0bb7a86ff844a34f129201dd7bfa9972298ca0d4dd4d412c46", "bundle_campaign": "multiplicative-convolution", "bundle_node_id": 385, "bundle_sha256": 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IntegerVectorZero(b,c,d,e,k) → IntegerVectorZero(d,e,f,g,k) → IntegerVectorZero(b,c,f,g,k)", "defined_statement_sha256": "25e205d7f57e2d1d6031c640c75791a6199b15e590609880eab7f26732982427", "definition_uses": { "ND0121": 3, "PD0013": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "c6acf382bf78b8dadc9d000dbb44fa59be1692ff73216b22d1927f1470911e4e", "free_names": [], "script_definition_uses": { "PD0013": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro hnext" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "have hm : " }, { "kind": "text", "text": "∃ t. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,t)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (i)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hm" } ], [ { "kind": "text", "text": "have hleft : a=x" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "specialize h (a)" } ], [ { "kind": "text", "text": "specialize h (x)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hm_witness" } ], [ { "kind": "text", "text": "have hright : x=z" } ], [ { "kind": "text", "text": "specialize hnext (i)" } ], [ { "kind": "text", "text": "specialize hnext (x)" } ], [ { "kind": "text", "text": "specialize hnext (z)" } ], [ { "kind": "text", "text": "apply hnext" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hm_witness" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "trans x" } ], [ { "kind": "text", "text": "exact hleft" } ], [ { "kind": "text", "text": "exact hright" } ] ], "statement_definition_uses": { "ND0121": 3 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(d,e,f,g,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,f,g,k)" } ] }, "dependencies": [ "beta_at_exists" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0003", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_trans", "original_ha_bundle_verified": true, "proof_bundle_node_id": 258, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro h", "intro hnext", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "have hm : exists t. ((exists fs_h_jt_middle. fs_h_jt_middle + S (t) = S ((S (i)) * e)) /\\ exists fs_q_jt_middle. d = fs_q_jt_middle * S ((S (i)) * e) + (t))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hm", "have hleft : a=x", "specialize h (i)", "specialize h (a)", "specialize h (x)", "apply h", "exact hi", "exact ha", "exact hm_witness", "have hright : x=z", "specialize hnext (i)", "specialize hnext (x)", "specialize hnext (z)", "apply hnext", "exact hi", "exact hm_witness", "exact hz", "trans x", "exact hleft", "exact hright" ], "script_sha256": "15eef5e2caf4c2c40e1e7b56ee580e7289a5f0eceb02d4546b2a5871ffe53444", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "15eef5e2caf4c2c40e1e7b56ee580e7289a5f0eceb02d4546b2a5871ffe53444", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "c6acf382bf78b8dadc9d000dbb44fa59be1692ff73216b22d1927f1470911e4e" } ], "stable_member": false, "statement": "forall b c d e f g k. (forall jt_index_transfirst jt_left_transfirst jt_right_transfirst. (exists jt_gap_transfirstindex. jt_gap_transfirstindex+S (jt_index_transfirst)=(k)) -> (((exists fs_h_jt_transfirstleft. fs_h_jt_transfirstleft + S (jt_left_transfirst) = S ((S (jt_index_transfirst)) * c)) /\\ exists fs_q_jt_transfirstleft. b = fs_q_jt_transfirstleft * S ((S (jt_index_transfirst)) * c) + (jt_left_transfirst))) -> (((exists fs_h_jt_transfirstright. fs_h_jt_transfirstright + S (jt_right_transfirst) = S ((S (jt_index_transfirst)) * e)) /\\ exists fs_q_jt_transfirstright. d = fs_q_jt_transfirstright * S ((S (jt_index_transfirst)) * e) + (jt_right_transfirst))) -> jt_left_transfirst=jt_right_transfirst) -> (forall jt_index_transsecond jt_left_transsecond jt_right_transsecond. (exists jt_gap_transsecondindex. jt_gap_transsecondindex+S (jt_index_transsecond)=(k)) -> (((exists fs_h_jt_transsecondleft. fs_h_jt_transsecondleft + S (jt_left_transsecond) = S ((S (jt_index_transsecond)) * e)) /\\ exists fs_q_jt_transsecondleft. d = fs_q_jt_transsecondleft * S ((S (jt_index_transsecond)) * e) + (jt_left_transsecond))) -> (((exists fs_h_jt_transsecondright. fs_h_jt_transsecondright + S (jt_right_transsecond) = S ((S (jt_index_transsecond)) * g)) /\\ exists fs_q_jt_transsecondright. f = fs_q_jt_transsecondright * S ((S (jt_index_transsecond)) * g) + (jt_right_transsecond))) -> jt_left_transsecond=jt_right_transsecond) -> (forall jt_index_transresult jt_left_transresult jt_right_transresult. (exists jt_gap_transresultindex. jt_gap_transresultindex+S (jt_index_transresult)=(k)) -> (((exists fs_h_jt_transresultleft. fs_h_jt_transresultleft + S (jt_left_transresult) = S ((S (jt_index_transresult)) * c)) /\\ exists fs_q_jt_transresultleft. b = fs_q_jt_transresultleft * S ((S (jt_index_transresult)) * c) + (jt_left_transresult))) -> (((exists fs_h_jt_transresultright. fs_h_jt_transresultright + S (jt_right_transresult) = S ((S (jt_index_transresult)) * g)) /\\ exists fs_q_jt_transresultright. f = fs_q_jt_transresultright * S ((S (jt_index_transresult)) * g) + (jt_right_transresult))) -> jt_left_transresult=jt_right_transresult)", "statement_sha256": "c6acf382bf78b8dadc9d000dbb44fa59be1692ff73216b22d1927f1470911e4e", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A real decoded middle coordinate witnesses transitivity." }, { "admission_dependencies": [ "beta_at_exists" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 22, "body_proof_nodes": 36, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro q", "intro b", "intro c", "intro d", "intro e", "intro k", "intro heq", "intro hdiv", "intro i", "intro z", "intro hi", "intro hz", "have ha : ∃ a. BetaAt(b,c,i,a)", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases ha", "have hval : x=z", "specialize heq (i)", "specialize heq (x)", "specialize heq (z)", "apply heq", "exact hi", "exact ha_witness", "exact hz", "rewrite <- hval", "specialize hdiv (i)", "specialize hdiv (x)", "apply hdiv", "exact hi", "exact ha_witness" ], "defined_statement": "∀ q. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. IntegerVectorZero(b,c,d,e,k) → JordanTupleAllDivisible(q,b,c,k) → JordanTupleAllDivisible(q,d,e,k)", "defined_statement_sha256": "10e8dc370daee00964e8e68f0cc573261bb1e521cc496840dda2efed144d7d8d", "definition_uses": { "ND0121": 1, "ND0371": 2, "PD0013": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "2e5333c6985ec080533a0a81356b17278f32d951270ea00f8bbf0205b1812cdb", "free_names": [], "script_definition_uses": { "PD0013": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ a. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,a)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (b)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (c)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (i)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "have hval : x=z" } ], [ { "kind": "text", "text": "specialize heq (i)" } ], [ { "kind": "text", "text": "specialize heq (x)" } ], [ { "kind": "text", "text": "specialize heq (z)" } ], [ { "kind": "text", "text": "apply heq" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha_witness" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "rewrite <- hval" } ], [ { "kind": "text", "text": "specialize hdiv (i)" } ], [ { "kind": "text", "text": "specialize hdiv (x)" } ], [ { "kind": "text", "text": "apply hdiv" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha_witness" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0371": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ q. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(q,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(q,d,e,k)" } ] }, "dependencies": [ "beta_at_exists" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0004", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_common_divisor_transport", "original_ha_bundle_verified": true, "proof_bundle_node_id": 259, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro q", "intro b", "intro c", "intro d", "intro e", "intro k", "intro heq", "intro hdiv", "intro i", "intro z", "intro hi", "intro hz", "have ha : exists a. ((exists fs_h_jt_dvdactual. fs_h_jt_dvdactual + S (a) = S ((S (i)) * c)) /\\ exists fs_q_jt_dvdactual. b = fs_q_jt_dvdactual * S ((S (i)) * c) + (a))", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases ha", "have hval : x=z", "specialize heq (i)", "specialize heq (x)", "specialize heq (z)", "apply heq", "exact hi", "exact ha_witness", "exact hz", "rewrite <- hval", "specialize hdiv (i)", "specialize hdiv (x)", "apply hdiv", "exact hi", "exact ha_witness" ], "script_sha256": "f7bdc5ad538ddaebc55f4f6309c0977b7311fd9adfa037324ef07e985347e9f8", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "f7bdc5ad538ddaebc55f4f6309c0977b7311fd9adfa037324ef07e985347e9f8", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "2e5333c6985ec080533a0a81356b17278f32d951270ea00f8bbf0205b1812cdb" } ], "stable_member": false, "statement": "forall q b c d e k. (forall jt_index_dvd_equal jt_left_dvd_equal jt_right_dvd_equal. (exists jt_gap_dvd_equalindex. jt_gap_dvd_equalindex+S (jt_index_dvd_equal)=(k)) -> (((exists fs_h_jt_dvd_equalleft. fs_h_jt_dvd_equalleft + S (jt_left_dvd_equal) = S ((S (jt_index_dvd_equal)) * c)) /\\ exists fs_q_jt_dvd_equalleft. b = fs_q_jt_dvd_equalleft * S ((S (jt_index_dvd_equal)) * c) + (jt_left_dvd_equal))) -> (((exists fs_h_jt_dvd_equalright. fs_h_jt_dvd_equalright + S (jt_right_dvd_equal) = S ((S (jt_index_dvd_equal)) * e)) /\\ exists fs_q_jt_dvd_equalright. d = fs_q_jt_dvd_equalright * S ((S (jt_index_dvd_equal)) * e) + (jt_right_dvd_equal))) -> jt_left_dvd_equal=jt_right_dvd_equal) -> (forall jt_index_dvdsource jt_value_dvdsource. (exists jt_gap_dvdsourceindex. jt_gap_dvdsourceindex+S (jt_index_dvdsource)=(k)) -> (((exists fs_h_jt_dvdsourceat. fs_h_jt_dvdsourceat + S (jt_value_dvdsource) = S ((S (jt_index_dvdsource)) * c)) /\\ exists fs_q_jt_dvdsourceat. b = fs_q_jt_dvdsourceat * S ((S (jt_index_dvdsource)) * c) + (jt_value_dvdsource))) -> (exists jt_factor_dvdsourcedivides. (jt_value_dvdsource)=(q)*jt_factor_dvdsourcedivides)) -> (forall jt_index_dvdtarget jt_value_dvdtarget. (exists jt_gap_dvdtargetindex. jt_gap_dvdtargetindex+S (jt_index_dvdtarget)=(k)) -> (((exists fs_h_jt_dvdtargetat. fs_h_jt_dvdtargetat + S (jt_value_dvdtarget) = S ((S (jt_index_dvdtarget)) * e)) /\\ exists fs_q_jt_dvdtargetat. d = fs_q_jt_dvdtargetat * S ((S (jt_index_dvdtarget)) * e) + (jt_value_dvdtarget))) -> (exists jt_factor_dvdtargetdivides. (jt_value_dvdtarget)=(q)*jt_factor_dvdtargetdivides))", "statement_sha256": "2e5333c6985ec080533a0a81356b17278f32d951270ea00f8bbf0205b1812cdb", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Divisibility of all coordinates is extensional across tuple encodings." }, { "admission_dependencies": [ "jordan_tuple_common_divisor_transport", "jordan_tuple_equal_symm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 42, "body_proof_nodes": 72, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro heq", "intro hp", "intro q", "intro hq", "intro hall", "specialize hp (q)", "apply hp", "exact hq", "specialize jordan_tuple_common_divisor_transport (q)", "specialize jordan_tuple_common_divisor_transport (d)", "specialize jordan_tuple_common_divisor_transport (e)", "specialize jordan_tuple_common_divisor_transport (b)", "specialize jordan_tuple_common_divisor_transport (c)", "specialize jordan_tuple_common_divisor_transport (k)", "apply jordan_tuple_common_divisor_transport", "specialize jordan_tuple_equal_symm (b)", "specialize jordan_tuple_equal_symm (c)", "specialize jordan_tuple_equal_symm (d)", "specialize jordan_tuple_equal_symm (e)", "specialize jordan_tuple_equal_symm (k)", "apply jordan_tuple_equal_symm", "exact heq", "exact hall" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. IntegerVectorZero(b,c,d,e,k) → JordanPrimitiveTuple(n,b,c,k) → JordanPrimitiveTuple(n,d,e,k)", "defined_statement_sha256": "e4f6d82c67bbdf7183f25f9b0396476ef0851bab4efdb555543e360c39c3567c", "definition_uses": { "ND0121": 1, "ND0372": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "9584f4d36d88327d95e013d376a3cd55e6cd1e26240f3ceb08ed97677ea03a0f", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize hp (q)" } ], [ { "kind": "text", "text": "apply hp" } ], [ { "kind": "text", "text": "exact hq" } ], [ { "kind": "text", "text": "specialize jordan_tuple_common_divisor_transport (q)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_common_divisor_transport (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_common_divisor_transport (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_common_divisor_transport (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_common_divisor_transport (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_common_divisor_transport (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_common_divisor_transport" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_symm" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0372": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,d,e,k)" } ] }, "dependencies": [ "jordan_tuple_common_divisor_transport", "jordan_tuple_equal_symm" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0005", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_transport", "original_ha_bundle_verified": true, "proof_bundle_node_id": 260, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro heq", "intro hp", "intro q", "intro hq", "intro hall", "specialize hp (q)", "apply hp", "exact hq", "specialize jordan_tuple_common_divisor_transport (q)", "specialize jordan_tuple_common_divisor_transport (d)", "specialize jordan_tuple_common_divisor_transport (e)", "specialize jordan_tuple_common_divisor_transport (b)", "specialize jordan_tuple_common_divisor_transport (c)", "specialize jordan_tuple_common_divisor_transport (k)", "apply jordan_tuple_common_divisor_transport", "specialize jordan_tuple_equal_symm (b)", "specialize jordan_tuple_equal_symm (c)", "specialize jordan_tuple_equal_symm (d)", "specialize jordan_tuple_equal_symm (e)", "specialize jordan_tuple_equal_symm (k)", "apply jordan_tuple_equal_symm", "exact heq", "exact hall" ], "script_sha256": "ab6fd3fb4a0a6ab352b8f441efea85fa4f45c0b194737e6ba60777c55a271f58", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "ab6fd3fb4a0a6ab352b8f441efea85fa4f45c0b194737e6ba60777c55a271f58", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "9584f4d36d88327d95e013d376a3cd55e6cd1e26240f3ceb08ed97677ea03a0f" } ], "stable_member": false, "statement": "forall n b c d e k. (forall jt_index_pr_equal jt_left_pr_equal jt_right_pr_equal. (exists jt_gap_pr_equalindex. jt_gap_pr_equalindex+S (jt_index_pr_equal)=(k)) -> (((exists fs_h_jt_pr_equalleft. fs_h_jt_pr_equalleft + S (jt_left_pr_equal) = S ((S (jt_index_pr_equal)) * c)) /\\ exists fs_q_jt_pr_equalleft. b = fs_q_jt_pr_equalleft * S ((S (jt_index_pr_equal)) * c) + (jt_left_pr_equal))) -> (((exists fs_h_jt_pr_equalright. fs_h_jt_pr_equalright + S (jt_right_pr_equal) = S ((S (jt_index_pr_equal)) * e)) /\\ exists fs_q_jt_pr_equalright. d = fs_q_jt_pr_equalright * S ((S (jt_index_pr_equal)) * e) + (jt_right_pr_equal))) -> jt_left_pr_equal=jt_right_pr_equal) -> (forall jt_divisor_prsource. (exists jt_factor_prsourcemodulus. (n)=(jt_divisor_prsource)*jt_factor_prsourcemodulus) -> (forall jt_index_prsourcecoordinates jt_value_prsourcecoordinates. (exists jt_gap_prsourcecoordinatesindex. jt_gap_prsourcecoordinatesindex+S (jt_index_prsourcecoordinates)=(k)) -> (((exists fs_h_jt_prsourcecoordinatesat. fs_h_jt_prsourcecoordinatesat + S (jt_value_prsourcecoordinates) = S ((S (jt_index_prsourcecoordinates)) * c)) /\\ exists fs_q_jt_prsourcecoordinatesat. b = fs_q_jt_prsourcecoordinatesat * S ((S (jt_index_prsourcecoordinates)) * c) + (jt_value_prsourcecoordinates))) -> (exists jt_factor_prsourcecoordinatesdivides. (jt_value_prsourcecoordinates)=(jt_divisor_prsource)*jt_factor_prsourcecoordinatesdivides)) -> jt_divisor_prsource=1) -> (forall jt_divisor_prtarget. (exists jt_factor_prtargetmodulus. (n)=(jt_divisor_prtarget)*jt_factor_prtargetmodulus) -> (forall jt_index_prtargetcoordinates jt_value_prtargetcoordinates. (exists jt_gap_prtargetcoordinatesindex. jt_gap_prtargetcoordinatesindex+S (jt_index_prtargetcoordinates)=(k)) -> (((exists fs_h_jt_prtargetcoordinatesat. fs_h_jt_prtargetcoordinatesat + S (jt_value_prtargetcoordinates) = S ((S (jt_index_prtargetcoordinates)) * e)) /\\ exists fs_q_jt_prtargetcoordinatesat. d = fs_q_jt_prtargetcoordinatesat * S ((S (jt_index_prtargetcoordinates)) * e) + (jt_value_prtargetcoordinates))) -> (exists jt_factor_prtargetcoordinatesdivides. (jt_value_prtargetcoordinates)=(jt_divisor_prtarget)*jt_factor_prtargetcoordinatesdivides)) -> jt_divisor_prtarget=1)", "statement_sha256": "9584f4d36d88327d95e013d376a3cd55e6cd1e26240f3ceb08ed97677ea03a0f", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Primitive means common-divisor one and is independent of beta presentation." }, { "admission_dependencies": [ "multiple_trans" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 24, "body_proof_nodes": 36, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro m", "intro b", "intro c", "intro k", "intro hnm", "intro hp", "intro q", "intro hqn", "intro hall", "specialize hp (q)", "apply hp", "specialize multiple_trans (n)", "specialize multiple_trans (q)", "specialize multiple_trans (m)", "apply multiple_trans", "exact hnm", "exact hqn", "exact hall" ], "defined_statement": "∀ n. ∀ m. ∀ b. ∀ c. ∀ k. Dvd(n,m) → JordanPrimitiveTuple(m,b,c,k) → JordanPrimitiveTuple(n,b,c,k)", "defined_statement_sha256": "f9679b74810c91578931b2894413557a2b2a56dfeb7f7cfc8b78b6b705b7f151", "definition_uses": { "ND0372": 2, "PD0003": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "74c075611a33db06f8907fc1da241e5bc81a7a124b89017e67f002c6d8730005", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hnm" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro hqn" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize hp (q)" } ], [ { "kind": "text", "text": "apply hp" } ], [ { "kind": "text", "text": "specialize multiple_trans (n)" } ], [ { "kind": "text", "text": "specialize multiple_trans (q)" } ], [ { "kind": "text", "text": "specialize multiple_trans (m)" } ], [ { "kind": "text", "text": "apply multiple_trans" } ], [ { "kind": "text", "text": "exact hnm" } ], [ { "kind": "text", "text": "exact hqn" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0372": 2, "PD0003": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ m. ∀ b. ∀ c. ∀ k. " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(n,m)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" } ] }, "dependencies": [ "multiple_trans" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0006", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_divisor_modulus", "original_ha_bundle_verified": true, "proof_bundle_node_id": 261, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro m", "intro b", "intro c", "intro k", "intro hnm", "intro hp", "intro q", "intro hqn", "intro hall", "specialize hp (q)", "apply hp", "specialize multiple_trans (n)", "specialize multiple_trans (q)", "specialize multiple_trans (m)", "apply multiple_trans", "exact hnm", "exact hqn", "exact hall" ], "script_sha256": "c3fe646b27e18e8d7796df1180668c6b97e2a7c222add944314816fb1be972ff", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "c3fe646b27e18e8d7796df1180668c6b97e2a7c222add944314816fb1be972ff", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "74c075611a33db06f8907fc1da241e5bc81a7a124b89017e67f002c6d8730005" } ], "stable_member": false, "statement": "forall n m b c k. (exists jt_factor_smalldivisor. (m)=(n)*jt_factor_smalldivisor) -> (forall jt_divisor_largemod. (exists jt_factor_largemodmodulus. (m)=(jt_divisor_largemod)*jt_factor_largemodmodulus) -> (forall jt_index_largemodcoordinates jt_value_largemodcoordinates. (exists jt_gap_largemodcoordinatesindex. jt_gap_largemodcoordinatesindex+S (jt_index_largemodcoordinates)=(k)) -> (((exists fs_h_jt_largemodcoordinatesat. fs_h_jt_largemodcoordinatesat + S (jt_value_largemodcoordinates) = S ((S (jt_index_largemodcoordinates)) * c)) /\\ exists fs_q_jt_largemodcoordinatesat. b = fs_q_jt_largemodcoordinatesat * S ((S (jt_index_largemodcoordinates)) * c) + (jt_value_largemodcoordinates))) -> (exists jt_factor_largemodcoordinatesdivides. (jt_value_largemodcoordinates)=(jt_divisor_largemod)*jt_factor_largemodcoordinatesdivides)) -> jt_divisor_largemod=1) -> (forall jt_divisor_smallmod. (exists jt_factor_smallmodmodulus. (n)=(jt_divisor_smallmod)*jt_factor_smallmodmodulus) -> (forall jt_index_smallmodcoordinates jt_value_smallmodcoordinates. (exists jt_gap_smallmodcoordinatesindex. jt_gap_smallmodcoordinatesindex+S (jt_index_smallmodcoordinates)=(k)) -> (((exists fs_h_jt_smallmodcoordinatesat. fs_h_jt_smallmodcoordinatesat + S (jt_value_smallmodcoordinates) = S ((S (jt_index_smallmodcoordinates)) * c)) /\\ exists fs_q_jt_smallmodcoordinatesat. b = fs_q_jt_smallmodcoordinatesat * S ((S (jt_index_smallmodcoordinates)) * c) + (jt_value_smallmodcoordinates))) -> (exists jt_factor_smallmodcoordinatesdivides. (jt_value_smallmodcoordinates)=(jt_divisor_smallmod)*jt_factor_smallmodcoordinatesdivides)) -> jt_divisor_smallmod=1)", "statement_sha256": "74c075611a33db06f8907fc1da241e5bc81a7a124b89017e67f002c6d8730005", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Primitivity descends from a modulus to a genuine divisor of it." }, { "admission_dependencies": [ "multiple_trans" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 27, "body_proof_nodes": 41, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro q", "intro r", "intro b", "intro c", "intro k", "intro hqr", "intro hall", "intro i", "intro a", "intro hi", "intro ha", "specialize multiple_trans (r)", "specialize multiple_trans (q)", "specialize multiple_trans (a)", "apply multiple_trans", "specialize hall (i)", "specialize hall (a)", "apply hall", "exact hi", "exact ha", "exact hqr" ], "defined_statement": "∀ q. ∀ r. ∀ b. ∀ c. ∀ k. Dvd(q,r) → JordanTupleAllDivisible(r,b,c,k) → JordanTupleAllDivisible(q,b,c,k)", "defined_statement_sha256": "5380462b86b9c4de4bd78352f26a88035e1d57c4ef2f049a433e9c80d282a8de", "definition_uses": { "ND0371": 2, "PD0003": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "8ece9b3ef504905df78fe53c71a9d4604ca720afbd49e5d502cce3520f623753", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro r" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hqr" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "specialize multiple_trans (r)" } ], [ { "kind": "text", "text": "specialize multiple_trans (q)" } ], [ { "kind": "text", "text": "specialize multiple_trans (a)" } ], [ { "kind": "text", "text": "apply multiple_trans" } ], [ { "kind": "text", "text": "specialize hall (i)" } ], [ { "kind": "text", "text": "specialize hall (a)" } ], [ { "kind": "text", "text": "apply hall" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hqr" } ] ], "statement_definition_uses": { "ND0371": 2, "PD0003": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ q. ∀ r. ∀ b. ∀ c. ∀ k. " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(q,r)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(r,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(q,b,c,k)" } ] }, "dependencies": [ "multiple_trans" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0007", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_divisor_downward", "original_ha_bundle_verified": true, "proof_bundle_node_id": 262, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro q", "intro r", "intro b", "intro c", "intro k", "intro hqr", "intro hall", "intro i", "intro a", "intro hi", "intro ha", "specialize multiple_trans (r)", "specialize multiple_trans (q)", "specialize multiple_trans (a)", "apply multiple_trans", "specialize hall (i)", "specialize hall (a)", "apply hall", "exact hi", "exact ha", "exact hqr" ], "script_sha256": "ced75b97dba6d47c9955a6af1c9edd65f9737fde7c1a7ebcf7c7539956a08abc", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "ced75b97dba6d47c9955a6af1c9edd65f9737fde7c1a7ebcf7c7539956a08abc", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "8ece9b3ef504905df78fe53c71a9d4604ca720afbd49e5d502cce3520f623753" } ], "stable_member": false, "statement": "forall q r b c k. (exists jt_factor_downfactor. (r)=(q)*jt_factor_downfactor) -> (forall jt_index_downsource jt_value_downsource. (exists jt_gap_downsourceindex. jt_gap_downsourceindex+S (jt_index_downsource)=(k)) -> (((exists fs_h_jt_downsourceat. fs_h_jt_downsourceat + S (jt_value_downsource) = S ((S (jt_index_downsource)) * c)) /\\ exists fs_q_jt_downsourceat. b = fs_q_jt_downsourceat * S ((S (jt_index_downsource)) * c) + (jt_value_downsource))) -> (exists jt_factor_downsourcedivides. (jt_value_downsource)=(r)*jt_factor_downsourcedivides)) -> (forall jt_index_downtarget jt_value_downtarget. (exists jt_gap_downtargetindex. jt_gap_downtargetindex+S (jt_index_downtarget)=(k)) -> (((exists fs_h_jt_downtargetat. fs_h_jt_downtargetat + S (jt_value_downtarget) = S ((S (jt_index_downtarget)) * c)) /\\ exists fs_q_jt_downtargetat. b = fs_q_jt_downtargetat * S ((S (jt_index_downtarget)) * c) + (jt_value_downtarget))) -> (exists jt_factor_downtargetdivides. (jt_value_downtarget)=(q)*jt_factor_downtargetdivides))", "statement_sha256": "8ece9b3ef504905df78fe53c71a9d4604ca720afbd49e5d502cce3520f623753", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A divisor of a common coordinate divisor is again a common divisor." }, { "admission_dependencies": [ "divisor_one" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 11, "body_proof_nodes": 14, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro k", "intro q", "intro hq", "intro hall", "specialize divisor_one (q)", "apply divisor_one", "exact hq" ], "defined_statement": "∀ b. ∀ c. ∀ k. JordanPrimitiveTuple(1,b,c,k)", "defined_statement_sha256": "638be357c58e8fbe3e1730bd651301f3b42d00175e545d992df16e36921257c4", "definition_uses": { "ND0372": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "10e32211389bdce65e761ae7f6cc5e76fd9bc205db29930912da9634dedcbdf1", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize divisor_one (q)" } ], [ { "kind": "text", "text": "apply divisor_one" } ], [ { "kind": "text", "text": "exact hq" } ] ], "statement_definition_uses": { "ND0372": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ k. " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(1,b,c,k)" } ] }, "dependencies": [ "divisor_one" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0008", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_modulus_one", "original_ha_bundle_verified": true, "proof_bundle_node_id": 263, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro k", "intro q", "intro hq", "intro hall", "specialize divisor_one (q)", "apply divisor_one", "exact hq" ], "script_sha256": "864876c4442142a8cb76922263470d5170ef33af8b902ef8523379c385f66b2c", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "864876c4442142a8cb76922263470d5170ef33af8b902ef8523379c385f66b2c", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "10e32211389bdce65e761ae7f6cc5e76fd9bc205db29930912da9634dedcbdf1" } ], "stable_member": false, "statement": "forall b c k. forall jt_divisor_one. (exists jt_factor_onemodulus. (1)=(jt_divisor_one)*jt_factor_onemodulus) -> (forall jt_index_onecoordinates jt_value_onecoordinates. (exists jt_gap_onecoordinatesindex. jt_gap_onecoordinatesindex+S (jt_index_onecoordinates)=(k)) -> (((exists fs_h_jt_onecoordinatesat. fs_h_jt_onecoordinatesat + S (jt_value_onecoordinates) = S ((S (jt_index_onecoordinates)) * c)) /\\ exists fs_q_jt_onecoordinatesat. b = fs_q_jt_onecoordinatesat * S ((S (jt_index_onecoordinates)) * c) + (jt_value_onecoordinates))) -> (exists jt_factor_onecoordinatesdivides. (jt_value_onecoordinates)=(jt_divisor_one)*jt_factor_onecoordinatesdivides)) -> jt_divisor_one=1", "statement_sha256": "10e32211389bdce65e761ae7f6cc5e76fd9bc205db29930912da9634dedcbdf1", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every finite tuple is primitive modulo one, including the zero tuple." }, { "admission_dependencies": [], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 9, "body_proof_nodes": 14, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro j", "intro h", "cases h", "apply h_left", "refl" ], "defined_statement": "∀ n. ∀ j. ¬JordanTotient(0,n,j)", "defined_statement_sha256": "4c57983785748b71f8706374ead27eada6f773dfcca9bb76b267008926ab1043", "definition_uses": { "ND0375": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "9c29eb8742288a4957bdafa4570f8c0abcf391f2a2ae34ac0823772b25bfeb8a", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "cases h" } ], [ { "kind": "text", "text": "apply h_left" } ], [ { "kind": "text", "text": "refl" } ] ], "statement_definition_uses": { "ND0375": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ j. ¬" }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(0,n,j)" } ] }, "dependencies": [], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0009", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_order_zero_excluded", "original_ha_bundle_verified": true, "proof_bundle_node_id": 264, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro j", "intro h", "cases h", "apply h_left", "refl" ], "script_sha256": "946e76013c812b43ed427d912a4fcee32ec5acc3f1e3689f135081feee4ddebd", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "946e76013c812b43ed427d912a4fcee32ec5acc3f1e3689f135081feee4ddebd", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "9c29eb8742288a4957bdafa4570f8c0abcf391f2a2ae34ac0823772b25bfeb8a" } ], "stable_member": false, "statement": "forall n j. ~(((~((0)=0)) /\\ (((~((n)=0)) /\\ (exists jt_codes_zeroorder jt_code_scale_zeroorder jt_scales_zeroorder jt_scale_scale_zeroorder. ((forall jt_i_zeroorderenum. (exists jt_gap_zeroorderenumsoundindex. jt_gap_zeroorderenumsoundindex+S (jt_i_zeroorderenum)=(j)) -> exists jt_b_zeroorderenum jt_c_zeroorderenum. ((((((exists fs_h_jt_zeroorderenumsoundcode. fs_h_jt_zeroorderenumsoundcode + S (jt_b_zeroorderenum) = S ((S (jt_i_zeroorderenum)) * jt_code_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumsoundcode. jt_codes_zeroorder = fs_q_jt_zeroorderenumsoundcode * S ((S (jt_i_zeroorderenum)) * jt_code_scale_zeroorder) + (jt_b_zeroorderenum))) /\\ (((exists fs_h_jt_zeroorderenumsoundscale. fs_h_jt_zeroorderenumsoundscale + S (jt_c_zeroorderenum) = S ((S (jt_i_zeroorderenum)) * jt_scale_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumsoundscale. jt_scales_zeroorder = fs_q_jt_zeroorderenumsoundscale * S ((S (jt_i_zeroorderenum)) * jt_scale_scale_zeroorder) + (jt_c_zeroorderenum))))) /\\ (((forall jt_index_zeroorderenumbound. (exists jt_gap_zeroorderenumboundindex. jt_gap_zeroorderenumboundindex+S (jt_index_zeroorderenumbound)=(0)) -> exists jt_value_zeroorderenumbound. ((((exists fs_h_jt_zeroorderenumboundat. fs_h_jt_zeroorderenumboundat + S (jt_value_zeroorderenumbound) = S ((S (jt_index_zeroorderenumbound)) * jt_c_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenumboundat. jt_b_zeroorderenum = fs_q_jt_zeroorderenumboundat * S ((S (jt_index_zeroorderenumbound)) * jt_c_zeroorderenum) + (jt_value_zeroorderenumbound))) /\\ (exists jt_gap_zeroorderenumboundvalue. jt_gap_zeroorderenumboundvalue+S (jt_value_zeroorderenumbound)=(n)))) /\\ (forall jt_divisor_zeroorderenumprimitive. (exists jt_factor_zeroorderenumprimitivemodulus. (n)=(jt_divisor_zeroorderenumprimitive)*jt_factor_zeroorderenumprimitivemodulus) -> (forall jt_index_zeroorderenumprimitivecoordinates jt_value_zeroorderenumprimitivecoordinates. (exists jt_gap_zeroorderenumprimitivecoordinatesindex. jt_gap_zeroorderenumprimitivecoordinatesindex+S (jt_index_zeroorderenumprimitivecoordinates)=(0)) -> (((exists fs_h_jt_zeroorderenumprimitivecoordinatesat. fs_h_jt_zeroorderenumprimitivecoordinatesat + S (jt_value_zeroorderenumprimitivecoordinates) = S ((S (jt_index_zeroorderenumprimitivecoordinates)) * jt_c_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenumprimitivecoordinatesat. jt_b_zeroorderenum = fs_q_jt_zeroorderenumprimitivecoordinatesat * S ((S (jt_index_zeroorderenumprimitivecoordinates)) * jt_c_zeroorderenum) + (jt_value_zeroorderenumprimitivecoordinates))) -> (exists jt_factor_zeroorderenumprimitivecoordinatesdivides. (jt_value_zeroorderenumprimitivecoordinates)=(jt_divisor_zeroorderenumprimitive)*jt_factor_zeroorderenumprimitivecoordinatesdivides)) -> jt_divisor_zeroorderenumprimitive=1))))) /\\ (((forall jt_b_zeroorderenum jt_c_zeroorderenum. (forall jt_index_zeroorderenuminputbound. (exists jt_gap_zeroorderenuminputboundindex. jt_gap_zeroorderenuminputboundindex+S (jt_index_zeroorderenuminputbound)=(0)) -> exists jt_value_zeroorderenuminputbound. ((((exists fs_h_jt_zeroorderenuminputboundat. fs_h_jt_zeroorderenuminputboundat + S (jt_value_zeroorderenuminputbound) = S ((S (jt_index_zeroorderenuminputbound)) * jt_c_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenuminputboundat. jt_b_zeroorderenum = fs_q_jt_zeroorderenuminputboundat * S ((S (jt_index_zeroorderenuminputbound)) * jt_c_zeroorderenum) + (jt_value_zeroorderenuminputbound))) /\\ (exists jt_gap_zeroorderenuminputboundvalue. jt_gap_zeroorderenuminputboundvalue+S (jt_value_zeroorderenuminputbound)=(n)))) -> (forall jt_divisor_zeroorderenuminputprimitive. (exists jt_factor_zeroorderenuminputprimitivemodulus. (n)=(jt_divisor_zeroorderenuminputprimitive)*jt_factor_zeroorderenuminputprimitivemodulus) -> (forall jt_index_zeroorderenuminputprimitivecoordinates jt_value_zeroorderenuminputprimitivecoordinates. (exists jt_gap_zeroorderenuminputprimitivecoordinatesindex. jt_gap_zeroorderenuminputprimitivecoordinatesindex+S (jt_index_zeroorderenuminputprimitivecoordinates)=(0)) -> (((exists fs_h_jt_zeroorderenuminputprimitivecoordinatesat. fs_h_jt_zeroorderenuminputprimitivecoordinatesat + S (jt_value_zeroorderenuminputprimitivecoordinates) = S ((S (jt_index_zeroorderenuminputprimitivecoordinates)) * jt_c_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenuminputprimitivecoordinatesat. jt_b_zeroorderenum = fs_q_jt_zeroorderenuminputprimitivecoordinatesat * S ((S (jt_index_zeroorderenuminputprimitivecoordinates)) * jt_c_zeroorderenum) + (jt_value_zeroorderenuminputprimitivecoordinates))) -> (exists jt_factor_zeroorderenuminputprimitivecoordinatesdivides. (jt_value_zeroorderenuminputprimitivecoordinates)=(jt_divisor_zeroorderenuminputprimitive)*jt_factor_zeroorderenuminputprimitivecoordinatesdivides)) -> jt_divisor_zeroorderenuminputprimitive=1) -> exists jt_i_zeroorderenum jt_d_zeroorderenum jt_e_zeroorderenum. ((exists jt_gap_zeroorderenumcompleteindex. jt_gap_zeroorderenumcompleteindex+S (jt_i_zeroorderenum)=(j)) /\\ (((((((exists fs_h_jt_zeroorderenumcompletecode. fs_h_jt_zeroorderenumcompletecode + S (jt_d_zeroorderenum) = S ((S (jt_i_zeroorderenum)) * jt_code_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumcompletecode. jt_codes_zeroorder = fs_q_jt_zeroorderenumcompletecode * S ((S (jt_i_zeroorderenum)) * jt_code_scale_zeroorder) + (jt_d_zeroorderenum))) /\\ (((exists fs_h_jt_zeroorderenumcompletescale. fs_h_jt_zeroorderenumcompletescale + S (jt_e_zeroorderenum) = S ((S (jt_i_zeroorderenum)) * jt_scale_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumcompletescale. jt_scales_zeroorder = fs_q_jt_zeroorderenumcompletescale * S ((S (jt_i_zeroorderenum)) * jt_scale_scale_zeroorder) + (jt_e_zeroorderenum))))) /\\ (forall jt_index_zeroorderenumrepresented jt_left_zeroorderenumrepresented jt_right_zeroorderenumrepresented. (exists jt_gap_zeroorderenumrepresentedindex. jt_gap_zeroorderenumrepresentedindex+S (jt_index_zeroorderenumrepresented)=(0)) -> (((exists fs_h_jt_zeroorderenumrepresentedleft. fs_h_jt_zeroorderenumrepresentedleft + S (jt_left_zeroorderenumrepresented) = S ((S (jt_index_zeroorderenumrepresented)) * jt_c_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenumrepresentedleft. jt_b_zeroorderenum = fs_q_jt_zeroorderenumrepresentedleft * S ((S (jt_index_zeroorderenumrepresented)) * jt_c_zeroorderenum) + (jt_left_zeroorderenumrepresented))) -> (((exists fs_h_jt_zeroorderenumrepresentedright. fs_h_jt_zeroorderenumrepresentedright + S (jt_right_zeroorderenumrepresented) = S ((S (jt_index_zeroorderenumrepresented)) * jt_e_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenumrepresentedright. jt_d_zeroorderenum = fs_q_jt_zeroorderenumrepresentedright * S ((S (jt_index_zeroorderenumrepresented)) * jt_e_zeroorderenum) + (jt_right_zeroorderenumrepresented))) -> jt_left_zeroorderenumrepresented=jt_right_zeroorderenumrepresented))))) /\\ (forall jt_i_zeroorderenum jt_h_zeroorderenum jt_b_zeroorderenum jt_c_zeroorderenum jt_d_zeroorderenum jt_e_zeroorderenum. (exists jt_gap_zeroorderenumfirstindex. jt_gap_zeroorderenumfirstindex+S (jt_i_zeroorderenum)=(j)) -> (exists jt_gap_zeroorderenumsecondindex. jt_gap_zeroorderenumsecondindex+S (jt_h_zeroorderenum)=(j)) -> (((((exists fs_h_jt_zeroorderenumfirstcode. fs_h_jt_zeroorderenumfirstcode + S (jt_b_zeroorderenum) = S ((S (jt_i_zeroorderenum)) * jt_code_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumfirstcode. jt_codes_zeroorder = fs_q_jt_zeroorderenumfirstcode * S ((S (jt_i_zeroorderenum)) * jt_code_scale_zeroorder) + (jt_b_zeroorderenum))) /\\ (((exists fs_h_jt_zeroorderenumfirstscale. fs_h_jt_zeroorderenumfirstscale + S (jt_c_zeroorderenum) = S ((S (jt_i_zeroorderenum)) * jt_scale_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumfirstscale. jt_scales_zeroorder = fs_q_jt_zeroorderenumfirstscale * S ((S (jt_i_zeroorderenum)) * jt_scale_scale_zeroorder) + (jt_c_zeroorderenum))))) -> (((((exists fs_h_jt_zeroorderenumsecondcode. fs_h_jt_zeroorderenumsecondcode + S (jt_d_zeroorderenum) = S ((S (jt_h_zeroorderenum)) * jt_code_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumsecondcode. jt_codes_zeroorder = fs_q_jt_zeroorderenumsecondcode * S ((S (jt_h_zeroorderenum)) * jt_code_scale_zeroorder) + (jt_d_zeroorderenum))) /\\ (((exists fs_h_jt_zeroorderenumsecondscale. fs_h_jt_zeroorderenumsecondscale + S (jt_e_zeroorderenum) = S ((S (jt_h_zeroorderenum)) * jt_scale_scale_zeroorder)) /\\ exists fs_q_jt_zeroorderenumsecondscale. jt_scales_zeroorder = fs_q_jt_zeroorderenumsecondscale * S ((S (jt_h_zeroorderenum)) * jt_scale_scale_zeroorder) + (jt_e_zeroorderenum))))) -> (forall jt_index_zeroorderenumsame jt_left_zeroorderenumsame jt_right_zeroorderenumsame. (exists jt_gap_zeroorderenumsameindex. jt_gap_zeroorderenumsameindex+S (jt_index_zeroorderenumsame)=(0)) -> (((exists fs_h_jt_zeroorderenumsameleft. fs_h_jt_zeroorderenumsameleft + S (jt_left_zeroorderenumsame) = S ((S (jt_index_zeroorderenumsame)) * jt_c_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenumsameleft. jt_b_zeroorderenum = fs_q_jt_zeroorderenumsameleft * S ((S (jt_index_zeroorderenumsame)) * jt_c_zeroorderenum) + (jt_left_zeroorderenumsame))) -> (((exists fs_h_jt_zeroorderenumsameright. fs_h_jt_zeroorderenumsameright + S (jt_right_zeroorderenumsame) = S ((S (jt_index_zeroorderenumsame)) * jt_e_zeroorderenum)) /\\ exists fs_q_jt_zeroorderenumsameright. jt_d_zeroorderenum = fs_q_jt_zeroorderenumsameright * S ((S (jt_index_zeroorderenumsame)) * jt_e_zeroorderenum) + (jt_right_zeroorderenumsame))) -> jt_left_zeroorderenumsame=jt_right_zeroorderenumsame) -> jt_i_zeroorderenum=jt_h_zeroorderenum)))))))))", "statement_sha256": "9c29eb8742288a4957bdafa4570f8c0abcf391f2a2ae34ac0823772b25bfeb8a", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Jordan is intentionally restricted to positive tuple order." }, { "admission_dependencies": [], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 13, "body_proof_nodes": 22, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro j", "intro h", "cases h", "cases h_right", "apply h_right_left", "refl" ], "defined_statement": "∀ k. ∀ j. ¬JordanTotient(k,0,j)", "defined_statement_sha256": "461d5877d5167d89c17614120a11d673c4bced10d5b232aefcd37d005fe52877", "definition_uses": { "ND0375": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "3cad9a4091db3795271f289596cd4daea0429e8e006b5e21503688bec46b29c9", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "cases h" } ], [ { "kind": "text", "text": "cases h_right" } ], [ { "kind": "text", "text": "apply h_right_left" } ], [ { "kind": "text", "text": "refl" } ] ], "statement_definition_uses": { "ND0375": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ j. ¬" }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,0,j)" } ] }, "dependencies": [], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT000A", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_modulus_zero_excluded", "original_ha_bundle_verified": true, "proof_bundle_node_id": 265, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro j", "intro h", "cases h", "cases h_right", "apply h_right_left", "refl" ], "script_sha256": "f1402151e3cf3bc68a37e903a4975d1520604373d3c5512c10f40155433dc2f1", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "f1402151e3cf3bc68a37e903a4975d1520604373d3c5512c10f40155433dc2f1", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "3cad9a4091db3795271f289596cd4daea0429e8e006b5e21503688bec46b29c9" } ], "stable_member": false, "statement": "forall k j. ~(((~((k)=0)) /\\ (((~((0)=0)) /\\ (exists jt_codes_zeromod jt_code_scale_zeromod jt_scales_zeromod jt_scale_scale_zeromod. ((forall jt_i_zeromodenum. (exists jt_gap_zeromodenumsoundindex. jt_gap_zeromodenumsoundindex+S (jt_i_zeromodenum)=(j)) -> exists jt_b_zeromodenum jt_c_zeromodenum. ((((((exists fs_h_jt_zeromodenumsoundcode. fs_h_jt_zeromodenumsoundcode + S (jt_b_zeromodenum) = S ((S (jt_i_zeromodenum)) * jt_code_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumsoundcode. jt_codes_zeromod = fs_q_jt_zeromodenumsoundcode * S ((S (jt_i_zeromodenum)) * jt_code_scale_zeromod) + (jt_b_zeromodenum))) /\\ (((exists fs_h_jt_zeromodenumsoundscale. fs_h_jt_zeromodenumsoundscale + S (jt_c_zeromodenum) = S ((S (jt_i_zeromodenum)) * jt_scale_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumsoundscale. jt_scales_zeromod = fs_q_jt_zeromodenumsoundscale * S ((S (jt_i_zeromodenum)) * jt_scale_scale_zeromod) + (jt_c_zeromodenum))))) /\\ (((forall jt_index_zeromodenumbound. (exists jt_gap_zeromodenumboundindex. jt_gap_zeromodenumboundindex+S (jt_index_zeromodenumbound)=(k)) -> exists jt_value_zeromodenumbound. ((((exists fs_h_jt_zeromodenumboundat. fs_h_jt_zeromodenumboundat + S (jt_value_zeromodenumbound) = S ((S (jt_index_zeromodenumbound)) * jt_c_zeromodenum)) /\\ exists fs_q_jt_zeromodenumboundat. jt_b_zeromodenum = fs_q_jt_zeromodenumboundat * S ((S (jt_index_zeromodenumbound)) * jt_c_zeromodenum) + (jt_value_zeromodenumbound))) /\\ (exists jt_gap_zeromodenumboundvalue. jt_gap_zeromodenumboundvalue+S (jt_value_zeromodenumbound)=(0)))) /\\ (forall jt_divisor_zeromodenumprimitive. (exists jt_factor_zeromodenumprimitivemodulus. (0)=(jt_divisor_zeromodenumprimitive)*jt_factor_zeromodenumprimitivemodulus) -> (forall jt_index_zeromodenumprimitivecoordinates jt_value_zeromodenumprimitivecoordinates. (exists jt_gap_zeromodenumprimitivecoordinatesindex. jt_gap_zeromodenumprimitivecoordinatesindex+S (jt_index_zeromodenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_zeromodenumprimitivecoordinatesat. fs_h_jt_zeromodenumprimitivecoordinatesat + S (jt_value_zeromodenumprimitivecoordinates) = S ((S (jt_index_zeromodenumprimitivecoordinates)) * jt_c_zeromodenum)) /\\ exists fs_q_jt_zeromodenumprimitivecoordinatesat. jt_b_zeromodenum = fs_q_jt_zeromodenumprimitivecoordinatesat * S ((S (jt_index_zeromodenumprimitivecoordinates)) * jt_c_zeromodenum) + (jt_value_zeromodenumprimitivecoordinates))) -> (exists jt_factor_zeromodenumprimitivecoordinatesdivides. (jt_value_zeromodenumprimitivecoordinates)=(jt_divisor_zeromodenumprimitive)*jt_factor_zeromodenumprimitivecoordinatesdivides)) -> jt_divisor_zeromodenumprimitive=1))))) /\\ (((forall jt_b_zeromodenum jt_c_zeromodenum. (forall jt_index_zeromodenuminputbound. (exists jt_gap_zeromodenuminputboundindex. jt_gap_zeromodenuminputboundindex+S (jt_index_zeromodenuminputbound)=(k)) -> exists jt_value_zeromodenuminputbound. ((((exists fs_h_jt_zeromodenuminputboundat. fs_h_jt_zeromodenuminputboundat + S (jt_value_zeromodenuminputbound) = S ((S (jt_index_zeromodenuminputbound)) * jt_c_zeromodenum)) /\\ exists fs_q_jt_zeromodenuminputboundat. jt_b_zeromodenum = fs_q_jt_zeromodenuminputboundat * S ((S (jt_index_zeromodenuminputbound)) * jt_c_zeromodenum) + (jt_value_zeromodenuminputbound))) /\\ (exists jt_gap_zeromodenuminputboundvalue. jt_gap_zeromodenuminputboundvalue+S (jt_value_zeromodenuminputbound)=(0)))) -> (forall jt_divisor_zeromodenuminputprimitive. (exists jt_factor_zeromodenuminputprimitivemodulus. (0)=(jt_divisor_zeromodenuminputprimitive)*jt_factor_zeromodenuminputprimitivemodulus) -> (forall jt_index_zeromodenuminputprimitivecoordinates jt_value_zeromodenuminputprimitivecoordinates. (exists jt_gap_zeromodenuminputprimitivecoordinatesindex. jt_gap_zeromodenuminputprimitivecoordinatesindex+S (jt_index_zeromodenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_zeromodenuminputprimitivecoordinatesat. fs_h_jt_zeromodenuminputprimitivecoordinatesat + S (jt_value_zeromodenuminputprimitivecoordinates) = S ((S (jt_index_zeromodenuminputprimitivecoordinates)) * jt_c_zeromodenum)) /\\ exists fs_q_jt_zeromodenuminputprimitivecoordinatesat. jt_b_zeromodenum = fs_q_jt_zeromodenuminputprimitivecoordinatesat * S ((S (jt_index_zeromodenuminputprimitivecoordinates)) * jt_c_zeromodenum) + (jt_value_zeromodenuminputprimitivecoordinates))) -> (exists jt_factor_zeromodenuminputprimitivecoordinatesdivides. (jt_value_zeromodenuminputprimitivecoordinates)=(jt_divisor_zeromodenuminputprimitive)*jt_factor_zeromodenuminputprimitivecoordinatesdivides)) -> jt_divisor_zeromodenuminputprimitive=1) -> exists jt_i_zeromodenum jt_d_zeromodenum jt_e_zeromodenum. ((exists jt_gap_zeromodenumcompleteindex. jt_gap_zeromodenumcompleteindex+S (jt_i_zeromodenum)=(j)) /\\ (((((((exists fs_h_jt_zeromodenumcompletecode. fs_h_jt_zeromodenumcompletecode + S (jt_d_zeromodenum) = S ((S (jt_i_zeromodenum)) * jt_code_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumcompletecode. jt_codes_zeromod = fs_q_jt_zeromodenumcompletecode * S ((S (jt_i_zeromodenum)) * jt_code_scale_zeromod) + (jt_d_zeromodenum))) /\\ (((exists fs_h_jt_zeromodenumcompletescale. fs_h_jt_zeromodenumcompletescale + S (jt_e_zeromodenum) = S ((S (jt_i_zeromodenum)) * jt_scale_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumcompletescale. jt_scales_zeromod = fs_q_jt_zeromodenumcompletescale * S ((S (jt_i_zeromodenum)) * jt_scale_scale_zeromod) + (jt_e_zeromodenum))))) /\\ (forall jt_index_zeromodenumrepresented jt_left_zeromodenumrepresented jt_right_zeromodenumrepresented. (exists jt_gap_zeromodenumrepresentedindex. jt_gap_zeromodenumrepresentedindex+S (jt_index_zeromodenumrepresented)=(k)) -> (((exists fs_h_jt_zeromodenumrepresentedleft. fs_h_jt_zeromodenumrepresentedleft + S (jt_left_zeromodenumrepresented) = S ((S (jt_index_zeromodenumrepresented)) * jt_c_zeromodenum)) /\\ exists fs_q_jt_zeromodenumrepresentedleft. jt_b_zeromodenum = fs_q_jt_zeromodenumrepresentedleft * S ((S (jt_index_zeromodenumrepresented)) * jt_c_zeromodenum) + (jt_left_zeromodenumrepresented))) -> (((exists fs_h_jt_zeromodenumrepresentedright. fs_h_jt_zeromodenumrepresentedright + S (jt_right_zeromodenumrepresented) = S ((S (jt_index_zeromodenumrepresented)) * jt_e_zeromodenum)) /\\ exists fs_q_jt_zeromodenumrepresentedright. jt_d_zeromodenum = fs_q_jt_zeromodenumrepresentedright * S ((S (jt_index_zeromodenumrepresented)) * jt_e_zeromodenum) + (jt_right_zeromodenumrepresented))) -> jt_left_zeromodenumrepresented=jt_right_zeromodenumrepresented))))) /\\ (forall jt_i_zeromodenum jt_h_zeromodenum jt_b_zeromodenum jt_c_zeromodenum jt_d_zeromodenum jt_e_zeromodenum. (exists jt_gap_zeromodenumfirstindex. jt_gap_zeromodenumfirstindex+S (jt_i_zeromodenum)=(j)) -> (exists jt_gap_zeromodenumsecondindex. jt_gap_zeromodenumsecondindex+S (jt_h_zeromodenum)=(j)) -> (((((exists fs_h_jt_zeromodenumfirstcode. fs_h_jt_zeromodenumfirstcode + S (jt_b_zeromodenum) = S ((S (jt_i_zeromodenum)) * jt_code_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumfirstcode. jt_codes_zeromod = fs_q_jt_zeromodenumfirstcode * S ((S (jt_i_zeromodenum)) * jt_code_scale_zeromod) + (jt_b_zeromodenum))) /\\ (((exists fs_h_jt_zeromodenumfirstscale. fs_h_jt_zeromodenumfirstscale + S (jt_c_zeromodenum) = S ((S (jt_i_zeromodenum)) * jt_scale_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumfirstscale. jt_scales_zeromod = fs_q_jt_zeromodenumfirstscale * S ((S (jt_i_zeromodenum)) * jt_scale_scale_zeromod) + (jt_c_zeromodenum))))) -> (((((exists fs_h_jt_zeromodenumsecondcode. fs_h_jt_zeromodenumsecondcode + S (jt_d_zeromodenum) = S ((S (jt_h_zeromodenum)) * jt_code_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumsecondcode. jt_codes_zeromod = fs_q_jt_zeromodenumsecondcode * S ((S (jt_h_zeromodenum)) * jt_code_scale_zeromod) + (jt_d_zeromodenum))) /\\ (((exists fs_h_jt_zeromodenumsecondscale. fs_h_jt_zeromodenumsecondscale + S (jt_e_zeromodenum) = S ((S (jt_h_zeromodenum)) * jt_scale_scale_zeromod)) /\\ exists fs_q_jt_zeromodenumsecondscale. jt_scales_zeromod = fs_q_jt_zeromodenumsecondscale * S ((S (jt_h_zeromodenum)) * jt_scale_scale_zeromod) + (jt_e_zeromodenum))))) -> (forall jt_index_zeromodenumsame jt_left_zeromodenumsame jt_right_zeromodenumsame. (exists jt_gap_zeromodenumsameindex. jt_gap_zeromodenumsameindex+S (jt_index_zeromodenumsame)=(k)) -> (((exists fs_h_jt_zeromodenumsameleft. fs_h_jt_zeromodenumsameleft + S (jt_left_zeromodenumsame) = S ((S (jt_index_zeromodenumsame)) * jt_c_zeromodenum)) /\\ exists fs_q_jt_zeromodenumsameleft. jt_b_zeromodenum = fs_q_jt_zeromodenumsameleft * S ((S (jt_index_zeromodenumsame)) * jt_c_zeromodenum) + (jt_left_zeromodenumsame))) -> (((exists fs_h_jt_zeromodenumsameright. fs_h_jt_zeromodenumsameright + S (jt_right_zeromodenumsame) = S ((S (jt_index_zeromodenumsame)) * jt_e_zeromodenum)) /\\ exists fs_q_jt_zeromodenumsameright. jt_d_zeromodenum = fs_q_jt_zeromodenumsameright * S ((S (jt_index_zeromodenumsame)) * jt_e_zeromodenum) + (jt_right_zeromodenumsame))) -> jt_left_zeromodenumsame=jt_right_zeromodenumsame) -> jt_i_zeromodenum=jt_h_zeromodenum)))))))))", "statement_sha256": "3cad9a4091db3795271f289596cd4daea0429e8e006b5e21503688bec46b29c9", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Jordan never counts a purported finite complete residue system modulo zero." }, { "admission_dependencies": [ "mul_ne_zero", "mul_zero_left", "coprime_divisor_factor_pair_exists", "jordan_tuple_divisor_downward", "mul_comm", "one_mul" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 35, "body_proof_nodes": 112, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro a", "intro b", "intro B", "intro C", "intro k", "intro ha", "intro hb", "intro hc", "intro hpa", "intro hpb", "intro d", "intro hd", "intro hall", "have hn : ~(a*b=0)", "intro hproductzero", "specialize mul_ne_zero (a)", "specialize mul_ne_zero (b)", "apply mul_ne_zero", "exact ha", "exact hb", "exact hproductzero", "have hdpos : ~(d=0)", "intro hz", "apply hn", "cases hd", "have hzero : d*x=0", "rewrite hz", "specialize mul_zero_left (x)", "apply mul_zero_left", "trans d*x", "exact hd_witness", "exact hzero", "have hp : ∃ r. ∃ s. DivisorFactorPair(a,b,d,r,s)", "specialize coprime_divisor_factor_pair_exists (a)", "specialize coprime_divisor_factor_pair_exists (b)", "specialize coprime_divisor_factor_pair_exists (d)", "apply coprime_divisor_factor_pair_exists", "exact hdpos", "exact hc", "exact hd", "cases hp", "cases hp_witness", "cases hp_witness_witness", "cases hp_witness_witness_right", "cases hp_witness_witness_right_right", "cases hp_witness_witness_right_right_right", "have hr : x=1", "specialize hpa (x)", "apply hpa", "exact hp_witness_witness_right_right_left", "specialize jordan_tuple_divisor_downward (x)", "specialize jordan_tuple_divisor_downward (d)", "specialize jordan_tuple_divisor_downward (B)", "specialize jordan_tuple_divisor_downward (C)", "specialize jordan_tuple_divisor_downward (k)", "apply jordan_tuple_divisor_downward", "exists x1", "exact hp_witness_witness_right_right_right_right", "exact hall", "have hs : x1=1", "specialize hpb (x1)", "apply hpb", "exact hp_witness_witness_right_right_right_left", "specialize jordan_tuple_divisor_downward (x1)", "specialize jordan_tuple_divisor_downward (d)", "specialize jordan_tuple_divisor_downward (B)", "specialize jordan_tuple_divisor_downward (C)", "specialize jordan_tuple_divisor_downward (k)", "apply jordan_tuple_divisor_downward", "exists x", "have hcomm : x*x1=x1*x", "specialize mul_comm (x)", "specialize mul_comm (x1)", "apply mul_comm", "trans x*x1", "exact hp_witness_witness_right_right_right_right", "exact hcomm", "exact hall", "trans x*x1", "exact hp_witness_witness_right_right_right_right", "rewrite hr", "rewrite hs", "specialize one_mul (1)", "apply one_mul" ], "defined_statement": "∀ a. ∀ b. ∀ B. ∀ C. ∀ k. ¬a = 0 → ¬b = 0 → Coprime(a,b) → JordanPrimitiveTuple(a,B,C,k) → JordanPrimitiveTuple(b,B,C,k) → JordanPrimitiveTuple(a · b,B,C,k)", "defined_statement_sha256": "2170c7c6aba63a69bdc938dda1e385cb099b880df89a9f89cf993003b85424fe", "definition_uses": { "ND0317": 1, "ND0372": 3, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "336cec60f52d3fec89b65ffbd204ed6c816d0d01ba84d0aa833179cb6c0b92df", "free_names": [], "script_definition_uses": { "ND0317": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hc" } ], [ { "kind": "text", "text": "intro hpa" } ], [ { "kind": "text", "text": "intro hpb" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "have hn : ~(a*b=0)" } ], [ { "kind": "text", "text": "intro hproductzero" } ], [ { "kind": "text", "text": "specialize mul_ne_zero (a)" } ], [ { "kind": "text", "text": "specialize mul_ne_zero (b)" } ], [ { "kind": "text", "text": "apply mul_ne_zero" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hproductzero" } ], [ { "kind": "text", "text": "have hdpos : ~(d=0)" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "apply hn" } ], [ { "kind": "text", "text": "cases hd" } ], [ { "kind": "text", "text": "have hzero : d*x=0" } ], [ { "kind": "text", "text": "rewrite hz" } ], [ { "kind": "text", "text": "specialize mul_zero_left (x)" } ], [ { "kind": "text", "text": "apply mul_zero_left" } ], [ { "kind": "text", "text": "trans d*x" } ], [ { "kind": "text", "text": "exact hd_witness" } ], [ { "kind": "text", "text": "exact hzero" } ], [ { "kind": "text", "text": "have hp : " }, { "kind": "text", "text": "∃ r. ∃ s. " }, { "definition": "ND0317", "kind": "definition", "text": "DivisorFactorPair(a,b,d,r,s)" } ], [ { "kind": "text", "text": "specialize coprime_divisor_factor_pair_exists (a)" } ], [ { "kind": "text", "text": "specialize coprime_divisor_factor_pair_exists (b)" } ], [ { "kind": "text", "text": "specialize coprime_divisor_factor_pair_exists (d)" } ], [ { "kind": "text", "text": "apply coprime_divisor_factor_pair_exists" } ], [ { "kind": "text", "text": "exact hdpos" } ], [ { "kind": "text", "text": "exact hc" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "cases hp" } ], [ { "kind": "text", "text": "cases hp_witness" } ], [ { "kind": "text", "text": "cases hp_witness_witness" } ], [ { "kind": "text", "text": "cases hp_witness_witness_right" } ], [ { "kind": "text", "text": "cases hp_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hp_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "have hr : x=1" } ], [ { "kind": "text", "text": "specialize hpa (x)" } ], [ { "kind": "text", "text": "apply hpa" } ], [ { "kind": "text", "text": "exact hp_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_divisor_downward" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exact hp_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "exact hall" } ], [ { "kind": "text", "text": "have hs : x1=1" } ], [ { "kind": "text", "text": "specialize hpb (x1)" } ], [ { "kind": "text", "text": "apply hpb" } ], [ { "kind": "text", "text": "exact hp_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_divisor_downward" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "have hcomm : x*x1=x1*x" } ], [ { "kind": "text", "text": "specialize mul_comm (x)" } ], [ { "kind": "text", "text": "specialize mul_comm (x1)" } ], [ { "kind": "text", "text": "apply mul_comm" } ], [ { "kind": "text", "text": "trans x*x1" } ], [ { "kind": "text", "text": "exact hp_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "exact hcomm" } ], [ { "kind": "text", "text": "exact hall" } ], [ { "kind": "text", "text": "trans x*x1" } ], [ { "kind": "text", "text": "exact hp_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "rewrite hr" } ], [ { "kind": "text", "text": "rewrite hs" } ], [ { "kind": "text", "text": "specialize one_mul (1)" } ], [ { "kind": "text", "text": "apply one_mul" } ] ], "statement_definition_uses": { "ND0372": 3, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ a. ∀ b. ∀ B. ∀ C. ∀ k. ¬a = 0 → ¬b = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(a,b)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(a,B,C,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(b,B,C,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(a · b,B,C,k)" } ] }, "dependencies": [ "mul_ne_zero", "mul_zero_left", "coprime_divisor_factor_pair_exists", "jordan_tuple_divisor_downward", "mul_comm", "one_mul" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT000B", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_coprime_product", "original_ha_bundle_verified": true, "proof_bundle_node_id": 268, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro a", "intro b", "intro B", "intro C", "intro k", "intro ha", "intro hb", "intro hc", "intro hpa", "intro hpb", "intro d", "intro hd", "intro hall", "have hn : ~(a*b=0)", "intro hproductzero", "specialize mul_ne_zero (a)", "specialize mul_ne_zero (b)", "apply mul_ne_zero", "exact ha", "exact hb", "exact hproductzero", "have hdpos : ~(d=0)", "intro hz", "apply hn", "cases hd", "have hzero : d*x=0", "rewrite hz", "specialize mul_zero_left (x)", "apply mul_zero_left", "trans d*x", "exact hd_witness", "exact hzero", "have hp : exists r s. ((~(r=0)) /\\ (((~(s=0)) /\\ (((exists jt_factor_paira. (a)=(r)*jt_factor_paira) /\\ (((exists jt_factor_pairb. (b)=(s)*jt_factor_pairb) /\\ (d=r*s))))))))", "specialize coprime_divisor_factor_pair_exists (a)", "specialize coprime_divisor_factor_pair_exists (b)", "specialize coprime_divisor_factor_pair_exists (d)", "apply coprime_divisor_factor_pair_exists", "exact hdpos", "exact hc", "exact hd", "cases hp", "cases hp_witness", "cases hp_witness_witness", "cases hp_witness_witness_right", "cases hp_witness_witness_right_right", "cases hp_witness_witness_right_right_right", "have hr : x=1", "specialize hpa (x)", "apply hpa", "exact hp_witness_witness_right_right_left", "specialize jordan_tuple_divisor_downward (x)", "specialize jordan_tuple_divisor_downward (d)", "specialize jordan_tuple_divisor_downward (B)", "specialize jordan_tuple_divisor_downward (C)", "specialize jordan_tuple_divisor_downward (k)", "apply jordan_tuple_divisor_downward", "exists x1", "exact hp_witness_witness_right_right_right_right", "exact hall", "have hs : x1=1", "specialize hpb (x1)", "apply hpb", "exact hp_witness_witness_right_right_right_left", "specialize jordan_tuple_divisor_downward (x1)", "specialize jordan_tuple_divisor_downward (d)", "specialize jordan_tuple_divisor_downward (B)", "specialize jordan_tuple_divisor_downward (C)", "specialize jordan_tuple_divisor_downward (k)", "apply jordan_tuple_divisor_downward", "exists x", "have hcomm : x*x1=x1*x", "specialize mul_comm (x)", "specialize mul_comm (x1)", "apply mul_comm", "trans x*x1", "exact hp_witness_witness_right_right_right_right", "exact hcomm", "exact hall", "trans x*x1", "exact hp_witness_witness_right_right_right_right", "rewrite hr", "rewrite hs", "specialize one_mul (1)", "apply one_mul" ], "script_sha256": "1b268215766637c9f511b2e3606bffb761802c05e4a2e3bfd2c511262dae76c4", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "1b268215766637c9f511b2e3606bffb761802c05e4a2e3bfd2c511262dae76c4", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "336cec60f52d3fec89b65ffbd204ed6c816d0d01ba84d0aa833179cb6c0b92df" } ], "stable_member": false, "statement": "forall a b B C k. ~(a=0) -> ~(b=0) -> (forall jt_divisor_productcop. (exists jt_factor_productcopa. (a)=(jt_divisor_productcop)*jt_factor_productcopa) -> (exists jt_factor_productcopb. (b)=(jt_divisor_productcop)*jt_factor_productcopb) -> jt_divisor_productcop=1) -> (forall jt_divisor_producta. (exists jt_factor_productamodulus. (a)=(jt_divisor_producta)*jt_factor_productamodulus) -> (forall jt_index_productacoordinates jt_value_productacoordinates. (exists jt_gap_productacoordinatesindex. jt_gap_productacoordinatesindex+S (jt_index_productacoordinates)=(k)) -> (((exists fs_h_jt_productacoordinatesat. fs_h_jt_productacoordinatesat + S (jt_value_productacoordinates) = S ((S (jt_index_productacoordinates)) * C)) /\\ exists fs_q_jt_productacoordinatesat. B = fs_q_jt_productacoordinatesat * S ((S (jt_index_productacoordinates)) * C) + (jt_value_productacoordinates))) -> (exists jt_factor_productacoordinatesdivides. (jt_value_productacoordinates)=(jt_divisor_producta)*jt_factor_productacoordinatesdivides)) -> jt_divisor_producta=1) -> (forall jt_divisor_productb. (exists jt_factor_productbmodulus. (b)=(jt_divisor_productb)*jt_factor_productbmodulus) -> (forall jt_index_productbcoordinates jt_value_productbcoordinates. (exists jt_gap_productbcoordinatesindex. jt_gap_productbcoordinatesindex+S (jt_index_productbcoordinates)=(k)) -> (((exists fs_h_jt_productbcoordinatesat. fs_h_jt_productbcoordinatesat + S (jt_value_productbcoordinates) = S ((S (jt_index_productbcoordinates)) * C)) /\\ exists fs_q_jt_productbcoordinatesat. B = fs_q_jt_productbcoordinatesat * S ((S (jt_index_productbcoordinates)) * C) + (jt_value_productbcoordinates))) -> (exists jt_factor_productbcoordinatesdivides. (jt_value_productbcoordinates)=(jt_divisor_productb)*jt_factor_productbcoordinatesdivides)) -> jt_divisor_productb=1) -> (forall jt_divisor_productab. (exists jt_factor_productabmodulus. (a*b)=(jt_divisor_productab)*jt_factor_productabmodulus) -> (forall jt_index_productabcoordinates jt_value_productabcoordinates. (exists jt_gap_productabcoordinatesindex. jt_gap_productabcoordinatesindex+S (jt_index_productabcoordinates)=(k)) -> (((exists fs_h_jt_productabcoordinatesat. fs_h_jt_productabcoordinatesat + S (jt_value_productabcoordinates) = S ((S (jt_index_productabcoordinates)) * C)) /\\ exists fs_q_jt_productabcoordinatesat. B = fs_q_jt_productabcoordinatesat * S ((S (jt_index_productabcoordinates)) * C) + (jt_value_productabcoordinates))) -> (exists jt_factor_productabcoordinatesdivides. (jt_value_productabcoordinates)=(jt_divisor_productab)*jt_factor_productabcoordinatesdivides)) -> jt_divisor_productab=1)", "statement_sha256": "336cec60f52d3fec89b65ffbd204ed6c816d0d01ba84d0aa833179cb6c0b92df", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Actual coprime divisor decomposition proves primitivity for the product modulus." }, { "admission_dependencies": [ "jordan_primitive_tuple_divisor_modulus", "mul_comm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 27, "body_proof_nodes": 69, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro a", "intro b", "intro B", "intro C", "intro k", "intro h", "split", "specialize jordan_primitive_tuple_divisor_modulus (a)", "specialize jordan_primitive_tuple_divisor_modulus (a*b)", "specialize jordan_primitive_tuple_divisor_modulus (B)", "specialize jordan_primitive_tuple_divisor_modulus (C)", "specialize jordan_primitive_tuple_divisor_modulus (k)", "apply jordan_primitive_tuple_divisor_modulus", "exists b", "refl", "exact h", "specialize jordan_primitive_tuple_divisor_modulus (b)", "specialize jordan_primitive_tuple_divisor_modulus (a*b)", "specialize jordan_primitive_tuple_divisor_modulus (B)", "specialize jordan_primitive_tuple_divisor_modulus (C)", "specialize jordan_primitive_tuple_divisor_modulus (k)", "apply jordan_primitive_tuple_divisor_modulus", "exists a", "specialize mul_comm (a)", "specialize mul_comm (b)", "apply mul_comm", "exact h" ], "defined_statement": "∀ a. ∀ b. ∀ B. ∀ C. ∀ k. JordanPrimitiveTuple(a · b,B,C,k) → JordanPrimitiveTuple(a,B,C,k) ∧ JordanPrimitiveTuple(b,B,C,k)", "defined_statement_sha256": "8f1eb83504591535d4c92082635687251ac9e08a4897d9c74ea4b36581f99b16", "definition_uses": { "ND0372": 3 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "3c35af1ab6d85da32bd7ccaa09b29b3aa111c7c09ebd34b7e2238169fc520be2", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (a)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (a*b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (B)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (C)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_divisor_modulus" } ], [ { "kind": "text", "text": "exists b" } ], [ { "kind": "text", "text": "refl" } ], [ { "kind": "text", "text": "exact h" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (a*b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (B)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (C)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_divisor_modulus (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_divisor_modulus" } ], [ { "kind": "text", "text": "exists a" } ], [ { "kind": "text", "text": "specialize mul_comm (a)" } ], [ { "kind": "text", "text": "specialize mul_comm (b)" } ], [ { "kind": "text", "text": "apply mul_comm" } ], [ { "kind": "text", "text": "exact h" } ] ], "statement_definition_uses": { "ND0372": 3 }, "statement_parts": [ { "kind": "text", "text": "∀ a. ∀ b. ∀ B. ∀ C. ∀ k. " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(a · b,B,C,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(a,B,C,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(b,B,C,k)" } ] }, "dependencies": [ "jordan_primitive_tuple_divisor_modulus", "mul_comm" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT000C", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_product_components", "original_ha_bundle_verified": true, "proof_bundle_node_id": 269, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro a", "intro b", "intro B", "intro C", "intro k", "intro h", "split", "specialize jordan_primitive_tuple_divisor_modulus (a)", "specialize jordan_primitive_tuple_divisor_modulus (a*b)", "specialize jordan_primitive_tuple_divisor_modulus (B)", "specialize jordan_primitive_tuple_divisor_modulus (C)", "specialize jordan_primitive_tuple_divisor_modulus (k)", "apply jordan_primitive_tuple_divisor_modulus", "exists b", "refl", "exact h", "specialize jordan_primitive_tuple_divisor_modulus (b)", "specialize jordan_primitive_tuple_divisor_modulus (a*b)", "specialize jordan_primitive_tuple_divisor_modulus (B)", "specialize jordan_primitive_tuple_divisor_modulus (C)", "specialize jordan_primitive_tuple_divisor_modulus (k)", "apply jordan_primitive_tuple_divisor_modulus", "exists a", "specialize mul_comm (a)", "specialize mul_comm (b)", "apply mul_comm", "exact h" ], "script_sha256": "8a9b9e6a6b4d02a5b62df58f56ccb9fba7a020efe9da279957d8e6ce85f4b4a6", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "8a9b9e6a6b4d02a5b62df58f56ccb9fba7a020efe9da279957d8e6ce85f4b4a6", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "3c35af1ab6d85da32bd7ccaa09b29b3aa111c7c09ebd34b7e2238169fc520be2" } ], "stable_member": false, "statement": "forall a b B C k. (forall jt_divisor_componentab. (exists jt_factor_componentabmodulus. (a*b)=(jt_divisor_componentab)*jt_factor_componentabmodulus) -> (forall jt_index_componentabcoordinates jt_value_componentabcoordinates. (exists jt_gap_componentabcoordinatesindex. jt_gap_componentabcoordinatesindex+S (jt_index_componentabcoordinates)=(k)) -> (((exists fs_h_jt_componentabcoordinatesat. fs_h_jt_componentabcoordinatesat + S (jt_value_componentabcoordinates) = S ((S (jt_index_componentabcoordinates)) * C)) /\\ exists fs_q_jt_componentabcoordinatesat. B = fs_q_jt_componentabcoordinatesat * S ((S (jt_index_componentabcoordinates)) * C) + (jt_value_componentabcoordinates))) -> (exists jt_factor_componentabcoordinatesdivides. (jt_value_componentabcoordinates)=(jt_divisor_componentab)*jt_factor_componentabcoordinatesdivides)) -> jt_divisor_componentab=1) -> ((forall jt_divisor_componenta. (exists jt_factor_componentamodulus. (a)=(jt_divisor_componenta)*jt_factor_componentamodulus) -> (forall jt_index_componentacoordinates jt_value_componentacoordinates. (exists jt_gap_componentacoordinatesindex. jt_gap_componentacoordinatesindex+S (jt_index_componentacoordinates)=(k)) -> (((exists fs_h_jt_componentacoordinatesat. fs_h_jt_componentacoordinatesat + S (jt_value_componentacoordinates) = S ((S (jt_index_componentacoordinates)) * C)) /\\ exists fs_q_jt_componentacoordinatesat. B = fs_q_jt_componentacoordinatesat * S ((S (jt_index_componentacoordinates)) * C) + (jt_value_componentacoordinates))) -> (exists jt_factor_componentacoordinatesdivides. (jt_value_componentacoordinates)=(jt_divisor_componenta)*jt_factor_componentacoordinatesdivides)) -> jt_divisor_componenta=1) /\\ (forall jt_divisor_componentb. (exists jt_factor_componentbmodulus. (b)=(jt_divisor_componentb)*jt_factor_componentbmodulus) -> (forall jt_index_componentbcoordinates jt_value_componentbcoordinates. (exists jt_gap_componentbcoordinatesindex. jt_gap_componentbcoordinatesindex+S (jt_index_componentbcoordinates)=(k)) -> (((exists fs_h_jt_componentbcoordinatesat. fs_h_jt_componentbcoordinatesat + S (jt_value_componentbcoordinates) = S ((S (jt_index_componentbcoordinates)) * C)) /\\ exists fs_q_jt_componentbcoordinatesat. B = fs_q_jt_componentbcoordinatesat * S ((S (jt_index_componentbcoordinates)) * C) + (jt_value_componentbcoordinates))) -> (exists jt_factor_componentbcoordinatesdivides. (jt_value_componentbcoordinates)=(jt_divisor_componentb)*jt_factor_componentbcoordinatesdivides)) -> jt_divisor_componentb=1))", "statement_sha256": "3c35af1ab6d85da32bd7ccaa09b29b3aa111c7c09ebd34b7e2238169fc520be2", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Both primitive projections follow even without coprimality of the moduli." }, { "admission_dependencies": [ "linear_congruence_zero_residue_divides", "mod_eq_trans", "mod_eq_symm", "mod_eq_of_mod_eq_multiple", "dvd_to_mod_zero" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 45, "body_proof_nodes": 87, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro d", "intro a", "intro z", "intro hdn", "intro hda", "intro hm", "specialize linear_congruence_zero_residue_divides (d)", "specialize linear_congruence_zero_residue_divides (z)", "apply linear_congruence_zero_residue_divides", "specialize mod_eq_trans (d)", "specialize mod_eq_trans (z)", "specialize mod_eq_trans (a)", "specialize mod_eq_trans (0)", "apply mod_eq_trans", "specialize mod_eq_symm (d)", "specialize mod_eq_symm (a)", "specialize mod_eq_symm (z)", "apply mod_eq_symm", "specialize mod_eq_of_mod_eq_multiple (d)", "specialize mod_eq_of_mod_eq_multiple (n)", "specialize mod_eq_of_mod_eq_multiple (a)", "specialize mod_eq_of_mod_eq_multiple (z)", "apply mod_eq_of_mod_eq_multiple", "exact hdn", "exact hm", "specialize dvd_to_mod_zero (d)", "specialize dvd_to_mod_zero (a)", "apply dvd_to_mod_zero", "exact hda" ], "defined_statement": "∀ n. ∀ d. ∀ a. ∀ z. 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" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,n)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,a,z)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,z)" } ] }, "dependencies": [ "linear_congruence_zero_residue_divides", "mod_eq_trans", "mod_eq_symm", "mod_eq_of_mod_eq_multiple", "dvd_to_mod_zero" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT000D", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_divisibility_congruence_transport", "original_ha_bundle_verified": true, "proof_bundle_node_id": 270, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro d", "intro a", "intro z", "intro hdn", "intro hda", "intro hm", "specialize linear_congruence_zero_residue_divides (d)", "specialize linear_congruence_zero_residue_divides (z)", "apply linear_congruence_zero_residue_divides", "specialize mod_eq_trans (d)", "specialize mod_eq_trans (z)", "specialize mod_eq_trans (a)", "specialize mod_eq_trans (0)", "apply mod_eq_trans", "specialize mod_eq_symm (d)", "specialize mod_eq_symm (a)", "specialize mod_eq_symm (z)", "apply mod_eq_symm", "specialize mod_eq_of_mod_eq_multiple (d)", "specialize mod_eq_of_mod_eq_multiple (n)", "specialize mod_eq_of_mod_eq_multiple (a)", "specialize mod_eq_of_mod_eq_multiple (z)", "apply mod_eq_of_mod_eq_multiple", "exact hdn", "exact hm", "specialize dvd_to_mod_zero (d)", "specialize dvd_to_mod_zero (a)", "apply dvd_to_mod_zero", "exact hda" ], "script_sha256": "f6383f30c134286732a019bb3561178404d74fbb7cb69e9cc3d7ab6330e4844a", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "f6383f30c134286732a019bb3561178404d74fbb7cb69e9cc3d7ab6330e4844a", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "d439c3d17372a538c62da4b20b35e4b41204acb2691360e685c61fd563e7c346" } ], "stable_member": false, "statement": "forall n d a z. (exists jt_factor_congdivisor. (n)=(d)*jt_factor_congdivisor) -> (exists jt_factor_congsource. (a)=(d)*jt_factor_congsource) -> (exists jt_left_congmod jt_right_congmod. (a)+(n)*jt_left_congmod=(z)+(n)*jt_right_congmod) -> (exists jt_factor_congtarget. (z)=(d)*jt_factor_congtarget)", "statement_sha256": "d439c3d17372a538c62da4b20b35e4b41204acb2691360e685c61fd563e7c346", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A genuine common modulus divisor survives balanced congruence, including divisor zero." }, { "admission_dependencies": [ "mod_eq_symm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 31, "body_proof_nodes": 47, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro h", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_symm (n)", "specialize mod_eq_symm (z)", "specialize mod_eq_symm (a)", "apply mod_eq_symm", "specialize h (i)", "specialize h (z)", "specialize h (a)", "apply h", "exact hi", "exact hz", "exact ha" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. 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" }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,d,e,b,c,k)" } ] }, "dependencies": [ "mod_eq_symm" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT000E", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_congruence_symm", "original_ha_bundle_verified": true, "proof_bundle_node_id": 271, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro h", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_symm (n)", "specialize mod_eq_symm (z)", "specialize mod_eq_symm (a)", "apply mod_eq_symm", "specialize h (i)", "specialize h (z)", "specialize h (a)", "apply h", "exact hi", "exact hz", "exact ha" ], "script_sha256": "ad50d65619512300acfc015426470f02d535bfa801f596c90bb8d152ecb09283", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "ad50d65619512300acfc015426470f02d535bfa801f596c90bb8d152ecb09283", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "2a886fbf1d9e39d01cd270e88633dee3614e2e936cb35e542b3b23d3eadcae0d" } ], "stable_member": false, "statement": "forall n b c d e k. (forall jt_index_msource jt_left_msource jt_right_msource. (exists jt_gap_msourceindex. jt_gap_msourceindex+S (jt_index_msource)=(k)) -> (((exists fs_h_jt_msourceleft. fs_h_jt_msourceleft + S (jt_left_msource) = S ((S (jt_index_msource)) * c)) /\\ exists fs_q_jt_msourceleft. b = fs_q_jt_msourceleft * S ((S (jt_index_msource)) * c) + (jt_left_msource))) -> (((exists fs_h_jt_msourceright. fs_h_jt_msourceright + S (jt_right_msource) = S ((S (jt_index_msource)) * e)) /\\ exists fs_q_jt_msourceright. d = fs_q_jt_msourceright * S ((S (jt_index_msource)) * e) + (jt_right_msource))) -> (exists jt_left_msourcemod jt_right_msourcemod. (jt_left_msource)+(n)*jt_left_msourcemod=(jt_right_msource)+(n)*jt_right_msourcemod)) -> (forall jt_index_mtarget jt_left_mtarget jt_right_mtarget. (exists jt_gap_mtargetindex. jt_gap_mtargetindex+S (jt_index_mtarget)=(k)) -> (((exists fs_h_jt_mtargetleft. fs_h_jt_mtargetleft + S (jt_left_mtarget) = S ((S (jt_index_mtarget)) * e)) /\\ exists fs_q_jt_mtargetleft. d = fs_q_jt_mtargetleft * S ((S (jt_index_mtarget)) * e) + (jt_left_mtarget))) -> (((exists fs_h_jt_mtargetright. fs_h_jt_mtargetright + S (jt_right_mtarget) = S ((S (jt_index_mtarget)) * c)) /\\ exists fs_q_jt_mtargetright. b = fs_q_jt_mtargetright * S ((S (jt_index_mtarget)) * c) + (jt_right_mtarget))) -> (exists jt_left_mtargetmod jt_right_mtargetmod. (jt_left_mtarget)+(n)*jt_left_mtargetmod=(jt_right_mtarget)+(n)*jt_right_mtargetmod))", "statement_sha256": "2a886fbf1d9e39d01cd270e88633dee3614e2e936cb35e542b3b23d3eadcae0d", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Pointwise balanced congruence is symmetric on every finite prefix." }, { "admission_dependencies": [ "beta_at_exists", "jordan_divisibility_congruence_transport", "mod_eq_symm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 29, "body_proof_nodes": 59, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hm", "intro hp", "intro q", "intro hqn", "intro hall", "specialize hp (q)", "apply hp", "exact hqn", "intro i", "intro a", "intro hi", "intro ha", "have hz : ∃ z. 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" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,z)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (i)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hz" } ], [ { "kind": "text", "text": "specialize jordan_divisibility_congruence_transport (n)" } ], [ { "kind": "text", "text": "specialize jordan_divisibility_congruence_transport (q)" } ], [ { "kind": "text", "text": "specialize jordan_divisibility_congruence_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_divisibility_congruence_transport (a)" } ], [ { "kind": "text", "text": "apply jordan_divisibility_congruence_transport" } ], [ { "kind": "text", "text": "exact hqn" } ], [ { "kind": "text", "text": "specialize hall (i)" } ], [ { "kind": "text", "text": "specialize hall (x)" } ], [ { "kind": "text", "text": "apply hall" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hz_witness" } ], [ { "kind": "text", "text": "specialize mod_eq_symm (n)" } ], [ { "kind": "text", "text": "specialize mod_eq_symm (a)" } ], [ { "kind": "text", "text": "specialize mod_eq_symm (x)" } ], [ { "kind": "text", "text": "apply mod_eq_symm" } ], [ { "kind": "text", "text": "specialize hm (i)" } ], [ { "kind": "text", "text": "specialize hm (a)" } ], [ { "kind": "text", "text": "specialize hm (x)" } ], [ { "kind": "text", "text": "apply hm" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz_witness" } ] ], "statement_definition_uses": { "ND0372": 2, "ND0373": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,d,e,k)" } ] }, "dependencies": [ "beta_at_exists", "jordan_divisibility_congruence_transport", "mod_eq_symm" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT000F", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_congruence_transport", "original_ha_bundle_verified": true, "proof_bundle_node_id": 272, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hm", "intro hp", "intro q", "intro hqn", "intro hall", "specialize hp (q)", "apply hp", "exact hqn", "intro i", "intro a", "intro hi", "intro ha", "have hz : exists z. ((exists fs_h_jt_primmodactual. fs_h_jt_primmodactual + S (z) = S ((S (i)) * e)) /\\ exists fs_q_jt_primmodactual. d = fs_q_jt_primmodactual * S ((S (i)) * e) + (z))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hz", "specialize jordan_divisibility_congruence_transport (n)", "specialize jordan_divisibility_congruence_transport (q)", "specialize jordan_divisibility_congruence_transport (x)", "specialize jordan_divisibility_congruence_transport (a)", "apply jordan_divisibility_congruence_transport", "exact hqn", "specialize hall (i)", "specialize hall (x)", "apply hall", "exact hi", "exact hz_witness", "specialize mod_eq_symm (n)", "specialize mod_eq_symm (a)", "specialize mod_eq_symm (x)", "apply mod_eq_symm", "specialize hm (i)", "specialize hm (a)", "specialize hm (x)", "apply hm", "exact hi", "exact ha", "exact hz_witness" ], "script_sha256": "37c3f067fa10a7addd993a5a5067c89afd40eaa86bbb266efcc1e45997700a35", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "37c3f067fa10a7addd993a5a5067c89afd40eaa86bbb266efcc1e45997700a35", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "c1054866cc6baf40e8eb2f61ae8ce8899d8312696287b2088facef0cad2f2ffe" } ], "stable_member": false, "statement": "forall n b c d e k. (forall jt_index_primmod jt_left_primmod jt_right_primmod. (exists jt_gap_primmodindex. jt_gap_primmodindex+S (jt_index_primmod)=(k)) -> (((exists fs_h_jt_primmodleft. fs_h_jt_primmodleft + S (jt_left_primmod) = S ((S (jt_index_primmod)) * c)) /\\ exists fs_q_jt_primmodleft. b = fs_q_jt_primmodleft * S ((S (jt_index_primmod)) * c) + (jt_left_primmod))) -> (((exists fs_h_jt_primmodright. fs_h_jt_primmodright + S (jt_right_primmod) = S ((S (jt_index_primmod)) * e)) /\\ exists fs_q_jt_primmodright. d = fs_q_jt_primmodright * S ((S (jt_index_primmod)) * e) + (jt_right_primmod))) -> (exists jt_left_primmodmod jt_right_primmodmod. (jt_left_primmod)+(n)*jt_left_primmodmod=(jt_right_primmod)+(n)*jt_right_primmodmod)) -> (forall jt_divisor_primmodsource. (exists jt_factor_primmodsourcemodulus. (n)=(jt_divisor_primmodsource)*jt_factor_primmodsourcemodulus) -> (forall jt_index_primmodsourcecoordinates jt_value_primmodsourcecoordinates. (exists jt_gap_primmodsourcecoordinatesindex. jt_gap_primmodsourcecoordinatesindex+S (jt_index_primmodsourcecoordinates)=(k)) -> (((exists fs_h_jt_primmodsourcecoordinatesat. fs_h_jt_primmodsourcecoordinatesat + S (jt_value_primmodsourcecoordinates) = S ((S (jt_index_primmodsourcecoordinates)) * c)) /\\ exists fs_q_jt_primmodsourcecoordinatesat. b = fs_q_jt_primmodsourcecoordinatesat * S ((S (jt_index_primmodsourcecoordinates)) * c) + (jt_value_primmodsourcecoordinates))) -> (exists jt_factor_primmodsourcecoordinatesdivides. (jt_value_primmodsourcecoordinates)=(jt_divisor_primmodsource)*jt_factor_primmodsourcecoordinatesdivides)) -> jt_divisor_primmodsource=1) -> (forall jt_divisor_primmodtarget. (exists jt_factor_primmodtargetmodulus. (n)=(jt_divisor_primmodtarget)*jt_factor_primmodtargetmodulus) -> (forall jt_index_primmodtargetcoordinates jt_value_primmodtargetcoordinates. (exists jt_gap_primmodtargetcoordinatesindex. jt_gap_primmodtargetcoordinatesindex+S (jt_index_primmodtargetcoordinates)=(k)) -> (((exists fs_h_jt_primmodtargetcoordinatesat. fs_h_jt_primmodtargetcoordinatesat + S (jt_value_primmodtargetcoordinates) = S ((S (jt_index_primmodtargetcoordinates)) * e)) /\\ exists fs_q_jt_primmodtargetcoordinatesat. d = fs_q_jt_primmodtargetcoordinatesat * S ((S (jt_index_primmodtargetcoordinates)) * e) + (jt_value_primmodtargetcoordinates))) -> (exists jt_factor_primmodtargetcoordinatesdivides. (jt_value_primmodtargetcoordinates)=(jt_divisor_primmodtarget)*jt_factor_primmodtargetcoordinatesdivides)) -> jt_divisor_primmodtarget=1)", "statement_sha256": "c1054866cc6baf40e8eb2f61ae8ce8899d8312696287b2088facef0cad2f2ffe", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Primitivity is invariant under actual coordinatewise residue transport, not field evaluation." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 18, "body_proof_nodes": 27, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro d", "intro b", "intro c", "intro i", "intro a", "intro hi", "intro ha", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "defined_statement": "∀ d. ∀ b. ∀ c. JordanTupleAllDivisible(d,b,c,0)", "defined_statement_sha256": "4f8e156297d76ff454b5e30f9fcd901db0cba88ed16e0bd8fb8f4ffa74e70330", "definition_uses": { "ND0371": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "859657c93d9cf84f8cc2d8311972be782a478031cf8d3c4741acbb8e5c10ebca", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (i)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "specialize zero_le (i)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "ND0371": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ d. ∀ b. ∀ c. " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,0)" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0010", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_all_divisible_empty", "original_ha_bundle_verified": true, "proof_bundle_node_id": 273, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro d", "intro b", "intro c", "intro i", "intro a", "intro hi", "intro ha", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "script_sha256": "572b608fc8f90236581fce2a05920d378001232ab9ef2e43ccc7b5924e246710", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "572b608fc8f90236581fce2a05920d378001232ab9ef2e43ccc7b5924e246710", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "859657c93d9cf84f8cc2d8311972be782a478031cf8d3c4741acbb8e5c10ebca" } ], "stable_member": false, "statement": "forall d b c. forall jt_index_allempty jt_value_allempty. (exists jt_gap_allemptyindex. jt_gap_allemptyindex+S (jt_index_allempty)=(0)) -> (((exists fs_h_jt_allemptyat. fs_h_jt_allemptyat + S (jt_value_allempty) = S ((S (jt_index_allempty)) * c)) /\\ exists fs_q_jt_allemptyat. b = fs_q_jt_allemptyat * S ((S (jt_index_allempty)) * c) + (jt_value_allempty))) -> (exists jt_factor_allemptydivides. (jt_value_allempty)=(d)*jt_factor_allemptydivides)", "statement_sha256": "859657c93d9cf84f8cc2d8311972be782a478031cf8d3c4741acbb8e5c10ebca", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every divisor vacuously divides all entries of an empty tuple." }, { "admission_dependencies": [ "finite_lt_succ_eq_or_lt", "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 25, "body_proof_nodes": 44, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro d", "intro b", "intro c", "intro k", "intro a", "intro hprefix", "intro ha", "intro hda", "intro i", "intro z", "intro hi", "intro hz", "have hcases : i = k ∨ Lt(i,k)", "specialize finite_lt_succ_eq_or_lt (k)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hcases", "have heq : z=a", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (k)", "specialize beta_at_unique (z)", "specialize beta_at_unique (a)", "apply beta_at_unique", "rewrite hcases_left at hz", "rewrite hcases_left at hz", "exact hz", "exact ha", "rewrite <- heq at hda", "exact hda", "specialize hprefix (i)", "specialize hprefix (z)", "apply hprefix", "exact hcases_right", "exact hz" ], "defined_statement": "∀ d. ∀ b. ∀ c. ∀ k. ∀ a. JordanTupleAllDivisible(d,b,c,k) → BetaAt(b,c,k,a) → Dvd(d,a) → JordanTupleAllDivisible(d,b,c,S k)", "defined_statement_sha256": "5d498ffb25105cf187a49867e2f2c41bdd9d70fce0526dc07c86a0b6dae8d3e4", "definition_uses": { "ND0371": 2, "PD0002": 1, "PD0003": 1, "PD0013": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "f04c24a1d4778893a13f1f86e41d594069a382f06b904e04732a2125c3992185", "free_names": [], "script_definition_uses": { "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hprefix" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hda" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "have hcases : " }, { "kind": "text", "text": "i = k ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,k)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (k)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (i)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hcases" } ], [ { "kind": "text", "text": "have heq : z=a" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (k)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (z)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hcases_left at hz" } ], [ { "kind": "text", "text": "rewrite hcases_left at hz" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "rewrite <- heq at hda" } ], [ { "kind": "text", "text": "exact hda" } ], [ { "kind": "text", "text": "specialize hprefix (i)" } ], [ { "kind": "text", "text": "specialize hprefix (z)" } ], [ { "kind": "text", "text": "apply hprefix" } ], [ { "kind": "text", "text": "exact hcases_right" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0371": 2, "PD0003": 1, "PD0013": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ d. ∀ b. ∀ c. ∀ k. ∀ a. " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,k,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,a)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,S k)" } ] }, "dependencies": [ "finite_lt_succ_eq_or_lt", "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0011", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_all_divisible_extend", "original_ha_bundle_verified": true, "proof_bundle_node_id": 274, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro d", "intro b", "intro c", "intro k", "intro a", "intro hprefix", "intro ha", "intro hda", "intro i", "intro z", "intro hi", "intro hz", "have hcases : i=k \\/ (exists jt_gap_allcases. jt_gap_allcases+S (i)=(k))", "specialize finite_lt_succ_eq_or_lt (k)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hcases", "have heq : z=a", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (k)", "specialize beta_at_unique (z)", "specialize beta_at_unique (a)", "apply beta_at_unique", "rewrite hcases_left at hz", "rewrite hcases_left at hz", "exact hz", "exact ha", "rewrite <- heq at hda", "exact hda", "specialize hprefix (i)", "specialize hprefix (z)", "apply hprefix", "exact hcases_right", "exact hz" ], "script_sha256": "2d1931365da0c9bbb9e0efbb98fabbfe20c9d2bc1b24c5af74d2c75900ade89c", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "2d1931365da0c9bbb9e0efbb98fabbfe20c9d2bc1b24c5af74d2c75900ade89c", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "f04c24a1d4778893a13f1f86e41d594069a382f06b904e04732a2125c3992185" } ], "stable_member": false, "statement": "forall d b c k a. (forall jt_index_allprefix jt_value_allprefix. (exists jt_gap_allprefixindex. jt_gap_allprefixindex+S (jt_index_allprefix)=(k)) -> (((exists fs_h_jt_allprefixat. fs_h_jt_allprefixat + S (jt_value_allprefix) = S ((S (jt_index_allprefix)) * c)) /\\ exists fs_q_jt_allprefixat. b = fs_q_jt_allprefixat * S ((S (jt_index_allprefix)) * c) + (jt_value_allprefix))) -> (exists jt_factor_allprefixdivides. (jt_value_allprefix)=(d)*jt_factor_allprefixdivides)) -> (((exists fs_h_jt_allentry. fs_h_jt_allentry + S (a) = S ((S (k)) * c)) /\\ exists fs_q_jt_allentry. b = fs_q_jt_allentry * S ((S (k)) * c) + (a))) -> (exists jt_factor_allvalue. (a)=(d)*jt_factor_allvalue) -> (forall jt_index_allsuccessor jt_value_allsuccessor. (exists jt_gap_allsuccessorindex. jt_gap_allsuccessorindex+S (jt_index_allsuccessor)=(S k)) -> (((exists fs_h_jt_allsuccessorat. fs_h_jt_allsuccessorat + S (jt_value_allsuccessor) = S ((S (jt_index_allsuccessor)) * c)) /\\ exists fs_q_jt_allsuccessorat. b = fs_q_jt_allsuccessorat * S ((S (jt_index_allsuccessor)) * c) + (jt_value_allsuccessor))) -> (exists jt_factor_allsuccessordivides. (jt_value_allsuccessor)=(d)*jt_factor_allsuccessordivides))", "statement_sha256": "f04c24a1d4778893a13f1f86e41d594069a382f06b904e04732a2125c3992185", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Adjoining an actually decoded divisible entry preserves common divisibility." }, { "admission_dependencies": [ "jordan_tuple_all_divisible_empty", "beta_at_exists", "multiple_decidable", "jordan_tuple_all_divisible_extend", "le_refl", "le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 27, "body_proof_nodes": 72, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro d", "intro b", "intro c", "induction k", "left", "specialize jordan_tuple_all_divisible_empty (d)", "specialize jordan_tuple_all_divisible_empty (b)", "specialize jordan_tuple_all_divisible_empty (c)", "apply jordan_tuple_all_divisible_empty", "have ha : ∃ a. BetaAt(b,c,k,a)", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases ha", "have hd : Dvd(d,x) ∨ ¬Dvd(d,x)", "specialize multiple_decidable (d)", "specialize multiple_decidable (x)", "apply multiple_decidable", "cases IH", "cases hd", "left", "specialize jordan_tuple_all_divisible_extend (d)", "specialize jordan_tuple_all_divisible_extend (b)", "specialize jordan_tuple_all_divisible_extend (c)", "specialize jordan_tuple_all_divisible_extend (k)", "specialize jordan_tuple_all_divisible_extend (x)", "apply jordan_tuple_all_divisible_extend", "exact IH_left", "exact ha_witness", "exact hd_left", "right", "intro h", "apply hd_right", "specialize h (k)", "specialize h (x)", "apply h", "specialize le_refl (S k)", "apply le_refl", "exact ha_witness", "right", "intro h", "apply IH_right", "intro i", "intro a", "intro hi", "intro hat", "specialize h (i)", "specialize h (a)", "apply h", "specialize le_succ (S i)", "specialize le_succ (k)", "apply le_succ", "exact hi", "exact hat" ], "defined_statement": "∀ d. ∀ b. ∀ c. ∀ k. JordanTupleAllDivisible(d,b,c,k) ∨ ¬JordanTupleAllDivisible(d,b,c,k)", "defined_statement_sha256": "dca5c1366523f141335f71199cb82ce28d51ad5adfc03f8f16fffee751a74bda", "definition_uses": { "ND0371": 2, "PD0003": 2, "PD0013": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "15d6707edb86e6ec0e9566edee470d63c926e402ae6cfe57ac5c8bca5dae0d80", "free_names": [], "script_definition_uses": { "PD0003": 2, "PD0013": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "induction k" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_empty (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_empty (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_empty (c)" } ], [ { "kind": "text", "text": "apply jordan_tuple_all_divisible_empty" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ a. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,k,a)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (b)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (c)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (k)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "have hd : " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,x)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,x)" } ], [ { "kind": "text", "text": "specialize multiple_decidable (d)" } ], [ { "kind": "text", "text": "specialize multiple_decidable (x)" } ], [ { "kind": "text", "text": "apply multiple_decidable" } ], [ { "kind": "text", "text": "cases IH" } ], [ { "kind": "text", "text": "cases hd" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_extend (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_extend (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_extend (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_extend (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_extend (x)" } ], [ { "kind": "text", "text": "apply jordan_tuple_all_divisible_extend" } ], [ { "kind": "text", "text": "exact IH_left" } ], [ { "kind": "text", "text": "exact ha_witness" } ], [ { "kind": "text", "text": "exact hd_left" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply hd_right" } ], [ { "kind": "text", "text": "specialize h (k)" } ], [ { "kind": "text", "text": "specialize h (x)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "specialize le_refl (S k)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "exact ha_witness" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply IH_right" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hat" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "specialize h (a)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "specialize le_succ (S i)" } ], [ { "kind": "text", "text": "specialize le_succ (k)" } ], [ { "kind": "text", "text": "apply le_succ" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hat" } ] ], "statement_definition_uses": { "ND0371": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ d. ∀ b. ∀ c. ∀ k. " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" } ] }, "dependencies": [ "jordan_tuple_all_divisible_empty", "beta_at_exists", "multiple_decidable", "jordan_tuple_all_divisible_extend", "le_refl", "le_succ" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0012", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_all_divisible_decidable", "original_ha_bundle_verified": true, "proof_bundle_node_id": 275, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro d", "intro b", "intro c", "induction k", "left", "specialize jordan_tuple_all_divisible_empty (d)", "specialize jordan_tuple_all_divisible_empty (b)", "specialize jordan_tuple_all_divisible_empty (c)", "apply jordan_tuple_all_divisible_empty", "have ha : exists a. ((exists fs_h_jt_alllast. fs_h_jt_alllast + S (a) = S ((S (k)) * c)) /\\ exists fs_q_jt_alllast. b = fs_q_jt_alllast * S ((S (k)) * c) + (a))", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases ha", "have hd : (exists jt_factor_allyes. (x)=(d)*jt_factor_allyes) \\/ ~(exists jt_factor_allno. (x)=(d)*jt_factor_allno)", "specialize multiple_decidable (d)", "specialize multiple_decidable (x)", "apply multiple_decidable", "cases IH", "cases hd", "left", "specialize jordan_tuple_all_divisible_extend (d)", "specialize jordan_tuple_all_divisible_extend (b)", "specialize jordan_tuple_all_divisible_extend (c)", "specialize jordan_tuple_all_divisible_extend (k)", "specialize jordan_tuple_all_divisible_extend (x)", "apply jordan_tuple_all_divisible_extend", "exact IH_left", "exact ha_witness", "exact hd_left", "right", "intro h", "apply hd_right", "specialize h (k)", "specialize h (x)", "apply h", "specialize le_refl (S k)", "apply le_refl", "exact ha_witness", "right", "intro h", "apply IH_right", "intro i", "intro a", "intro hi", "intro hat", "specialize h (i)", "specialize h (a)", "apply h", "specialize le_succ (S i)", "specialize le_succ (k)", "apply le_succ", "exact hi", "exact hat" ], "script_sha256": "190645d0d0560e2aa461589ad5e8c1c528b25be93478e17191b07184708f269a", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "190645d0d0560e2aa461589ad5e8c1c528b25be93478e17191b07184708f269a", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "15d6707edb86e6ec0e9566edee470d63c926e402ae6cfe57ac5c8bca5dae0d80" } ], "stable_member": false, "statement": "forall d b c k. (forall jt_index_alldecyes jt_value_alldecyes. (exists jt_gap_alldecyesindex. jt_gap_alldecyesindex+S (jt_index_alldecyes)=(k)) -> (((exists fs_h_jt_alldecyesat. fs_h_jt_alldecyesat + S (jt_value_alldecyes) = S ((S (jt_index_alldecyes)) * c)) /\\ exists fs_q_jt_alldecyesat. b = fs_q_jt_alldecyesat * S ((S (jt_index_alldecyes)) * c) + (jt_value_alldecyes))) -> (exists jt_factor_alldecyesdivides. (jt_value_alldecyes)=(d)*jt_factor_alldecyesdivides)) \\/ ~(forall jt_index_alldecno jt_value_alldecno. (exists jt_gap_alldecnoindex. jt_gap_alldecnoindex+S (jt_index_alldecno)=(k)) -> (((exists fs_h_jt_alldecnoat. fs_h_jt_alldecnoat + S (jt_value_alldecno) = S ((S (jt_index_alldecno)) * c)) /\\ exists fs_q_jt_alldecnoat. b = fs_q_jt_alldecnoat * S ((S (jt_index_alldecno)) * c) + (jt_value_alldecno))) -> (exists jt_factor_alldecnodivides. (jt_value_alldecno)=(d)*jt_factor_alldecnodivides))", "statement_sha256": "15d6707edb86e6ec0e9566edee470d63c926e402ae6cfe57ac5c8bca5dae0d80", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Finite induction decides common divisibility from genuine beta entries; no bounded-code oracle." }, { "admission_dependencies": [ "eq_decidable", "multiple_decidable", "jordan_tuple_all_divisible_decidable" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 17, "body_proof_nodes": 49, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro k", "intro d", "have heq : d=1 \\/ ~(d=1)", "specialize eq_decidable (d)", "specialize eq_decidable (1)", "apply eq_decidable", "cases heq", "left", "intro hd", "intro hall", "exact heq_left", "have hd : Dvd(d,n) ∨ ¬Dvd(d,n)", "specialize multiple_decidable (d)", "specialize multiple_decidable (n)", "apply multiple_decidable", "cases hd", "have hall : JordanTupleAllDivisible(d,b,c,k) ∨ ¬JordanTupleAllDivisible(d,b,c,k)", "specialize jordan_tuple_all_divisible_decidable (d)", "specialize jordan_tuple_all_divisible_decidable (b)", "specialize jordan_tuple_all_divisible_decidable (c)", "specialize jordan_tuple_all_divisible_decidable (k)", "apply jordan_tuple_all_divisible_decidable", "cases hall", "right", "intro h", "apply heq_right", "apply h", "exact hd_left", "exact hall_left", "left", "intro hdiv", "intro hcoords", "exfalso", "apply hall_right", "exact hcoords", "left", "intro hdiv", "intro hcoords", "exfalso", "apply hd_right", "exact hdiv" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ k. ∀ d. (Dvd(d,n) → JordanTupleAllDivisible(d,b,c,k) → d = 1) ∨ ¬(Dvd(d,n) → JordanTupleAllDivisible(d,b,c,k) → d = 1)", "defined_statement_sha256": "42fff438bed90b391ab09b58023a1c7e98db32c9cb56efe873a6cd28e0a54956", "definition_uses": { "ND0371": 4, "PD0003": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "cbfa0bf321b6354435258d1434bb2f175a61274beae731fd6225b8ea4d3fed40", "free_names": [], "script_definition_uses": { "ND0371": 2, "PD0003": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "have heq : d=1 \\/ ~(d=1)" } ], [ { "kind": "text", "text": "specialize eq_decidable (d)" } ], [ { "kind": "text", "text": "specialize eq_decidable (1)" } ], [ { "kind": "text", "text": "apply eq_decidable" } ], [ { "kind": "text", "text": "cases heq" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "exact heq_left" } ], [ { "kind": "text", "text": "have hd : " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,n)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,n)" } ], [ { "kind": "text", "text": "specialize multiple_decidable (d)" } ], [ { "kind": "text", "text": "specialize multiple_decidable (n)" } ], [ { "kind": "text", "text": "apply multiple_decidable" } ], [ { "kind": "text", "text": "cases hd" } ], [ { "kind": "text", "text": "have hall : " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_decidable (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_decidable (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_decidable (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_all_divisible_decidable (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_all_divisible_decidable" } ], [ { "kind": "text", "text": "cases hall" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply heq_right" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "exact hd_left" } ], [ { "kind": "text", "text": "exact hall_left" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hcoords" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hall_right" } ], [ { "kind": "text", "text": "exact hcoords" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hcoords" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hd_right" } ], [ { "kind": "text", "text": "exact hdiv" } ] ], "statement_definition_uses": { "ND0371": 2, "PD0003": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ k. ∀ d. (" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" }, { "kind": "text", "text": " → d = 1) ∨ ¬(" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(d,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(d,b,c,k)" }, { "kind": "text", "text": " → d = 1)" } ] }, "dependencies": [ "eq_decidable", "multiple_decidable", "jordan_tuple_all_divisible_decidable" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0013", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_divisor_test_decidable", "original_ha_bundle_verified": true, "proof_bundle_node_id": 276, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro k", "intro d", "have heq : d=1 \\/ ~(d=1)", "specialize eq_decidable (d)", "specialize eq_decidable (1)", "apply eq_decidable", "cases heq", "left", "intro hd", "intro hall", "exact heq_left", "have hd : (exists jt_factor_leafyes. (n)=(d)*jt_factor_leafyes) \\/ ~(exists jt_factor_leafno. (n)=(d)*jt_factor_leafno)", "specialize multiple_decidable (d)", "specialize multiple_decidable (n)", "apply multiple_decidable", "cases hd", "have hall : (forall jt_index_leafallyes jt_value_leafallyes. (exists jt_gap_leafallyesindex. jt_gap_leafallyesindex+S (jt_index_leafallyes)=(k)) -> (((exists fs_h_jt_leafallyesat. fs_h_jt_leafallyesat + S (jt_value_leafallyes) = S ((S (jt_index_leafallyes)) * c)) /\\ exists fs_q_jt_leafallyesat. b = fs_q_jt_leafallyesat * S ((S (jt_index_leafallyes)) * c) + (jt_value_leafallyes))) -> (exists jt_factor_leafallyesdivides. (jt_value_leafallyes)=(d)*jt_factor_leafallyesdivides)) \\/ ~(forall jt_index_leafallno jt_value_leafallno. (exists jt_gap_leafallnoindex. jt_gap_leafallnoindex+S (jt_index_leafallno)=(k)) -> (((exists fs_h_jt_leafallnoat. fs_h_jt_leafallnoat + S (jt_value_leafallno) = S ((S (jt_index_leafallno)) * c)) /\\ exists fs_q_jt_leafallnoat. b = fs_q_jt_leafallnoat * S ((S (jt_index_leafallno)) * c) + (jt_value_leafallno))) -> (exists jt_factor_leafallnodivides. 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((exists jt_factor_leaftestyesmodulus. (n)=(d)*jt_factor_leaftestyesmodulus) -> (forall jt_index_leaftestyescoordinates jt_value_leaftestyescoordinates. (exists jt_gap_leaftestyescoordinatesindex. jt_gap_leaftestyescoordinatesindex+S (jt_index_leaftestyescoordinates)=(k)) -> (((exists fs_h_jt_leaftestyescoordinatesat. fs_h_jt_leaftestyescoordinatesat + S (jt_value_leaftestyescoordinates) = S ((S (jt_index_leaftestyescoordinates)) * c)) /\\ exists fs_q_jt_leaftestyescoordinatesat. b = fs_q_jt_leaftestyescoordinatesat * S ((S (jt_index_leaftestyescoordinates)) * c) + (jt_value_leaftestyescoordinates))) -> (exists jt_factor_leaftestyescoordinatesdivides. (jt_value_leaftestyescoordinates)=(d)*jt_factor_leaftestyescoordinatesdivides)) -> (d)=1) \\/ ~((exists jt_factor_leaftestnomodulus. (n)=(d)*jt_factor_leaftestnomodulus) -> (forall jt_index_leaftestnocoordinates jt_value_leaftestnocoordinates. (exists jt_gap_leaftestnocoordinatesindex. jt_gap_leaftestnocoordinatesindex+S (jt_index_leaftestnocoordinates)=(k)) -> (((exists fs_h_jt_leaftestnocoordinatesat. fs_h_jt_leaftestnocoordinatesat + S (jt_value_leaftestnocoordinates) = S ((S (jt_index_leaftestnocoordinates)) * c)) /\\ exists fs_q_jt_leaftestnocoordinatesat. b = fs_q_jt_leaftestnocoordinatesat * S ((S (jt_index_leaftestnocoordinates)) * c) + (jt_value_leaftestnocoordinates))) -> (exists jt_factor_leaftestnocoordinatesdivides. (jt_value_leaftestnocoordinates)=(d)*jt_factor_leaftestnocoordinatesdivides)) -> (d)=1)", "statement_sha256": "cbfa0bf321b6354435258d1434bb2f175a61274beae731fd6225b8ea4d3fed40", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Decide each actual common-divisor-one implication constructively." }, { "admission_dependencies": [ "lt_not_le", "zero_le", "jordan_tuple_divisor_test_decidable", "finite_lt_succ_eq_or_lt", "le_refl", "le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 33, "body_proof_nodes": 113, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro k", "induction L", "left", "intro d", "intro hd", "intro hdiv", "intro hall", "exfalso", "specialize lt_not_le (d)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hd", "specialize zero_le (d)", "apply zero_le", "have ht : (Dvd(L,n) → JordanTupleAllDivisible(L,b,c,k) → L = 1) ∨ ¬(Dvd(L,n) → JordanTupleAllDivisible(L,b,c,k) → L = 1)", "specialize jordan_tuple_divisor_test_decidable (n)", "specialize jordan_tuple_divisor_test_decidable (b)", "specialize jordan_tuple_divisor_test_decidable (c)", "specialize jordan_tuple_divisor_test_decidable (k)", "specialize jordan_tuple_divisor_test_decidable (L)", "apply jordan_tuple_divisor_test_decidable", "cases IH", "cases ht", "left", "intro d", "intro hd", "have hc : d = L ∨ Lt(d,L)", "specialize finite_lt_succ_eq_or_lt (L)", "specialize finite_lt_succ_eq_or_lt (d)", "apply finite_lt_succ_eq_or_lt", "exact hd", "cases hc", "intro hdiv", "intro hall", "have heq : L=1", "apply ht_left", "rewrite <- hc_left", "exact hdiv", "intro i", "intro a", "intro hi", "intro ha", "rewrite <- hc_left", "specialize hall (i)", "specialize hall (a)", "apply hall", "exact hi", "exact ha", "trans L", "exact hc_left", "exact heq", "intro hdiv", "intro hall", "specialize IH_left (d)", "apply IH_left", "exact hc_right", "exact hdiv", "exact hall", "right", "intro h", "apply ht_right", "intro hdiv", "intro hall", "specialize h (L)", "apply h", "specialize le_refl (S L)", "apply le_refl", "exact hdiv", "exact hall", "right", "intro h", "apply IH_right", "intro d", "intro hd", "intro hdiv", "intro hall", "specialize h (d)", "apply h", "specialize le_succ (S d)", "specialize le_succ (L)", "apply le_succ", "exact hd", "exact hdiv", "exact hall" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ k. ∀ L. (∀ x. Lt(x,L) → Dvd(x,n) → JordanTupleAllDivisible(x,b,c,k) → x = 1) ∨ ¬(∀ x. Lt(x,L) → Dvd(x,n) → JordanTupleAllDivisible(x,b,c,k) → x = 1)", "defined_statement_sha256": "00833f7bff53bec538a1519f90331a9f96cb6c42efa5c23f2fec7292aaa427f2", "definition_uses": { "ND0371": 4, "PD0002": 3, "PD0003": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "9ab63be1cf6614eee122aabeabb2f3f4e3927cad520a2ddc5bb4cebea7e43766", "free_names": [], "script_definition_uses": { "ND0371": 2, "PD0002": 1, "PD0003": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "induction L" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (d)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "specialize zero_le (d)" } ], [ { "kind": "text", "text": "apply zero_le" } ], [ { "kind": "text", "text": "have ht : " }, { "kind": "text", "text": "(" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(L,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(L,b,c,k)" }, { "kind": "text", "text": " → L = 1) ∨ ¬(" }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(L,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(L,b,c,k)" }, { "kind": "text", "text": " → L = 1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_test_decidable (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_test_decidable (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_test_decidable (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_test_decidable (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_test_decidable (L)" } ], [ { "kind": "text", "text": "apply jordan_tuple_divisor_test_decidable" } ], [ { "kind": "text", "text": "cases IH" } ], [ { "kind": "text", "text": "cases ht" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "d = L ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(d,L)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (L)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (d)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "have heq : L=1" } ], [ { "kind": "text", "text": "apply ht_left" } ], [ { "kind": "text", "text": "rewrite <- hc_left" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "rewrite <- hc_left" } ], [ { "kind": "text", "text": "specialize hall (i)" } ], [ { "kind": "text", "text": "specialize hall (a)" } ], [ { "kind": "text", "text": "apply hall" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "trans L" } ], [ { "kind": "text", "text": "exact hc_left" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize IH_left (d)" } ], [ { "kind": "text", "text": "apply IH_left" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "exact hall" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply ht_right" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize h (L)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "specialize le_refl (S L)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "exact hall" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply IH_right" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize h (d)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "specialize le_succ (S d)" } ], [ { "kind": "text", "text": "specialize le_succ (L)" } ], [ { "kind": "text", "text": "apply le_succ" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0371": 2, "PD0002": 2, "PD0003": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ k. ∀ L. (∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,L)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(x,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(x,b,c,k)" }, { "kind": "text", "text": " → x = 1) ∨ ¬(∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,L)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(x,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(x,b,c,k)" }, { "kind": "text", "text": " → x = 1)" } ] }, "dependencies": [ "lt_not_le", "zero_le", "jordan_tuple_divisor_test_decidable", "finite_lt_succ_eq_or_lt", "le_refl", "le_succ" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0014", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_primitive_bounded_decidable", "original_ha_bundle_verified": true, "proof_bundle_node_id": 277, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro k", "induction L", "left", "intro d", "intro hd", "intro hdiv", "intro hall", "exfalso", "specialize lt_not_le (d)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hd", "specialize zero_le (d)", "apply zero_le", "have ht : ((exists jt_factor_boundedyesmodulus. (n)=(L)*jt_factor_boundedyesmodulus) -> (forall jt_index_boundedyescoordinates jt_value_boundedyescoordinates. (exists jt_gap_boundedyescoordinatesindex. jt_gap_boundedyescoordinatesindex+S (jt_index_boundedyescoordinates)=(k)) -> (((exists fs_h_jt_boundedyescoordinatesat. fs_h_jt_boundedyescoordinatesat + S (jt_value_boundedyescoordinates) = S ((S (jt_index_boundedyescoordinates)) * c)) /\\ exists fs_q_jt_boundedyescoordinatesat. b = fs_q_jt_boundedyescoordinatesat * S ((S (jt_index_boundedyescoordinates)) * c) + (jt_value_boundedyescoordinates))) -> (exists jt_factor_boundedyescoordinatesdivides. (jt_value_boundedyescoordinates)=(L)*jt_factor_boundedyescoordinatesdivides)) -> (L)=1) \\/ ~((exists jt_factor_boundednomodulus. (n)=(L)*jt_factor_boundednomodulus) -> (forall jt_index_boundednocoordinates jt_value_boundednocoordinates. (exists jt_gap_boundednocoordinatesindex. jt_gap_boundednocoordinatesindex+S (jt_index_boundednocoordinates)=(k)) -> (((exists fs_h_jt_boundednocoordinatesat. fs_h_jt_boundednocoordinatesat + S (jt_value_boundednocoordinates) = S ((S (jt_index_boundednocoordinates)) * c)) /\\ exists fs_q_jt_boundednocoordinatesat. b = fs_q_jt_boundednocoordinatesat * S ((S (jt_index_boundednocoordinates)) * c) + (jt_value_boundednocoordinates))) -> (exists jt_factor_boundednocoordinatesdivides. (jt_value_boundednocoordinates)=(L)*jt_factor_boundednocoordinatesdivides)) -> (L)=1)", "specialize jordan_tuple_divisor_test_decidable (n)", "specialize jordan_tuple_divisor_test_decidable (b)", "specialize jordan_tuple_divisor_test_decidable (c)", "specialize jordan_tuple_divisor_test_decidable (k)", "specialize jordan_tuple_divisor_test_decidable (L)", "apply jordan_tuple_divisor_test_decidable", "cases IH", "cases ht", "left", "intro d", "intro hd", "have hc : d=L \\/ (exists jt_gap_boundedcases. jt_gap_boundedcases+S (d)=(L))", "specialize finite_lt_succ_eq_or_lt (L)", "specialize finite_lt_succ_eq_or_lt (d)", "apply finite_lt_succ_eq_or_lt", "exact hd", "cases hc", "intro hdiv", "intro hall", "have heq : L=1", "apply ht_left", "rewrite <- hc_left", "exact hdiv", "intro i", "intro a", "intro hi", "intro ha", "rewrite <- hc_left", "specialize hall (i)", "specialize hall (a)", "apply hall", "exact hi", "exact ha", "trans L", "exact hc_left", "exact heq", "intro hdiv", "intro hall", "specialize IH_left (d)", "apply IH_left", "exact hc_right", "exact hdiv", "exact hall", "right", "intro h", "apply ht_right", "intro hdiv", "intro hall", "specialize h (L)", "apply h", "specialize le_refl (S L)", "apply le_refl", "exact hdiv", "exact hall", "right", "intro h", "apply IH_right", "intro d", "intro hd", "intro hdiv", "intro hall", "specialize h (d)", "apply h", "specialize le_succ (S d)", "specialize le_succ (L)", "apply le_succ", "exact hd", "exact hdiv", "exact hall" ], "script_sha256": "a966edea80470ff6445f2c9f12f5b05deb4f2811d75c4e0750144c078255b35a", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "a966edea80470ff6445f2c9f12f5b05deb4f2811d75c4e0750144c078255b35a", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "9ab63be1cf6614eee122aabeabb2f3f4e3927cad520a2ddc5bb4cebea7e43766" } ], "stable_member": false, "statement": "forall n b c k L. (forall jt_divisor_bounddec_yes. (exists jt_gap_bounddec_yesbound. jt_gap_bounddec_yesbound+S (jt_divisor_bounddec_yes)=(L)) -> ((exists jt_factor_bounddec_yestestmodulus. (n)=(jt_divisor_bounddec_yes)*jt_factor_bounddec_yestestmodulus) -> (forall jt_index_bounddec_yestestcoordinates jt_value_bounddec_yestestcoordinates. (exists jt_gap_bounddec_yestestcoordinatesindex. jt_gap_bounddec_yestestcoordinatesindex+S (jt_index_bounddec_yestestcoordinates)=(k)) -> (((exists fs_h_jt_bounddec_yestestcoordinatesat. fs_h_jt_bounddec_yestestcoordinatesat + S (jt_value_bounddec_yestestcoordinates) = S ((S (jt_index_bounddec_yestestcoordinates)) * c)) /\\ exists fs_q_jt_bounddec_yestestcoordinatesat. b = fs_q_jt_bounddec_yestestcoordinatesat * S ((S (jt_index_bounddec_yestestcoordinates)) * c) + (jt_value_bounddec_yestestcoordinates))) -> (exists jt_factor_bounddec_yestestcoordinatesdivides. (jt_value_bounddec_yestestcoordinates)=(jt_divisor_bounddec_yes)*jt_factor_bounddec_yestestcoordinatesdivides)) -> (jt_divisor_bounddec_yes)=1)) \\/ ~(forall jt_divisor_bounddec_no. (exists jt_gap_bounddec_nobound. jt_gap_bounddec_nobound+S (jt_divisor_bounddec_no)=(L)) -> ((exists jt_factor_bounddec_notestmodulus. (n)=(jt_divisor_bounddec_no)*jt_factor_bounddec_notestmodulus) -> (forall jt_index_bounddec_notestcoordinates jt_value_bounddec_notestcoordinates. (exists jt_gap_bounddec_notestcoordinatesindex. jt_gap_bounddec_notestcoordinatesindex+S (jt_index_bounddec_notestcoordinates)=(k)) -> (((exists fs_h_jt_bounddec_notestcoordinatesat. fs_h_jt_bounddec_notestcoordinatesat + S (jt_value_bounddec_notestcoordinates) = S ((S (jt_index_bounddec_notestcoordinates)) * c)) /\\ exists fs_q_jt_bounddec_notestcoordinatesat. b = fs_q_jt_bounddec_notestcoordinatesat * S ((S (jt_index_bounddec_notestcoordinates)) * c) + (jt_value_bounddec_notestcoordinates))) -> (exists jt_factor_bounddec_notestcoordinatesdivides. (jt_value_bounddec_notestcoordinates)=(jt_divisor_bounddec_no)*jt_factor_bounddec_notestcoordinatesdivides)) -> (jt_divisor_bounddec_no)=1))", "statement_sha256": "9ab63be1cf6614eee122aabeabb2f3f4e3927cad520a2ddc5bb4cebea7e43766", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A finite sweep decides the primitive common-divisor condition up to any natural bound." }, { "admission_dependencies": [ "jordan_tuple_primitive_bounded_decidable", "divisor_le_nonzero" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 23, "body_proof_nodes": 57, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro k", "intro hn", "have hd : (∀ x. Lt(x,S n) → Dvd(x,n) → JordanTupleAllDivisible(x,b,c,k) → x = 1) ∨ ¬(∀ x. Lt(x,S n) → Dvd(x,n) → JordanTupleAllDivisible(x,b,c,k) → x = 1)", "specialize jordan_tuple_primitive_bounded_decidable (n)", "specialize jordan_tuple_primitive_bounded_decidable (b)", "specialize jordan_tuple_primitive_bounded_decidable (c)", "specialize jordan_tuple_primitive_bounded_decidable (k)", "specialize jordan_tuple_primitive_bounded_decidable (S n)", "apply jordan_tuple_primitive_bounded_decidable", "cases hd", "left", "intro d", "intro hdiv", "intro hall", "specialize hd_left (d)", "apply hd_left", "have hb : Le(d,n)", "specialize divisor_le_nonzero (d)", "specialize divisor_le_nonzero (n)", "apply divisor_le_nonzero", "exact hn", "exact hdiv", "cases hb", "exists x", "trans S (x+d)", "rewrite PA4", "refl", "congr", "exact hb_witness", "exact hdiv", "exact hall", "right", "intro hp", "apply hd_right", "intro d", "intro hbound", "intro hdiv", "intro hall", "specialize hp (d)", "apply hp", "exact hdiv", "exact hall" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ k. ¬n = 0 → JordanPrimitiveTuple(n,b,c,k) ∨ ¬JordanPrimitiveTuple(n,b,c,k)", "defined_statement_sha256": "5262df96c4cdc7eda70f04688af25911206ce7e4f164605c0746909f44850af3", "definition_uses": { "ND0371": 2, "ND0372": 2, "PD0001": 1, "PD0002": 2, "PD0003": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "3aff4616b95e229cc90e41220a94bafcd2492c670272843ee7b8181f143eac4b", "free_names": [], "script_definition_uses": { "ND0371": 2, "PD0001": 1, "PD0002": 2, "PD0003": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "have hd : " }, { "kind": "text", "text": "(∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,S n)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(x,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(x,b,c,k)" }, { "kind": "text", "text": " → x = 1) ∨ ¬(∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,S n)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(x,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(x,b,c,k)" }, { "kind": "text", "text": " → x = 1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_primitive_bounded_decidable (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_primitive_bounded_decidable (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_primitive_bounded_decidable (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_primitive_bounded_decidable (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_primitive_bounded_decidable (S n)" } ], [ { "kind": "text", "text": "apply jordan_tuple_primitive_bounded_decidable" } ], [ { "kind": "text", "text": "cases hd" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize hd_left (d)" } ], [ { "kind": "text", "text": "apply hd_left" } ], [ { "kind": "text", "text": "have hb : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(d,n)" } ], [ { "kind": "text", "text": "specialize divisor_le_nonzero (d)" } ], [ { "kind": "text", "text": "specialize divisor_le_nonzero (n)" } ], [ { "kind": "text", "text": "apply divisor_le_nonzero" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "cases hb" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "trans S (x+d)" } ], [ { "kind": "text", "text": "rewrite PA4" } ], [ { "kind": "text", "text": "refl" } ], [ { "kind": "text", "text": "congr" } ], [ { "kind": "text", "text": "exact hb_witness" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "exact hall" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "apply hd_right" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hbound" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize hp (d)" } ], [ { "kind": "text", "text": "apply hp" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0372": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ k. ¬n = 0 → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" } ] }, "dependencies": [ "jordan_tuple_primitive_bounded_decidable", "divisor_le_nonzero" ], "enrolled_in_alpha": true, "factory": "make_jordan_totient_candidate_theorems", "first_admitted_version": "v35", "id": "JT0015", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_decidable", "original_ha_bundle_verified": true, "proof_bundle_node_id": 278, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro k", "intro hn", "have hd : (forall jt_divisor_primitiveyes. (exists jt_gap_primitiveyesbound. jt_gap_primitiveyesbound+S (jt_divisor_primitiveyes)=(S n)) -> ((exists jt_factor_primitiveyestestmodulus. (n)=(jt_divisor_primitiveyes)*jt_factor_primitiveyestestmodulus) -> (forall jt_index_primitiveyestestcoordinates jt_value_primitiveyestestcoordinates. (exists jt_gap_primitiveyestestcoordinatesindex. jt_gap_primitiveyestestcoordinatesindex+S (jt_index_primitiveyestestcoordinates)=(k)) -> (((exists fs_h_jt_primitiveyestestcoordinatesat. fs_h_jt_primitiveyestestcoordinatesat + S (jt_value_primitiveyestestcoordinates) = S ((S (jt_index_primitiveyestestcoordinates)) * c)) /\\ exists fs_q_jt_primitiveyestestcoordinatesat. b = fs_q_jt_primitiveyestestcoordinatesat * S ((S (jt_index_primitiveyestestcoordinates)) * c) + (jt_value_primitiveyestestcoordinates))) -> (exists jt_factor_primitiveyestestcoordinatesdivides. (jt_value_primitiveyestestcoordinates)=(jt_divisor_primitiveyes)*jt_factor_primitiveyestestcoordinatesdivides)) -> (jt_divisor_primitiveyes)=1)) \\/ ~(forall jt_divisor_primitiveno. (exists jt_gap_primitivenobound. jt_gap_primitivenobound+S (jt_divisor_primitiveno)=(S n)) -> ((exists jt_factor_primitivenotestmodulus. (n)=(jt_divisor_primitiveno)*jt_factor_primitivenotestmodulus) -> (forall jt_index_primitivenotestcoordinates jt_value_primitivenotestcoordinates. (exists jt_gap_primitivenotestcoordinatesindex. jt_gap_primitivenotestcoordinatesindex+S (jt_index_primitivenotestcoordinates)=(k)) -> (((exists fs_h_jt_primitivenotestcoordinatesat. fs_h_jt_primitivenotestcoordinatesat + S (jt_value_primitivenotestcoordinates) = S ((S (jt_index_primitivenotestcoordinates)) * c)) /\\ exists fs_q_jt_primitivenotestcoordinatesat. b = fs_q_jt_primitivenotestcoordinatesat * S ((S (jt_index_primitivenotestcoordinates)) * c) + (jt_value_primitivenotestcoordinates))) -> (exists jt_factor_primitivenotestcoordinatesdivides. (jt_value_primitivenotestcoordinates)=(jt_divisor_primitiveno)*jt_factor_primitivenotestcoordinatesdivides)) -> (jt_divisor_primitiveno)=1))", "specialize jordan_tuple_primitive_bounded_decidable (n)", "specialize jordan_tuple_primitive_bounded_decidable (b)", "specialize jordan_tuple_primitive_bounded_decidable (c)", "specialize jordan_tuple_primitive_bounded_decidable (k)", "specialize jordan_tuple_primitive_bounded_decidable (S n)", "apply jordan_tuple_primitive_bounded_decidable", "cases hd", "left", "intro d", "intro hdiv", "intro hall", "specialize hd_left (d)", "apply hd_left", "have hb : exists g. g+d=n", "specialize divisor_le_nonzero (d)", "specialize divisor_le_nonzero (n)", "apply divisor_le_nonzero", "exact hn", "exact hdiv", "cases hb", "exists x", "trans S (x+d)", "rewrite PA4", "refl", "congr", "exact hb_witness", "exact hdiv", "exact hall", "right", "intro hp", "apply hd_right", "intro d", "intro hbound", "intro hdiv", "intro hall", "specialize hp (d)", "apply hp", "exact hdiv", "exact hall" ], "script_sha256": "eb6d3ec18f08426710b1a591af6e765f02127fcbf2cc060fa56290928a4ffd77", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_totient_candidate_theorems", "script_sha256": "eb6d3ec18f08426710b1a591af6e765f02127fcbf2cc060fa56290928a4ffd77", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "3aff4616b95e229cc90e41220a94bafcd2492c670272843ee7b8181f143eac4b" } ], "stable_member": false, "statement": "forall n b c k. ~(n=0) -> (forall jt_divisor_decprimitiveyes. (exists jt_factor_decprimitiveyesmodulus. (n)=(jt_divisor_decprimitiveyes)*jt_factor_decprimitiveyesmodulus) -> (forall jt_index_decprimitiveyescoordinates jt_value_decprimitiveyescoordinates. (exists jt_gap_decprimitiveyescoordinatesindex. jt_gap_decprimitiveyescoordinatesindex+S (jt_index_decprimitiveyescoordinates)=(k)) -> (((exists fs_h_jt_decprimitiveyescoordinatesat. fs_h_jt_decprimitiveyescoordinatesat + S (jt_value_decprimitiveyescoordinates) = S ((S (jt_index_decprimitiveyescoordinates)) * c)) /\\ exists fs_q_jt_decprimitiveyescoordinatesat. b = fs_q_jt_decprimitiveyescoordinatesat * S ((S (jt_index_decprimitiveyescoordinates)) * c) + (jt_value_decprimitiveyescoordinates))) -> (exists jt_factor_decprimitiveyescoordinatesdivides. (jt_value_decprimitiveyescoordinates)=(jt_divisor_decprimitiveyes)*jt_factor_decprimitiveyescoordinatesdivides)) -> jt_divisor_decprimitiveyes=1) \\/ ~(forall jt_divisor_decprimitiveno. (exists jt_factor_decprimitivenomodulus. (n)=(jt_divisor_decprimitiveno)*jt_factor_decprimitivenomodulus) -> (forall jt_index_decprimitivenocoordinates jt_value_decprimitivenocoordinates. (exists jt_gap_decprimitivenocoordinatesindex. jt_gap_decprimitivenocoordinatesindex+S (jt_index_decprimitivenocoordinates)=(k)) -> (((exists fs_h_jt_decprimitivenocoordinatesat. fs_h_jt_decprimitivenocoordinatesat + S (jt_value_decprimitivenocoordinates) = S ((S (jt_index_decprimitivenocoordinates)) * c)) /\\ exists fs_q_jt_decprimitivenocoordinatesat. b = fs_q_jt_decprimitivenocoordinatesat * S ((S (jt_index_decprimitivenocoordinates)) * c) + (jt_value_decprimitivenocoordinates))) -> (exists jt_factor_decprimitivenocoordinatesdivides. (jt_value_decprimitivenocoordinates)=(jt_divisor_decprimitiveno)*jt_factor_decprimitivenocoordinatesdivides)) -> jt_divisor_decprimitiveno=1)", "statement_sha256": "3aff4616b95e229cc90e41220a94bafcd2492c670272843ee7b8181f143eac4b", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "At positive modulus every possible common divisor lies below S n, giving genuine tuple-predicate decidability." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 21, "body_proof_nodes": 30, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. IntegerVectorZero(b,c,d,e,0)", "defined_statement_sha256": "9ebe9ac0afa8130351fa97afaeef03799d1ea7727230c3f3117c4082f6b87e0c", "definition_uses": { "ND0121": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "4b0e08fed0b4fdad2b1cad83d00d3c22ce0666966bc42f39c6e3c6c6bf21ae7c", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (i)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "specialize zero_le (i)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "ND0121": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,0)" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT0016", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_empty", "original_ha_bundle_verified": true, "proof_bundle_node_id": 279, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "script_sha256": "06b34ae2a3a272ad11f6866818d1c86345107f914591b001a2be14fbbc9fcc88", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "06b34ae2a3a272ad11f6866818d1c86345107f914591b001a2be14fbbc9fcc88", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "4b0e08fed0b4fdad2b1cad83d00d3c22ce0666966bc42f39c6e3c6c6bf21ae7c" } ], "stable_member": false, "statement": "forall b c d e. forall jt_index_eqempty jt_left_eqempty jt_right_eqempty. (exists jt_gap_eqemptyindex. jt_gap_eqemptyindex+S (jt_index_eqempty)=(0)) -> (((exists fs_h_jt_eqemptyleft. fs_h_jt_eqemptyleft + S (jt_left_eqempty) = S ((S (jt_index_eqempty)) * c)) /\\ exists fs_q_jt_eqemptyleft. b = fs_q_jt_eqemptyleft * S ((S (jt_index_eqempty)) * c) + (jt_left_eqempty))) -> (((exists fs_h_jt_eqemptyright. fs_h_jt_eqemptyright + S (jt_right_eqempty) = S ((S (jt_index_eqempty)) * e)) /\\ exists fs_q_jt_eqemptyright. d = fs_q_jt_eqemptyright * S ((S (jt_index_eqempty)) * e) + (jt_right_eqempty))) -> jt_left_eqempty=jt_right_eqempty", "statement_sha256": "4b0e08fed0b4fdad2b1cad83d00d3c22ce0666966bc42f39c6e3c6c6bf21ae7c", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "All actual empty tuples are coordinatewise equal." }, { "admission_dependencies": [ "le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 28, "body_proof_nodes": 42, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro h", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize h (i)", "specialize h (a)", "specialize h (z)", "apply h", "specialize le_succ (S i)", "specialize le_succ (k)", "apply le_succ", "exact hi", "exact ha", "exact hz" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. IntegerVectorZero(b,c,d,e,S k) → IntegerVectorZero(b,c,d,e,k)", "defined_statement_sha256": "be08274974e91b18b79a38467519dfc432a49364041283ca0725c414516da2a5", "definition_uses": { "ND0121": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "f3ea7e5fb7596cc173a54fccc4bcfc2a2633a56c4aa06d29d35ae10e6ef427d9", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "specialize h (a)" } ], [ { "kind": "text", "text": "specialize h (z)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "specialize le_succ (S i)" } ], [ { "kind": "text", "text": "specialize le_succ (k)" } ], [ { "kind": "text", "text": "apply le_succ" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0121": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,S k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" } ] }, "dependencies": [ "le_succ" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT0017", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_drop_last", "original_ha_bundle_verified": true, "proof_bundle_node_id": 280, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro h", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize h (i)", "specialize h (a)", "specialize h (z)", "apply h", "specialize le_succ (S i)", "specialize le_succ (k)", "apply le_succ", "exact hi", "exact ha", "exact hz" ], "script_sha256": "5ccfaeb269ad262f95b9cf46da351d698cad0672e368bd9a9ae9d5ce2c813cbe", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "5ccfaeb269ad262f95b9cf46da351d698cad0672e368bd9a9ae9d5ce2c813cbe", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "f3ea7e5fb7596cc173a54fccc4bcfc2a2633a56c4aa06d29d35ae10e6ef427d9" } ], "stable_member": false, "statement": "forall b c d e k. (forall jt_index_eqdropsource jt_left_eqdropsource jt_right_eqdropsource. (exists jt_gap_eqdropsourceindex. jt_gap_eqdropsourceindex+S (jt_index_eqdropsource)=(S k)) -> (((exists fs_h_jt_eqdropsourceleft. fs_h_jt_eqdropsourceleft + S (jt_left_eqdropsource) = S ((S (jt_index_eqdropsource)) * c)) /\\ exists fs_q_jt_eqdropsourceleft. b = fs_q_jt_eqdropsourceleft * S ((S (jt_index_eqdropsource)) * c) + (jt_left_eqdropsource))) -> (((exists fs_h_jt_eqdropsourceright. fs_h_jt_eqdropsourceright + S (jt_right_eqdropsource) = S ((S (jt_index_eqdropsource)) * e)) /\\ exists fs_q_jt_eqdropsourceright. d = fs_q_jt_eqdropsourceright * S ((S (jt_index_eqdropsource)) * e) + (jt_right_eqdropsource))) -> jt_left_eqdropsource=jt_right_eqdropsource) -> (forall jt_index_eqdroptarget jt_left_eqdroptarget jt_right_eqdroptarget. (exists jt_gap_eqdroptargetindex. jt_gap_eqdroptargetindex+S (jt_index_eqdroptarget)=(k)) -> (((exists fs_h_jt_eqdroptargetleft. fs_h_jt_eqdroptargetleft + S (jt_left_eqdroptarget) = S ((S (jt_index_eqdroptarget)) * c)) /\\ exists fs_q_jt_eqdroptargetleft. b = fs_q_jt_eqdroptargetleft * S ((S (jt_index_eqdroptarget)) * c) + (jt_left_eqdroptarget))) -> (((exists fs_h_jt_eqdroptargetright. fs_h_jt_eqdroptargetright + S (jt_right_eqdroptarget) = S ((S (jt_index_eqdroptarget)) * e)) /\\ exists fs_q_jt_eqdroptargetright. d = fs_q_jt_eqdroptargetright * S ((S (jt_index_eqdroptarget)) * e) + (jt_right_eqdroptarget))) -> jt_left_eqdroptarget=jt_right_eqdroptarget)", "statement_sha256": "f3ea7e5fb7596cc173a54fccc4bcfc2a2633a56c4aa06d29d35ae10e6ef427d9", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Restrict coordinate equality to the actual predecessor prefix." }, { "admission_dependencies": [ "finite_lt_succ_eq_or_lt", "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 31, "body_proof_nodes": 68, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro a", "intro z", "intro hp", "intro ha", "intro hz", "intro heq", "intro i", "intro r", "intro s", "intro hi", "intro hr", "intro hs", "have hc : i = k ∨ Lt(i,k)", "specialize finite_lt_succ_eq_or_lt (k)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hc", "have hleft : r=a", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (k)", "specialize beta_at_unique (r)", "specialize beta_at_unique (a)", "apply beta_at_unique", "rewrite hc_left at hr", "rewrite hc_left at hr", "exact hr", "exact ha", "have hright : s=z", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (k)", "specialize beta_at_unique (s)", "specialize beta_at_unique (z)", "apply beta_at_unique", "rewrite hc_left at hs", "rewrite hc_left at hs", "exact hs", "exact hz", "rewrite hleft", "rewrite hright", "exact heq", "specialize hp (i)", "specialize hp (r)", "specialize hp (s)", "apply hp", "exact hc_right", "exact hr", "exact hs" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ a. ∀ z. IntegerVectorZero(b,c,d,e,k) → BetaAt(b,c,k,a) → BetaAt(d,e,k,z) → a = z → IntegerVectorZero(b,c,d,e,S k)", "defined_statement_sha256": "09f869e9ef848aaddd6ce16244804215579110b2b7fb3cbe2d62ed239ce98e1f", "definition_uses": { "ND0121": 2, "PD0002": 1, "PD0013": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "d6811cfaf8d3cd24e20742255df5a45d465e244e956bdfe9ceca1ca6fd92d6e3", "free_names": [], "script_definition_uses": { "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro r" } ], [ { "kind": "text", "text": "intro s" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hs" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "i = k ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,k)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (k)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (i)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "have hleft : r=a" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (k)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (r)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hc_left at hr" } ], [ { "kind": "text", "text": "rewrite hc_left at hr" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "have hright : s=z" } ], [ { "kind": "text", "text": "specialize beta_at_unique (d)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (e)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (k)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (s)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (z)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hc_left at hs" } ], [ { "kind": "text", "text": "rewrite hc_left at hs" } ], [ { "kind": "text", "text": "exact hs" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "rewrite hleft" } ], [ { "kind": "text", "text": "rewrite hright" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "specialize hp (i)" } ], [ { "kind": "text", "text": "specialize hp (r)" } ], [ { "kind": "text", "text": "specialize hp (s)" } ], [ { "kind": "text", "text": "apply hp" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hs" } ] ], "statement_definition_uses": { "ND0121": 2, "PD0013": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ a. ∀ z. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,k,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,k,z)" }, { "kind": "text", "text": " → a = z → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,S k)" } ] }, "dependencies": [ "finite_lt_succ_eq_or_lt", "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT0018", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_extend", "original_ha_bundle_verified": true, "proof_bundle_node_id": 281, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro a", "intro z", "intro hp", "intro ha", "intro hz", "intro heq", "intro i", "intro r", "intro s", "intro hi", "intro hr", "intro hs", "have hc : i=k \\/ (exists jt_gap_eqextendcases. jt_gap_eqextendcases+S (i)=(k))", "specialize finite_lt_succ_eq_or_lt (k)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hc", "have hleft : r=a", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (k)", "specialize beta_at_unique (r)", "specialize beta_at_unique (a)", "apply beta_at_unique", "rewrite hc_left at hr", "rewrite hc_left at hr", "exact hr", "exact ha", "have hright : s=z", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (k)", "specialize beta_at_unique (s)", "specialize beta_at_unique (z)", "apply beta_at_unique", "rewrite hc_left at hs", "rewrite hc_left at hs", "exact hs", "exact hz", "rewrite hleft", "rewrite hright", "exact heq", "specialize hp (i)", "specialize hp (r)", "specialize hp (s)", "apply hp", "exact hc_right", "exact hr", "exact hs" ], "script_sha256": "67daaeb625b08afb40feaddc144b39c1807fecaef2b664e04f1d512b7bb97d37", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "67daaeb625b08afb40feaddc144b39c1807fecaef2b664e04f1d512b7bb97d37", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "d6811cfaf8d3cd24e20742255df5a45d465e244e956bdfe9ceca1ca6fd92d6e3" } ], "stable_member": false, "statement": "forall b c d e k a z. (forall jt_index_eqextendprefix jt_left_eqextendprefix jt_right_eqextendprefix. (exists jt_gap_eqextendprefixindex. jt_gap_eqextendprefixindex+S (jt_index_eqextendprefix)=(k)) -> (((exists fs_h_jt_eqextendprefixleft. fs_h_jt_eqextendprefixleft + S (jt_left_eqextendprefix) = S ((S (jt_index_eqextendprefix)) * c)) /\\ exists fs_q_jt_eqextendprefixleft. b = fs_q_jt_eqextendprefixleft * S ((S (jt_index_eqextendprefix)) * c) + (jt_left_eqextendprefix))) -> (((exists fs_h_jt_eqextendprefixright. fs_h_jt_eqextendprefixright + S (jt_right_eqextendprefix) = S ((S (jt_index_eqextendprefix)) * e)) /\\ exists fs_q_jt_eqextendprefixright. d = fs_q_jt_eqextendprefixright * S ((S (jt_index_eqextendprefix)) * e) + (jt_right_eqextendprefix))) -> jt_left_eqextendprefix=jt_right_eqextendprefix) -> (((exists fs_h_jt_eqextendleft. fs_h_jt_eqextendleft + S (a) = S ((S (k)) * c)) /\\ exists fs_q_jt_eqextendleft. b = fs_q_jt_eqextendleft * S ((S (k)) * c) + (a))) -> (((exists fs_h_jt_eqextendright. fs_h_jt_eqextendright + S (z) = S ((S (k)) * e)) /\\ exists fs_q_jt_eqextendright. d = fs_q_jt_eqextendright * S ((S (k)) * e) + (z))) -> a=z -> (forall jt_index_eqextendtarget jt_left_eqextendtarget jt_right_eqextendtarget. (exists jt_gap_eqextendtargetindex. jt_gap_eqextendtargetindex+S (jt_index_eqextendtarget)=(S k)) -> (((exists fs_h_jt_eqextendtargetleft. fs_h_jt_eqextendtargetleft + S (jt_left_eqextendtarget) = S ((S (jt_index_eqextendtarget)) * c)) /\\ exists fs_q_jt_eqextendtargetleft. b = fs_q_jt_eqextendtargetleft * S ((S (jt_index_eqextendtarget)) * c) + (jt_left_eqextendtarget))) -> (((exists fs_h_jt_eqextendtargetright. fs_h_jt_eqextendtargetright + S (jt_right_eqextendtarget) = S ((S (jt_index_eqextendtarget)) * e)) /\\ exists fs_q_jt_eqextendtargetright. d = fs_q_jt_eqextendtargetright * S ((S (jt_index_eqextendtarget)) * e) + (jt_right_eqextendtarget))) -> jt_left_eqextendtarget=jt_right_eqextendtarget)", "statement_sha256": "d6811cfaf8d3cd24e20742255df5a45d465e244e956bdfe9ceca1ca6fd92d6e3", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Extend equal prefixes using two actual equal last entries." }, { "admission_dependencies": [ "jordan_tuple_equal_empty", "beta_at_exists", "eq_decidable", "jordan_tuple_equal_extend", "le_refl", "jordan_tuple_equal_drop_last" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 30, "body_proof_nodes": 79, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "induction k", "left", "specialize jordan_tuple_equal_empty (b)", "specialize jordan_tuple_equal_empty (c)", "specialize jordan_tuple_equal_empty (d)", "specialize jordan_tuple_equal_empty (e)", "apply jordan_tuple_equal_empty", "have ha : ∃ a. BetaAt(b,c,k,a)", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases ha", "have hz : ∃ z. BetaAt(d,e,k,z)", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases hz", "have heq : x=x1 \\/ ~(x=x1)", "specialize eq_decidable (x)", "specialize eq_decidable (x1)", "apply eq_decidable", "cases IH", "cases heq", "left", "specialize jordan_tuple_equal_extend (b)", "specialize jordan_tuple_equal_extend (c)", "specialize jordan_tuple_equal_extend (d)", "specialize jordan_tuple_equal_extend (e)", "specialize jordan_tuple_equal_extend (k)", "specialize jordan_tuple_equal_extend (x)", "specialize jordan_tuple_equal_extend (x1)", "apply jordan_tuple_equal_extend", "exact IH_left", "exact ha_witness", "exact hz_witness", "exact heq_left", "right", "intro h", "apply heq_right", "specialize h (k)", "specialize h (x)", "specialize h (x1)", "apply h", "specialize le_refl (S k)", "apply le_refl", "exact ha_witness", "exact hz_witness", "right", "intro h", "apply IH_right", "specialize jordan_tuple_equal_drop_last (b)", "specialize jordan_tuple_equal_drop_last (c)", "specialize jordan_tuple_equal_drop_last (d)", "specialize jordan_tuple_equal_drop_last (e)", "specialize jordan_tuple_equal_drop_last (k)", "apply jordan_tuple_equal_drop_last", "exact h" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. IntegerVectorZero(b,c,d,e,k) ∨ ¬IntegerVectorZero(b,c,d,e,k)", "defined_statement_sha256": "4c43b09a2c881e0240a0f6df65271f6d247ce0699134dceb14982969366f5ba5", "definition_uses": { "ND0121": 2, "PD0013": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "38ad9034f9e94976b73822fac5e0e2dff77887683e6b78209ca1c159a0ac05ff", "free_names": [], "script_definition_uses": { "PD0013": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "induction k" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_empty (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_empty (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_empty (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_empty (e)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_empty" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ a. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,k,a)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (b)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (c)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (k)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "have hz : " }, { "kind": "text", "text": "∃ z. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,k,z)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (k)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hz" } ], [ { "kind": "text", "text": "have heq : x=x1 \\/ ~(x=x1)" } ], [ { "kind": "text", "text": "specialize eq_decidable (x)" } ], [ { "kind": "text", "text": "specialize eq_decidable (x1)" } ], [ { "kind": "text", "text": "apply eq_decidable" } ], [ { "kind": "text", "text": "cases IH" } ], [ { "kind": "text", "text": "cases heq" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_extend (x1)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_extend" } ], [ { "kind": "text", "text": "exact IH_left" } ], [ { "kind": "text", "text": "exact ha_witness" } ], [ { "kind": "text", "text": "exact hz_witness" } ], [ { "kind": "text", "text": "exact heq_left" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply heq_right" } ], [ { "kind": "text", "text": "specialize h (k)" } ], [ { "kind": "text", "text": "specialize h (x)" } ], [ { "kind": "text", "text": "specialize h (x1)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "specialize le_refl (S k)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "exact ha_witness" } ], [ { "kind": "text", "text": "exact hz_witness" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "apply IH_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_drop_last (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_drop_last (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_drop_last (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_drop_last (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_drop_last (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_drop_last" } ], [ { "kind": "text", "text": "exact h" } ] ], "statement_definition_uses": { "ND0121": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" } ] }, "dependencies": [ "jordan_tuple_equal_empty", "beta_at_exists", "eq_decidable", "jordan_tuple_equal_extend", "le_refl", "jordan_tuple_equal_drop_last" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT0019", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_decidable", "original_ha_bundle_verified": true, "proof_bundle_node_id": 282, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "induction k", "left", "specialize jordan_tuple_equal_empty (b)", "specialize jordan_tuple_equal_empty (c)", "specialize jordan_tuple_equal_empty (d)", "specialize jordan_tuple_equal_empty (e)", "apply jordan_tuple_equal_empty", "have ha : exists a. ((exists fs_h_jt_eqdecisiona. fs_h_jt_eqdecisiona + S (a) = S ((S (k)) * c)) /\\ exists fs_q_jt_eqdecisiona. b = fs_q_jt_eqdecisiona * S ((S (k)) * c) + (a))", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases ha", "have hz : exists z. ((exists fs_h_jt_eqdecisionz. fs_h_jt_eqdecisionz + S (z) = S ((S (k)) * e)) /\\ exists fs_q_jt_eqdecisionz. d = fs_q_jt_eqdecisionz * S ((S (k)) * e) + (z))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases hz", "have heq : x=x1 \\/ ~(x=x1)", "specialize eq_decidable (x)", "specialize eq_decidable (x1)", "apply eq_decidable", "cases IH", "cases heq", "left", "specialize jordan_tuple_equal_extend (b)", "specialize jordan_tuple_equal_extend (c)", "specialize jordan_tuple_equal_extend (d)", "specialize jordan_tuple_equal_extend (e)", "specialize jordan_tuple_equal_extend (k)", "specialize jordan_tuple_equal_extend (x)", "specialize jordan_tuple_equal_extend (x1)", "apply jordan_tuple_equal_extend", "exact IH_left", "exact ha_witness", "exact hz_witness", "exact heq_left", "right", "intro h", "apply heq_right", "specialize h (k)", "specialize h (x)", "specialize h (x1)", "apply h", "specialize le_refl (S k)", "apply le_refl", "exact ha_witness", "exact hz_witness", "right", "intro h", "apply IH_right", "specialize jordan_tuple_equal_drop_last (b)", "specialize jordan_tuple_equal_drop_last (c)", "specialize jordan_tuple_equal_drop_last (d)", "specialize jordan_tuple_equal_drop_last (e)", "specialize jordan_tuple_equal_drop_last (k)", "apply jordan_tuple_equal_drop_last", "exact h" ], "script_sha256": "ba06d2e7e7c370bd2fc306b50583027c2d81af4e9d73346a40893d8d35d21d21", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "ba06d2e7e7c370bd2fc306b50583027c2d81af4e9d73346a40893d8d35d21d21", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "38ad9034f9e94976b73822fac5e0e2dff77887683e6b78209ca1c159a0ac05ff" } ], "stable_member": false, "statement": "forall b c d e k. (forall jt_index_eqyes jt_left_eqyes jt_right_eqyes. (exists jt_gap_eqyesindex. jt_gap_eqyesindex+S (jt_index_eqyes)=(k)) -> (((exists fs_h_jt_eqyesleft. fs_h_jt_eqyesleft + S (jt_left_eqyes) = S ((S (jt_index_eqyes)) * c)) /\\ exists fs_q_jt_eqyesleft. b = fs_q_jt_eqyesleft * S ((S (jt_index_eqyes)) * c) + (jt_left_eqyes))) -> (((exists fs_h_jt_eqyesright. fs_h_jt_eqyesright + S (jt_right_eqyes) = S ((S (jt_index_eqyes)) * e)) /\\ exists fs_q_jt_eqyesright. d = fs_q_jt_eqyesright * S ((S (jt_index_eqyes)) * e) + (jt_right_eqyes))) -> jt_left_eqyes=jt_right_eqyes) \\/ ~(forall jt_index_eqno jt_left_eqno jt_right_eqno. (exists jt_gap_eqnoindex. jt_gap_eqnoindex+S (jt_index_eqno)=(k)) -> (((exists fs_h_jt_eqnoleft. fs_h_jt_eqnoleft + S (jt_left_eqno) = S ((S (jt_index_eqno)) * c)) /\\ exists fs_q_jt_eqnoleft. b = fs_q_jt_eqnoleft * S ((S (jt_index_eqno)) * c) + (jt_left_eqno))) -> (((exists fs_h_jt_eqnoright. fs_h_jt_eqnoright + S (jt_right_eqno) = S ((S (jt_index_eqno)) * e)) /\\ exists fs_q_jt_eqnoright. d = fs_q_jt_eqnoright * S ((S (jt_index_eqno)) * e) + (jt_right_eqno))) -> jt_left_eqno=jt_right_eqno)", "statement_sha256": "38ad9034f9e94976b73822fac5e0e2dff77887683e6b78209ca1c159a0ac05ff", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Inductively decide decoded coordinate equality, never equality of beta codes." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 29, "body_proof_nodes": 49, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro k", "intro B", "intro C", "intro D", "intro E", "intro h", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_right", "specialize lt_not_le (x)", "specialize lt_not_le (0)", "apply lt_not_le", "exact h_witness_witness_witness_left", "specialize zero_le (x)", "apply zero_le" ], "defined_statement": "∀ b. ∀ c. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ¬JordanTupleListed(b,c,k,B,C,D,E,0)", "defined_statement_sha256": "ab89df19822132ad7d8243ccdbb5e7e47d262461650e624067d08e0d2a0639ac", "definition_uses": { "ND0376": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "e4abd379e0ca483ebfa211980529adf379bb982d3a1a1f72e05646b85e15bc84", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "cases h" } ], [ { "kind": "text", "text": "cases h_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness_right" } ], [ { "kind": "text", "text": "specialize lt_not_le (x)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_left" } ], [ { "kind": "text", "text": "specialize zero_le (x)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "ND0376": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ¬" }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,B,C,D,E,0)" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT001A", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_listed_empty", "original_ha_bundle_verified": true, "proof_bundle_node_id": 283, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro k", "intro B", "intro C", "intro D", "intro E", "intro h", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_right", "specialize lt_not_le (x)", "specialize lt_not_le (0)", "apply lt_not_le", "exact h_witness_witness_witness_left", "specialize zero_le (x)", "apply zero_le" ], "script_sha256": "94c35c1eed51634a98d53e8092b0016c80a8bf9885c8b8760c46296d6356b0b1", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "94c35c1eed51634a98d53e8092b0016c80a8bf9885c8b8760c46296d6356b0b1", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "e4abd379e0ca483ebfa211980529adf379bb982d3a1a1f72e05646b85e15bc84" } ], "stable_member": false, "statement": "forall b c k B C D E. ~(exists jt_index_listedempty jt_code_listedempty jt_scale_listedempty. ((exists jt_gap_listedemptyindex. jt_gap_listedemptyindex+S (jt_index_listedempty)=(0)) /\\ (((((((exists fs_h_jt_listedemptycode. fs_h_jt_listedemptycode + S (jt_code_listedempty) = S ((S (jt_index_listedempty)) * C)) /\\ exists fs_q_jt_listedemptycode. B = fs_q_jt_listedemptycode * S ((S (jt_index_listedempty)) * C) + (jt_code_listedempty))) /\\ (((exists fs_h_jt_listedemptyscale. fs_h_jt_listedemptyscale + S (jt_scale_listedempty) = S ((S (jt_index_listedempty)) * E)) /\\ exists fs_q_jt_listedemptyscale. D = fs_q_jt_listedemptyscale * S ((S (jt_index_listedempty)) * E) + (jt_scale_listedempty))))) /\\ (forall jt_index_listedemptyequal jt_left_listedemptyequal jt_right_listedemptyequal. (exists jt_gap_listedemptyequalindex. jt_gap_listedemptyequalindex+S (jt_index_listedemptyequal)=(k)) -> (((exists fs_h_jt_listedemptyequalleft. fs_h_jt_listedemptyequalleft + S (jt_left_listedemptyequal) = S ((S (jt_index_listedemptyequal)) * c)) /\\ exists fs_q_jt_listedemptyequalleft. b = fs_q_jt_listedemptyequalleft * S ((S (jt_index_listedemptyequal)) * c) + (jt_left_listedemptyequal))) -> (((exists fs_h_jt_listedemptyequalright. fs_h_jt_listedemptyequalright + S (jt_right_listedemptyequal) = S ((S (jt_index_listedemptyequal)) * jt_scale_listedempty)) /\\ exists fs_q_jt_listedemptyequalright. jt_code_listedempty = fs_q_jt_listedemptyequalright * S ((S (jt_index_listedemptyequal)) * jt_scale_listedempty) + (jt_right_listedemptyequal))) -> jt_left_listedemptyequal=jt_right_listedemptyequal)))))", "statement_sha256": "e4abd379e0ca483ebfa211980529adf379bb982d3a1a1f72e05646b85e15bc84", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "An empty actual outer list contains no representative." }, { "admission_dependencies": [ "le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 31, "body_proof_nodes": 50, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro k", "intro B", "intro C", "intro D", "intro E", "intro j", "intro h", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "split", "specialize le_succ (S x)", "specialize le_succ (j)", "apply le_succ", "exact h_witness_witness_witness_left", "split", "exact h_witness_witness_witness_right_left", "exact h_witness_witness_witness_right_right" ], "defined_statement": "∀ b. ∀ c. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. JordanTupleListed(b,c,k,B,C,D,E,j) → JordanTupleListed(b,c,k,B,C,D,E,S j)", "defined_statement_sha256": "74215accd3c11d66f4e591353e4573fd3513ede9dc6c4b0f9a1f7489d04f88d4", "definition_uses": { "ND0376": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "893243ef415155238ee8aa4f6c9044332c5c42f5193a3513815a1a0e02d3d9be", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "cases h" } ], [ { "kind": "text", "text": "cases h_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness_right" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize le_succ (S x)" } ], [ { "kind": "text", "text": "specialize le_succ (j)" } ], [ { "kind": "text", "text": "apply le_succ" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0376": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,B,C,D,E,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,B,C,D,E,S j)" } ] }, "dependencies": [ "le_succ" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT001B", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_listed_lift", "original_ha_bundle_verified": true, "proof_bundle_node_id": 284, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro k", "intro B", "intro C", "intro D", "intro E", "intro j", "intro h", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "split", "specialize le_succ (S x)", "specialize le_succ (j)", "apply le_succ", "exact h_witness_witness_witness_left", "split", "exact h_witness_witness_witness_right_left", "exact h_witness_witness_witness_right_right" ], "script_sha256": "7272b31b72ff5c52d2ab33569a41807237d1c63dc75750ec22d57d14dc4785c7", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "7272b31b72ff5c52d2ab33569a41807237d1c63dc75750ec22d57d14dc4785c7", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "893243ef415155238ee8aa4f6c9044332c5c42f5193a3513815a1a0e02d3d9be" } ], "stable_member": false, "statement": "forall b c k B C D E j. (exists jt_index_listedliftold jt_code_listedliftold jt_scale_listedliftold. ((exists jt_gap_listedliftoldindex. jt_gap_listedliftoldindex+S (jt_index_listedliftold)=(j)) /\\ (((((((exists fs_h_jt_listedliftoldcode. fs_h_jt_listedliftoldcode + S (jt_code_listedliftold) = S ((S (jt_index_listedliftold)) * C)) /\\ exists fs_q_jt_listedliftoldcode. B = fs_q_jt_listedliftoldcode * S ((S (jt_index_listedliftold)) * C) + (jt_code_listedliftold))) /\\ (((exists fs_h_jt_listedliftoldscale. fs_h_jt_listedliftoldscale + S (jt_scale_listedliftold) = S ((S (jt_index_listedliftold)) * E)) /\\ exists fs_q_jt_listedliftoldscale. D = fs_q_jt_listedliftoldscale * S ((S (jt_index_listedliftold)) * E) + (jt_scale_listedliftold))))) /\\ (forall jt_index_listedliftoldequal jt_left_listedliftoldequal jt_right_listedliftoldequal. (exists jt_gap_listedliftoldequalindex. jt_gap_listedliftoldequalindex+S (jt_index_listedliftoldequal)=(k)) -> (((exists fs_h_jt_listedliftoldequalleft. fs_h_jt_listedliftoldequalleft + S (jt_left_listedliftoldequal) = S ((S (jt_index_listedliftoldequal)) * c)) /\\ exists fs_q_jt_listedliftoldequalleft. b = fs_q_jt_listedliftoldequalleft * S ((S (jt_index_listedliftoldequal)) * c) + (jt_left_listedliftoldequal))) -> (((exists fs_h_jt_listedliftoldequalright. fs_h_jt_listedliftoldequalright + S (jt_right_listedliftoldequal) = S ((S (jt_index_listedliftoldequal)) * jt_scale_listedliftold)) /\\ exists fs_q_jt_listedliftoldequalright. jt_code_listedliftold = fs_q_jt_listedliftoldequalright * S ((S (jt_index_listedliftoldequal)) * jt_scale_listedliftold) + (jt_right_listedliftoldequal))) -> jt_left_listedliftoldequal=jt_right_listedliftoldequal))))) -> (exists jt_index_listedliftnew jt_code_listedliftnew jt_scale_listedliftnew. ((exists jt_gap_listedliftnewindex. jt_gap_listedliftnewindex+S (jt_index_listedliftnew)=(S j)) /\\ (((((((exists fs_h_jt_listedliftnewcode. fs_h_jt_listedliftnewcode + S (jt_code_listedliftnew) = S ((S (jt_index_listedliftnew)) * C)) /\\ exists fs_q_jt_listedliftnewcode. B = fs_q_jt_listedliftnewcode * S ((S (jt_index_listedliftnew)) * C) + (jt_code_listedliftnew))) /\\ (((exists fs_h_jt_listedliftnewscale. fs_h_jt_listedliftnewscale + S (jt_scale_listedliftnew) = S ((S (jt_index_listedliftnew)) * E)) /\\ exists fs_q_jt_listedliftnewscale. D = fs_q_jt_listedliftnewscale * S ((S (jt_index_listedliftnew)) * E) + (jt_scale_listedliftnew))))) /\\ (forall jt_index_listedliftnewequal jt_left_listedliftnewequal jt_right_listedliftnewequal. (exists jt_gap_listedliftnewequalindex. jt_gap_listedliftnewequalindex+S (jt_index_listedliftnewequal)=(k)) -> (((exists fs_h_jt_listedliftnewequalleft. fs_h_jt_listedliftnewequalleft + S (jt_left_listedliftnewequal) = S ((S (jt_index_listedliftnewequal)) * c)) /\\ exists fs_q_jt_listedliftnewequalleft. b = fs_q_jt_listedliftnewequalleft * S ((S (jt_index_listedliftnewequal)) * c) + (jt_left_listedliftnewequal))) -> (((exists fs_h_jt_listedliftnewequalright. fs_h_jt_listedliftnewequalright + S (jt_right_listedliftnewequal) = S ((S (jt_index_listedliftnewequal)) * jt_scale_listedliftnew)) /\\ exists fs_q_jt_listedliftnewequalright. jt_code_listedliftnew = fs_q_jt_listedliftnewequalright * S ((S (jt_index_listedliftnewequal)) * jt_scale_listedliftnew) + (jt_right_listedliftnewequal))) -> jt_left_listedliftnewequal=jt_right_listedliftnewequal)))))", "statement_sha256": "893243ef415155238ee8aa4f6c9044332c5c42f5193a3513815a1a0e02d3d9be", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "An existing list position remains below the successor bound." }, { "admission_dependencies": [ "jordan_tuple_listed_empty", "jordan_tuple_listed_lift", "beta_at_exists", "jordan_tuple_equal_decidable", "le_refl", "finite_lt_succ_eq_or_lt", "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 39, "body_proof_nodes": 160, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro k", "intro B", "intro C", "intro D", "intro E", "induction j", "right", "intro hempty", "specialize jordan_tuple_listed_empty (b)", "specialize jordan_tuple_listed_empty (c)", "specialize jordan_tuple_listed_empty (k)", "specialize jordan_tuple_listed_empty (B)", "specialize jordan_tuple_listed_empty (C)", "specialize jordan_tuple_listed_empty (D)", "specialize jordan_tuple_listed_empty (E)", "apply jordan_tuple_listed_empty", "exact hempty", "cases IH", "left", "specialize jordan_tuple_listed_lift (b)", "specialize jordan_tuple_listed_lift (c)", "specialize jordan_tuple_listed_lift (k)", "specialize jordan_tuple_listed_lift (B)", "specialize jordan_tuple_listed_lift (C)", "specialize jordan_tuple_listed_lift (D)", "specialize jordan_tuple_listed_lift (E)", "specialize jordan_tuple_listed_lift (j)", "apply jordan_tuple_listed_lift", "exact IH_left", "have hd : ∃ d. BetaAt(B,C,j,d)", "specialize beta_at_exists (B)", "specialize beta_at_exists (C)", "specialize beta_at_exists (j)", "apply beta_at_exists", "cases hd", "have he : ∃ e. BetaAt(D,E,j,e)", "specialize beta_at_exists (D)", "specialize beta_at_exists (E)", "specialize beta_at_exists (j)", "apply beta_at_exists", "cases he", "have heq : IntegerVectorZero(b,c,x,x1,k) ∨ ¬IntegerVectorZero(b,c,x,x1,k)", "specialize jordan_tuple_equal_decidable (b)", "specialize jordan_tuple_equal_decidable (c)", "specialize jordan_tuple_equal_decidable (x)", "specialize jordan_tuple_equal_decidable (x1)", "specialize jordan_tuple_equal_decidable (k)", "apply jordan_tuple_equal_decidable", "cases heq", "left", "exists j", "exists x", "exists x1", "split", "specialize le_refl (S j)", "apply le_refl", "split", "split", "exact hd_witness", "exact he_witness", "exact heq_left", "right", "intro h", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_right", "have hc : x2 = j ∨ Lt(x2,j)", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (x2)", "apply finite_lt_succ_eq_or_lt", "exact h_witness_witness_witness_left", "cases hc", "cases h_witness_witness_witness_right_left", "have hdval : x3=x", "specialize beta_at_unique (B)", "specialize beta_at_unique (C)", "specialize beta_at_unique (j)", "specialize beta_at_unique (x3)", "specialize beta_at_unique (x)", "apply beta_at_unique", "rewrite hc_left at h_witness_witness_witness_right_left_left", "rewrite hc_left at h_witness_witness_witness_right_left_left", "exact h_witness_witness_witness_right_left_left", "exact hd_witness", "have heval : x4=x1", "specialize beta_at_unique (D)", "specialize beta_at_unique (E)", "specialize beta_at_unique (j)", "specialize beta_at_unique (x4)", "specialize beta_at_unique (x1)", "apply beta_at_unique", "rewrite hc_left at h_witness_witness_witness_right_left_right", "rewrite hc_left at h_witness_witness_witness_right_left_right", "exact h_witness_witness_witness_right_left_right", "exact he_witness", "apply heq_right", "rewrite hdval at h_witness_witness_witness_right_right", "rewrite heval at h_witness_witness_witness_right_right", "rewrite heval at h_witness_witness_witness_right_right", "exact h_witness_witness_witness_right_right", "apply IH_right", "exists x2", "exists x3", "exists x4", "split", "exact hc_right", "split", "exact h_witness_witness_witness_right_left", "exact h_witness_witness_witness_right_right" ], "defined_statement": "∀ b. ∀ c. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. JordanTupleListed(b,c,k,B,C,D,E,j) ∨ ¬JordanTupleListed(b,c,k,B,C,D,E,j)", "defined_statement_sha256": "57cd7d3a83bcd6dcdab8c4a3d03191570f1c7856e713d3e95044d46085040d1c", "definition_uses": { "ND0121": 2, "ND0376": 2, "PD0002": 1, "PD0013": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "7f7471d21130de91047e38f2628eb1277f4cc6b4483dc948f69ea7339329ef2a", "free_names": [], "script_definition_uses": { "ND0121": 2, "PD0002": 1, "PD0013": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "induction j" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro hempty" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_empty (E)" } ], [ { "kind": "text", "text": "apply jordan_tuple_listed_empty" } ], [ { "kind": "text", "text": "exact hempty" } ], [ { "kind": "text", "text": "cases IH" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_lift (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_listed_lift" } ], [ { "kind": "text", "text": "exact IH_left" } ], [ { "kind": "text", "text": "have hd : " }, { "kind": "text", "text": "∃ d. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,j,d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (B)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (C)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (j)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hd" } ], [ { "kind": "text", "text": "have he : " }, { "kind": "text", "text": "∃ e. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,j,e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (D)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (E)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (j)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases he" } ], [ { "kind": "text", "text": "have heq : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,x,x1,k)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,x,x1,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_decidable (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_decidable (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_decidable (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_decidable (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_decidable (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_decidable" } ], [ { "kind": "text", "text": "cases heq" } ], [ { "kind": "text", "text": "left" } ], [ { "kind": "text", "text": "exists j" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize le_refl (S j)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hd_witness" } ], [ { "kind": "text", "text": "exact he_witness" } ], [ { "kind": "text", "text": "exact heq_left" } ], [ { "kind": "text", "text": "right" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "cases h" } ], [ { "kind": "text", "text": "cases h_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness_right" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "x2 = j ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x2,j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (x2)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_left" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "cases h_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "have hdval : x3=x" } ], [ { "kind": "text", "text": "specialize beta_at_unique (B)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (C)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x3)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hc_left at h_witness_witness_witness_right_left_left" } ], [ { "kind": "text", "text": "rewrite hc_left at h_witness_witness_witness_right_left_left" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_left_left" } ], [ { "kind": "text", "text": "exact hd_witness" } ], [ { "kind": "text", "text": "have heval : x4=x1" } ], [ { "kind": "text", "text": "specialize beta_at_unique (D)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (E)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x4)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hc_left at h_witness_witness_witness_right_left_right" } ], [ { "kind": "text", "text": "rewrite hc_left at h_witness_witness_witness_right_left_right" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_left_right" } ], [ { "kind": "text", "text": "exact he_witness" } ], [ { "kind": "text", "text": "apply heq_right" } ], [ { "kind": "text", "text": "rewrite hdval at h_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "rewrite heval at h_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "rewrite heval at h_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "apply IH_right" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact h_witness_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0376": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,B,C,D,E,j)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,B,C,D,E,j)" } ] }, "dependencies": [ "jordan_tuple_listed_empty", "jordan_tuple_listed_lift", "beta_at_exists", "jordan_tuple_equal_decidable", "le_refl", "finite_lt_succ_eq_or_lt", "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT001C", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_listed_decidable", "original_ha_bundle_verified": true, "proof_bundle_node_id": 285, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro k", "intro B", "intro C", "intro D", "intro E", "induction j", "right", "intro hempty", "specialize jordan_tuple_listed_empty (b)", "specialize jordan_tuple_listed_empty (c)", "specialize jordan_tuple_listed_empty (k)", "specialize jordan_tuple_listed_empty (B)", "specialize jordan_tuple_listed_empty (C)", "specialize jordan_tuple_listed_empty (D)", "specialize jordan_tuple_listed_empty (E)", "apply jordan_tuple_listed_empty", "exact hempty", "cases IH", "left", "specialize jordan_tuple_listed_lift (b)", "specialize jordan_tuple_listed_lift (c)", "specialize jordan_tuple_listed_lift (k)", "specialize jordan_tuple_listed_lift (B)", "specialize jordan_tuple_listed_lift (C)", "specialize jordan_tuple_listed_lift (D)", "specialize jordan_tuple_listed_lift (E)", "specialize jordan_tuple_listed_lift (j)", "apply jordan_tuple_listed_lift", "exact IH_left", "have hd : exists d. ((exists fs_h_jt_listedlastcode. fs_h_jt_listedlastcode + S (d) = S ((S (j)) * C)) /\\ exists fs_q_jt_listedlastcode. B = fs_q_jt_listedlastcode * S ((S (j)) * C) + (d))", "specialize beta_at_exists (B)", "specialize beta_at_exists (C)", "specialize beta_at_exists (j)", "apply beta_at_exists", "cases hd", "have he : exists e. ((exists fs_h_jt_listedlastscale. fs_h_jt_listedlastscale + S (e) = S ((S (j)) * E)) /\\ exists fs_q_jt_listedlastscale. D = fs_q_jt_listedlastscale * S ((S (j)) * E) + (e))", "specialize beta_at_exists (D)", "specialize beta_at_exists (E)", "specialize beta_at_exists (j)", "apply beta_at_exists", "cases he", "have heq : (forall jt_index_listedyes jt_left_listedyes jt_right_listedyes. (exists jt_gap_listedyesindex. jt_gap_listedyesindex+S (jt_index_listedyes)=(k)) -> (((exists fs_h_jt_listedyesleft. fs_h_jt_listedyesleft + S (jt_left_listedyes) = S ((S (jt_index_listedyes)) * c)) /\\ exists fs_q_jt_listedyesleft. b = fs_q_jt_listedyesleft * S ((S (jt_index_listedyes)) * c) + (jt_left_listedyes))) -> (((exists fs_h_jt_listedyesright. fs_h_jt_listedyesright + S (jt_right_listedyes) = S ((S (jt_index_listedyes)) * x1)) /\\ exists fs_q_jt_listedyesright. x = fs_q_jt_listedyesright * S ((S (jt_index_listedyes)) * x1) + (jt_right_listedyes))) -> jt_left_listedyes=jt_right_listedyes) \\/ ~(forall jt_index_listedno jt_left_listedno jt_right_listedno. (exists jt_gap_listednoindex. jt_gap_listednoindex+S (jt_index_listedno)=(k)) -> (((exists fs_h_jt_listednoleft. fs_h_jt_listednoleft + S (jt_left_listedno) = S ((S (jt_index_listedno)) * c)) /\\ exists fs_q_jt_listednoleft. b = fs_q_jt_listednoleft * S ((S (jt_index_listedno)) * c) + (jt_left_listedno))) -> (((exists fs_h_jt_listednoright. fs_h_jt_listednoright + S (jt_right_listedno) = S ((S (jt_index_listedno)) * x1)) /\\ exists fs_q_jt_listednoright. x = fs_q_jt_listednoright * S ((S (jt_index_listedno)) * x1) + (jt_right_listedno))) -> jt_left_listedno=jt_right_listedno)", "specialize jordan_tuple_equal_decidable (b)", "specialize jordan_tuple_equal_decidable (c)", "specialize jordan_tuple_equal_decidable (x)", "specialize jordan_tuple_equal_decidable (x1)", "specialize jordan_tuple_equal_decidable (k)", "apply jordan_tuple_equal_decidable", "cases heq", "left", "exists j", "exists x", "exists x1", "split", "specialize le_refl (S j)", "apply le_refl", "split", "split", "exact hd_witness", "exact he_witness", "exact heq_left", "right", "intro h", "cases h", "cases h_witness", "cases h_witness_witness", "cases h_witness_witness_witness", "cases h_witness_witness_witness_right", "have hc : x2=j \\/ (exists jt_gap_listedcase. jt_gap_listedcase+S (x2)=(j))", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (x2)", "apply finite_lt_succ_eq_or_lt", "exact h_witness_witness_witness_left", "cases hc", "cases h_witness_witness_witness_right_left", "have hdval : x3=x", "specialize beta_at_unique (B)", "specialize beta_at_unique (C)", "specialize beta_at_unique (j)", "specialize beta_at_unique (x3)", "specialize beta_at_unique (x)", "apply beta_at_unique", "rewrite hc_left at h_witness_witness_witness_right_left_left", "rewrite hc_left at h_witness_witness_witness_right_left_left", "exact h_witness_witness_witness_right_left_left", "exact hd_witness", "have heval : x4=x1", "specialize beta_at_unique (D)", "specialize beta_at_unique (E)", "specialize beta_at_unique (j)", "specialize beta_at_unique (x4)", "specialize beta_at_unique (x1)", "apply beta_at_unique", "rewrite hc_left at h_witness_witness_witness_right_left_right", "rewrite hc_left at h_witness_witness_witness_right_left_right", "exact h_witness_witness_witness_right_left_right", "exact he_witness", "apply heq_right", "rewrite hdval at h_witness_witness_witness_right_right", "rewrite heval at h_witness_witness_witness_right_right", "rewrite heval at h_witness_witness_witness_right_right", "exact h_witness_witness_witness_right_right", "apply IH_right", "exists x2", "exists x3", "exists x4", "split", "exact hc_right", "split", "exact h_witness_witness_witness_right_left", "exact h_witness_witness_witness_right_right" ], "script_sha256": "a4542bfff0057ba4cccd6c2eaee41774be1377dc41b997c20b952e1be80fcb73", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "a4542bfff0057ba4cccd6c2eaee41774be1377dc41b997c20b952e1be80fcb73", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "7f7471d21130de91047e38f2628eb1277f4cc6b4483dc948f69ea7339329ef2a" } ], "stable_member": false, "statement": "forall b c k B C D E j. (exists jt_index_listedyes jt_code_listedyes jt_scale_listedyes. ((exists jt_gap_listedyesindex. jt_gap_listedyesindex+S (jt_index_listedyes)=(j)) /\\ (((((((exists fs_h_jt_listedyescode. fs_h_jt_listedyescode + S (jt_code_listedyes) = S ((S (jt_index_listedyes)) * C)) /\\ exists fs_q_jt_listedyescode. B = fs_q_jt_listedyescode * S ((S (jt_index_listedyes)) * C) + (jt_code_listedyes))) /\\ (((exists fs_h_jt_listedyesscale. fs_h_jt_listedyesscale + S (jt_scale_listedyes) = S ((S (jt_index_listedyes)) * E)) /\\ exists fs_q_jt_listedyesscale. D = fs_q_jt_listedyesscale * S ((S (jt_index_listedyes)) * E) + (jt_scale_listedyes))))) /\\ (forall jt_index_listedyesequal jt_left_listedyesequal jt_right_listedyesequal. (exists jt_gap_listedyesequalindex. jt_gap_listedyesequalindex+S (jt_index_listedyesequal)=(k)) -> (((exists fs_h_jt_listedyesequalleft. fs_h_jt_listedyesequalleft + S (jt_left_listedyesequal) = S ((S (jt_index_listedyesequal)) * c)) /\\ exists fs_q_jt_listedyesequalleft. b = fs_q_jt_listedyesequalleft * S ((S (jt_index_listedyesequal)) * c) + (jt_left_listedyesequal))) -> (((exists fs_h_jt_listedyesequalright. fs_h_jt_listedyesequalright + S (jt_right_listedyesequal) = S ((S (jt_index_listedyesequal)) * jt_scale_listedyes)) /\\ exists fs_q_jt_listedyesequalright. jt_code_listedyes = fs_q_jt_listedyesequalright * S ((S (jt_index_listedyesequal)) * jt_scale_listedyes) + (jt_right_listedyesequal))) -> jt_left_listedyesequal=jt_right_listedyesequal))))) \\/ ~(exists jt_index_listedno jt_code_listedno jt_scale_listedno. ((exists jt_gap_listednoindex. jt_gap_listednoindex+S (jt_index_listedno)=(j)) /\\ (((((((exists fs_h_jt_listednocode. fs_h_jt_listednocode + S (jt_code_listedno) = S ((S (jt_index_listedno)) * C)) /\\ exists fs_q_jt_listednocode. B = fs_q_jt_listednocode * S ((S (jt_index_listedno)) * C) + (jt_code_listedno))) /\\ (((exists fs_h_jt_listednoscale. fs_h_jt_listednoscale + S (jt_scale_listedno) = S ((S (jt_index_listedno)) * E)) /\\ exists fs_q_jt_listednoscale. D = fs_q_jt_listednoscale * S ((S (jt_index_listedno)) * E) + (jt_scale_listedno))))) /\\ (forall jt_index_listednoequal jt_left_listednoequal jt_right_listednoequal. (exists jt_gap_listednoequalindex. jt_gap_listednoequalindex+S (jt_index_listednoequal)=(k)) -> (((exists fs_h_jt_listednoequalleft. fs_h_jt_listednoequalleft + S (jt_left_listednoequal) = S ((S (jt_index_listednoequal)) * c)) /\\ exists fs_q_jt_listednoequalleft. b = fs_q_jt_listednoequalleft * S ((S (jt_index_listednoequal)) * c) + (jt_left_listednoequal))) -> (((exists fs_h_jt_listednoequalright. fs_h_jt_listednoequalright + S (jt_right_listednoequal) = S ((S (jt_index_listednoequal)) * jt_scale_listedno)) /\\ exists fs_q_jt_listednoequalright. jt_code_listedno = fs_q_jt_listednoequalright * S ((S (jt_index_listednoequal)) * jt_scale_listedno) + (jt_right_listednoequal))) -> jt_left_listednoequal=jt_right_listednoequal)))))", "statement_sha256": "7f7471d21130de91047e38f2628eb1277f4cc6b4483dc948f69ea7339329ef2a", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Finite list membership is decided by actual outer entries and coordinate equality." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 31, "body_proof_nodes": 82, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro c", "intro B", "intro C", "intro D", "intro E", "split", "intro i", "intro hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "split", "intro i", "intro h", "intro b", "intro e", "intro d", "intro f", "intro hi", "intro hh", "intro he", "intro hf", "intro heq", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "intro z", "intro hz", "intro hb", "intro hp", "exfalso", "specialize lt_not_le (z)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hz", "specialize zero_le (z)", "apply zero_le" ], "defined_statement": "∀ k. ∀ n. ∀ c. ∀ B. ∀ C. ∀ D. ∀ E. JordanTupleScan(k,n,c,0,B,C,D,E,0)", "defined_statement_sha256": "c1225ec34798140a1e1ecb61e2773db90498be71336391cccb8ce1c724169509", "definition_uses": { "ND0377": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "dcf562fc09f24131baad8869c9e1216a08d10ae7b032b3a2acec2b72da2f0e17", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (i)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "specialize zero_le (i)" } ], [ { "kind": "text", "text": "apply zero_le" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hf" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (i)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "specialize zero_le (i)" } ], [ { "kind": "text", "text": "apply zero_le" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (z)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "specialize zero_le (z)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "ND0377": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ c. ∀ B. ∀ C. ∀ D. ∀ E. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,0,B,C,D,E,0)" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_candidate_theorems", "first_admitted_version": "v35", "id": "JT001D", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_scan_empty", "original_ha_bundle_verified": true, "proof_bundle_node_id": 286, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro c", "intro B", "intro C", "intro D", "intro E", "split", "intro i", "intro hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "split", "intro i", "intro h", "intro b", "intro e", "intro d", "intro f", "intro hi", "intro hh", "intro he", "intro hf", "intro heq", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le", "intro z", "intro hz", "intro hb", "intro hp", "exfalso", "specialize lt_not_le (z)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hz", "specialize zero_le (z)", "apply zero_le" ], "script_sha256": "4dfd72c6e84927205ff7d94da63738705e1f48fceea7a2092720b9cdcbec9b3b", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_candidate_theorems", "script_sha256": "4dfd72c6e84927205ff7d94da63738705e1f48fceea7a2092720b9cdcbec9b3b", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "dcf562fc09f24131baad8869c9e1216a08d10ae7b032b3a2acec2b72da2f0e17" } ], "stable_member": false, "statement": "forall k n c B C D E. ((forall jt_i_scanempty. (exists jt_gap_scanemptysoundindex. jt_gap_scanemptysoundindex+S (jt_i_scanempty)=(0)) -> exists jt_b_scanempty jt_e_scanempty. ((((((exists fs_h_jt_scanemptysoundcode. fs_h_jt_scanemptysoundcode + S (jt_b_scanempty) = S ((S (jt_i_scanempty)) * C)) /\\ exists fs_q_jt_scanemptysoundcode. B = fs_q_jt_scanemptysoundcode * S ((S (jt_i_scanempty)) * C) + (jt_b_scanempty))) /\\ (((exists fs_h_jt_scanemptysoundscale. fs_h_jt_scanemptysoundscale + S (jt_e_scanempty) = S ((S (jt_i_scanempty)) * E)) /\\ exists fs_q_jt_scanemptysoundscale. D = fs_q_jt_scanemptysoundscale * S ((S (jt_i_scanempty)) * E) + (jt_e_scanempty))))) /\\ (((forall jt_index_scanemptybound. (exists jt_gap_scanemptyboundindex. jt_gap_scanemptyboundindex+S (jt_index_scanemptybound)=(k)) -> exists jt_value_scanemptybound. ((((exists fs_h_jt_scanemptyboundat. fs_h_jt_scanemptyboundat + S (jt_value_scanemptybound) = S ((S (jt_index_scanemptybound)) * jt_e_scanempty)) /\\ exists fs_q_jt_scanemptyboundat. jt_b_scanempty = fs_q_jt_scanemptyboundat * S ((S (jt_index_scanemptybound)) * jt_e_scanempty) + (jt_value_scanemptybound))) /\\ (exists jt_gap_scanemptyboundvalue. jt_gap_scanemptyboundvalue+S (jt_value_scanemptybound)=(n)))) /\\ (forall jt_divisor_scanemptyprimitive. (exists jt_factor_scanemptyprimitivemodulus. (n)=(jt_divisor_scanemptyprimitive)*jt_factor_scanemptyprimitivemodulus) -> (forall jt_index_scanemptyprimitivecoordinates jt_value_scanemptyprimitivecoordinates. (exists jt_gap_scanemptyprimitivecoordinatesindex. jt_gap_scanemptyprimitivecoordinatesindex+S (jt_index_scanemptyprimitivecoordinates)=(k)) -> (((exists fs_h_jt_scanemptyprimitivecoordinatesat. fs_h_jt_scanemptyprimitivecoordinatesat + S (jt_value_scanemptyprimitivecoordinates) = S ((S (jt_index_scanemptyprimitivecoordinates)) * jt_e_scanempty)) /\\ exists fs_q_jt_scanemptyprimitivecoordinatesat. jt_b_scanempty = fs_q_jt_scanemptyprimitivecoordinatesat * S ((S (jt_index_scanemptyprimitivecoordinates)) * jt_e_scanempty) + (jt_value_scanemptyprimitivecoordinates))) -> (exists jt_factor_scanemptyprimitivecoordinatesdivides. (jt_value_scanemptyprimitivecoordinates)=(jt_divisor_scanemptyprimitive)*jt_factor_scanemptyprimitivecoordinatesdivides)) -> jt_divisor_scanemptyprimitive=1))))) /\\ (((forall jt_i_scanempty jt_h_scanempty jt_b_scanempty jt_e_scanempty jt_d_scanempty jt_f_scanempty. (exists jt_gap_scanemptyfirstindex. jt_gap_scanemptyfirstindex+S (jt_i_scanempty)=(0)) -> (exists jt_gap_scanemptysecondindex. jt_gap_scanemptysecondindex+S (jt_h_scanempty)=(0)) -> (((((exists fs_h_jt_scanemptyfirstcode. fs_h_jt_scanemptyfirstcode + S (jt_b_scanempty) = S ((S (jt_i_scanempty)) * C)) /\\ exists fs_q_jt_scanemptyfirstcode. B = fs_q_jt_scanemptyfirstcode * S ((S (jt_i_scanempty)) * C) + (jt_b_scanempty))) /\\ (((exists fs_h_jt_scanemptyfirstscale. fs_h_jt_scanemptyfirstscale + S (jt_e_scanempty) = S ((S (jt_i_scanempty)) * E)) /\\ exists fs_q_jt_scanemptyfirstscale. D = fs_q_jt_scanemptyfirstscale * S ((S (jt_i_scanempty)) * E) + (jt_e_scanempty))))) -> (((((exists fs_h_jt_scanemptysecondcode. fs_h_jt_scanemptysecondcode + S (jt_d_scanempty) = S ((S (jt_h_scanempty)) * C)) /\\ exists fs_q_jt_scanemptysecondcode. B = fs_q_jt_scanemptysecondcode * S ((S (jt_h_scanempty)) * C) + (jt_d_scanempty))) /\\ (((exists fs_h_jt_scanemptysecondscale. fs_h_jt_scanemptysecondscale + S (jt_f_scanempty) = S ((S (jt_h_scanempty)) * E)) /\\ exists fs_q_jt_scanemptysecondscale. D = fs_q_jt_scanemptysecondscale * S ((S (jt_h_scanempty)) * E) + (jt_f_scanempty))))) -> (forall jt_index_scanemptysame jt_left_scanemptysame jt_right_scanemptysame. (exists jt_gap_scanemptysameindex. jt_gap_scanemptysameindex+S (jt_index_scanemptysame)=(k)) -> (((exists fs_h_jt_scanemptysameleft. fs_h_jt_scanemptysameleft + S (jt_left_scanemptysame) = S ((S (jt_index_scanemptysame)) * jt_e_scanempty)) /\\ exists fs_q_jt_scanemptysameleft. jt_b_scanempty = fs_q_jt_scanemptysameleft * S ((S (jt_index_scanemptysame)) * jt_e_scanempty) + (jt_left_scanemptysame))) -> (((exists fs_h_jt_scanemptysameright. fs_h_jt_scanemptysameright + S (jt_right_scanemptysame) = S ((S (jt_index_scanemptysame)) * jt_f_scanempty)) /\\ exists fs_q_jt_scanemptysameright. jt_d_scanempty = fs_q_jt_scanemptysameright * S ((S (jt_index_scanemptysame)) * jt_f_scanempty) + (jt_right_scanemptysame))) -> jt_left_scanemptysame=jt_right_scanemptysame) -> jt_i_scanempty=jt_h_scanempty) /\\ (forall jt_z_scanempty. (exists jt_gap_scanemptycodeindex. jt_gap_scanemptycodeindex+S (jt_z_scanempty)=(0)) -> (forall jt_index_scanemptyinputbound. (exists jt_gap_scanemptyinputboundindex. jt_gap_scanemptyinputboundindex+S (jt_index_scanemptyinputbound)=(k)) -> exists jt_value_scanemptyinputbound. ((((exists fs_h_jt_scanemptyinputboundat. fs_h_jt_scanemptyinputboundat + S (jt_value_scanemptyinputbound) = S ((S (jt_index_scanemptyinputbound)) * c)) /\\ exists fs_q_jt_scanemptyinputboundat. jt_z_scanempty = fs_q_jt_scanemptyinputboundat * S ((S (jt_index_scanemptyinputbound)) * c) + (jt_value_scanemptyinputbound))) /\\ (exists jt_gap_scanemptyinputboundvalue. jt_gap_scanemptyinputboundvalue+S (jt_value_scanemptyinputbound)=(n)))) -> (forall jt_divisor_scanemptyinputprimitive. (exists jt_factor_scanemptyinputprimitivemodulus. (n)=(jt_divisor_scanemptyinputprimitive)*jt_factor_scanemptyinputprimitivemodulus) -> (forall jt_index_scanemptyinputprimitivecoordinates jt_value_scanemptyinputprimitivecoordinates. (exists jt_gap_scanemptyinputprimitivecoordinatesindex. jt_gap_scanemptyinputprimitivecoordinatesindex+S (jt_index_scanemptyinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_scanemptyinputprimitivecoordinatesat. fs_h_jt_scanemptyinputprimitivecoordinatesat + S (jt_value_scanemptyinputprimitivecoordinates) = S ((S (jt_index_scanemptyinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_scanemptyinputprimitivecoordinatesat. jt_z_scanempty = fs_q_jt_scanemptyinputprimitivecoordinatesat * S ((S (jt_index_scanemptyinputprimitivecoordinates)) * c) + (jt_value_scanemptyinputprimitivecoordinates))) -> (exists jt_factor_scanemptyinputprimitivecoordinatesdivides. (jt_value_scanemptyinputprimitivecoordinates)=(jt_divisor_scanemptyinputprimitive)*jt_factor_scanemptyinputprimitivecoordinatesdivides)) -> jt_divisor_scanemptyinputprimitive=1) -> (exists jt_index_scanemptylisted jt_code_scanemptylisted jt_scale_scanemptylisted. ((exists jt_gap_scanemptylistedindex. jt_gap_scanemptylistedindex+S (jt_index_scanemptylisted)=(0)) /\\ (((((((exists fs_h_jt_scanemptylistedcode. fs_h_jt_scanemptylistedcode + S (jt_code_scanemptylisted) = S ((S (jt_index_scanemptylisted)) * C)) /\\ exists fs_q_jt_scanemptylistedcode. B = fs_q_jt_scanemptylistedcode * S ((S (jt_index_scanemptylisted)) * C) + (jt_code_scanemptylisted))) /\\ (((exists fs_h_jt_scanemptylistedscale. fs_h_jt_scanemptylistedscale + S (jt_scale_scanemptylisted) = S ((S (jt_index_scanemptylisted)) * E)) /\\ exists fs_q_jt_scanemptylistedscale. D = fs_q_jt_scanemptylistedscale * S ((S (jt_index_scanemptylisted)) * E) + (jt_scale_scanemptylisted))))) /\\ (forall jt_index_scanemptylistedequal jt_left_scanemptylistedequal jt_right_scanemptylistedequal. (exists jt_gap_scanemptylistedequalindex. jt_gap_scanemptylistedequalindex+S (jt_index_scanemptylistedequal)=(k)) -> (((exists fs_h_jt_scanemptylistedequalleft. fs_h_jt_scanemptylistedequalleft + S (jt_left_scanemptylistedequal) = S ((S (jt_index_scanemptylistedequal)) * c)) /\\ exists fs_q_jt_scanemptylistedequalleft. jt_z_scanempty = fs_q_jt_scanemptylistedequalleft * S ((S (jt_index_scanemptylistedequal)) * c) + (jt_left_scanemptylistedequal))) -> (((exists fs_h_jt_scanemptylistedequalright. fs_h_jt_scanemptylistedequalright + S (jt_right_scanemptylistedequal) = S ((S (jt_index_scanemptylistedequal)) * jt_scale_scanemptylisted)) /\\ exists fs_q_jt_scanemptylistedequalright. jt_code_scanemptylisted = fs_q_jt_scanemptylistedequalright * S ((S (jt_index_scanemptylistedequal)) * jt_scale_scanemptylisted) + (jt_right_scanemptylistedequal))) -> jt_left_scanemptylistedequal=jt_right_scanemptylistedequal)))))))))", "statement_sha256": "dcf562fc09f24131baad8869c9e1216a08d10ae7b032b3a2acec2b72da2f0e17", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Empty scan has soundness, no duplicate positions, and vacuous code coverage." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 32, "body_proof_nodes": 50, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro h", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (i)", "specialize beta_at_unique (a)", "specialize beta_at_unique (z)", "apply beta_at_unique", "specialize h (i)", "specialize h (a)", "apply h", "exact hi", "exact ha", "exact hz" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. BetaPrefixEqual(b,c,d,e,k) → IntegerVectorZero(b,c,d,e,k)", "defined_statement_sha256": "aac559247cc154969db9e0e87dd2d8dee8abb0e2d753383764cf44ed53a4371e", "definition_uses": { "ND0121": 1, "ND0263": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "130648aa073905c8a960cb16940d1cad6438ee787540326ea52d1d4d770414b0", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize beta_at_unique (d)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (e)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (z)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "specialize h (a)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0263": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0263", "kind": "definition", "text": "BetaPrefixEqual(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT001E", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_prefix_equal", "original_ha_bundle_verified": true, "proof_bundle_node_id": 287, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro h", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (i)", "specialize beta_at_unique (a)", "specialize beta_at_unique (z)", "apply beta_at_unique", "specialize h (i)", "specialize h (a)", "apply h", "exact hi", "exact ha", "exact hz" ], "script_sha256": "c6bf30f6e455b8a85a231aa05f04e45e75c00bb224f29212bccae6b7edf76164", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "c6bf30f6e455b8a85a231aa05f04e45e75c00bb224f29212bccae6b7edf76164", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "130648aa073905c8a960cb16940d1cad6438ee787540326ea52d1d4d770414b0" } ], "stable_member": false, "statement": "forall b c d e k. (forall jt_index_prefixsource jt_value_prefixsource. (exists jt_gap_prefixsourceindex. jt_gap_prefixsourceindex+S (jt_index_prefixsource)=(k)) -> (((exists fs_h_jt_prefixsourceold. fs_h_jt_prefixsourceold + S (jt_value_prefixsource) = S ((S (jt_index_prefixsource)) * c)) /\\ exists fs_q_jt_prefixsourceold. b = fs_q_jt_prefixsourceold * S ((S (jt_index_prefixsource)) * c) + (jt_value_prefixsource))) -> (((exists fs_h_jt_prefixsourcenew. fs_h_jt_prefixsourcenew + S (jt_value_prefixsource) = S ((S (jt_index_prefixsource)) * e)) /\\ exists fs_q_jt_prefixsourcenew. d = fs_q_jt_prefixsourcenew * S ((S (jt_index_prefixsource)) * e) + (jt_value_prefixsource)))) -> (forall jt_index_prefixtarget jt_left_prefixtarget jt_right_prefixtarget. (exists jt_gap_prefixtargetindex. jt_gap_prefixtargetindex+S (jt_index_prefixtarget)=(k)) -> (((exists fs_h_jt_prefixtargetleft. fs_h_jt_prefixtargetleft + S (jt_left_prefixtarget) = S ((S (jt_index_prefixtarget)) * c)) /\\ exists fs_q_jt_prefixtargetleft. b = fs_q_jt_prefixtargetleft * S ((S (jt_index_prefixtarget)) * c) + (jt_left_prefixtarget))) -> (((exists fs_h_jt_prefixtargetright. fs_h_jt_prefixtargetright + S (jt_right_prefixtarget) = S ((S (jt_index_prefixtarget)) * e)) /\\ exists fs_q_jt_prefixtargetright. d = fs_q_jt_prefixtargetright * S ((S (jt_index_prefixtarget)) * e) + (jt_right_prefixtarget))) -> jt_left_prefixtarget=jt_right_prefixtarget)", "statement_sha256": "130648aa073905c8a960cb16940d1cad6438ee787540326ea52d1d4d770414b0", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "An actual one-way beta prefix extension preserves decoded coordinates." }, { "admission_dependencies": [ "beta_at_exists" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 22, "body_proof_nodes": 41, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro i", "intro a", "intro h", "intro hi", "intro ha", "have hz : ∃ z. BetaAt(d,e,i,z)", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hz", "have heq : a=x", "specialize h (i)", "specialize h (a)", "specialize h (x)", "apply h", "exact hi", "exact ha", "exact hz_witness", "rewrite heq", "rewrite heq", "exact hz_witness" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ i. ∀ a. IntegerVectorZero(b,c,d,e,k) → Lt(i,k) → BetaAt(b,c,i,a) → BetaAt(d,e,i,a)", "defined_statement_sha256": "b9302c190c4659593778ce4885642b036dcea4a49f408f2e8b2c1d40dd7ca9db", "definition_uses": { "ND0121": 1, "PD0002": 1, "PD0013": 3 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "2b4ea486e160cac69c60697ce91996b3404519b8d3c91cdf06a8a9e0adc2dcef", "free_names": [], "script_definition_uses": { "PD0013": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "have hz : " }, { "kind": "text", "text": "∃ z. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,z)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (i)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hz" } ], [ { "kind": "text", "text": "have heq : a=x" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "specialize h (a)" } ], [ { "kind": "text", "text": "specialize h (x)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz_witness" } ], [ { "kind": "text", "text": "rewrite heq" } ], [ { "kind": "text", "text": "rewrite heq" } ], [ { "kind": "text", "text": "exact hz_witness" } ] ], "statement_definition_uses": { "ND0121": 1, "PD0002": 1, "PD0013": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ i. ∀ a. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,a)" } ] }, "dependencies": [ "beta_at_exists" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT001F", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_entry", "original_ha_bundle_verified": true, "proof_bundle_node_id": 288, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro i", "intro a", "intro h", "intro hi", "intro ha", "have hz : exists z. ((exists fs_h_jt_equalentry. fs_h_jt_equalentry + S (z) = S ((S (i)) * e)) /\\ exists fs_q_jt_equalentry. d = fs_q_jt_equalentry * S ((S (i)) * e) + (z))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hz", "have heq : a=x", "specialize h (i)", "specialize h (a)", "specialize h (x)", "apply h", "exact hi", "exact ha", "exact hz_witness", "rewrite heq", "rewrite heq", "exact hz_witness" ], "script_sha256": "9b3447ace6d5920075960146ec984ffc58d31be5ea059ad365d379b6be4df31a", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "9b3447ace6d5920075960146ec984ffc58d31be5ea059ad365d379b6be4df31a", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "2b4ea486e160cac69c60697ce91996b3404519b8d3c91cdf06a8a9e0adc2dcef" } ], "stable_member": false, "statement": "forall b c d e k i a. (forall jt_index_entryequal jt_left_entryequal jt_right_entryequal. (exists jt_gap_entryequalindex. jt_gap_entryequalindex+S (jt_index_entryequal)=(k)) -> (((exists fs_h_jt_entryequalleft. fs_h_jt_entryequalleft + S (jt_left_entryequal) = S ((S (jt_index_entryequal)) * c)) /\\ exists fs_q_jt_entryequalleft. b = fs_q_jt_entryequalleft * S ((S (jt_index_entryequal)) * c) + (jt_left_entryequal))) -> (((exists fs_h_jt_entryequalright. fs_h_jt_entryequalright + S (jt_right_entryequal) = S ((S (jt_index_entryequal)) * e)) /\\ exists fs_q_jt_entryequalright. d = fs_q_jt_entryequalright * S ((S (jt_index_entryequal)) * e) + (jt_right_entryequal))) -> jt_left_entryequal=jt_right_entryequal) -> (exists jt_gap_entryindex. jt_gap_entryindex+S (i)=(k)) -> (((exists fs_h_jt_entrysource. fs_h_jt_entrysource + S (a) = S ((S (i)) * c)) /\\ exists fs_q_jt_entrysource. b = fs_q_jt_entrysource * S ((S (i)) * c) + (a))) -> (((exists fs_h_jt_entrytarget. fs_h_jt_entrytarget + S (a) = S ((S (i)) * e)) /\\ exists fs_q_jt_entrytarget. d = fs_q_jt_entrytarget * S ((S (i)) * e) + (a)))", "statement_sha256": "2b4ea486e160cac69c60697ce91996b3404519b8d3c91cdf06a8a9e0adc2dcef", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Coordinate equality transports an actual beta entry with unchanged value." }, { "admission_dependencies": [ "jordan_tuple_equal_entry" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 25, "body_proof_nodes": 35, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro n", "intro heq", "intro hb", "intro i", "intro hi", "have ha : ∃ a. BetaAt(b,c,i,a) ∧ Lt(a,n)", "specialize hb (i)", "apply hb", "exact hi", "cases ha", "cases ha_witness", "exists x", "split", "specialize jordan_tuple_equal_entry (b)", "specialize jordan_tuple_equal_entry (c)", "specialize jordan_tuple_equal_entry (d)", "specialize jordan_tuple_equal_entry (e)", "specialize jordan_tuple_equal_entry (k)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (x)", "apply jordan_tuple_equal_entry", "exact heq", "exact hi", "exact ha_witness_left", "exact ha_witness_right" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ n. IntegerVectorZero(b,c,d,e,k) → BetaPrefixInto(b,c,k,n) → BetaPrefixInto(d,e,k,n)", "defined_statement_sha256": "f2cb7dd31dd480283e871529b9c4b3099d3533a9bc0c4cea53049ff9f9855968", "definition_uses": { "ND0121": 1, "ND0262": 2, "PD0002": 1, "PD0013": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "9e8158eefd12777db17090ea69eff48227039a9e862f503d0d696dcc42455d32", "free_names": [], "script_definition_uses": { "PD0002": 1, "PD0013": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ a. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,a)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(a,n)" } ], [ { "kind": "text", "text": "specialize hb (i)" } ], [ { "kind": "text", "text": "apply hb" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "cases ha_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha_witness_left" } ], [ { "kind": "text", "text": "exact ha_witness_right" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0262": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ n. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(d,e,k,n)" } ] }, "dependencies": [ "jordan_tuple_equal_entry" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT0020", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_bounded_transport", "original_ha_bundle_verified": true, "proof_bundle_node_id": 289, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro n", "intro heq", "intro hb", "intro i", "intro hi", "have ha : exists a. ((((exists fs_h_jt_boundtransport. fs_h_jt_boundtransport + S (a) = S ((S (i)) * c)) /\\ exists fs_q_jt_boundtransport. b = fs_q_jt_boundtransport * S ((S (i)) * c) + (a))) /\\ (exists jt_gap_boundvalue. jt_gap_boundvalue+S (a)=(n)))", "specialize hb (i)", "apply hb", "exact hi", "cases ha", "cases ha_witness", "exists x", "split", "specialize jordan_tuple_equal_entry (b)", "specialize jordan_tuple_equal_entry (c)", "specialize jordan_tuple_equal_entry (d)", "specialize jordan_tuple_equal_entry (e)", "specialize jordan_tuple_equal_entry (k)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (x)", "apply jordan_tuple_equal_entry", "exact heq", "exact hi", "exact ha_witness_left", "exact ha_witness_right" ], "script_sha256": "4ba1a9b62d4961977e5e01b6469526ff437bd1fa16abe59432560ce6dbd50fb2", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "4ba1a9b62d4961977e5e01b6469526ff437bd1fa16abe59432560ce6dbd50fb2", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "9e8158eefd12777db17090ea69eff48227039a9e862f503d0d696dcc42455d32" } ], "stable_member": false, "statement": "forall b c d e k n. (forall jt_index_boundequal jt_left_boundequal jt_right_boundequal. (exists jt_gap_boundequalindex. jt_gap_boundequalindex+S (jt_index_boundequal)=(k)) -> (((exists fs_h_jt_boundequalleft. fs_h_jt_boundequalleft + S (jt_left_boundequal) = S ((S (jt_index_boundequal)) * c)) /\\ exists fs_q_jt_boundequalleft. b = fs_q_jt_boundequalleft * S ((S (jt_index_boundequal)) * c) + (jt_left_boundequal))) -> (((exists fs_h_jt_boundequalright. fs_h_jt_boundequalright + S (jt_right_boundequal) = S ((S (jt_index_boundequal)) * e)) /\\ exists fs_q_jt_boundequalright. d = fs_q_jt_boundequalright * S ((S (jt_index_boundequal)) * e) + (jt_right_boundequal))) -> jt_left_boundequal=jt_right_boundequal) -> (forall jt_index_boundsource. (exists jt_gap_boundsourceindex. jt_gap_boundsourceindex+S (jt_index_boundsource)=(k)) -> exists jt_value_boundsource. ((((exists fs_h_jt_boundsourceat. fs_h_jt_boundsourceat + S (jt_value_boundsource) = S ((S (jt_index_boundsource)) * c)) /\\ exists fs_q_jt_boundsourceat. b = fs_q_jt_boundsourceat * S ((S (jt_index_boundsource)) * c) + (jt_value_boundsource))) /\\ (exists jt_gap_boundsourcevalue. jt_gap_boundsourcevalue+S (jt_value_boundsource)=(n)))) -> (forall jt_index_boundtarget. (exists jt_gap_boundtargetindex. jt_gap_boundtargetindex+S (jt_index_boundtarget)=(k)) -> exists jt_value_boundtarget. ((((exists fs_h_jt_boundtargetat. fs_h_jt_boundtargetat + S (jt_value_boundtarget) = S ((S (jt_index_boundtarget)) * e)) /\\ exists fs_q_jt_boundtargetat. d = fs_q_jt_boundtargetat * S ((S (jt_index_boundtarget)) * e) + (jt_value_boundtarget))) /\\ (exists jt_gap_boundtargetvalue. jt_gap_boundtargetvalue+S (jt_value_boundtarget)=(n))))", "statement_sha256": "9e8158eefd12777db17090ea69eff48227039a9e862f503d0d696dcc42455d32", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The canonical coordinate bound is independent of tuple encoding." }, { "admission_dependencies": [ "beta_prefix_extend", "jordan_tuple_prefix_equal" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 26, "body_proof_nodes": 54, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro B", "intro C", "intro D", "intro E", "intro j", "intro b", "intro c", "have hc : ∃ u. ∃ v. BetaAt(u,v,j,b) ∧ BetaPrefixEqual(B,C,u,v,j)", "specialize beta_prefix_extend (j)", "specialize beta_prefix_extend (B)", "specialize beta_prefix_extend (C)", "specialize beta_prefix_extend (b)", "apply beta_prefix_extend", "cases hc", "cases hc_witness", "cases hc_witness_witness", "have hs : ∃ u. ∃ v. BetaAt(u,v,j,c) ∧ BetaPrefixEqual(D,E,u,v,j)", "specialize beta_prefix_extend (j)", "specialize beta_prefix_extend (D)", "specialize beta_prefix_extend (E)", "specialize beta_prefix_extend (c)", "apply beta_prefix_extend", "cases hs", "cases hs_witness", "cases hs_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "split", "specialize jordan_tuple_prefix_equal (B)", "specialize jordan_tuple_prefix_equal (C)", "specialize jordan_tuple_prefix_equal (x)", "specialize jordan_tuple_prefix_equal (x1)", "specialize jordan_tuple_prefix_equal (j)", "apply jordan_tuple_prefix_equal", "exact hc_witness_witness_right", "split", "specialize jordan_tuple_prefix_equal (D)", "specialize jordan_tuple_prefix_equal (E)", "specialize jordan_tuple_prefix_equal (x2)", "specialize jordan_tuple_prefix_equal (x3)", "specialize jordan_tuple_prefix_equal (j)", "apply jordan_tuple_prefix_equal", "exact hs_witness_witness_right", "split", "exact hc_witness_witness_left", "exact hs_witness_witness_left" ], "defined_statement": "∀ B. ∀ C. ∀ D. ∀ E. ∀ j. ∀ b. ∀ c. ∃ U. ∃ V. ∃ W. ∃ X. IntegerVectorZero(B,C,U,V,j) ∧ (IntegerVectorZero(D,E,W,X,j) ∧ (BetaAt(U,V,j,b) ∧ BetaAt(W,X,j,c)))", "defined_statement_sha256": "d93166fff4d24e4a97ae46ebdf299dbc08d296b98010b6cc697f3d1fadc19f79", "definition_uses": { "ND0121": 2, "ND0263": 2, "PD0013": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "ef2c6370871d31f3515c6c432f2ee6f851fe7ba8580095f131a4ce17ab600d01", "free_names": [], "script_definition_uses": { "ND0263": 2, "PD0013": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "∃ u. ∃ v. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(u,v,j,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0263", "kind": "definition", "text": "BetaPrefixEqual(B,C,u,v,j)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (j)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (B)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (C)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (b)" } ], [ { "kind": "text", "text": "apply beta_prefix_extend" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "cases hc_witness" } ], [ { "kind": "text", "text": "cases hc_witness_witness" } ], [ { "kind": "text", "text": "have hs : " }, { "kind": "text", "text": "∃ u. ∃ v. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(u,v,j,c)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0263", "kind": "definition", "text": "BetaPrefixEqual(D,E,u,v,j)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (j)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (D)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (E)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (c)" } ], [ { "kind": "text", "text": "apply beta_prefix_extend" } ], [ { "kind": "text", "text": "cases hs" } ], [ { "kind": "text", "text": "cases hs_witness" } ], [ { "kind": "text", "text": "cases hs_witness_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_prefix_equal" } ], [ { "kind": "text", "text": "exact hc_witness_witness_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_prefix_equal" } ], [ { "kind": "text", "text": "exact hs_witness_witness_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hc_witness_witness_left" } ], [ { "kind": "text", "text": "exact hs_witness_witness_left" } ] ], "statement_definition_uses": { "ND0121": 2, "PD0013": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ B. ∀ C. ∀ D. ∀ E. ∀ j. ∀ b. ∀ c. ∃ U. ∃ V. ∃ W. ∃ X. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(B,C,U,V,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(D,E,W,X,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(U,V,j,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(W,X,j,c)" }, { "kind": "text", "text": "))" } ] }, "dependencies": [ "beta_prefix_extend", "jordan_tuple_prefix_equal" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT0021", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_outer_append_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 290, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro B", "intro C", "intro D", "intro E", "intro j", "intro b", "intro c", "have hc : exists u v. ((((exists fs_h_jt_appendcode. fs_h_jt_appendcode + S (b) = S ((S (j)) * v)) /\\ exists fs_q_jt_appendcode. u = fs_q_jt_appendcode * S ((S (j)) * v) + (b))) /\\ (forall jt_index_appendcodeprefix jt_value_appendcodeprefix. (exists jt_gap_appendcodeprefixindex. jt_gap_appendcodeprefixindex+S (jt_index_appendcodeprefix)=(j)) -> (((exists fs_h_jt_appendcodeprefixold. fs_h_jt_appendcodeprefixold + S (jt_value_appendcodeprefix) = S ((S (jt_index_appendcodeprefix)) * C)) /\\ exists fs_q_jt_appendcodeprefixold. B = fs_q_jt_appendcodeprefixold * S ((S (jt_index_appendcodeprefix)) * C) + (jt_value_appendcodeprefix))) -> (((exists fs_h_jt_appendcodeprefixnew. fs_h_jt_appendcodeprefixnew + S (jt_value_appendcodeprefix) = S ((S (jt_index_appendcodeprefix)) * v)) /\\ exists fs_q_jt_appendcodeprefixnew. u = fs_q_jt_appendcodeprefixnew * S ((S (jt_index_appendcodeprefix)) * v) + (jt_value_appendcodeprefix)))))", "specialize beta_prefix_extend (j)", "specialize beta_prefix_extend (B)", "specialize beta_prefix_extend (C)", "specialize beta_prefix_extend (b)", "apply beta_prefix_extend", "cases hc", "cases hc_witness", "cases hc_witness_witness", "have hs : exists u v. ((((exists fs_h_jt_appendscale. fs_h_jt_appendscale + S (c) = S ((S (j)) * v)) /\\ exists fs_q_jt_appendscale. u = fs_q_jt_appendscale * S ((S (j)) * v) + (c))) /\\ (forall jt_index_appendscaleprefix jt_value_appendscaleprefix. (exists jt_gap_appendscaleprefixindex. jt_gap_appendscaleprefixindex+S (jt_index_appendscaleprefix)=(j)) -> (((exists fs_h_jt_appendscaleprefixold. fs_h_jt_appendscaleprefixold + S (jt_value_appendscaleprefix) = S ((S (jt_index_appendscaleprefix)) * E)) /\\ exists fs_q_jt_appendscaleprefixold. D = fs_q_jt_appendscaleprefixold * S ((S (jt_index_appendscaleprefix)) * E) + (jt_value_appendscaleprefix))) -> (((exists fs_h_jt_appendscaleprefixnew. fs_h_jt_appendscaleprefixnew + S (jt_value_appendscaleprefix) = S ((S (jt_index_appendscaleprefix)) * v)) /\\ exists fs_q_jt_appendscaleprefixnew. u = fs_q_jt_appendscaleprefixnew * S ((S (jt_index_appendscaleprefix)) * v) + (jt_value_appendscaleprefix)))))", "specialize beta_prefix_extend (j)", "specialize beta_prefix_extend (D)", "specialize beta_prefix_extend (E)", "specialize beta_prefix_extend (c)", "apply beta_prefix_extend", "cases hs", "cases hs_witness", "cases hs_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "split", "specialize jordan_tuple_prefix_equal (B)", "specialize jordan_tuple_prefix_equal (C)", "specialize jordan_tuple_prefix_equal (x)", "specialize jordan_tuple_prefix_equal (x1)", "specialize jordan_tuple_prefix_equal (j)", "apply jordan_tuple_prefix_equal", "exact hc_witness_witness_right", "split", "specialize jordan_tuple_prefix_equal (D)", "specialize jordan_tuple_prefix_equal (E)", "specialize jordan_tuple_prefix_equal (x2)", "specialize jordan_tuple_prefix_equal (x3)", "specialize jordan_tuple_prefix_equal (j)", "apply jordan_tuple_prefix_equal", "exact hs_witness_witness_right", "split", "exact hc_witness_witness_left", "exact hs_witness_witness_left" ], "script_sha256": "26196fd9adaada2d205e850428ab003ca95e3cfc1eeede53853a89fa22bd4008", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "26196fd9adaada2d205e850428ab003ca95e3cfc1eeede53853a89fa22bd4008", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "ef2c6370871d31f3515c6c432f2ee6f851fe7ba8580095f131a4ce17ab600d01" } ], "stable_member": false, "statement": "forall B C D E j b c. exists U V W X. ((forall jt_index_outercodes jt_left_outercodes jt_right_outercodes. (exists jt_gap_outercodesindex. jt_gap_outercodesindex+S (jt_index_outercodes)=(j)) -> (((exists fs_h_jt_outercodesleft. fs_h_jt_outercodesleft + S (jt_left_outercodes) = S ((S (jt_index_outercodes)) * C)) /\\ exists fs_q_jt_outercodesleft. B = fs_q_jt_outercodesleft * S ((S (jt_index_outercodes)) * C) + (jt_left_outercodes))) -> (((exists fs_h_jt_outercodesright. fs_h_jt_outercodesright + S (jt_right_outercodes) = S ((S (jt_index_outercodes)) * V)) /\\ exists fs_q_jt_outercodesright. U = fs_q_jt_outercodesright * S ((S (jt_index_outercodes)) * V) + (jt_right_outercodes))) -> jt_left_outercodes=jt_right_outercodes) /\\ (((forall jt_index_outerscales jt_left_outerscales jt_right_outerscales. (exists jt_gap_outerscalesindex. jt_gap_outerscalesindex+S (jt_index_outerscales)=(j)) -> (((exists fs_h_jt_outerscalesleft. fs_h_jt_outerscalesleft + S (jt_left_outerscales) = S ((S (jt_index_outerscales)) * E)) /\\ exists fs_q_jt_outerscalesleft. D = fs_q_jt_outerscalesleft * S ((S (jt_index_outerscales)) * E) + (jt_left_outerscales))) -> (((exists fs_h_jt_outerscalesright. fs_h_jt_outerscalesright + S (jt_right_outerscales) = S ((S (jt_index_outerscales)) * X)) /\\ exists fs_q_jt_outerscalesright. W = fs_q_jt_outerscalesright * S ((S (jt_index_outerscales)) * X) + (jt_right_outerscales))) -> jt_left_outerscales=jt_right_outerscales) /\\ (((((exists fs_h_jt_outerlastcode. fs_h_jt_outerlastcode + S (b) = S ((S (j)) * V)) /\\ exists fs_q_jt_outerlastcode. U = fs_q_jt_outerlastcode * S ((S (j)) * V) + (b))) /\\ (((exists fs_h_jt_outerlastscale. fs_h_jt_outerlastscale + S (c) = S ((S (j)) * X)) /\\ exists fs_q_jt_outerlastscale. W = fs_q_jt_outerlastscale * S ((S (j)) * X) + (c))))))))", "statement_sha256": "ef2c6370871d31f3515c6c432f2ee6f851fe7ba8580095f131a4ce17ab600d01", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Construct both outer beta streams after an append; no list witness is assumed." }, { "admission_dependencies": [ "jordan_tuple_equal_trans" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 46, "body_proof_nodes": 75, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro B", "intro C", "intro D", "intro E", "intro j", "intro heq", "intro hl", "cases hl", "cases hl_witness", "cases hl_witness_witness", "cases hl_witness_witness_witness", "cases hl_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "split", "exact hl_witness_witness_witness_left", "split", "exact hl_witness_witness_witness_right_left", "specialize jordan_tuple_equal_trans (b)", "specialize jordan_tuple_equal_trans (c)", "specialize jordan_tuple_equal_trans (d)", "specialize jordan_tuple_equal_trans (e)", "specialize jordan_tuple_equal_trans (x1)", "specialize jordan_tuple_equal_trans (x2)", "specialize jordan_tuple_equal_trans (k)", "apply jordan_tuple_equal_trans", "exact heq", "exact hl_witness_witness_witness_right_right" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. IntegerVectorZero(b,c,d,e,k) → JordanTupleListed(d,e,k,B,C,D,E,j) → JordanTupleListed(b,c,k,B,C,D,E,j)", "defined_statement_sha256": "18375bb2df20d3698cd2d8c18b08e3005f31c52781cf6c1eae83f736e6347534", "definition_uses": { "ND0121": 1, "ND0376": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "a97e6df915eeebb9bd4e61c7998cbd88a12a0edee03fcf2a255e04cb23148bea", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hl_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness_witness_right" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hl_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hl_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_trans" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "exact hl_witness_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0376": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(d,e,k,B,C,D,E,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,B,C,D,E,j)" } ] }, "dependencies": [ "jordan_tuple_equal_trans" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT0022", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_listed_equal_transport", "original_ha_bundle_verified": true, "proof_bundle_node_id": 291, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro B", "intro C", "intro D", "intro E", "intro j", "intro heq", "intro hl", "cases hl", "cases hl_witness", "cases hl_witness_witness", "cases hl_witness_witness_witness", "cases hl_witness_witness_witness_right", "exists x", "exists x1", "exists x2", "split", "exact hl_witness_witness_witness_left", "split", "exact hl_witness_witness_witness_right_left", "specialize jordan_tuple_equal_trans (b)", "specialize jordan_tuple_equal_trans (c)", "specialize jordan_tuple_equal_trans (d)", "specialize jordan_tuple_equal_trans (e)", "specialize jordan_tuple_equal_trans (x1)", "specialize jordan_tuple_equal_trans (x2)", "specialize jordan_tuple_equal_trans (k)", "apply jordan_tuple_equal_trans", "exact heq", "exact hl_witness_witness_witness_right_right" ], "script_sha256": "a3a59bb80de75c8c6dd15ab6514a7dd3a19ed8a78bded52b6d451713957464d8", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "a3a59bb80de75c8c6dd15ab6514a7dd3a19ed8a78bded52b6d451713957464d8", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "a97e6df915eeebb9bd4e61c7998cbd88a12a0edee03fcf2a255e04cb23148bea" } ], "stable_member": false, "statement": "forall b c d e k B C D E j. (forall jt_index_listedtransport jt_left_listedtransport jt_right_listedtransport. (exists jt_gap_listedtransportindex. jt_gap_listedtransportindex+S (jt_index_listedtransport)=(k)) -> (((exists fs_h_jt_listedtransportleft. fs_h_jt_listedtransportleft + S (jt_left_listedtransport) = S ((S (jt_index_listedtransport)) * c)) /\\ exists fs_q_jt_listedtransportleft. b = fs_q_jt_listedtransportleft * S ((S (jt_index_listedtransport)) * c) + (jt_left_listedtransport))) -> (((exists fs_h_jt_listedtransportright. fs_h_jt_listedtransportright + S (jt_right_listedtransport) = S ((S (jt_index_listedtransport)) * e)) /\\ exists fs_q_jt_listedtransportright. d = fs_q_jt_listedtransportright * S ((S (jt_index_listedtransport)) * e) + (jt_right_listedtransport))) -> jt_left_listedtransport=jt_right_listedtransport) -> (exists jt_index_listedsource jt_code_listedsource jt_scale_listedsource. ((exists jt_gap_listedsourceindex. jt_gap_listedsourceindex+S (jt_index_listedsource)=(j)) /\\ (((((((exists fs_h_jt_listedsourcecode. fs_h_jt_listedsourcecode + S (jt_code_listedsource) = S ((S (jt_index_listedsource)) * C)) /\\ exists fs_q_jt_listedsourcecode. B = fs_q_jt_listedsourcecode * S ((S (jt_index_listedsource)) * C) + (jt_code_listedsource))) /\\ (((exists fs_h_jt_listedsourcescale. fs_h_jt_listedsourcescale + S (jt_scale_listedsource) = S ((S (jt_index_listedsource)) * E)) /\\ exists fs_q_jt_listedsourcescale. D = fs_q_jt_listedsourcescale * S ((S (jt_index_listedsource)) * E) + (jt_scale_listedsource))))) /\\ (forall jt_index_listedsourceequal jt_left_listedsourceequal jt_right_listedsourceequal. (exists jt_gap_listedsourceequalindex. jt_gap_listedsourceequalindex+S (jt_index_listedsourceequal)=(k)) -> (((exists fs_h_jt_listedsourceequalleft. fs_h_jt_listedsourceequalleft + S (jt_left_listedsourceequal) = S ((S (jt_index_listedsourceequal)) * e)) /\\ exists fs_q_jt_listedsourceequalleft. d = fs_q_jt_listedsourceequalleft * S ((S (jt_index_listedsourceequal)) * e) + (jt_left_listedsourceequal))) -> (((exists fs_h_jt_listedsourceequalright. fs_h_jt_listedsourceequalright + S (jt_right_listedsourceequal) = S ((S (jt_index_listedsourceequal)) * jt_scale_listedsource)) /\\ exists fs_q_jt_listedsourceequalright. jt_code_listedsource = fs_q_jt_listedsourceequalright * S ((S (jt_index_listedsourceequal)) * jt_scale_listedsource) + (jt_right_listedsourceequal))) -> jt_left_listedsourceequal=jt_right_listedsourceequal))))) -> (exists jt_index_listedtarget jt_code_listedtarget jt_scale_listedtarget. ((exists jt_gap_listedtargetindex. jt_gap_listedtargetindex+S (jt_index_listedtarget)=(j)) /\\ (((((((exists fs_h_jt_listedtargetcode. fs_h_jt_listedtargetcode + S (jt_code_listedtarget) = S ((S (jt_index_listedtarget)) * C)) /\\ exists fs_q_jt_listedtargetcode. B = fs_q_jt_listedtargetcode * S ((S (jt_index_listedtarget)) * C) + (jt_code_listedtarget))) /\\ (((exists fs_h_jt_listedtargetscale. fs_h_jt_listedtargetscale + S (jt_scale_listedtarget) = S ((S (jt_index_listedtarget)) * E)) /\\ exists fs_q_jt_listedtargetscale. D = fs_q_jt_listedtargetscale * S ((S (jt_index_listedtarget)) * E) + (jt_scale_listedtarget))))) /\\ (forall jt_index_listedtargetequal jt_left_listedtargetequal jt_right_listedtargetequal. (exists jt_gap_listedtargetequalindex. jt_gap_listedtargetequalindex+S (jt_index_listedtargetequal)=(k)) -> (((exists fs_h_jt_listedtargetequalleft. fs_h_jt_listedtargetequalleft + S (jt_left_listedtargetequal) = S ((S (jt_index_listedtargetequal)) * c)) /\\ exists fs_q_jt_listedtargetequalleft. b = fs_q_jt_listedtargetequalleft * S ((S (jt_index_listedtargetequal)) * c) + (jt_left_listedtargetequal))) -> (((exists fs_h_jt_listedtargetequalright. fs_h_jt_listedtargetequalright + S (jt_right_listedtargetequal) = S ((S (jt_index_listedtargetequal)) * jt_scale_listedtarget)) /\\ exists fs_q_jt_listedtargetequalright. jt_code_listedtarget = fs_q_jt_listedtargetequalright * S ((S (jt_index_listedtargetequal)) * jt_scale_listedtarget) + (jt_right_listedtargetequal))) -> jt_left_listedtargetequal=jt_right_listedtargetequal)))))", "statement_sha256": "a97e6df915eeebb9bd4e61c7998cbd88a12a0edee03fcf2a255e04cb23148bea", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "List membership is a property of the represented coordinate tuple." }, { "admission_dependencies": [ "finite_lt_succ_eq_or_lt" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 26, "body_proof_nodes": 51, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro c", "intro t", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hscan", "intro hcurrent", "cases hscan", "cases hscan_right", "split", "exact hscan_left", "split", "exact hscan_right_left", "intro z", "intro hz", "intro hb", "intro hp", "have hc : z = t ∨ Lt(z,t)", "specialize finite_lt_succ_eq_or_lt (t)", "specialize finite_lt_succ_eq_or_lt (z)", "apply finite_lt_succ_eq_or_lt", "exact hz", "cases hc", "rewrite hc_left", "apply hcurrent", "rewrite hc_left at hb", "exact hb", "rewrite hc_left at hp", "exact hp", "specialize hscan_right_right (z)", "apply hscan_right_right", "exact hc_right", "exact hb", "exact hp" ], "defined_statement": "∀ k. ∀ n. ∀ c. ∀ t. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. JordanTupleScan(k,n,c,t,B,C,D,E,j) → (BetaPrefixInto(t,c,k,n) → JordanPrimitiveTuple(n,t,c,k) → JordanTupleListed(t,c,k,B,C,D,E,j)) → JordanTupleScan(k,n,c,S t,B,C,D,E,j)", "defined_statement_sha256": "699c6ddb814a460e9613f6cb31ac684bcdb2b1ea0dee69dfe08679d0ad7a620d", "definition_uses": { "ND0262": 1, "ND0372": 1, "ND0376": 1, "ND0377": 2, "PD0002": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "3cdba818e1aef043d7d0ba76cc3e51590a3aa16377e9d39291701fc60964f9cf", "free_names": [], "script_definition_uses": { "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro t" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hscan" } ], [ { "kind": "text", "text": "intro hcurrent" } ], [ { "kind": "text", "text": "cases hscan" } ], [ { "kind": "text", "text": "cases hscan_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hscan_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hscan_right_left" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "z = t ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,t)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (t)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (z)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "apply hcurrent" } ], [ { "kind": "text", "text": "rewrite hc_left at hb" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "rewrite hc_left at hp" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "specialize hscan_right_right (z)" } ], [ { "kind": "text", "text": "apply hscan_right_right" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hp" } ] ], "statement_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0376": 1, "ND0377": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ c. ∀ t. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,t,B,C,D,E,j)" }, { "kind": "text", "text": " → (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(t,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,t,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(t,c,k,B,C,D,E,j)" }, { "kind": "text", "text": ") → " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,S t,B,C,D,E,j)" } ] }, "dependencies": [ "finite_lt_succ_eq_or_lt" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT0023", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_scan_skip", "original_ha_bundle_verified": true, "proof_bundle_node_id": 292, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro c", "intro t", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hscan", "intro hcurrent", "cases hscan", "cases hscan_right", "split", "exact hscan_left", "split", "exact hscan_right_left", "intro z", "intro hz", "intro hb", "intro hp", "have hc : z=t \\/ (exists jt_gap_scanskipcase. jt_gap_scanskipcase+S (z)=(t))", "specialize finite_lt_succ_eq_or_lt (t)", "specialize finite_lt_succ_eq_or_lt (z)", "apply finite_lt_succ_eq_or_lt", "exact hz", "cases hc", "rewrite hc_left", "apply hcurrent", "rewrite hc_left at hb", "exact hb", "rewrite hc_left at hp", "exact hp", "specialize hscan_right_right (z)", "apply hscan_right_right", "exact hc_right", "exact hb", "exact hp" ], "script_sha256": "13304ffa7c3a853ef0c4d7a522866fcd79415e68384300b39de4277dba69347b", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "13304ffa7c3a853ef0c4d7a522866fcd79415e68384300b39de4277dba69347b", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "3cdba818e1aef043d7d0ba76cc3e51590a3aa16377e9d39291701fc60964f9cf" } ], "stable_member": false, "statement": "forall k n c t B C D E j. (((forall jt_i_skipsource. (exists jt_gap_skipsourcesoundindex. jt_gap_skipsourcesoundindex+S (jt_i_skipsource)=(j)) -> exists jt_b_skipsource jt_e_skipsource. ((((((exists fs_h_jt_skipsourcesoundcode. fs_h_jt_skipsourcesoundcode + S (jt_b_skipsource) = S ((S (jt_i_skipsource)) * C)) /\\ exists fs_q_jt_skipsourcesoundcode. B = fs_q_jt_skipsourcesoundcode * S ((S (jt_i_skipsource)) * C) + (jt_b_skipsource))) /\\ (((exists fs_h_jt_skipsourcesoundscale. fs_h_jt_skipsourcesoundscale + S (jt_e_skipsource) = S ((S (jt_i_skipsource)) * E)) /\\ exists fs_q_jt_skipsourcesoundscale. D = fs_q_jt_skipsourcesoundscale * S ((S (jt_i_skipsource)) * E) + (jt_e_skipsource))))) /\\ (((forall jt_index_skipsourcebound. (exists jt_gap_skipsourceboundindex. jt_gap_skipsourceboundindex+S (jt_index_skipsourcebound)=(k)) -> exists jt_value_skipsourcebound. ((((exists fs_h_jt_skipsourceboundat. fs_h_jt_skipsourceboundat + S (jt_value_skipsourcebound) = S ((S (jt_index_skipsourcebound)) * jt_e_skipsource)) /\\ exists fs_q_jt_skipsourceboundat. jt_b_skipsource = fs_q_jt_skipsourceboundat * S ((S (jt_index_skipsourcebound)) * jt_e_skipsource) + (jt_value_skipsourcebound))) /\\ (exists jt_gap_skipsourceboundvalue. jt_gap_skipsourceboundvalue+S (jt_value_skipsourcebound)=(n)))) /\\ (forall jt_divisor_skipsourceprimitive. (exists jt_factor_skipsourceprimitivemodulus. (n)=(jt_divisor_skipsourceprimitive)*jt_factor_skipsourceprimitivemodulus) -> (forall jt_index_skipsourceprimitivecoordinates jt_value_skipsourceprimitivecoordinates. (exists jt_gap_skipsourceprimitivecoordinatesindex. jt_gap_skipsourceprimitivecoordinatesindex+S (jt_index_skipsourceprimitivecoordinates)=(k)) -> (((exists fs_h_jt_skipsourceprimitivecoordinatesat. fs_h_jt_skipsourceprimitivecoordinatesat + S (jt_value_skipsourceprimitivecoordinates) = S ((S (jt_index_skipsourceprimitivecoordinates)) * jt_e_skipsource)) /\\ exists fs_q_jt_skipsourceprimitivecoordinatesat. jt_b_skipsource = fs_q_jt_skipsourceprimitivecoordinatesat * S ((S (jt_index_skipsourceprimitivecoordinates)) * jt_e_skipsource) + (jt_value_skipsourceprimitivecoordinates))) -> (exists jt_factor_skipsourceprimitivecoordinatesdivides. (jt_value_skipsourceprimitivecoordinates)=(jt_divisor_skipsourceprimitive)*jt_factor_skipsourceprimitivecoordinatesdivides)) -> jt_divisor_skipsourceprimitive=1))))) /\\ (((forall jt_i_skipsource jt_h_skipsource jt_b_skipsource jt_e_skipsource jt_d_skipsource jt_f_skipsource. (exists jt_gap_skipsourcefirstindex. jt_gap_skipsourcefirstindex+S (jt_i_skipsource)=(j)) -> (exists jt_gap_skipsourcesecondindex. jt_gap_skipsourcesecondindex+S (jt_h_skipsource)=(j)) -> (((((exists fs_h_jt_skipsourcefirstcode. fs_h_jt_skipsourcefirstcode + S (jt_b_skipsource) = S ((S (jt_i_skipsource)) * C)) /\\ exists fs_q_jt_skipsourcefirstcode. B = fs_q_jt_skipsourcefirstcode * S ((S (jt_i_skipsource)) * C) + (jt_b_skipsource))) /\\ (((exists fs_h_jt_skipsourcefirstscale. fs_h_jt_skipsourcefirstscale + S (jt_e_skipsource) = S ((S (jt_i_skipsource)) * E)) /\\ exists fs_q_jt_skipsourcefirstscale. D = fs_q_jt_skipsourcefirstscale * S ((S (jt_i_skipsource)) * E) + (jt_e_skipsource))))) -> (((((exists fs_h_jt_skipsourcesecondcode. fs_h_jt_skipsourcesecondcode + S (jt_d_skipsource) = S ((S (jt_h_skipsource)) * C)) /\\ exists fs_q_jt_skipsourcesecondcode. B = fs_q_jt_skipsourcesecondcode * S ((S (jt_h_skipsource)) * C) + (jt_d_skipsource))) /\\ (((exists fs_h_jt_skipsourcesecondscale. fs_h_jt_skipsourcesecondscale + S (jt_f_skipsource) = S ((S (jt_h_skipsource)) * E)) /\\ exists fs_q_jt_skipsourcesecondscale. D = fs_q_jt_skipsourcesecondscale * S ((S (jt_h_skipsource)) * E) + (jt_f_skipsource))))) -> (forall jt_index_skipsourcesame jt_left_skipsourcesame jt_right_skipsourcesame. (exists jt_gap_skipsourcesameindex. jt_gap_skipsourcesameindex+S (jt_index_skipsourcesame)=(k)) -> (((exists fs_h_jt_skipsourcesameleft. fs_h_jt_skipsourcesameleft + S (jt_left_skipsourcesame) = S ((S (jt_index_skipsourcesame)) * jt_e_skipsource)) /\\ exists fs_q_jt_skipsourcesameleft. jt_b_skipsource = fs_q_jt_skipsourcesameleft * S ((S (jt_index_skipsourcesame)) * jt_e_skipsource) + (jt_left_skipsourcesame))) -> (((exists fs_h_jt_skipsourcesameright. fs_h_jt_skipsourcesameright + S (jt_right_skipsourcesame) = S ((S (jt_index_skipsourcesame)) * jt_f_skipsource)) /\\ exists fs_q_jt_skipsourcesameright. jt_d_skipsource = fs_q_jt_skipsourcesameright * S ((S (jt_index_skipsourcesame)) * jt_f_skipsource) + (jt_right_skipsourcesame))) -> jt_left_skipsourcesame=jt_right_skipsourcesame) -> jt_i_skipsource=jt_h_skipsource) /\\ (forall jt_z_skipsource. (exists jt_gap_skipsourcecodeindex. jt_gap_skipsourcecodeindex+S (jt_z_skipsource)=(t)) -> (forall jt_index_skipsourceinputbound. (exists jt_gap_skipsourceinputboundindex. jt_gap_skipsourceinputboundindex+S (jt_index_skipsourceinputbound)=(k)) -> exists jt_value_skipsourceinputbound. ((((exists fs_h_jt_skipsourceinputboundat. fs_h_jt_skipsourceinputboundat + S (jt_value_skipsourceinputbound) = S ((S (jt_index_skipsourceinputbound)) * c)) /\\ exists fs_q_jt_skipsourceinputboundat. jt_z_skipsource = fs_q_jt_skipsourceinputboundat * S ((S (jt_index_skipsourceinputbound)) * c) + (jt_value_skipsourceinputbound))) /\\ (exists jt_gap_skipsourceinputboundvalue. jt_gap_skipsourceinputboundvalue+S (jt_value_skipsourceinputbound)=(n)))) -> (forall jt_divisor_skipsourceinputprimitive. (exists jt_factor_skipsourceinputprimitivemodulus. (n)=(jt_divisor_skipsourceinputprimitive)*jt_factor_skipsourceinputprimitivemodulus) -> (forall jt_index_skipsourceinputprimitivecoordinates jt_value_skipsourceinputprimitivecoordinates. (exists jt_gap_skipsourceinputprimitivecoordinatesindex. jt_gap_skipsourceinputprimitivecoordinatesindex+S (jt_index_skipsourceinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_skipsourceinputprimitivecoordinatesat. fs_h_jt_skipsourceinputprimitivecoordinatesat + S (jt_value_skipsourceinputprimitivecoordinates) = S ((S (jt_index_skipsourceinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_skipsourceinputprimitivecoordinatesat. jt_z_skipsource = fs_q_jt_skipsourceinputprimitivecoordinatesat * S ((S (jt_index_skipsourceinputprimitivecoordinates)) * c) + (jt_value_skipsourceinputprimitivecoordinates))) -> (exists jt_factor_skipsourceinputprimitivecoordinatesdivides. (jt_value_skipsourceinputprimitivecoordinates)=(jt_divisor_skipsourceinputprimitive)*jt_factor_skipsourceinputprimitivecoordinatesdivides)) -> jt_divisor_skipsourceinputprimitive=1) -> (exists jt_index_skipsourcelisted jt_code_skipsourcelisted jt_scale_skipsourcelisted. ((exists jt_gap_skipsourcelistedindex. jt_gap_skipsourcelistedindex+S (jt_index_skipsourcelisted)=(j)) /\\ (((((((exists fs_h_jt_skipsourcelistedcode. fs_h_jt_skipsourcelistedcode + S (jt_code_skipsourcelisted) = S ((S (jt_index_skipsourcelisted)) * C)) /\\ exists fs_q_jt_skipsourcelistedcode. B = fs_q_jt_skipsourcelistedcode * S ((S (jt_index_skipsourcelisted)) * C) + (jt_code_skipsourcelisted))) /\\ (((exists fs_h_jt_skipsourcelistedscale. fs_h_jt_skipsourcelistedscale + S (jt_scale_skipsourcelisted) = S ((S (jt_index_skipsourcelisted)) * E)) /\\ exists fs_q_jt_skipsourcelistedscale. D = fs_q_jt_skipsourcelistedscale * S ((S (jt_index_skipsourcelisted)) * E) + (jt_scale_skipsourcelisted))))) /\\ (forall jt_index_skipsourcelistedequal jt_left_skipsourcelistedequal jt_right_skipsourcelistedequal. (exists jt_gap_skipsourcelistedequalindex. jt_gap_skipsourcelistedequalindex+S (jt_index_skipsourcelistedequal)=(k)) -> (((exists fs_h_jt_skipsourcelistedequalleft. fs_h_jt_skipsourcelistedequalleft + S (jt_left_skipsourcelistedequal) = S ((S (jt_index_skipsourcelistedequal)) * c)) /\\ exists fs_q_jt_skipsourcelistedequalleft. jt_z_skipsource = fs_q_jt_skipsourcelistedequalleft * S ((S (jt_index_skipsourcelistedequal)) * c) + (jt_left_skipsourcelistedequal))) -> (((exists fs_h_jt_skipsourcelistedequalright. fs_h_jt_skipsourcelistedequalright + S (jt_right_skipsourcelistedequal) = S ((S (jt_index_skipsourcelistedequal)) * jt_scale_skipsourcelisted)) /\\ exists fs_q_jt_skipsourcelistedequalright. jt_code_skipsourcelisted = fs_q_jt_skipsourcelistedequalright * S ((S (jt_index_skipsourcelistedequal)) * jt_scale_skipsourcelisted) + (jt_right_skipsourcelistedequal))) -> jt_left_skipsourcelistedequal=jt_right_skipsourcelistedequal)))))))))) -> ((forall jt_index_skipcurrentbound. (exists jt_gap_skipcurrentboundindex. jt_gap_skipcurrentboundindex+S (jt_index_skipcurrentbound)=(k)) -> exists jt_value_skipcurrentbound. ((((exists fs_h_jt_skipcurrentboundat. fs_h_jt_skipcurrentboundat + S (jt_value_skipcurrentbound) = S ((S (jt_index_skipcurrentbound)) * c)) /\\ exists fs_q_jt_skipcurrentboundat. t = fs_q_jt_skipcurrentboundat * S ((S (jt_index_skipcurrentbound)) * c) + (jt_value_skipcurrentbound))) /\\ (exists jt_gap_skipcurrentboundvalue. jt_gap_skipcurrentboundvalue+S (jt_value_skipcurrentbound)=(n)))) -> (forall jt_divisor_skipcurrentprimitive. (exists jt_factor_skipcurrentprimitivemodulus. (n)=(jt_divisor_skipcurrentprimitive)*jt_factor_skipcurrentprimitivemodulus) -> (forall jt_index_skipcurrentprimitivecoordinates jt_value_skipcurrentprimitivecoordinates. (exists jt_gap_skipcurrentprimitivecoordinatesindex. jt_gap_skipcurrentprimitivecoordinatesindex+S (jt_index_skipcurrentprimitivecoordinates)=(k)) -> (((exists fs_h_jt_skipcurrentprimitivecoordinatesat. fs_h_jt_skipcurrentprimitivecoordinatesat + S (jt_value_skipcurrentprimitivecoordinates) = S ((S (jt_index_skipcurrentprimitivecoordinates)) * c)) /\\ exists fs_q_jt_skipcurrentprimitivecoordinatesat. t = fs_q_jt_skipcurrentprimitivecoordinatesat * S ((S (jt_index_skipcurrentprimitivecoordinates)) * c) + (jt_value_skipcurrentprimitivecoordinates))) -> (exists jt_factor_skipcurrentprimitivecoordinatesdivides. (jt_value_skipcurrentprimitivecoordinates)=(jt_divisor_skipcurrentprimitive)*jt_factor_skipcurrentprimitivecoordinatesdivides)) -> jt_divisor_skipcurrentprimitive=1) -> (exists jt_index_skipcurrentlisted jt_code_skipcurrentlisted jt_scale_skipcurrentlisted. ((exists jt_gap_skipcurrentlistedindex. jt_gap_skipcurrentlistedindex+S (jt_index_skipcurrentlisted)=(j)) /\\ (((((((exists fs_h_jt_skipcurrentlistedcode. fs_h_jt_skipcurrentlistedcode + S (jt_code_skipcurrentlisted) = S ((S (jt_index_skipcurrentlisted)) * C)) /\\ exists fs_q_jt_skipcurrentlistedcode. B = fs_q_jt_skipcurrentlistedcode * S ((S (jt_index_skipcurrentlisted)) * C) + (jt_code_skipcurrentlisted))) /\\ (((exists fs_h_jt_skipcurrentlistedscale. fs_h_jt_skipcurrentlistedscale + S (jt_scale_skipcurrentlisted) = S ((S (jt_index_skipcurrentlisted)) * E)) /\\ exists fs_q_jt_skipcurrentlistedscale. D = fs_q_jt_skipcurrentlistedscale * S ((S (jt_index_skipcurrentlisted)) * E) + (jt_scale_skipcurrentlisted))))) /\\ (forall jt_index_skipcurrentlistedequal jt_left_skipcurrentlistedequal jt_right_skipcurrentlistedequal. (exists jt_gap_skipcurrentlistedequalindex. jt_gap_skipcurrentlistedequalindex+S (jt_index_skipcurrentlistedequal)=(k)) -> (((exists fs_h_jt_skipcurrentlistedequalleft. fs_h_jt_skipcurrentlistedequalleft + S (jt_left_skipcurrentlistedequal) = S ((S (jt_index_skipcurrentlistedequal)) * c)) /\\ exists fs_q_jt_skipcurrentlistedequalleft. t = fs_q_jt_skipcurrentlistedequalleft * S ((S (jt_index_skipcurrentlistedequal)) * c) + (jt_left_skipcurrentlistedequal))) -> (((exists fs_h_jt_skipcurrentlistedequalright. fs_h_jt_skipcurrentlistedequalright + S (jt_right_skipcurrentlistedequal) = S ((S (jt_index_skipcurrentlistedequal)) * jt_scale_skipcurrentlisted)) /\\ exists fs_q_jt_skipcurrentlistedequalright. jt_code_skipcurrentlisted = fs_q_jt_skipcurrentlistedequalright * S ((S (jt_index_skipcurrentlistedequal)) * jt_scale_skipcurrentlisted) + (jt_right_skipcurrentlistedequal))) -> jt_left_skipcurrentlistedequal=jt_right_skipcurrentlistedequal)))))) -> (((forall jt_i_skiptarget. (exists jt_gap_skiptargetsoundindex. jt_gap_skiptargetsoundindex+S (jt_i_skiptarget)=(j)) -> exists jt_b_skiptarget jt_e_skiptarget. ((((((exists fs_h_jt_skiptargetsoundcode. fs_h_jt_skiptargetsoundcode + S (jt_b_skiptarget) = S ((S (jt_i_skiptarget)) * C)) /\\ exists fs_q_jt_skiptargetsoundcode. B = fs_q_jt_skiptargetsoundcode * S ((S (jt_i_skiptarget)) * C) + (jt_b_skiptarget))) /\\ (((exists fs_h_jt_skiptargetsoundscale. fs_h_jt_skiptargetsoundscale + S (jt_e_skiptarget) = S ((S (jt_i_skiptarget)) * E)) /\\ exists fs_q_jt_skiptargetsoundscale. D = fs_q_jt_skiptargetsoundscale * S ((S (jt_i_skiptarget)) * E) + (jt_e_skiptarget))))) /\\ (((forall jt_index_skiptargetbound. (exists jt_gap_skiptargetboundindex. jt_gap_skiptargetboundindex+S (jt_index_skiptargetbound)=(k)) -> exists jt_value_skiptargetbound. ((((exists fs_h_jt_skiptargetboundat. fs_h_jt_skiptargetboundat + S (jt_value_skiptargetbound) = S ((S (jt_index_skiptargetbound)) * jt_e_skiptarget)) /\\ exists fs_q_jt_skiptargetboundat. jt_b_skiptarget = fs_q_jt_skiptargetboundat * S ((S (jt_index_skiptargetbound)) * jt_e_skiptarget) + (jt_value_skiptargetbound))) /\\ (exists jt_gap_skiptargetboundvalue. jt_gap_skiptargetboundvalue+S (jt_value_skiptargetbound)=(n)))) /\\ (forall jt_divisor_skiptargetprimitive. (exists jt_factor_skiptargetprimitivemodulus. (n)=(jt_divisor_skiptargetprimitive)*jt_factor_skiptargetprimitivemodulus) -> (forall jt_index_skiptargetprimitivecoordinates jt_value_skiptargetprimitivecoordinates. (exists jt_gap_skiptargetprimitivecoordinatesindex. jt_gap_skiptargetprimitivecoordinatesindex+S (jt_index_skiptargetprimitivecoordinates)=(k)) -> (((exists fs_h_jt_skiptargetprimitivecoordinatesat. fs_h_jt_skiptargetprimitivecoordinatesat + S (jt_value_skiptargetprimitivecoordinates) = S ((S (jt_index_skiptargetprimitivecoordinates)) * jt_e_skiptarget)) /\\ exists fs_q_jt_skiptargetprimitivecoordinatesat. jt_b_skiptarget = fs_q_jt_skiptargetprimitivecoordinatesat * S ((S (jt_index_skiptargetprimitivecoordinates)) * jt_e_skiptarget) + (jt_value_skiptargetprimitivecoordinates))) -> (exists jt_factor_skiptargetprimitivecoordinatesdivides. (jt_value_skiptargetprimitivecoordinates)=(jt_divisor_skiptargetprimitive)*jt_factor_skiptargetprimitivecoordinatesdivides)) -> jt_divisor_skiptargetprimitive=1))))) /\\ (((forall jt_i_skiptarget jt_h_skiptarget jt_b_skiptarget jt_e_skiptarget jt_d_skiptarget jt_f_skiptarget. (exists jt_gap_skiptargetfirstindex. jt_gap_skiptargetfirstindex+S (jt_i_skiptarget)=(j)) -> (exists jt_gap_skiptargetsecondindex. jt_gap_skiptargetsecondindex+S (jt_h_skiptarget)=(j)) -> (((((exists fs_h_jt_skiptargetfirstcode. fs_h_jt_skiptargetfirstcode + S (jt_b_skiptarget) = S ((S (jt_i_skiptarget)) * C)) /\\ exists fs_q_jt_skiptargetfirstcode. B = fs_q_jt_skiptargetfirstcode * S ((S (jt_i_skiptarget)) * C) + (jt_b_skiptarget))) /\\ (((exists fs_h_jt_skiptargetfirstscale. fs_h_jt_skiptargetfirstscale + S (jt_e_skiptarget) = S ((S (jt_i_skiptarget)) * E)) /\\ exists fs_q_jt_skiptargetfirstscale. D = fs_q_jt_skiptargetfirstscale * S ((S (jt_i_skiptarget)) * E) + (jt_e_skiptarget))))) -> (((((exists fs_h_jt_skiptargetsecondcode. fs_h_jt_skiptargetsecondcode + S (jt_d_skiptarget) = S ((S (jt_h_skiptarget)) * C)) /\\ exists fs_q_jt_skiptargetsecondcode. B = fs_q_jt_skiptargetsecondcode * S ((S (jt_h_skiptarget)) * C) + (jt_d_skiptarget))) /\\ (((exists fs_h_jt_skiptargetsecondscale. fs_h_jt_skiptargetsecondscale + S (jt_f_skiptarget) = S ((S (jt_h_skiptarget)) * E)) /\\ exists fs_q_jt_skiptargetsecondscale. D = fs_q_jt_skiptargetsecondscale * S ((S (jt_h_skiptarget)) * E) + (jt_f_skiptarget))))) -> (forall jt_index_skiptargetsame jt_left_skiptargetsame jt_right_skiptargetsame. (exists jt_gap_skiptargetsameindex. jt_gap_skiptargetsameindex+S (jt_index_skiptargetsame)=(k)) -> (((exists fs_h_jt_skiptargetsameleft. fs_h_jt_skiptargetsameleft + S (jt_left_skiptargetsame) = S ((S (jt_index_skiptargetsame)) * jt_e_skiptarget)) /\\ exists fs_q_jt_skiptargetsameleft. jt_b_skiptarget = fs_q_jt_skiptargetsameleft * S ((S (jt_index_skiptargetsame)) * jt_e_skiptarget) + (jt_left_skiptargetsame))) -> (((exists fs_h_jt_skiptargetsameright. fs_h_jt_skiptargetsameright + S (jt_right_skiptargetsame) = S ((S (jt_index_skiptargetsame)) * jt_f_skiptarget)) /\\ exists fs_q_jt_skiptargetsameright. jt_d_skiptarget = fs_q_jt_skiptargetsameright * S ((S (jt_index_skiptargetsame)) * jt_f_skiptarget) + (jt_right_skiptargetsame))) -> jt_left_skiptargetsame=jt_right_skiptargetsame) -> jt_i_skiptarget=jt_h_skiptarget) /\\ (forall jt_z_skiptarget. (exists jt_gap_skiptargetcodeindex. jt_gap_skiptargetcodeindex+S (jt_z_skiptarget)=(S t)) -> (forall jt_index_skiptargetinputbound. (exists jt_gap_skiptargetinputboundindex. jt_gap_skiptargetinputboundindex+S (jt_index_skiptargetinputbound)=(k)) -> exists jt_value_skiptargetinputbound. ((((exists fs_h_jt_skiptargetinputboundat. fs_h_jt_skiptargetinputboundat + S (jt_value_skiptargetinputbound) = S ((S (jt_index_skiptargetinputbound)) * c)) /\\ exists fs_q_jt_skiptargetinputboundat. jt_z_skiptarget = fs_q_jt_skiptargetinputboundat * S ((S (jt_index_skiptargetinputbound)) * c) + (jt_value_skiptargetinputbound))) /\\ (exists jt_gap_skiptargetinputboundvalue. jt_gap_skiptargetinputboundvalue+S (jt_value_skiptargetinputbound)=(n)))) -> (forall jt_divisor_skiptargetinputprimitive. (exists jt_factor_skiptargetinputprimitivemodulus. (n)=(jt_divisor_skiptargetinputprimitive)*jt_factor_skiptargetinputprimitivemodulus) -> (forall jt_index_skiptargetinputprimitivecoordinates jt_value_skiptargetinputprimitivecoordinates. (exists jt_gap_skiptargetinputprimitivecoordinatesindex. jt_gap_skiptargetinputprimitivecoordinatesindex+S (jt_index_skiptargetinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_skiptargetinputprimitivecoordinatesat. fs_h_jt_skiptargetinputprimitivecoordinatesat + S (jt_value_skiptargetinputprimitivecoordinates) = S ((S (jt_index_skiptargetinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_skiptargetinputprimitivecoordinatesat. jt_z_skiptarget = fs_q_jt_skiptargetinputprimitivecoordinatesat * S ((S (jt_index_skiptargetinputprimitivecoordinates)) * c) + (jt_value_skiptargetinputprimitivecoordinates))) -> (exists jt_factor_skiptargetinputprimitivecoordinatesdivides. (jt_value_skiptargetinputprimitivecoordinates)=(jt_divisor_skiptargetinputprimitive)*jt_factor_skiptargetinputprimitivecoordinatesdivides)) -> jt_divisor_skiptargetinputprimitive=1) -> (exists jt_index_skiptargetlisted jt_code_skiptargetlisted jt_scale_skiptargetlisted. ((exists jt_gap_skiptargetlistedindex. jt_gap_skiptargetlistedindex+S (jt_index_skiptargetlisted)=(j)) /\\ (((((((exists fs_h_jt_skiptargetlistedcode. fs_h_jt_skiptargetlistedcode + S (jt_code_skiptargetlisted) = S ((S (jt_index_skiptargetlisted)) * C)) /\\ exists fs_q_jt_skiptargetlistedcode. B = fs_q_jt_skiptargetlistedcode * S ((S (jt_index_skiptargetlisted)) * C) + (jt_code_skiptargetlisted))) /\\ (((exists fs_h_jt_skiptargetlistedscale. fs_h_jt_skiptargetlistedscale + S (jt_scale_skiptargetlisted) = S ((S (jt_index_skiptargetlisted)) * E)) /\\ exists fs_q_jt_skiptargetlistedscale. D = fs_q_jt_skiptargetlistedscale * S ((S (jt_index_skiptargetlisted)) * E) + (jt_scale_skiptargetlisted))))) /\\ (forall jt_index_skiptargetlistedequal jt_left_skiptargetlistedequal jt_right_skiptargetlistedequal. (exists jt_gap_skiptargetlistedequalindex. jt_gap_skiptargetlistedequalindex+S (jt_index_skiptargetlistedequal)=(k)) -> (((exists fs_h_jt_skiptargetlistedequalleft. fs_h_jt_skiptargetlistedequalleft + S (jt_left_skiptargetlistedequal) = S ((S (jt_index_skiptargetlistedequal)) * c)) /\\ exists fs_q_jt_skiptargetlistedequalleft. jt_z_skiptarget = fs_q_jt_skiptargetlistedequalleft * S ((S (jt_index_skiptargetlistedequal)) * c) + (jt_left_skiptargetlistedequal))) -> (((exists fs_h_jt_skiptargetlistedequalright. fs_h_jt_skiptargetlistedequalright + S (jt_right_skiptargetlistedequal) = S ((S (jt_index_skiptargetlistedequal)) * jt_scale_skiptargetlisted)) /\\ exists fs_q_jt_skiptargetlistedequalright. jt_code_skiptargetlisted = fs_q_jt_skiptargetlistedequalright * S ((S (jt_index_skiptargetlistedequal)) * jt_scale_skiptargetlisted) + (jt_right_skiptargetlistedequal))) -> jt_left_skiptargetlistedequal=jt_right_skiptargetlistedequal))))))))))", "statement_sha256": "3cdba818e1aef043d7d0ba76cc3e51590a3aa16377e9d39291701fc60964f9cf", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Skip precisely when the current admissible tuple is already represented." }, { "admission_dependencies": [ "jordan_tuple_listed_equal_transport", "jordan_tuple_bounded_transport", "jordan_primitive_tuple_transport" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 34, "body_proof_nodes": 79, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro c", "intro T", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hbox", "intro hscan", "cases hscan", "cases hscan_right", "split", "exact hscan_left", "split", "intro b", "intro e", "intro hb", "intro hp", "have hz : ∃ z. Lt(z,T) ∧ IntegerVectorZero(b,e,z,c,k)", "specialize hbox (b)", "specialize hbox (e)", "apply hbox", "exact hb", "cases hz", "cases hz_witness", "specialize jordan_tuple_listed_equal_transport (b)", "specialize jordan_tuple_listed_equal_transport (e)", "specialize jordan_tuple_listed_equal_transport (x)", "specialize jordan_tuple_listed_equal_transport (c)", "specialize jordan_tuple_listed_equal_transport (k)", "specialize jordan_tuple_listed_equal_transport (B)", "specialize jordan_tuple_listed_equal_transport (C)", "specialize jordan_tuple_listed_equal_transport (D)", "specialize jordan_tuple_listed_equal_transport (E)", "specialize jordan_tuple_listed_equal_transport (j)", "apply jordan_tuple_listed_equal_transport", "exact hz_witness_right", "specialize hscan_right_right (x)", "apply hscan_right_right", "exact hz_witness_left", "specialize jordan_tuple_bounded_transport (b)", "specialize jordan_tuple_bounded_transport (e)", "specialize jordan_tuple_bounded_transport (x)", "specialize jordan_tuple_bounded_transport (c)", "specialize jordan_tuple_bounded_transport (k)", "specialize jordan_tuple_bounded_transport (n)", "apply jordan_tuple_bounded_transport", "exact hz_witness_right", "exact hb", "specialize jordan_primitive_tuple_transport (n)", "specialize jordan_primitive_tuple_transport (b)", "specialize jordan_primitive_tuple_transport (e)", "specialize jordan_primitive_tuple_transport (x)", "specialize jordan_primitive_tuple_transport (c)", "specialize jordan_primitive_tuple_transport (k)", "apply jordan_primitive_tuple_transport", "exact hz_witness_right", "exact hp", "exact hscan_right_left" ], "defined_statement": "∀ k. ∀ n. ∀ c. ∀ T. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. JordanTupleRepresentatives(k,n,c,T) → JordanTupleScan(k,n,c,T,B,C,D,E,j) → JordanTupleEnumeration(k,n,B,C,D,E,j)", "defined_statement_sha256": "65243d0fa07c2b97bfa31a0ff14f0645adb46751a2ba2ff8d18187e7943d019e", "definition_uses": { "ND0121": 1, "ND0374": 1, "ND0377": 1, "ND0378": 1, "PD0002": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "86d98af7dc2c78453a607d0ec0b5db7ea1ffd3a9f2590eda3851538ae12cf74d", "free_names": [], "script_definition_uses": { "ND0121": 1, "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hbox" } ], [ { "kind": "text", "text": "intro hscan" } ], [ { "kind": "text", "text": "cases hscan" } ], [ { "kind": "text", "text": "cases hscan_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hscan_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "have hz : " }, { "kind": "text", "text": "∃ z. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,T)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,e,z,c,k)" } ], [ { "kind": "text", "text": "specialize hbox (b)" } ], [ { "kind": "text", "text": "specialize hbox (e)" } ], [ { "kind": "text", "text": "apply hbox" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "cases hz" } ], [ { "kind": "text", "text": "cases hz_witness" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_equal_transport (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_listed_equal_transport" } ], [ { "kind": "text", "text": "exact hz_witness_right" } ], [ { "kind": "text", "text": "specialize hscan_right_right (x)" } ], [ { "kind": "text", "text": "apply hscan_right_right" } ], [ { "kind": "text", "text": "exact hz_witness_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_transport (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_transport (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_transport (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_transport (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_transport (n)" } ], [ { "kind": "text", "text": "apply jordan_tuple_bounded_transport" } ], [ { "kind": "text", "text": "exact hz_witness_right" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_transport (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_transport (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_transport (e)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_transport (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_transport (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_transport" } ], [ { "kind": "text", "text": "exact hz_witness_right" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "exact hscan_right_left" } ] ], "statement_definition_uses": { "ND0374": 1, "ND0377": 1, "ND0378": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ c. ∀ T. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. " }, { "definition": "ND0378", "kind": "definition", "text": "JordanTupleRepresentatives(k,n,c,T)" }, { "kind": "text", "text": " → " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,T,B,C,D,E,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,B,C,D,E,j)" } ] }, "dependencies": [ "jordan_tuple_listed_equal_transport", "jordan_tuple_bounded_transport", "jordan_primitive_tuple_transport" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT0024", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_scan_complete", "original_ha_bundle_verified": true, "proof_bundle_node_id": 293, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro c", "intro T", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hbox", "intro hscan", "cases hscan", "cases hscan_right", "split", "exact hscan_left", "split", "intro b", "intro e", "intro hb", "intro hp", "have hz : exists z. ((exists jt_gap_finishbound. jt_gap_finishbound+S (z)=(T)) /\\ (forall jt_index_finishequal jt_left_finishequal jt_right_finishequal. (exists jt_gap_finishequalindex. jt_gap_finishequalindex+S (jt_index_finishequal)=(k)) -> (((exists fs_h_jt_finishequalleft. fs_h_jt_finishequalleft + S (jt_left_finishequal) = S ((S (jt_index_finishequal)) * e)) /\\ exists fs_q_jt_finishequalleft. b = fs_q_jt_finishequalleft * S ((S (jt_index_finishequal)) * e) + (jt_left_finishequal))) -> (((exists fs_h_jt_finishequalright. fs_h_jt_finishequalright + S (jt_right_finishequal) = S ((S (jt_index_finishequal)) * c)) /\\ exists fs_q_jt_finishequalright. z = fs_q_jt_finishequalright * S ((S (jt_index_finishequal)) * c) + (jt_right_finishequal))) -> jt_left_finishequal=jt_right_finishequal))", "specialize hbox (b)", "specialize hbox (e)", "apply hbox", "exact hb", "cases hz", "cases hz_witness", "specialize jordan_tuple_listed_equal_transport (b)", "specialize jordan_tuple_listed_equal_transport (e)", "specialize jordan_tuple_listed_equal_transport (x)", "specialize jordan_tuple_listed_equal_transport (c)", "specialize jordan_tuple_listed_equal_transport (k)", "specialize jordan_tuple_listed_equal_transport (B)", "specialize jordan_tuple_listed_equal_transport (C)", "specialize jordan_tuple_listed_equal_transport (D)", "specialize jordan_tuple_listed_equal_transport (E)", "specialize jordan_tuple_listed_equal_transport (j)", "apply jordan_tuple_listed_equal_transport", "exact hz_witness_right", "specialize hscan_right_right (x)", "apply hscan_right_right", "exact hz_witness_left", "specialize jordan_tuple_bounded_transport (b)", "specialize jordan_tuple_bounded_transport (e)", "specialize jordan_tuple_bounded_transport (x)", "specialize jordan_tuple_bounded_transport (c)", "specialize jordan_tuple_bounded_transport (k)", "specialize jordan_tuple_bounded_transport (n)", "apply jordan_tuple_bounded_transport", "exact hz_witness_right", "exact hb", "specialize jordan_primitive_tuple_transport (n)", "specialize jordan_primitive_tuple_transport (b)", "specialize jordan_primitive_tuple_transport (e)", "specialize jordan_primitive_tuple_transport (x)", "specialize jordan_primitive_tuple_transport (c)", "specialize jordan_primitive_tuple_transport (k)", "apply jordan_primitive_tuple_transport", "exact hz_witness_right", "exact hp", "exact hscan_right_left" ], "script_sha256": "59aebcb2841724a204569504bed09557ab275f9be9eab8cfb84ed345fd0dd3c7", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "59aebcb2841724a204569504bed09557ab275f9be9eab8cfb84ed345fd0dd3c7", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "86d98af7dc2c78453a607d0ec0b5db7ea1ffd3a9f2590eda3851538ae12cf74d" } ], "stable_member": false, "statement": "forall k n c T B C D E j. (forall jt_code_completebox jt_scale_completebox. (forall jt_index_completeboxbound. (exists jt_gap_completeboxboundindex. jt_gap_completeboxboundindex+S (jt_index_completeboxbound)=(k)) -> exists jt_value_completeboxbound. ((((exists fs_h_jt_completeboxboundat. fs_h_jt_completeboxboundat + S (jt_value_completeboxbound) = S ((S (jt_index_completeboxbound)) * jt_scale_completebox)) /\\ exists fs_q_jt_completeboxboundat. jt_code_completebox = fs_q_jt_completeboxboundat * S ((S (jt_index_completeboxbound)) * jt_scale_completebox) + (jt_value_completeboxbound))) /\\ (exists jt_gap_completeboxboundvalue. jt_gap_completeboxboundvalue+S (jt_value_completeboxbound)=(n)))) -> exists jt_representative_completebox. ((exists jt_gap_completeboxindex. jt_gap_completeboxindex+S (jt_representative_completebox)=(T)) /\\ (forall jt_index_completeboxequal jt_left_completeboxequal jt_right_completeboxequal. (exists jt_gap_completeboxequalindex. jt_gap_completeboxequalindex+S (jt_index_completeboxequal)=(k)) -> (((exists fs_h_jt_completeboxequalleft. fs_h_jt_completeboxequalleft + S (jt_left_completeboxequal) = S ((S (jt_index_completeboxequal)) * jt_scale_completebox)) /\\ exists fs_q_jt_completeboxequalleft. jt_code_completebox = fs_q_jt_completeboxequalleft * S ((S (jt_index_completeboxequal)) * jt_scale_completebox) + (jt_left_completeboxequal))) -> (((exists fs_h_jt_completeboxequalright. fs_h_jt_completeboxequalright + S (jt_right_completeboxequal) = S ((S (jt_index_completeboxequal)) * c)) /\\ exists fs_q_jt_completeboxequalright. jt_representative_completebox = fs_q_jt_completeboxequalright * S ((S (jt_index_completeboxequal)) * c) + (jt_right_completeboxequal))) -> jt_left_completeboxequal=jt_right_completeboxequal))) -> (((forall jt_i_completescan. (exists jt_gap_completescansoundindex. jt_gap_completescansoundindex+S (jt_i_completescan)=(j)) -> exists jt_b_completescan jt_e_completescan. ((((((exists fs_h_jt_completescansoundcode. fs_h_jt_completescansoundcode + S (jt_b_completescan) = S ((S (jt_i_completescan)) * C)) /\\ exists fs_q_jt_completescansoundcode. B = fs_q_jt_completescansoundcode * S ((S (jt_i_completescan)) * C) + (jt_b_completescan))) /\\ (((exists fs_h_jt_completescansoundscale. fs_h_jt_completescansoundscale + S (jt_e_completescan) = S ((S (jt_i_completescan)) * E)) /\\ exists fs_q_jt_completescansoundscale. D = fs_q_jt_completescansoundscale * S ((S (jt_i_completescan)) * E) + (jt_e_completescan))))) /\\ (((forall jt_index_completescanbound. (exists jt_gap_completescanboundindex. jt_gap_completescanboundindex+S (jt_index_completescanbound)=(k)) -> exists jt_value_completescanbound. ((((exists fs_h_jt_completescanboundat. fs_h_jt_completescanboundat + S (jt_value_completescanbound) = S ((S (jt_index_completescanbound)) * jt_e_completescan)) /\\ exists fs_q_jt_completescanboundat. jt_b_completescan = fs_q_jt_completescanboundat * S ((S (jt_index_completescanbound)) * jt_e_completescan) + (jt_value_completescanbound))) /\\ (exists jt_gap_completescanboundvalue. jt_gap_completescanboundvalue+S (jt_value_completescanbound)=(n)))) /\\ (forall jt_divisor_completescanprimitive. (exists jt_factor_completescanprimitivemodulus. (n)=(jt_divisor_completescanprimitive)*jt_factor_completescanprimitivemodulus) -> (forall jt_index_completescanprimitivecoordinates jt_value_completescanprimitivecoordinates. (exists jt_gap_completescanprimitivecoordinatesindex. jt_gap_completescanprimitivecoordinatesindex+S (jt_index_completescanprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completescanprimitivecoordinatesat. fs_h_jt_completescanprimitivecoordinatesat + S (jt_value_completescanprimitivecoordinates) = S ((S (jt_index_completescanprimitivecoordinates)) * jt_e_completescan)) /\\ exists fs_q_jt_completescanprimitivecoordinatesat. jt_b_completescan = fs_q_jt_completescanprimitivecoordinatesat * S ((S (jt_index_completescanprimitivecoordinates)) * jt_e_completescan) + (jt_value_completescanprimitivecoordinates))) -> (exists jt_factor_completescanprimitivecoordinatesdivides. (jt_value_completescanprimitivecoordinates)=(jt_divisor_completescanprimitive)*jt_factor_completescanprimitivecoordinatesdivides)) -> jt_divisor_completescanprimitive=1))))) /\\ (((forall jt_i_completescan jt_h_completescan jt_b_completescan jt_e_completescan jt_d_completescan jt_f_completescan. (exists jt_gap_completescanfirstindex. jt_gap_completescanfirstindex+S (jt_i_completescan)=(j)) -> (exists jt_gap_completescansecondindex. jt_gap_completescansecondindex+S (jt_h_completescan)=(j)) -> (((((exists fs_h_jt_completescanfirstcode. fs_h_jt_completescanfirstcode + S (jt_b_completescan) = S ((S (jt_i_completescan)) * C)) /\\ exists fs_q_jt_completescanfirstcode. B = fs_q_jt_completescanfirstcode * S ((S (jt_i_completescan)) * C) + (jt_b_completescan))) /\\ (((exists fs_h_jt_completescanfirstscale. fs_h_jt_completescanfirstscale + S (jt_e_completescan) = S ((S (jt_i_completescan)) * E)) /\\ exists fs_q_jt_completescanfirstscale. D = fs_q_jt_completescanfirstscale * S ((S (jt_i_completescan)) * E) + (jt_e_completescan))))) -> (((((exists fs_h_jt_completescansecondcode. fs_h_jt_completescansecondcode + S (jt_d_completescan) = S ((S (jt_h_completescan)) * C)) /\\ exists fs_q_jt_completescansecondcode. B = fs_q_jt_completescansecondcode * S ((S (jt_h_completescan)) * C) + (jt_d_completescan))) /\\ (((exists fs_h_jt_completescansecondscale. fs_h_jt_completescansecondscale + S (jt_f_completescan) = S ((S (jt_h_completescan)) * E)) /\\ exists fs_q_jt_completescansecondscale. D = fs_q_jt_completescansecondscale * S ((S (jt_h_completescan)) * E) + (jt_f_completescan))))) -> (forall jt_index_completescansame jt_left_completescansame jt_right_completescansame. (exists jt_gap_completescansameindex. jt_gap_completescansameindex+S (jt_index_completescansame)=(k)) -> (((exists fs_h_jt_completescansameleft. fs_h_jt_completescansameleft + S (jt_left_completescansame) = S ((S (jt_index_completescansame)) * jt_e_completescan)) /\\ exists fs_q_jt_completescansameleft. jt_b_completescan = fs_q_jt_completescansameleft * S ((S (jt_index_completescansame)) * jt_e_completescan) + (jt_left_completescansame))) -> (((exists fs_h_jt_completescansameright. fs_h_jt_completescansameright + S (jt_right_completescansame) = S ((S (jt_index_completescansame)) * jt_f_completescan)) /\\ exists fs_q_jt_completescansameright. jt_d_completescan = fs_q_jt_completescansameright * S ((S (jt_index_completescansame)) * jt_f_completescan) + (jt_right_completescansame))) -> jt_left_completescansame=jt_right_completescansame) -> jt_i_completescan=jt_h_completescan) /\\ (forall jt_z_completescan. (exists jt_gap_completescancodeindex. jt_gap_completescancodeindex+S (jt_z_completescan)=(T)) -> (forall jt_index_completescaninputbound. (exists jt_gap_completescaninputboundindex. jt_gap_completescaninputboundindex+S (jt_index_completescaninputbound)=(k)) -> exists jt_value_completescaninputbound. ((((exists fs_h_jt_completescaninputboundat. fs_h_jt_completescaninputboundat + S (jt_value_completescaninputbound) = S ((S (jt_index_completescaninputbound)) * c)) /\\ exists fs_q_jt_completescaninputboundat. jt_z_completescan = fs_q_jt_completescaninputboundat * S ((S (jt_index_completescaninputbound)) * c) + (jt_value_completescaninputbound))) /\\ (exists jt_gap_completescaninputboundvalue. jt_gap_completescaninputboundvalue+S (jt_value_completescaninputbound)=(n)))) -> (forall jt_divisor_completescaninputprimitive. (exists jt_factor_completescaninputprimitivemodulus. (n)=(jt_divisor_completescaninputprimitive)*jt_factor_completescaninputprimitivemodulus) -> (forall jt_index_completescaninputprimitivecoordinates jt_value_completescaninputprimitivecoordinates. (exists jt_gap_completescaninputprimitivecoordinatesindex. jt_gap_completescaninputprimitivecoordinatesindex+S (jt_index_completescaninputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completescaninputprimitivecoordinatesat. fs_h_jt_completescaninputprimitivecoordinatesat + S (jt_value_completescaninputprimitivecoordinates) = S ((S (jt_index_completescaninputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_completescaninputprimitivecoordinatesat. jt_z_completescan = fs_q_jt_completescaninputprimitivecoordinatesat * S ((S (jt_index_completescaninputprimitivecoordinates)) * c) + (jt_value_completescaninputprimitivecoordinates))) -> (exists jt_factor_completescaninputprimitivecoordinatesdivides. (jt_value_completescaninputprimitivecoordinates)=(jt_divisor_completescaninputprimitive)*jt_factor_completescaninputprimitivecoordinatesdivides)) -> jt_divisor_completescaninputprimitive=1) -> (exists jt_index_completescanlisted jt_code_completescanlisted jt_scale_completescanlisted. ((exists jt_gap_completescanlistedindex. jt_gap_completescanlistedindex+S (jt_index_completescanlisted)=(j)) /\\ (((((((exists fs_h_jt_completescanlistedcode. fs_h_jt_completescanlistedcode + S (jt_code_completescanlisted) = S ((S (jt_index_completescanlisted)) * C)) /\\ exists fs_q_jt_completescanlistedcode. B = fs_q_jt_completescanlistedcode * S ((S (jt_index_completescanlisted)) * C) + (jt_code_completescanlisted))) /\\ (((exists fs_h_jt_completescanlistedscale. fs_h_jt_completescanlistedscale + S (jt_scale_completescanlisted) = S ((S (jt_index_completescanlisted)) * E)) /\\ exists fs_q_jt_completescanlistedscale. D = fs_q_jt_completescanlistedscale * S ((S (jt_index_completescanlisted)) * E) + (jt_scale_completescanlisted))))) /\\ (forall jt_index_completescanlistedequal jt_left_completescanlistedequal jt_right_completescanlistedequal. (exists jt_gap_completescanlistedequalindex. jt_gap_completescanlistedequalindex+S (jt_index_completescanlistedequal)=(k)) -> (((exists fs_h_jt_completescanlistedequalleft. fs_h_jt_completescanlistedequalleft + S (jt_left_completescanlistedequal) = S ((S (jt_index_completescanlistedequal)) * c)) /\\ exists fs_q_jt_completescanlistedequalleft. jt_z_completescan = fs_q_jt_completescanlistedequalleft * S ((S (jt_index_completescanlistedequal)) * c) + (jt_left_completescanlistedequal))) -> (((exists fs_h_jt_completescanlistedequalright. fs_h_jt_completescanlistedequalright + S (jt_right_completescanlistedequal) = S ((S (jt_index_completescanlistedequal)) * jt_scale_completescanlisted)) /\\ exists fs_q_jt_completescanlistedequalright. jt_code_completescanlisted = fs_q_jt_completescanlistedequalright * S ((S (jt_index_completescanlistedequal)) * jt_scale_completescanlisted) + (jt_right_completescanlistedequal))) -> jt_left_completescanlistedequal=jt_right_completescanlistedequal)))))))))) -> (((forall jt_i_completeenum. (exists jt_gap_completeenumsoundindex. jt_gap_completeenumsoundindex+S (jt_i_completeenum)=(j)) -> exists jt_b_completeenum jt_c_completeenum. ((((((exists fs_h_jt_completeenumsoundcode. fs_h_jt_completeenumsoundcode + S (jt_b_completeenum) = S ((S (jt_i_completeenum)) * C)) /\\ exists fs_q_jt_completeenumsoundcode. B = fs_q_jt_completeenumsoundcode * S ((S (jt_i_completeenum)) * C) + (jt_b_completeenum))) /\\ (((exists fs_h_jt_completeenumsoundscale. fs_h_jt_completeenumsoundscale + S (jt_c_completeenum) = S ((S (jt_i_completeenum)) * E)) /\\ exists fs_q_jt_completeenumsoundscale. D = fs_q_jt_completeenumsoundscale * S ((S (jt_i_completeenum)) * E) + (jt_c_completeenum))))) /\\ (((forall jt_index_completeenumbound. (exists jt_gap_completeenumboundindex. jt_gap_completeenumboundindex+S (jt_index_completeenumbound)=(k)) -> exists jt_value_completeenumbound. ((((exists fs_h_jt_completeenumboundat. fs_h_jt_completeenumboundat + S (jt_value_completeenumbound) = S ((S (jt_index_completeenumbound)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumboundat. jt_b_completeenum = fs_q_jt_completeenumboundat * S ((S (jt_index_completeenumbound)) * jt_c_completeenum) + (jt_value_completeenumbound))) /\\ (exists jt_gap_completeenumboundvalue. jt_gap_completeenumboundvalue+S (jt_value_completeenumbound)=(n)))) /\\ (forall jt_divisor_completeenumprimitive. (exists jt_factor_completeenumprimitivemodulus. (n)=(jt_divisor_completeenumprimitive)*jt_factor_completeenumprimitivemodulus) -> (forall jt_index_completeenumprimitivecoordinates jt_value_completeenumprimitivecoordinates. (exists jt_gap_completeenumprimitivecoordinatesindex. jt_gap_completeenumprimitivecoordinatesindex+S (jt_index_completeenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completeenumprimitivecoordinatesat. fs_h_jt_completeenumprimitivecoordinatesat + S (jt_value_completeenumprimitivecoordinates) = S ((S (jt_index_completeenumprimitivecoordinates)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumprimitivecoordinatesat. jt_b_completeenum = fs_q_jt_completeenumprimitivecoordinatesat * S ((S (jt_index_completeenumprimitivecoordinates)) * jt_c_completeenum) + (jt_value_completeenumprimitivecoordinates))) -> (exists jt_factor_completeenumprimitivecoordinatesdivides. (jt_value_completeenumprimitivecoordinates)=(jt_divisor_completeenumprimitive)*jt_factor_completeenumprimitivecoordinatesdivides)) -> jt_divisor_completeenumprimitive=1))))) /\\ (((forall jt_b_completeenum jt_c_completeenum. (forall jt_index_completeenuminputbound. (exists jt_gap_completeenuminputboundindex. jt_gap_completeenuminputboundindex+S (jt_index_completeenuminputbound)=(k)) -> exists jt_value_completeenuminputbound. ((((exists fs_h_jt_completeenuminputboundat. fs_h_jt_completeenuminputboundat + S (jt_value_completeenuminputbound) = S ((S (jt_index_completeenuminputbound)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenuminputboundat. jt_b_completeenum = fs_q_jt_completeenuminputboundat * S ((S (jt_index_completeenuminputbound)) * jt_c_completeenum) + (jt_value_completeenuminputbound))) /\\ (exists jt_gap_completeenuminputboundvalue. jt_gap_completeenuminputboundvalue+S (jt_value_completeenuminputbound)=(n)))) -> (forall jt_divisor_completeenuminputprimitive. (exists jt_factor_completeenuminputprimitivemodulus. (n)=(jt_divisor_completeenuminputprimitive)*jt_factor_completeenuminputprimitivemodulus) -> (forall jt_index_completeenuminputprimitivecoordinates jt_value_completeenuminputprimitivecoordinates. (exists jt_gap_completeenuminputprimitivecoordinatesindex. jt_gap_completeenuminputprimitivecoordinatesindex+S (jt_index_completeenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completeenuminputprimitivecoordinatesat. fs_h_jt_completeenuminputprimitivecoordinatesat + S (jt_value_completeenuminputprimitivecoordinates) = S ((S (jt_index_completeenuminputprimitivecoordinates)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenuminputprimitivecoordinatesat. jt_b_completeenum = fs_q_jt_completeenuminputprimitivecoordinatesat * S ((S (jt_index_completeenuminputprimitivecoordinates)) * jt_c_completeenum) + (jt_value_completeenuminputprimitivecoordinates))) -> (exists jt_factor_completeenuminputprimitivecoordinatesdivides. (jt_value_completeenuminputprimitivecoordinates)=(jt_divisor_completeenuminputprimitive)*jt_factor_completeenuminputprimitivecoordinatesdivides)) -> jt_divisor_completeenuminputprimitive=1) -> exists jt_i_completeenum jt_d_completeenum jt_e_completeenum. ((exists jt_gap_completeenumcompleteindex. jt_gap_completeenumcompleteindex+S (jt_i_completeenum)=(j)) /\\ (((((((exists fs_h_jt_completeenumcompletecode. fs_h_jt_completeenumcompletecode + S (jt_d_completeenum) = S ((S (jt_i_completeenum)) * C)) /\\ exists fs_q_jt_completeenumcompletecode. B = fs_q_jt_completeenumcompletecode * S ((S (jt_i_completeenum)) * C) + (jt_d_completeenum))) /\\ (((exists fs_h_jt_completeenumcompletescale. fs_h_jt_completeenumcompletescale + S (jt_e_completeenum) = S ((S (jt_i_completeenum)) * E)) /\\ exists fs_q_jt_completeenumcompletescale. D = fs_q_jt_completeenumcompletescale * S ((S (jt_i_completeenum)) * E) + (jt_e_completeenum))))) /\\ (forall jt_index_completeenumrepresented jt_left_completeenumrepresented jt_right_completeenumrepresented. (exists jt_gap_completeenumrepresentedindex. jt_gap_completeenumrepresentedindex+S (jt_index_completeenumrepresented)=(k)) -> (((exists fs_h_jt_completeenumrepresentedleft. fs_h_jt_completeenumrepresentedleft + S (jt_left_completeenumrepresented) = S ((S (jt_index_completeenumrepresented)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumrepresentedleft. jt_b_completeenum = fs_q_jt_completeenumrepresentedleft * S ((S (jt_index_completeenumrepresented)) * jt_c_completeenum) + (jt_left_completeenumrepresented))) -> (((exists fs_h_jt_completeenumrepresentedright. fs_h_jt_completeenumrepresentedright + S (jt_right_completeenumrepresented) = S ((S (jt_index_completeenumrepresented)) * jt_e_completeenum)) /\\ exists fs_q_jt_completeenumrepresentedright. jt_d_completeenum = fs_q_jt_completeenumrepresentedright * S ((S (jt_index_completeenumrepresented)) * jt_e_completeenum) + (jt_right_completeenumrepresented))) -> jt_left_completeenumrepresented=jt_right_completeenumrepresented))))) /\\ (forall jt_i_completeenum jt_h_completeenum jt_b_completeenum jt_c_completeenum jt_d_completeenum jt_e_completeenum. (exists jt_gap_completeenumfirstindex. jt_gap_completeenumfirstindex+S (jt_i_completeenum)=(j)) -> (exists jt_gap_completeenumsecondindex. jt_gap_completeenumsecondindex+S (jt_h_completeenum)=(j)) -> (((((exists fs_h_jt_completeenumfirstcode. fs_h_jt_completeenumfirstcode + S (jt_b_completeenum) = S ((S (jt_i_completeenum)) * C)) /\\ exists fs_q_jt_completeenumfirstcode. B = fs_q_jt_completeenumfirstcode * S ((S (jt_i_completeenum)) * C) + (jt_b_completeenum))) /\\ (((exists fs_h_jt_completeenumfirstscale. fs_h_jt_completeenumfirstscale + S (jt_c_completeenum) = S ((S (jt_i_completeenum)) * E)) /\\ exists fs_q_jt_completeenumfirstscale. D = fs_q_jt_completeenumfirstscale * S ((S (jt_i_completeenum)) * E) + (jt_c_completeenum))))) -> (((((exists fs_h_jt_completeenumsecondcode. fs_h_jt_completeenumsecondcode + S (jt_d_completeenum) = S ((S (jt_h_completeenum)) * C)) /\\ exists fs_q_jt_completeenumsecondcode. B = fs_q_jt_completeenumsecondcode * S ((S (jt_h_completeenum)) * C) + (jt_d_completeenum))) /\\ (((exists fs_h_jt_completeenumsecondscale. fs_h_jt_completeenumsecondscale + S (jt_e_completeenum) = S ((S (jt_h_completeenum)) * E)) /\\ exists fs_q_jt_completeenumsecondscale. D = fs_q_jt_completeenumsecondscale * S ((S (jt_h_completeenum)) * E) + (jt_e_completeenum))))) -> (forall jt_index_completeenumsame jt_left_completeenumsame jt_right_completeenumsame. (exists jt_gap_completeenumsameindex. jt_gap_completeenumsameindex+S (jt_index_completeenumsame)=(k)) -> (((exists fs_h_jt_completeenumsameleft. fs_h_jt_completeenumsameleft + S (jt_left_completeenumsame) = S ((S (jt_index_completeenumsame)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumsameleft. jt_b_completeenum = fs_q_jt_completeenumsameleft * S ((S (jt_index_completeenumsame)) * jt_c_completeenum) + (jt_left_completeenumsame))) -> (((exists fs_h_jt_completeenumsameright. fs_h_jt_completeenumsameright + S (jt_right_completeenumsame) = S ((S (jt_index_completeenumsame)) * jt_e_completeenum)) /\\ exists fs_q_jt_completeenumsameright. jt_d_completeenum = fs_q_jt_completeenumsameright * S ((S (jt_index_completeenumsame)) * jt_e_completeenum) + (jt_right_completeenumsame))) -> jt_left_completeenumsame=jt_right_completeenumsame) -> jt_i_completeenum=jt_h_completeenum)))))", "statement_sha256": "86d98af7dc2c78453a607d0ec0b5db7ea1ffd3a9f2590eda3851538ae12cf74d", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A completed duplicate-free scan of a genuine representative box is the independent enumeration graph." }, { "admission_dependencies": [ "jordan_tuple_scan_complete" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 41, "body_proof_nodes": 63, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro c", "intro T", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hk", "intro hn", "intro hbox", "intro hscan", "split", "exact hk", "split", "exact hn", "exists B", "exists C", "exists D", "exists E", "specialize jordan_tuple_scan_complete (k)", "specialize jordan_tuple_scan_complete (n)", "specialize jordan_tuple_scan_complete (c)", "specialize jordan_tuple_scan_complete (T)", "specialize jordan_tuple_scan_complete (B)", "specialize jordan_tuple_scan_complete (C)", "specialize jordan_tuple_scan_complete (D)", "specialize jordan_tuple_scan_complete (E)", "specialize jordan_tuple_scan_complete (j)", "apply jordan_tuple_scan_complete", "exact hbox", "exact hscan" ], "defined_statement": "∀ k. ∀ n. ∀ c. ∀ T. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. ¬k = 0 → ¬n = 0 → JordanTupleRepresentatives(k,n,c,T) → JordanTupleScan(k,n,c,T,B,C,D,E,j) → JordanTotient(k,n,j)", "defined_statement_sha256": "ee5480c969b83a74bfee5d87957fb6f466f129feb0839ff5690ea0d7fb8581c6", "definition_uses": { "ND0375": 1, "ND0377": 1, "ND0378": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "57ea34f6695be33fd68f3f759d7a1cde9b32f76a5794b34f3d9b05f6d8060340", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hk" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hbox" } ], [ { "kind": "text", "text": "intro hscan" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hk" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exists B" } ], [ { "kind": "text", "text": "exists C" } ], [ { "kind": "text", "text": "exists D" } ], [ { "kind": "text", "text": "exists E" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (T)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_complete (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_complete" } ], [ { "kind": "text", "text": "exact hbox" } ], [ { "kind": "text", "text": "exact hscan" } ] ], "statement_definition_uses": { "ND0375": 1, "ND0377": 1, "ND0378": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ c. ∀ T. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. ¬k = 0 → ¬n = 0 → " }, { "definition": "ND0378", "kind": "definition", "text": "JordanTupleRepresentatives(k,n,c,T)" }, { "kind": "text", "text": " → " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,T,B,C,D,E,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,n,j)" } ] }, "dependencies": [ "jordan_tuple_scan_complete" ], "enrolled_in_alpha": true, "factory": "make_jordan_enumeration_bridge_candidate_theorems", "first_admitted_version": "v35", "id": "JT0025", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_from_complete_scan", "original_ha_bundle_verified": true, "proof_bundle_node_id": 294, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro c", "intro T", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hk", "intro hn", "intro hbox", "intro hscan", "split", "exact hk", "split", "exact hn", "exists B", "exists C", "exists D", "exists E", "specialize jordan_tuple_scan_complete (k)", "specialize jordan_tuple_scan_complete (n)", "specialize jordan_tuple_scan_complete (c)", "specialize jordan_tuple_scan_complete (T)", "specialize jordan_tuple_scan_complete (B)", "specialize jordan_tuple_scan_complete (C)", "specialize jordan_tuple_scan_complete (D)", "specialize jordan_tuple_scan_complete (E)", "specialize jordan_tuple_scan_complete (j)", "apply jordan_tuple_scan_complete", "exact hbox", "exact hscan" ], "script_sha256": "53e4d3788ab749bda1a71984c82a7d39fc6ad6a9e36a70e5431f788e5535b8e5", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_enumeration_bridge_candidate_theorems", "script_sha256": "53e4d3788ab749bda1a71984c82a7d39fc6ad6a9e36a70e5431f788e5535b8e5", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "57ea34f6695be33fd68f3f759d7a1cde9b32f76a5794b34f3d9b05f6d8060340" } ], "stable_member": false, "statement": "forall k n c T B C D E j. ~(k=0) -> ~(n=0) -> (forall jt_code_jordanbox jt_scale_jordanbox. (forall jt_index_jordanboxbound. (exists jt_gap_jordanboxboundindex. jt_gap_jordanboxboundindex+S (jt_index_jordanboxbound)=(k)) -> exists jt_value_jordanboxbound. ((((exists fs_h_jt_jordanboxboundat. fs_h_jt_jordanboxboundat + S (jt_value_jordanboxbound) = S ((S (jt_index_jordanboxbound)) * jt_scale_jordanbox)) /\\ exists fs_q_jt_jordanboxboundat. jt_code_jordanbox = fs_q_jt_jordanboxboundat * S ((S (jt_index_jordanboxbound)) * jt_scale_jordanbox) + (jt_value_jordanboxbound))) /\\ (exists jt_gap_jordanboxboundvalue. jt_gap_jordanboxboundvalue+S (jt_value_jordanboxbound)=(n)))) -> exists jt_representative_jordanbox. ((exists jt_gap_jordanboxindex. jt_gap_jordanboxindex+S (jt_representative_jordanbox)=(T)) /\\ (forall jt_index_jordanboxequal jt_left_jordanboxequal jt_right_jordanboxequal. (exists jt_gap_jordanboxequalindex. jt_gap_jordanboxequalindex+S (jt_index_jordanboxequal)=(k)) -> (((exists fs_h_jt_jordanboxequalleft. fs_h_jt_jordanboxequalleft + S (jt_left_jordanboxequal) = S ((S (jt_index_jordanboxequal)) * jt_scale_jordanbox)) /\\ exists fs_q_jt_jordanboxequalleft. jt_code_jordanbox = fs_q_jt_jordanboxequalleft * S ((S (jt_index_jordanboxequal)) * jt_scale_jordanbox) + (jt_left_jordanboxequal))) -> (((exists fs_h_jt_jordanboxequalright. fs_h_jt_jordanboxequalright + S (jt_right_jordanboxequal) = S ((S (jt_index_jordanboxequal)) * c)) /\\ exists fs_q_jt_jordanboxequalright. jt_representative_jordanbox = fs_q_jt_jordanboxequalright * S ((S (jt_index_jordanboxequal)) * c) + (jt_right_jordanboxequal))) -> jt_left_jordanboxequal=jt_right_jordanboxequal))) -> (((forall jt_i_jordanscan. (exists jt_gap_jordanscansoundindex. jt_gap_jordanscansoundindex+S (jt_i_jordanscan)=(j)) -> exists jt_b_jordanscan jt_e_jordanscan. ((((((exists fs_h_jt_jordanscansoundcode. fs_h_jt_jordanscansoundcode + S (jt_b_jordanscan) = S ((S (jt_i_jordanscan)) * C)) /\\ exists fs_q_jt_jordanscansoundcode. B = fs_q_jt_jordanscansoundcode * S ((S (jt_i_jordanscan)) * C) + (jt_b_jordanscan))) /\\ (((exists fs_h_jt_jordanscansoundscale. fs_h_jt_jordanscansoundscale + S (jt_e_jordanscan) = S ((S (jt_i_jordanscan)) * E)) /\\ exists fs_q_jt_jordanscansoundscale. D = fs_q_jt_jordanscansoundscale * S ((S (jt_i_jordanscan)) * E) + (jt_e_jordanscan))))) /\\ (((forall jt_index_jordanscanbound. (exists jt_gap_jordanscanboundindex. jt_gap_jordanscanboundindex+S (jt_index_jordanscanbound)=(k)) -> exists jt_value_jordanscanbound. ((((exists fs_h_jt_jordanscanboundat. fs_h_jt_jordanscanboundat + S (jt_value_jordanscanbound) = S ((S (jt_index_jordanscanbound)) * jt_e_jordanscan)) /\\ exists fs_q_jt_jordanscanboundat. jt_b_jordanscan = fs_q_jt_jordanscanboundat * S ((S (jt_index_jordanscanbound)) * jt_e_jordanscan) + (jt_value_jordanscanbound))) /\\ (exists jt_gap_jordanscanboundvalue. jt_gap_jordanscanboundvalue+S (jt_value_jordanscanbound)=(n)))) /\\ (forall jt_divisor_jordanscanprimitive. (exists jt_factor_jordanscanprimitivemodulus. (n)=(jt_divisor_jordanscanprimitive)*jt_factor_jordanscanprimitivemodulus) -> (forall jt_index_jordanscanprimitivecoordinates jt_value_jordanscanprimitivecoordinates. (exists jt_gap_jordanscanprimitivecoordinatesindex. jt_gap_jordanscanprimitivecoordinatesindex+S (jt_index_jordanscanprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jordanscanprimitivecoordinatesat. fs_h_jt_jordanscanprimitivecoordinatesat + S (jt_value_jordanscanprimitivecoordinates) = S ((S (jt_index_jordanscanprimitivecoordinates)) * jt_e_jordanscan)) /\\ exists fs_q_jt_jordanscanprimitivecoordinatesat. jt_b_jordanscan = fs_q_jt_jordanscanprimitivecoordinatesat * S ((S (jt_index_jordanscanprimitivecoordinates)) * jt_e_jordanscan) + (jt_value_jordanscanprimitivecoordinates))) -> (exists jt_factor_jordanscanprimitivecoordinatesdivides. (jt_value_jordanscanprimitivecoordinates)=(jt_divisor_jordanscanprimitive)*jt_factor_jordanscanprimitivecoordinatesdivides)) -> jt_divisor_jordanscanprimitive=1))))) /\\ (((forall jt_i_jordanscan jt_h_jordanscan jt_b_jordanscan jt_e_jordanscan jt_d_jordanscan jt_f_jordanscan. (exists jt_gap_jordanscanfirstindex. jt_gap_jordanscanfirstindex+S (jt_i_jordanscan)=(j)) -> (exists jt_gap_jordanscansecondindex. jt_gap_jordanscansecondindex+S (jt_h_jordanscan)=(j)) -> (((((exists fs_h_jt_jordanscanfirstcode. fs_h_jt_jordanscanfirstcode + S (jt_b_jordanscan) = S ((S (jt_i_jordanscan)) * C)) /\\ exists fs_q_jt_jordanscanfirstcode. B = fs_q_jt_jordanscanfirstcode * S ((S (jt_i_jordanscan)) * C) + (jt_b_jordanscan))) /\\ (((exists fs_h_jt_jordanscanfirstscale. fs_h_jt_jordanscanfirstscale + S (jt_e_jordanscan) = S ((S (jt_i_jordanscan)) * E)) /\\ exists fs_q_jt_jordanscanfirstscale. D = fs_q_jt_jordanscanfirstscale * S ((S (jt_i_jordanscan)) * E) + (jt_e_jordanscan))))) -> (((((exists fs_h_jt_jordanscansecondcode. fs_h_jt_jordanscansecondcode + S (jt_d_jordanscan) = S ((S (jt_h_jordanscan)) * C)) /\\ exists fs_q_jt_jordanscansecondcode. B = fs_q_jt_jordanscansecondcode * S ((S (jt_h_jordanscan)) * C) + (jt_d_jordanscan))) /\\ (((exists fs_h_jt_jordanscansecondscale. fs_h_jt_jordanscansecondscale + S (jt_f_jordanscan) = S ((S (jt_h_jordanscan)) * E)) /\\ exists fs_q_jt_jordanscansecondscale. D = fs_q_jt_jordanscansecondscale * S ((S (jt_h_jordanscan)) * E) + (jt_f_jordanscan))))) -> (forall jt_index_jordanscansame jt_left_jordanscansame jt_right_jordanscansame. (exists jt_gap_jordanscansameindex. jt_gap_jordanscansameindex+S (jt_index_jordanscansame)=(k)) -> (((exists fs_h_jt_jordanscansameleft. fs_h_jt_jordanscansameleft + S (jt_left_jordanscansame) = S ((S (jt_index_jordanscansame)) * jt_e_jordanscan)) /\\ exists fs_q_jt_jordanscansameleft. jt_b_jordanscan = fs_q_jt_jordanscansameleft * S ((S (jt_index_jordanscansame)) * jt_e_jordanscan) + (jt_left_jordanscansame))) -> (((exists fs_h_jt_jordanscansameright. fs_h_jt_jordanscansameright + S (jt_right_jordanscansame) = S ((S (jt_index_jordanscansame)) * jt_f_jordanscan)) /\\ exists fs_q_jt_jordanscansameright. jt_d_jordanscan = fs_q_jt_jordanscansameright * S ((S (jt_index_jordanscansame)) * jt_f_jordanscan) + (jt_right_jordanscansame))) -> jt_left_jordanscansame=jt_right_jordanscansame) -> jt_i_jordanscan=jt_h_jordanscan) /\\ (forall jt_z_jordanscan. (exists jt_gap_jordanscancodeindex. jt_gap_jordanscancodeindex+S (jt_z_jordanscan)=(T)) -> (forall jt_index_jordanscaninputbound. (exists jt_gap_jordanscaninputboundindex. jt_gap_jordanscaninputboundindex+S (jt_index_jordanscaninputbound)=(k)) -> exists jt_value_jordanscaninputbound. ((((exists fs_h_jt_jordanscaninputboundat. fs_h_jt_jordanscaninputboundat + S (jt_value_jordanscaninputbound) = S ((S (jt_index_jordanscaninputbound)) * c)) /\\ exists fs_q_jt_jordanscaninputboundat. jt_z_jordanscan = fs_q_jt_jordanscaninputboundat * S ((S (jt_index_jordanscaninputbound)) * c) + (jt_value_jordanscaninputbound))) /\\ (exists jt_gap_jordanscaninputboundvalue. jt_gap_jordanscaninputboundvalue+S (jt_value_jordanscaninputbound)=(n)))) -> (forall jt_divisor_jordanscaninputprimitive. (exists jt_factor_jordanscaninputprimitivemodulus. (n)=(jt_divisor_jordanscaninputprimitive)*jt_factor_jordanscaninputprimitivemodulus) -> (forall jt_index_jordanscaninputprimitivecoordinates jt_value_jordanscaninputprimitivecoordinates. (exists jt_gap_jordanscaninputprimitivecoordinatesindex. jt_gap_jordanscaninputprimitivecoordinatesindex+S (jt_index_jordanscaninputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jordanscaninputprimitivecoordinatesat. fs_h_jt_jordanscaninputprimitivecoordinatesat + S (jt_value_jordanscaninputprimitivecoordinates) = S ((S (jt_index_jordanscaninputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_jordanscaninputprimitivecoordinatesat. jt_z_jordanscan = fs_q_jt_jordanscaninputprimitivecoordinatesat * S ((S (jt_index_jordanscaninputprimitivecoordinates)) * c) + (jt_value_jordanscaninputprimitivecoordinates))) -> (exists jt_factor_jordanscaninputprimitivecoordinatesdivides. (jt_value_jordanscaninputprimitivecoordinates)=(jt_divisor_jordanscaninputprimitive)*jt_factor_jordanscaninputprimitivecoordinatesdivides)) -> jt_divisor_jordanscaninputprimitive=1) -> (exists jt_index_jordanscanlisted jt_code_jordanscanlisted jt_scale_jordanscanlisted. ((exists jt_gap_jordanscanlistedindex. jt_gap_jordanscanlistedindex+S (jt_index_jordanscanlisted)=(j)) /\\ (((((((exists fs_h_jt_jordanscanlistedcode. fs_h_jt_jordanscanlistedcode + S (jt_code_jordanscanlisted) = S ((S (jt_index_jordanscanlisted)) * C)) /\\ exists fs_q_jt_jordanscanlistedcode. B = fs_q_jt_jordanscanlistedcode * S ((S (jt_index_jordanscanlisted)) * C) + (jt_code_jordanscanlisted))) /\\ (((exists fs_h_jt_jordanscanlistedscale. fs_h_jt_jordanscanlistedscale + S (jt_scale_jordanscanlisted) = S ((S (jt_index_jordanscanlisted)) * E)) /\\ exists fs_q_jt_jordanscanlistedscale. D = fs_q_jt_jordanscanlistedscale * S ((S (jt_index_jordanscanlisted)) * E) + (jt_scale_jordanscanlisted))))) /\\ (forall jt_index_jordanscanlistedequal jt_left_jordanscanlistedequal jt_right_jordanscanlistedequal. (exists jt_gap_jordanscanlistedequalindex. jt_gap_jordanscanlistedequalindex+S (jt_index_jordanscanlistedequal)=(k)) -> (((exists fs_h_jt_jordanscanlistedequalleft. fs_h_jt_jordanscanlistedequalleft + S (jt_left_jordanscanlistedequal) = S ((S (jt_index_jordanscanlistedequal)) * c)) /\\ exists fs_q_jt_jordanscanlistedequalleft. jt_z_jordanscan = fs_q_jt_jordanscanlistedequalleft * S ((S (jt_index_jordanscanlistedequal)) * c) + (jt_left_jordanscanlistedequal))) -> (((exists fs_h_jt_jordanscanlistedequalright. fs_h_jt_jordanscanlistedequalright + S (jt_right_jordanscanlistedequal) = S ((S (jt_index_jordanscanlistedequal)) * jt_scale_jordanscanlisted)) /\\ exists fs_q_jt_jordanscanlistedequalright. jt_code_jordanscanlisted = fs_q_jt_jordanscanlistedequalright * S ((S (jt_index_jordanscanlistedequal)) * jt_scale_jordanscanlisted) + (jt_right_jordanscanlistedequal))) -> jt_left_jordanscanlistedequal=jt_right_jordanscanlistedequal)))))))))) -> (((~((k)=0)) /\\ (((~((n)=0)) /\\ (exists jt_codes_jordancount jt_code_scale_jordancount jt_scales_jordancount jt_scale_scale_jordancount. ((forall jt_i_jordancountenum. (exists jt_gap_jordancountenumsoundindex. jt_gap_jordancountenumsoundindex+S (jt_i_jordancountenum)=(j)) -> exists jt_b_jordancountenum jt_c_jordancountenum. ((((((exists fs_h_jt_jordancountenumsoundcode. fs_h_jt_jordancountenumsoundcode + S (jt_b_jordancountenum) = S ((S (jt_i_jordancountenum)) * jt_code_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumsoundcode. jt_codes_jordancount = fs_q_jt_jordancountenumsoundcode * S ((S (jt_i_jordancountenum)) * jt_code_scale_jordancount) + (jt_b_jordancountenum))) /\\ (((exists fs_h_jt_jordancountenumsoundscale. fs_h_jt_jordancountenumsoundscale + S (jt_c_jordancountenum) = S ((S (jt_i_jordancountenum)) * jt_scale_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumsoundscale. jt_scales_jordancount = fs_q_jt_jordancountenumsoundscale * S ((S (jt_i_jordancountenum)) * jt_scale_scale_jordancount) + (jt_c_jordancountenum))))) /\\ (((forall jt_index_jordancountenumbound. (exists jt_gap_jordancountenumboundindex. jt_gap_jordancountenumboundindex+S (jt_index_jordancountenumbound)=(k)) -> exists jt_value_jordancountenumbound. ((((exists fs_h_jt_jordancountenumboundat. fs_h_jt_jordancountenumboundat + S (jt_value_jordancountenumbound) = S ((S (jt_index_jordancountenumbound)) * jt_c_jordancountenum)) /\\ exists fs_q_jt_jordancountenumboundat. jt_b_jordancountenum = fs_q_jt_jordancountenumboundat * S ((S (jt_index_jordancountenumbound)) * jt_c_jordancountenum) + (jt_value_jordancountenumbound))) /\\ (exists jt_gap_jordancountenumboundvalue. jt_gap_jordancountenumboundvalue+S (jt_value_jordancountenumbound)=(n)))) /\\ (forall jt_divisor_jordancountenumprimitive. (exists jt_factor_jordancountenumprimitivemodulus. (n)=(jt_divisor_jordancountenumprimitive)*jt_factor_jordancountenumprimitivemodulus) -> (forall jt_index_jordancountenumprimitivecoordinates jt_value_jordancountenumprimitivecoordinates. (exists jt_gap_jordancountenumprimitivecoordinatesindex. jt_gap_jordancountenumprimitivecoordinatesindex+S (jt_index_jordancountenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jordancountenumprimitivecoordinatesat. fs_h_jt_jordancountenumprimitivecoordinatesat + S (jt_value_jordancountenumprimitivecoordinates) = S ((S (jt_index_jordancountenumprimitivecoordinates)) * jt_c_jordancountenum)) /\\ exists fs_q_jt_jordancountenumprimitivecoordinatesat. jt_b_jordancountenum = fs_q_jt_jordancountenumprimitivecoordinatesat * S ((S (jt_index_jordancountenumprimitivecoordinates)) * jt_c_jordancountenum) + (jt_value_jordancountenumprimitivecoordinates))) -> (exists jt_factor_jordancountenumprimitivecoordinatesdivides. (jt_value_jordancountenumprimitivecoordinates)=(jt_divisor_jordancountenumprimitive)*jt_factor_jordancountenumprimitivecoordinatesdivides)) -> jt_divisor_jordancountenumprimitive=1))))) /\\ (((forall jt_b_jordancountenum jt_c_jordancountenum. (forall jt_index_jordancountenuminputbound. (exists jt_gap_jordancountenuminputboundindex. jt_gap_jordancountenuminputboundindex+S (jt_index_jordancountenuminputbound)=(k)) -> exists jt_value_jordancountenuminputbound. ((((exists fs_h_jt_jordancountenuminputboundat. fs_h_jt_jordancountenuminputboundat + S (jt_value_jordancountenuminputbound) = S ((S (jt_index_jordancountenuminputbound)) * jt_c_jordancountenum)) /\\ exists fs_q_jt_jordancountenuminputboundat. jt_b_jordancountenum = fs_q_jt_jordancountenuminputboundat * S ((S (jt_index_jordancountenuminputbound)) * jt_c_jordancountenum) + (jt_value_jordancountenuminputbound))) /\\ (exists jt_gap_jordancountenuminputboundvalue. jt_gap_jordancountenuminputboundvalue+S (jt_value_jordancountenuminputbound)=(n)))) -> (forall jt_divisor_jordancountenuminputprimitive. (exists jt_factor_jordancountenuminputprimitivemodulus. (n)=(jt_divisor_jordancountenuminputprimitive)*jt_factor_jordancountenuminputprimitivemodulus) -> (forall jt_index_jordancountenuminputprimitivecoordinates jt_value_jordancountenuminputprimitivecoordinates. (exists jt_gap_jordancountenuminputprimitivecoordinatesindex. jt_gap_jordancountenuminputprimitivecoordinatesindex+S (jt_index_jordancountenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jordancountenuminputprimitivecoordinatesat. fs_h_jt_jordancountenuminputprimitivecoordinatesat + S (jt_value_jordancountenuminputprimitivecoordinates) = S ((S (jt_index_jordancountenuminputprimitivecoordinates)) * jt_c_jordancountenum)) /\\ exists fs_q_jt_jordancountenuminputprimitivecoordinatesat. jt_b_jordancountenum = fs_q_jt_jordancountenuminputprimitivecoordinatesat * S ((S (jt_index_jordancountenuminputprimitivecoordinates)) * jt_c_jordancountenum) + (jt_value_jordancountenuminputprimitivecoordinates))) -> (exists jt_factor_jordancountenuminputprimitivecoordinatesdivides. (jt_value_jordancountenuminputprimitivecoordinates)=(jt_divisor_jordancountenuminputprimitive)*jt_factor_jordancountenuminputprimitivecoordinatesdivides)) -> jt_divisor_jordancountenuminputprimitive=1) -> exists jt_i_jordancountenum jt_d_jordancountenum jt_e_jordancountenum. ((exists jt_gap_jordancountenumcompleteindex. jt_gap_jordancountenumcompleteindex+S (jt_i_jordancountenum)=(j)) /\\ (((((((exists fs_h_jt_jordancountenumcompletecode. fs_h_jt_jordancountenumcompletecode + S (jt_d_jordancountenum) = S ((S (jt_i_jordancountenum)) * jt_code_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumcompletecode. jt_codes_jordancount = fs_q_jt_jordancountenumcompletecode * S ((S (jt_i_jordancountenum)) * jt_code_scale_jordancount) + (jt_d_jordancountenum))) /\\ (((exists fs_h_jt_jordancountenumcompletescale. fs_h_jt_jordancountenumcompletescale + S (jt_e_jordancountenum) = S ((S (jt_i_jordancountenum)) * jt_scale_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumcompletescale. jt_scales_jordancount = fs_q_jt_jordancountenumcompletescale * S ((S (jt_i_jordancountenum)) * jt_scale_scale_jordancount) + (jt_e_jordancountenum))))) /\\ (forall jt_index_jordancountenumrepresented jt_left_jordancountenumrepresented jt_right_jordancountenumrepresented. (exists jt_gap_jordancountenumrepresentedindex. jt_gap_jordancountenumrepresentedindex+S (jt_index_jordancountenumrepresented)=(k)) -> (((exists fs_h_jt_jordancountenumrepresentedleft. fs_h_jt_jordancountenumrepresentedleft + S (jt_left_jordancountenumrepresented) = S ((S (jt_index_jordancountenumrepresented)) * jt_c_jordancountenum)) /\\ exists fs_q_jt_jordancountenumrepresentedleft. jt_b_jordancountenum = fs_q_jt_jordancountenumrepresentedleft * S ((S (jt_index_jordancountenumrepresented)) * jt_c_jordancountenum) + (jt_left_jordancountenumrepresented))) -> (((exists fs_h_jt_jordancountenumrepresentedright. fs_h_jt_jordancountenumrepresentedright + S (jt_right_jordancountenumrepresented) = S ((S (jt_index_jordancountenumrepresented)) * jt_e_jordancountenum)) /\\ exists fs_q_jt_jordancountenumrepresentedright. jt_d_jordancountenum = fs_q_jt_jordancountenumrepresentedright * S ((S (jt_index_jordancountenumrepresented)) * jt_e_jordancountenum) + (jt_right_jordancountenumrepresented))) -> jt_left_jordancountenumrepresented=jt_right_jordancountenumrepresented))))) /\\ (forall jt_i_jordancountenum jt_h_jordancountenum jt_b_jordancountenum jt_c_jordancountenum jt_d_jordancountenum jt_e_jordancountenum. (exists jt_gap_jordancountenumfirstindex. jt_gap_jordancountenumfirstindex+S (jt_i_jordancountenum)=(j)) -> (exists jt_gap_jordancountenumsecondindex. jt_gap_jordancountenumsecondindex+S (jt_h_jordancountenum)=(j)) -> (((((exists fs_h_jt_jordancountenumfirstcode. fs_h_jt_jordancountenumfirstcode + S (jt_b_jordancountenum) = S ((S (jt_i_jordancountenum)) * jt_code_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumfirstcode. jt_codes_jordancount = fs_q_jt_jordancountenumfirstcode * S ((S (jt_i_jordancountenum)) * jt_code_scale_jordancount) + (jt_b_jordancountenum))) /\\ (((exists fs_h_jt_jordancountenumfirstscale. fs_h_jt_jordancountenumfirstscale + S (jt_c_jordancountenum) = S ((S (jt_i_jordancountenum)) * jt_scale_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumfirstscale. jt_scales_jordancount = fs_q_jt_jordancountenumfirstscale * S ((S (jt_i_jordancountenum)) * jt_scale_scale_jordancount) + (jt_c_jordancountenum))))) -> (((((exists fs_h_jt_jordancountenumsecondcode. fs_h_jt_jordancountenumsecondcode + S (jt_d_jordancountenum) = S ((S (jt_h_jordancountenum)) * jt_code_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumsecondcode. jt_codes_jordancount = fs_q_jt_jordancountenumsecondcode * S ((S (jt_h_jordancountenum)) * jt_code_scale_jordancount) + (jt_d_jordancountenum))) /\\ (((exists fs_h_jt_jordancountenumsecondscale. fs_h_jt_jordancountenumsecondscale + S (jt_e_jordancountenum) = S ((S (jt_h_jordancountenum)) * jt_scale_scale_jordancount)) /\\ exists fs_q_jt_jordancountenumsecondscale. jt_scales_jordancount = fs_q_jt_jordancountenumsecondscale * S ((S (jt_h_jordancountenum)) * jt_scale_scale_jordancount) + (jt_e_jordancountenum))))) -> (forall jt_index_jordancountenumsame jt_left_jordancountenumsame jt_right_jordancountenumsame. (exists jt_gap_jordancountenumsameindex. jt_gap_jordancountenumsameindex+S (jt_index_jordancountenumsame)=(k)) -> (((exists fs_h_jt_jordancountenumsameleft. fs_h_jt_jordancountenumsameleft + S (jt_left_jordancountenumsame) = S ((S (jt_index_jordancountenumsame)) * jt_c_jordancountenum)) /\\ exists fs_q_jt_jordancountenumsameleft. jt_b_jordancountenum = fs_q_jt_jordancountenumsameleft * S ((S (jt_index_jordancountenumsame)) * jt_c_jordancountenum) + (jt_left_jordancountenumsame))) -> (((exists fs_h_jt_jordancountenumsameright. fs_h_jt_jordancountenumsameright + S (jt_right_jordancountenumsame) = S ((S (jt_index_jordancountenumsame)) * jt_e_jordancountenum)) /\\ exists fs_q_jt_jordancountenumsameright. jt_d_jordancountenum = fs_q_jt_jordancountenumsameright * S ((S (jt_index_jordancountenumsame)) * jt_e_jordancountenum) + (jt_right_jordancountenumsame))) -> jt_left_jordancountenumsame=jt_right_jordancountenumsame) -> jt_i_jordancountenum=jt_h_jordancountenum)))))))))", "statement_sha256": "57ea34f6695be33fd68f3f759d7a1cde9b32f76a5794b34f3d9b05f6d8060340", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Package a genuinely completed scan as a Jordan cardinality, without assuming totality." }, { "admission_dependencies": [ "jordan_tuple_outer_append_exists", "jordan_tuple_equal_symm", "finite_lt_succ_eq_or_lt", "jordan_tuple_equal_entry", "beta_at_unique", "le_refl", "integer_vector_equal_components_zero", "le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 63, "body_proof_nodes": 616, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro c", "intro t", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hscan", "intro hb", "intro hp", "intro hfresh", "have hext : ∃ U. ∃ V. ∃ W. ∃ X. IntegerVectorZero(B,C,U,V,j) ∧ (IntegerVectorZero(D,E,W,X,j) ∧ (BetaAt(U,V,j,t) ∧ BetaAt(W,X,j,c)))", "specialize jordan_tuple_outer_append_exists (B)", "specialize jordan_tuple_outer_append_exists (C)", "specialize jordan_tuple_outer_append_exists (D)", "specialize jordan_tuple_outer_append_exists (E)", "specialize jordan_tuple_outer_append_exists (j)", "specialize jordan_tuple_outer_append_exists (t)", "specialize jordan_tuple_outer_append_exists (c)", "apply jordan_tuple_outer_append_exists", "cases hext", "cases hext_witness", "cases hext_witness_witness", "cases hext_witness_witness_witness", "cases hext_witness_witness_witness_witness", "cases hext_witness_witness_witness_witness_right", "cases hext_witness_witness_witness_witness_right_right", "cases hscan", "cases hscan_right", "have hcodesback : IntegerVectorZero(x,x1,B,C,j)", "specialize jordan_tuple_equal_symm (B)", "specialize jordan_tuple_equal_symm (C)", "specialize jordan_tuple_equal_symm (x)", "specialize jordan_tuple_equal_symm (x1)", "specialize jordan_tuple_equal_symm (j)", "apply jordan_tuple_equal_symm", "exact hext_witness_witness_witness_witness_left", "have hscalesback : IntegerVectorZero(x2,x3,D,E,j)", "specialize jordan_tuple_equal_symm (D)", "specialize jordan_tuple_equal_symm (E)", "specialize jordan_tuple_equal_symm (x2)", "specialize jordan_tuple_equal_symm (x3)", "specialize jordan_tuple_equal_symm (j)", "apply jordan_tuple_equal_symm", "exact hext_witness_witness_witness_witness_right_left", "exists x", "exists x1", "exists x2", "exists x3", "split", "intro i", "intro hi", "have hic : i = j ∨ Lt(i,j)", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hic", "exists t", "exists c", "split", "split", "rewrite hic_left", "rewrite hic_left", "exact hext_witness_witness_witness_witness_right_right_left", "rewrite hic_left", "rewrite hic_left", "exact hext_witness_witness_witness_witness_right_right_right", "split", "exact hb", "exact hp", "have hvalue : ∃ b. ∃ e. BetaAt(B,C,i,b) ∧ BetaAt(D,E,i,e) ∧ (BetaPrefixInto(b,e,k,n) ∧ JordanPrimitiveTuple(n,b,e,k))", "specialize hscan_left (i)", "apply hscan_left", "exact hic_right", "cases hvalue", "cases hvalue_witness", "cases hvalue_witness_witness", "cases hvalue_witness_witness_right", "cases hvalue_witness_witness_left", "exists x4", "exists x5", "split", "split", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (x4)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_left", "exact hic_right", "exact hvalue_witness_witness_left_left", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (x5)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_right_left", "exact hic_right", "exact hvalue_witness_witness_left_right", "split", "exact hvalue_witness_witness_right_left", "exact hvalue_witness_witness_right_right", "split", "intro i", "intro h", "intro b", "intro e", "intro d", "intro f", "intro hi", "intro hh", "intro hfirst", "intro hsecond", "intro heq", "cases hfirst", "cases hsecond", "have hic : i = j ∨ Lt(i,j)", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "have hhc : h = j ∨ Lt(h,j)", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (h)", "apply finite_lt_succ_eq_or_lt", "exact hh", "cases hic", "cases hhc", "trans j", "exact hic_left", "symm", "exact hhc_left", "have hbval : b=t", "specialize beta_at_unique (x)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (j)", "specialize beta_at_unique (b)", "specialize beta_at_unique (t)", "apply beta_at_unique", "rewrite hic_left at hfirst_left", "rewrite hic_left at hfirst_left", "exact hfirst_left", "exact hext_witness_witness_witness_witness_right_right_left", "have heval : e=c", "specialize beta_at_unique (x2)", "specialize beta_at_unique (x3)", "specialize beta_at_unique (j)", "specialize beta_at_unique (e)", "specialize beta_at_unique (c)", "apply beta_at_unique", "rewrite hic_left at hfirst_right", "rewrite hic_left at hfirst_right", "exact hfirst_right", "exact hext_witness_witness_witness_witness_right_right_right", "exfalso", "apply hfresh", "exists h", "exists d", "exists f", "split", "exact hhc_right", "split", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (d)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hhc_right", "exact hsecond_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (f)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hhc_right", "exact hsecond_right", "rewrite hbval at heq", "rewrite heval at heq", "rewrite heval at heq", "exact heq", "cases hhc", "have hdval : d=t", "specialize beta_at_unique (x)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (j)", "specialize beta_at_unique (d)", "specialize beta_at_unique (t)", "apply beta_at_unique", "rewrite hhc_left at hsecond_left", "rewrite hhc_left at hsecond_left", "exact hsecond_left", "exact hext_witness_witness_witness_witness_right_right_left", "have hfval : f=c", "specialize beta_at_unique (x2)", "specialize beta_at_unique (x3)", "specialize beta_at_unique (j)", "specialize beta_at_unique (f)", "specialize beta_at_unique (c)", "apply beta_at_unique", "rewrite hhc_left at hsecond_right", "rewrite hhc_left at hsecond_right", "exact hsecond_right", "exact hext_witness_witness_witness_witness_right_right_right", "exfalso", "apply hfresh", "exists i", "exists b", "exists e", "split", "exact hic_right", "split", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (b)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hic_right", "exact hfirst_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (e)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hic_right", "exact hfirst_right", "specialize jordan_tuple_equal_symm (b)", "specialize jordan_tuple_equal_symm (e)", "specialize jordan_tuple_equal_symm (t)", "specialize jordan_tuple_equal_symm (c)", "specialize jordan_tuple_equal_symm (k)", "apply jordan_tuple_equal_symm", "rewrite hdval at heq", "rewrite hfval at heq", "rewrite hfval at heq", "exact heq", "specialize hscan_right_left (i)", "specialize hscan_right_left (h)", "specialize hscan_right_left (b)", "specialize hscan_right_left (e)", "specialize hscan_right_left (d)", "specialize hscan_right_left (f)", "apply hscan_right_left", "exact hic_right", "exact hhc_right", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (b)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hic_right", "exact hfirst_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (e)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hic_right", "exact hfirst_right", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (d)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hhc_right", "exact hsecond_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (f)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hhc_right", "exact hsecond_right", "exact heq", "intro z", "intro hz", "intro hzb", "intro hzp", "have hzc : z = t ∨ Lt(z,t)", "specialize finite_lt_succ_eq_or_lt (t)", "specialize finite_lt_succ_eq_or_lt (z)", "apply finite_lt_succ_eq_or_lt", "exact hz", "cases hzc", "rewrite hzc_left", "exists j", "exists t", "exists c", "split", "specialize le_refl (S j)", "apply le_refl", "split", "split", "exact hext_witness_witness_witness_witness_right_right_left", "exact hext_witness_witness_witness_witness_right_right_right", "specialize jordan_tuple_equal_refl (t)", "specialize jordan_tuple_equal_refl (c)", "specialize jordan_tuple_equal_refl (k)", "apply jordan_tuple_equal_refl", "have hold : JordanTupleListed(z,c,k,B,C,D,E,j)", "specialize hscan_right_right (z)", "apply hscan_right_right", "exact hzc_right", "exact hzb", "exact hzp", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "cases hold_witness_witness_witness_right", "cases hold_witness_witness_witness_right_left", "exists x4", "exists x5", "exists x6", "split", "specialize le_succ (S x4)", "specialize le_succ (j)", "apply le_succ", "exact hold_witness_witness_witness_left", "split", "split", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (x4)", "specialize jordan_tuple_equal_entry (x5)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_left", "exact hold_witness_witness_witness_left", "exact hold_witness_witness_witness_right_left_left", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (x4)", "specialize jordan_tuple_equal_entry (x6)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_right_left", "exact hold_witness_witness_witness_left", "exact hold_witness_witness_witness_right_left_right", "exact hold_witness_witness_witness_right_right" ], "defined_statement": "∀ k. ∀ n. ∀ c. ∀ t. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. JordanTupleScan(k,n,c,t,B,C,D,E,j) → BetaPrefixInto(t,c,k,n) → JordanPrimitiveTuple(n,t,c,k) → ¬JordanTupleListed(t,c,k,B,C,D,E,j) → ∃ x. ∃ y. ∃ z. ∃ m. JordanTupleScan(k,n,c,S t,x,y,z,m,S j)", "defined_statement_sha256": "4c7df108a75eb313f95900c280cd03eb7279318c8da4c3ef5b7690ee2e99eaae", "definition_uses": { "ND0121": 4, "ND0262": 2, "ND0372": 2, "ND0376": 2, "ND0377": 2, "PD0002": 4, "PD0013": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "ac5d2d8b85e501803ff4d2130e971b02aa950449ffbdc6444027712912c72a0b", "free_names": [], "script_definition_uses": { "ND0121": 4, "ND0262": 1, "ND0372": 1, "ND0376": 1, "PD0002": 4, "PD0013": 4 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro t" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hscan" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hfresh" } ], [ { "kind": "text", "text": "have hext : " }, { "kind": "text", "text": "∃ U. ∃ V. ∃ W. ∃ X. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(B,C,U,V,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(D,E,W,X,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(U,V,j,t)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(W,X,j,c)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (c)" } ], [ { "kind": "text", "text": "apply jordan_tuple_outer_append_exists" } ], [ { "kind": "text", "text": "cases hext" } ], [ { "kind": "text", "text": "cases hext_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hscan" } ], [ { "kind": "text", "text": "cases hscan_right" } ], [ { "kind": "text", "text": "have hcodesback : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,x1,B,C,j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_symm" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "have hscalesback : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x2,x3,D,E,j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (j)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_symm" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "have hic : " }, { "kind": "text", "text": "i = j ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (i)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hic" } ], [ { "kind": "text", "text": "exists t" } ], [ { "kind": "text", "text": "exists c" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hic_left" } ], [ { "kind": "text", "text": "rewrite hic_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "rewrite hic_left" } ], [ { "kind": "text", "text": "rewrite hic_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "have hvalue : " }, { "kind": "text", "text": "∃ b. ∃ e. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(B,C,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(D,E,i,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,e,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,e,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hscan_left (i)" } ], [ { "kind": "text", "text": "apply hscan_left" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "cases hvalue" } ], [ { "kind": "text", "text": "cases hvalue_witness" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness_right" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness_left" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "exists x5" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x4)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_left_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x5)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_left_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_right_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "intro hfirst" } ], [ { "kind": "text", "text": "intro hsecond" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "cases hfirst" } ], [ { "kind": "text", "text": "cases hsecond" } ], [ { "kind": "text", "text": "have hic : " }, { "kind": "text", "text": "i = j ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (i)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "have hhc : " }, { "kind": "text", "text": "h = j ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(h,j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (j)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (h)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "cases hic" } ], [ { "kind": "text", "text": "cases hhc" } ], [ { "kind": "text", "text": "trans j" } ], [ { "kind": "text", "text": "exact hic_left" } ], [ { "kind": "text", "text": "symm" } ], [ { "kind": "text", "text": "exact hhc_left" } ], [ { "kind": "text", "text": "have hbval : b=t" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (t)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hic_left at hfirst_left" } ], [ { "kind": "text", "text": "rewrite hic_left at hfirst_left" } ], [ { "kind": "text", "text": "exact hfirst_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "have heval : e=c" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x2)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x3)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (e)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hic_left at hfirst_right" } ], [ { "kind": "text", "text": "rewrite hic_left at hfirst_right" } ], [ { "kind": "text", "text": "exact hfirst_right" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hfresh" } ], [ { "kind": "text", "text": "exists h" } ], [ { "kind": "text", "text": "exists d" } ], [ { "kind": "text", "text": "exists f" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hhc_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (d)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hcodesback" } ], [ { "kind": "text", "text": "exact hhc_right" } ], [ { "kind": "text", "text": "exact hsecond_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (f)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hscalesback" } ], [ { "kind": "text", "text": "exact hhc_right" } ], [ { "kind": "text", "text": "exact hsecond_right" } ], [ { "kind": "text", "text": "rewrite hbval at heq" } ], [ { "kind": "text", "text": "rewrite heval at heq" } ], [ { "kind": "text", "text": "rewrite heval at heq" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "cases hhc" } ], [ { "kind": "text", "text": "have hdval : d=t" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (d)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (t)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hhc_left at hsecond_left" } ], [ { "kind": "text", "text": "rewrite hhc_left at hsecond_left" } ], [ { "kind": "text", "text": "exact hsecond_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "have hfval : f=c" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x2)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x3)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (f)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hhc_left at hsecond_right" } ], [ { "kind": "text", "text": "rewrite hhc_left at hsecond_right" } ], [ { "kind": "text", "text": "exact hsecond_right" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hfresh" } ], [ { "kind": "text", "text": "exists i" } ], [ { "kind": "text", "text": "exists b" } ], [ { "kind": "text", "text": "exists e" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (b)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hcodesback" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hfirst_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (e)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hscalesback" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hfirst_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_symm" } ], [ { "kind": "text", "text": "rewrite hdval at heq" } ], [ { "kind": "text", "text": "rewrite hfval at heq" } ], [ { "kind": "text", "text": "rewrite hfval at heq" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "specialize hscan_right_left (i)" } ], [ { "kind": "text", "text": "specialize hscan_right_left (h)" } ], [ { "kind": "text", "text": "specialize hscan_right_left (b)" } ], [ { "kind": "text", "text": "specialize hscan_right_left (e)" } ], [ { "kind": "text", "text": "specialize hscan_right_left (d)" } ], [ { "kind": "text", "text": "specialize hscan_right_left (f)" } ], [ { "kind": "text", "text": "apply hscan_right_left" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hhc_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (b)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hcodesback" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hfirst_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (e)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hscalesback" } ], [ { "kind": "text", "text": "exact hic_right" } ], [ { "kind": "text", "text": "exact hfirst_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (d)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hcodesback" } ], [ { "kind": "text", "text": "exact hhc_right" } ], [ { "kind": "text", "text": "exact hsecond_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (f)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hscalesback" } ], [ { "kind": "text", "text": "exact hhc_right" } ], [ { "kind": "text", "text": "exact hsecond_right" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "intro hzb" } ], [ { "kind": "text", "text": "intro hzp" } ], [ { "kind": "text", "text": "have hzc : " }, { "kind": "text", "text": "z = t ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,t)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (t)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (z)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "cases hzc" } ], [ { "kind": "text", "text": "rewrite hzc_left" } ], [ { "kind": "text", "text": "exists j" } ], [ { "kind": "text", "text": "exists t" } ], [ { "kind": "text", "text": "exists c" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize le_refl (S j)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_refl (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_refl (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_refl (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_refl" } ], [ { "kind": "text", "text": "have hold : " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(z,c,k,B,C,D,E,j)" } ], [ { "kind": "text", "text": "specialize hscan_right_right (z)" } ], [ { "kind": "text", "text": "apply hscan_right_right" } ], [ { "kind": "text", "text": "exact hzc_right" } ], [ { "kind": "text", "text": "exact hzb" } ], [ { "kind": "text", "text": "exact hzp" } ], [ { "kind": "text", "text": "cases hold" } ], [ { "kind": "text", "text": "cases hold_witness" } ], [ { "kind": "text", "text": "cases hold_witness_witness" } ], [ { "kind": "text", "text": "cases hold_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hold_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hold_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "exists x5" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize le_succ (S x4)" } ], [ { "kind": "text", "text": "specialize le_succ (j)" } ], [ { "kind": "text", "text": "apply le_succ" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x5)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_right_left_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x6)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_right_left_right" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0376": 1, "ND0377": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ c. ∀ t. ∀ B. ∀ C. ∀ D. ∀ E. ∀ j. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,t,B,C,D,E,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(t,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,t,c,k)" }, { "kind": "text", "text": " → ¬" }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(t,c,k,B,C,D,E,j)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ m. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,S t,x,y,z,m,S j)" } ] }, "dependencies": [ "jordan_tuple_outer_append_exists", "jordan_tuple_equal_symm", "finite_lt_succ_eq_or_lt", "jordan_tuple_equal_entry", "beta_at_unique", "le_refl", "jordan_tuple_equal_refl", "le_succ" ], "enrolled_in_alpha": true, "factory": "make_jordan_scan_candidate_theorems", "first_admitted_version": "v35", "id": "JT0026", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_scan_append", "original_ha_bundle_verified": true, "proof_bundle_node_id": 295, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro c", "intro t", "intro B", "intro C", "intro D", "intro E", "intro j", "intro hscan", "intro hb", "intro hp", "intro hfresh", "have hext : exists U V W X. ((forall jt_index_scanappendcodes jt_left_scanappendcodes jt_right_scanappendcodes. (exists jt_gap_scanappendcodesindex. jt_gap_scanappendcodesindex+S (jt_index_scanappendcodes)=(j)) -> (((exists fs_h_jt_scanappendcodesleft. fs_h_jt_scanappendcodesleft + S (jt_left_scanappendcodes) = S ((S (jt_index_scanappendcodes)) * C)) /\\ exists fs_q_jt_scanappendcodesleft. B = fs_q_jt_scanappendcodesleft * S ((S (jt_index_scanappendcodes)) * C) + (jt_left_scanappendcodes))) -> (((exists fs_h_jt_scanappendcodesright. fs_h_jt_scanappendcodesright + S (jt_right_scanappendcodes) = S ((S (jt_index_scanappendcodes)) * V)) /\\ exists fs_q_jt_scanappendcodesright. U = fs_q_jt_scanappendcodesright * S ((S (jt_index_scanappendcodes)) * V) + (jt_right_scanappendcodes))) -> jt_left_scanappendcodes=jt_right_scanappendcodes) /\\ (((forall jt_index_scanappendscales jt_left_scanappendscales jt_right_scanappendscales. (exists jt_gap_scanappendscalesindex. jt_gap_scanappendscalesindex+S (jt_index_scanappendscales)=(j)) -> (((exists fs_h_jt_scanappendscalesleft. fs_h_jt_scanappendscalesleft + S (jt_left_scanappendscales) = S ((S (jt_index_scanappendscales)) * E)) /\\ exists fs_q_jt_scanappendscalesleft. D = fs_q_jt_scanappendscalesleft * S ((S (jt_index_scanappendscales)) * E) + (jt_left_scanappendscales))) -> (((exists fs_h_jt_scanappendscalesright. fs_h_jt_scanappendscalesright + S (jt_right_scanappendscales) = S ((S (jt_index_scanappendscales)) * X)) /\\ exists fs_q_jt_scanappendscalesright. W = fs_q_jt_scanappendscalesright * S ((S (jt_index_scanappendscales)) * X) + (jt_right_scanappendscales))) -> jt_left_scanappendscales=jt_right_scanappendscales) /\\ (((((exists fs_h_jt_scanappendlastcode. fs_h_jt_scanappendlastcode + S (t) = S ((S (j)) * V)) /\\ exists fs_q_jt_scanappendlastcode. U = fs_q_jt_scanappendlastcode * S ((S (j)) * V) + (t))) /\\ (((exists fs_h_jt_scanappendlastscale. fs_h_jt_scanappendlastscale + S (c) = S ((S (j)) * X)) /\\ exists fs_q_jt_scanappendlastscale. W = fs_q_jt_scanappendlastscale * S ((S (j)) * X) + (c))))))))", "specialize jordan_tuple_outer_append_exists (B)", "specialize jordan_tuple_outer_append_exists (C)", "specialize jordan_tuple_outer_append_exists (D)", "specialize jordan_tuple_outer_append_exists (E)", "specialize jordan_tuple_outer_append_exists (j)", "specialize jordan_tuple_outer_append_exists (t)", "specialize jordan_tuple_outer_append_exists (c)", "apply jordan_tuple_outer_append_exists", "cases hext", "cases hext_witness", "cases hext_witness_witness", "cases hext_witness_witness_witness", "cases hext_witness_witness_witness_witness", "cases hext_witness_witness_witness_witness_right", "cases hext_witness_witness_witness_witness_right_right", "cases hscan", "cases hscan_right", "have hcodesback : forall jt_index_scanappendcodesback jt_left_scanappendcodesback jt_right_scanappendcodesback. (exists jt_gap_scanappendcodesbackindex. jt_gap_scanappendcodesbackindex+S (jt_index_scanappendcodesback)=(j)) -> (((exists fs_h_jt_scanappendcodesbackleft. fs_h_jt_scanappendcodesbackleft + S (jt_left_scanappendcodesback) = S ((S (jt_index_scanappendcodesback)) * x1)) /\\ exists fs_q_jt_scanappendcodesbackleft. x = fs_q_jt_scanappendcodesbackleft * S ((S (jt_index_scanappendcodesback)) * x1) + (jt_left_scanappendcodesback))) -> (((exists fs_h_jt_scanappendcodesbackright. fs_h_jt_scanappendcodesbackright + S (jt_right_scanappendcodesback) = S ((S (jt_index_scanappendcodesback)) * C)) /\\ exists fs_q_jt_scanappendcodesbackright. B = fs_q_jt_scanappendcodesbackright * S ((S (jt_index_scanappendcodesback)) * C) + (jt_right_scanappendcodesback))) -> jt_left_scanappendcodesback=jt_right_scanappendcodesback", "specialize jordan_tuple_equal_symm (B)", "specialize jordan_tuple_equal_symm (C)", "specialize jordan_tuple_equal_symm (x)", "specialize jordan_tuple_equal_symm (x1)", "specialize jordan_tuple_equal_symm (j)", "apply jordan_tuple_equal_symm", "exact hext_witness_witness_witness_witness_left", "have hscalesback : forall jt_index_scanappendscalesback jt_left_scanappendscalesback jt_right_scanappendscalesback. (exists jt_gap_scanappendscalesbackindex. jt_gap_scanappendscalesbackindex+S (jt_index_scanappendscalesback)=(j)) -> (((exists fs_h_jt_scanappendscalesbackleft. fs_h_jt_scanappendscalesbackleft + S (jt_left_scanappendscalesback) = S ((S (jt_index_scanappendscalesback)) * x3)) /\\ exists fs_q_jt_scanappendscalesbackleft. x2 = fs_q_jt_scanappendscalesbackleft * S ((S (jt_index_scanappendscalesback)) * x3) + (jt_left_scanappendscalesback))) -> (((exists fs_h_jt_scanappendscalesbackright. fs_h_jt_scanappendscalesbackright + S (jt_right_scanappendscalesback) = S ((S (jt_index_scanappendscalesback)) * E)) /\\ exists fs_q_jt_scanappendscalesbackright. D = fs_q_jt_scanappendscalesbackright * S ((S (jt_index_scanappendscalesback)) * E) + (jt_right_scanappendscalesback))) -> jt_left_scanappendscalesback=jt_right_scanappendscalesback", "specialize jordan_tuple_equal_symm (D)", "specialize jordan_tuple_equal_symm (E)", "specialize jordan_tuple_equal_symm (x2)", "specialize jordan_tuple_equal_symm (x3)", "specialize jordan_tuple_equal_symm (j)", "apply jordan_tuple_equal_symm", "exact hext_witness_witness_witness_witness_right_left", "exists x", "exists x1", "exists x2", "exists x3", "split", "intro i", "intro hi", "have hic : i=j \\/ (exists jt_gap_appendindex. jt_gap_appendindex+S (i)=(j))", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hic", "exists t", "exists c", "split", "split", "rewrite hic_left", "rewrite hic_left", "exact hext_witness_witness_witness_witness_right_right_left", "rewrite hic_left", "rewrite hic_left", "exact hext_witness_witness_witness_witness_right_right_right", "split", "exact hb", "exact hp", "have hvalue : exists b e. ((((((exists fs_h_jt_appendoldcode. fs_h_jt_appendoldcode + S (b) = S ((S (i)) * C)) /\\ exists fs_q_jt_appendoldcode. B = fs_q_jt_appendoldcode * S ((S (i)) * C) + (b))) /\\ (((exists fs_h_jt_appendoldscale. fs_h_jt_appendoldscale + S (e) = S ((S (i)) * E)) /\\ exists fs_q_jt_appendoldscale. D = fs_q_jt_appendoldscale * S ((S (i)) * E) + (e))))) /\\ (((forall jt_index_appendoldbound. (exists jt_gap_appendoldboundindex. jt_gap_appendoldboundindex+S (jt_index_appendoldbound)=(k)) -> exists jt_value_appendoldbound. ((((exists fs_h_jt_appendoldboundat. fs_h_jt_appendoldboundat + S (jt_value_appendoldbound) = S ((S (jt_index_appendoldbound)) * e)) /\\ exists fs_q_jt_appendoldboundat. b = fs_q_jt_appendoldboundat * S ((S (jt_index_appendoldbound)) * e) + (jt_value_appendoldbound))) /\\ (exists jt_gap_appendoldboundvalue. jt_gap_appendoldboundvalue+S (jt_value_appendoldbound)=(n)))) /\\ (forall jt_divisor_appendoldprimitive. (exists jt_factor_appendoldprimitivemodulus. (n)=(jt_divisor_appendoldprimitive)*jt_factor_appendoldprimitivemodulus) -> (forall jt_index_appendoldprimitivecoordinates jt_value_appendoldprimitivecoordinates. (exists jt_gap_appendoldprimitivecoordinatesindex. jt_gap_appendoldprimitivecoordinatesindex+S (jt_index_appendoldprimitivecoordinates)=(k)) -> (((exists fs_h_jt_appendoldprimitivecoordinatesat. fs_h_jt_appendoldprimitivecoordinatesat + S (jt_value_appendoldprimitivecoordinates) = S ((S (jt_index_appendoldprimitivecoordinates)) * e)) /\\ exists fs_q_jt_appendoldprimitivecoordinatesat. b = fs_q_jt_appendoldprimitivecoordinatesat * S ((S (jt_index_appendoldprimitivecoordinates)) * e) + (jt_value_appendoldprimitivecoordinates))) -> (exists jt_factor_appendoldprimitivecoordinatesdivides. (jt_value_appendoldprimitivecoordinates)=(jt_divisor_appendoldprimitive)*jt_factor_appendoldprimitivecoordinatesdivides)) -> jt_divisor_appendoldprimitive=1))))", "specialize hscan_left (i)", "apply hscan_left", "exact hic_right", "cases hvalue", "cases hvalue_witness", "cases hvalue_witness_witness", "cases hvalue_witness_witness_right", "cases hvalue_witness_witness_left", "exists x4", "exists x5", "split", "split", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (x4)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_left", "exact hic_right", "exact hvalue_witness_witness_left_left", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (x5)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_right_left", "exact hic_right", "exact hvalue_witness_witness_left_right", "split", "exact hvalue_witness_witness_right_left", "exact hvalue_witness_witness_right_right", "split", "intro i", "intro h", "intro b", "intro e", "intro d", "intro f", "intro hi", "intro hh", "intro hfirst", "intro hsecond", "intro heq", "cases hfirst", "cases hsecond", "have hic : i=j \\/ (exists jt_gap_appendfirstcase. jt_gap_appendfirstcase+S (i)=(j))", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "have hhc : h=j \\/ (exists jt_gap_appendsecondcase. jt_gap_appendsecondcase+S (h)=(j))", "specialize finite_lt_succ_eq_or_lt (j)", "specialize finite_lt_succ_eq_or_lt (h)", "apply finite_lt_succ_eq_or_lt", "exact hh", "cases hic", "cases hhc", "trans j", "exact hic_left", "symm", "exact hhc_left", "have hbval : b=t", "specialize beta_at_unique (x)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (j)", "specialize beta_at_unique (b)", "specialize beta_at_unique (t)", "apply beta_at_unique", "rewrite hic_left at hfirst_left", "rewrite hic_left at hfirst_left", "exact hfirst_left", "exact hext_witness_witness_witness_witness_right_right_left", "have heval : e=c", "specialize beta_at_unique (x2)", "specialize beta_at_unique (x3)", "specialize beta_at_unique (j)", "specialize beta_at_unique (e)", "specialize beta_at_unique (c)", "apply beta_at_unique", "rewrite hic_left at hfirst_right", "rewrite hic_left at hfirst_right", "exact hfirst_right", "exact hext_witness_witness_witness_witness_right_right_right", "exfalso", "apply hfresh", "exists h", "exists d", "exists f", "split", "exact hhc_right", "split", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (d)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hhc_right", "exact hsecond_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (f)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hhc_right", "exact hsecond_right", "rewrite hbval at heq", "rewrite heval at heq", "rewrite heval at heq", "exact heq", "cases hhc", "have hdval : d=t", "specialize beta_at_unique (x)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (j)", "specialize beta_at_unique (d)", "specialize beta_at_unique (t)", "apply beta_at_unique", "rewrite hhc_left at hsecond_left", "rewrite hhc_left at hsecond_left", "exact hsecond_left", "exact hext_witness_witness_witness_witness_right_right_left", "have hfval : f=c", "specialize beta_at_unique (x2)", "specialize beta_at_unique (x3)", "specialize beta_at_unique (j)", "specialize beta_at_unique (f)", "specialize beta_at_unique (c)", "apply beta_at_unique", "rewrite hhc_left at hsecond_right", "rewrite hhc_left at hsecond_right", "exact hsecond_right", "exact hext_witness_witness_witness_witness_right_right_right", "exfalso", "apply hfresh", "exists i", "exists b", "exists e", "split", "exact hic_right", "split", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (b)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hic_right", "exact hfirst_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (e)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hic_right", "exact hfirst_right", "specialize jordan_tuple_equal_symm (b)", "specialize jordan_tuple_equal_symm (e)", "specialize jordan_tuple_equal_symm (t)", "specialize jordan_tuple_equal_symm (c)", "specialize jordan_tuple_equal_symm (k)", "apply jordan_tuple_equal_symm", "rewrite hdval at heq", "rewrite hfval at heq", "rewrite hfval at heq", "exact heq", "specialize hscan_right_left (i)", "specialize hscan_right_left (h)", "specialize hscan_right_left (b)", "specialize hscan_right_left (e)", "specialize hscan_right_left (d)", "specialize hscan_right_left (f)", "apply hscan_right_left", "exact hic_right", "exact hhc_right", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (b)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hic_right", "exact hfirst_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (i)", "specialize jordan_tuple_equal_entry (e)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hic_right", "exact hfirst_right", "split", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (d)", "apply jordan_tuple_equal_entry", "exact hcodesback", "exact hhc_right", "exact hsecond_left", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (h)", "specialize jordan_tuple_equal_entry (f)", "apply jordan_tuple_equal_entry", "exact hscalesback", "exact hhc_right", "exact hsecond_right", "exact heq", "intro z", "intro hz", "intro hzb", "intro hzp", "have hzc : z=t \\/ (exists jt_gap_appendcovercase. jt_gap_appendcovercase+S (z)=(t))", "specialize finite_lt_succ_eq_or_lt (t)", "specialize finite_lt_succ_eq_or_lt (z)", "apply finite_lt_succ_eq_or_lt", "exact hz", "cases hzc", "rewrite hzc_left", "exists j", "exists t", "exists c", "split", "specialize le_refl (S j)", "apply le_refl", "split", "split", "exact hext_witness_witness_witness_witness_right_right_left", "exact hext_witness_witness_witness_witness_right_right_right", "specialize jordan_tuple_equal_refl (t)", "specialize jordan_tuple_equal_refl (c)", "specialize jordan_tuple_equal_refl (k)", "apply jordan_tuple_equal_refl", "have hold : exists jt_index_appendoldlisted jt_code_appendoldlisted jt_scale_appendoldlisted. ((exists jt_gap_appendoldlistedindex. jt_gap_appendoldlistedindex+S (jt_index_appendoldlisted)=(j)) /\\ (((((((exists fs_h_jt_appendoldlistedcode. fs_h_jt_appendoldlistedcode + S (jt_code_appendoldlisted) = S ((S (jt_index_appendoldlisted)) * C)) /\\ exists fs_q_jt_appendoldlistedcode. B = fs_q_jt_appendoldlistedcode * S ((S (jt_index_appendoldlisted)) * C) + (jt_code_appendoldlisted))) /\\ (((exists fs_h_jt_appendoldlistedscale. fs_h_jt_appendoldlistedscale + S (jt_scale_appendoldlisted) = S ((S (jt_index_appendoldlisted)) * E)) /\\ exists fs_q_jt_appendoldlistedscale. D = fs_q_jt_appendoldlistedscale * S ((S (jt_index_appendoldlisted)) * E) + (jt_scale_appendoldlisted))))) /\\ (forall jt_index_appendoldlistedequal jt_left_appendoldlistedequal jt_right_appendoldlistedequal. (exists jt_gap_appendoldlistedequalindex. jt_gap_appendoldlistedequalindex+S (jt_index_appendoldlistedequal)=(k)) -> (((exists fs_h_jt_appendoldlistedequalleft. fs_h_jt_appendoldlistedequalleft + S (jt_left_appendoldlistedequal) = S ((S (jt_index_appendoldlistedequal)) * c)) /\\ exists fs_q_jt_appendoldlistedequalleft. z = fs_q_jt_appendoldlistedequalleft * S ((S (jt_index_appendoldlistedequal)) * c) + (jt_left_appendoldlistedequal))) -> (((exists fs_h_jt_appendoldlistedequalright. fs_h_jt_appendoldlistedequalright + S (jt_right_appendoldlistedequal) = S ((S (jt_index_appendoldlistedequal)) * jt_scale_appendoldlisted)) /\\ exists fs_q_jt_appendoldlistedequalright. jt_code_appendoldlisted = fs_q_jt_appendoldlistedequalright * S ((S (jt_index_appendoldlistedequal)) * jt_scale_appendoldlisted) + (jt_right_appendoldlistedequal))) -> jt_left_appendoldlistedequal=jt_right_appendoldlistedequal))))", "specialize hscan_right_right (z)", "apply hscan_right_right", "exact hzc_right", "exact hzb", "exact hzp", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "cases hold_witness_witness_witness_right", "cases hold_witness_witness_witness_right_left", "exists x4", "exists x5", "exists x6", "split", "specialize le_succ (S x4)", "specialize le_succ (j)", "apply le_succ", "exact hold_witness_witness_witness_left", "split", "split", "specialize jordan_tuple_equal_entry (B)", "specialize jordan_tuple_equal_entry (C)", "specialize jordan_tuple_equal_entry (x)", "specialize jordan_tuple_equal_entry (x1)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (x4)", "specialize jordan_tuple_equal_entry (x5)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_left", "exact hold_witness_witness_witness_left", "exact hold_witness_witness_witness_right_left_left", "specialize jordan_tuple_equal_entry (D)", "specialize jordan_tuple_equal_entry (E)", "specialize jordan_tuple_equal_entry (x2)", "specialize jordan_tuple_equal_entry (x3)", "specialize jordan_tuple_equal_entry (j)", "specialize jordan_tuple_equal_entry (x4)", "specialize jordan_tuple_equal_entry (x6)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_right_left", "exact hold_witness_witness_witness_left", "exact hold_witness_witness_witness_right_left_right", "exact hold_witness_witness_witness_right_right" ], "script_sha256": "ffc43ec1e78dc1c6d31f5e6adc9648f32d278585e7f6de010fd3037dbe3fd2c0", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_scan_candidate_theorems", "script_sha256": "ffc43ec1e78dc1c6d31f5e6adc9648f32d278585e7f6de010fd3037dbe3fd2c0", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "ac5d2d8b85e501803ff4d2130e971b02aa950449ffbdc6444027712912c72a0b" } ], "stable_member": false, "statement": "forall k n c t B C D E j. (((forall jt_i_appendinvariant. (exists jt_gap_appendinvariantsoundindex. jt_gap_appendinvariantsoundindex+S (jt_i_appendinvariant)=(j)) -> exists jt_b_appendinvariant jt_e_appendinvariant. ((((((exists fs_h_jt_appendinvariantsoundcode. fs_h_jt_appendinvariantsoundcode + S (jt_b_appendinvariant) = S ((S (jt_i_appendinvariant)) * C)) /\\ exists fs_q_jt_appendinvariantsoundcode. B = fs_q_jt_appendinvariantsoundcode * S ((S (jt_i_appendinvariant)) * C) + (jt_b_appendinvariant))) /\\ (((exists fs_h_jt_appendinvariantsoundscale. fs_h_jt_appendinvariantsoundscale + S (jt_e_appendinvariant) = S ((S (jt_i_appendinvariant)) * E)) /\\ exists fs_q_jt_appendinvariantsoundscale. D = fs_q_jt_appendinvariantsoundscale * S ((S (jt_i_appendinvariant)) * E) + (jt_e_appendinvariant))))) /\\ (((forall jt_index_appendinvariantbound. (exists jt_gap_appendinvariantboundindex. jt_gap_appendinvariantboundindex+S (jt_index_appendinvariantbound)=(k)) -> exists jt_value_appendinvariantbound. ((((exists fs_h_jt_appendinvariantboundat. fs_h_jt_appendinvariantboundat + S (jt_value_appendinvariantbound) = S ((S (jt_index_appendinvariantbound)) * jt_e_appendinvariant)) /\\ exists fs_q_jt_appendinvariantboundat. jt_b_appendinvariant = fs_q_jt_appendinvariantboundat * S ((S (jt_index_appendinvariantbound)) * jt_e_appendinvariant) + (jt_value_appendinvariantbound))) /\\ (exists jt_gap_appendinvariantboundvalue. jt_gap_appendinvariantboundvalue+S (jt_value_appendinvariantbound)=(n)))) /\\ (forall jt_divisor_appendinvariantprimitive. (exists jt_factor_appendinvariantprimitivemodulus. (n)=(jt_divisor_appendinvariantprimitive)*jt_factor_appendinvariantprimitivemodulus) -> (forall jt_index_appendinvariantprimitivecoordinates jt_value_appendinvariantprimitivecoordinates. (exists jt_gap_appendinvariantprimitivecoordinatesindex. jt_gap_appendinvariantprimitivecoordinatesindex+S (jt_index_appendinvariantprimitivecoordinates)=(k)) -> (((exists fs_h_jt_appendinvariantprimitivecoordinatesat. fs_h_jt_appendinvariantprimitivecoordinatesat + S (jt_value_appendinvariantprimitivecoordinates) = S ((S (jt_index_appendinvariantprimitivecoordinates)) * jt_e_appendinvariant)) /\\ exists fs_q_jt_appendinvariantprimitivecoordinatesat. jt_b_appendinvariant = fs_q_jt_appendinvariantprimitivecoordinatesat * S ((S (jt_index_appendinvariantprimitivecoordinates)) * jt_e_appendinvariant) + (jt_value_appendinvariantprimitivecoordinates))) -> (exists jt_factor_appendinvariantprimitivecoordinatesdivides. (jt_value_appendinvariantprimitivecoordinates)=(jt_divisor_appendinvariantprimitive)*jt_factor_appendinvariantprimitivecoordinatesdivides)) -> jt_divisor_appendinvariantprimitive=1))))) /\\ (((forall jt_i_appendinvariant jt_h_appendinvariant jt_b_appendinvariant jt_e_appendinvariant jt_d_appendinvariant jt_f_appendinvariant. (exists jt_gap_appendinvariantfirstindex. jt_gap_appendinvariantfirstindex+S (jt_i_appendinvariant)=(j)) -> (exists jt_gap_appendinvariantsecondindex. jt_gap_appendinvariantsecondindex+S (jt_h_appendinvariant)=(j)) -> (((((exists fs_h_jt_appendinvariantfirstcode. fs_h_jt_appendinvariantfirstcode + S (jt_b_appendinvariant) = S ((S (jt_i_appendinvariant)) * C)) /\\ exists fs_q_jt_appendinvariantfirstcode. B = fs_q_jt_appendinvariantfirstcode * S ((S (jt_i_appendinvariant)) * C) + (jt_b_appendinvariant))) /\\ (((exists fs_h_jt_appendinvariantfirstscale. fs_h_jt_appendinvariantfirstscale + S (jt_e_appendinvariant) = S ((S (jt_i_appendinvariant)) * E)) /\\ exists fs_q_jt_appendinvariantfirstscale. D = fs_q_jt_appendinvariantfirstscale * S ((S (jt_i_appendinvariant)) * E) + (jt_e_appendinvariant))))) -> (((((exists fs_h_jt_appendinvariantsecondcode. fs_h_jt_appendinvariantsecondcode + S (jt_d_appendinvariant) = S ((S (jt_h_appendinvariant)) * C)) /\\ exists fs_q_jt_appendinvariantsecondcode. B = fs_q_jt_appendinvariantsecondcode * S ((S (jt_h_appendinvariant)) * C) + (jt_d_appendinvariant))) /\\ (((exists fs_h_jt_appendinvariantsecondscale. fs_h_jt_appendinvariantsecondscale + S (jt_f_appendinvariant) = S ((S (jt_h_appendinvariant)) * E)) /\\ exists fs_q_jt_appendinvariantsecondscale. D = fs_q_jt_appendinvariantsecondscale * S ((S (jt_h_appendinvariant)) * E) + (jt_f_appendinvariant))))) -> (forall jt_index_appendinvariantsame jt_left_appendinvariantsame jt_right_appendinvariantsame. (exists jt_gap_appendinvariantsameindex. jt_gap_appendinvariantsameindex+S (jt_index_appendinvariantsame)=(k)) -> (((exists fs_h_jt_appendinvariantsameleft. fs_h_jt_appendinvariantsameleft + S (jt_left_appendinvariantsame) = S ((S (jt_index_appendinvariantsame)) * jt_e_appendinvariant)) /\\ exists fs_q_jt_appendinvariantsameleft. jt_b_appendinvariant = fs_q_jt_appendinvariantsameleft * S ((S (jt_index_appendinvariantsame)) * jt_e_appendinvariant) + (jt_left_appendinvariantsame))) -> (((exists fs_h_jt_appendinvariantsameright. fs_h_jt_appendinvariantsameright + S (jt_right_appendinvariantsame) = S ((S (jt_index_appendinvariantsame)) * jt_f_appendinvariant)) /\\ exists fs_q_jt_appendinvariantsameright. jt_d_appendinvariant = fs_q_jt_appendinvariantsameright * S ((S (jt_index_appendinvariantsame)) * jt_f_appendinvariant) + (jt_right_appendinvariantsame))) -> jt_left_appendinvariantsame=jt_right_appendinvariantsame) -> jt_i_appendinvariant=jt_h_appendinvariant) /\\ (forall jt_z_appendinvariant. (exists jt_gap_appendinvariantcodeindex. jt_gap_appendinvariantcodeindex+S (jt_z_appendinvariant)=(t)) -> (forall jt_index_appendinvariantinputbound. (exists jt_gap_appendinvariantinputboundindex. jt_gap_appendinvariantinputboundindex+S (jt_index_appendinvariantinputbound)=(k)) -> exists jt_value_appendinvariantinputbound. ((((exists fs_h_jt_appendinvariantinputboundat. fs_h_jt_appendinvariantinputboundat + S (jt_value_appendinvariantinputbound) = S ((S (jt_index_appendinvariantinputbound)) * c)) /\\ exists fs_q_jt_appendinvariantinputboundat. jt_z_appendinvariant = fs_q_jt_appendinvariantinputboundat * S ((S (jt_index_appendinvariantinputbound)) * c) + (jt_value_appendinvariantinputbound))) /\\ (exists jt_gap_appendinvariantinputboundvalue. jt_gap_appendinvariantinputboundvalue+S (jt_value_appendinvariantinputbound)=(n)))) -> (forall jt_divisor_appendinvariantinputprimitive. (exists jt_factor_appendinvariantinputprimitivemodulus. (n)=(jt_divisor_appendinvariantinputprimitive)*jt_factor_appendinvariantinputprimitivemodulus) -> (forall jt_index_appendinvariantinputprimitivecoordinates jt_value_appendinvariantinputprimitivecoordinates. (exists jt_gap_appendinvariantinputprimitivecoordinatesindex. jt_gap_appendinvariantinputprimitivecoordinatesindex+S (jt_index_appendinvariantinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_appendinvariantinputprimitivecoordinatesat. fs_h_jt_appendinvariantinputprimitivecoordinatesat + S (jt_value_appendinvariantinputprimitivecoordinates) = S ((S (jt_index_appendinvariantinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_appendinvariantinputprimitivecoordinatesat. jt_z_appendinvariant = fs_q_jt_appendinvariantinputprimitivecoordinatesat * S ((S (jt_index_appendinvariantinputprimitivecoordinates)) * c) + (jt_value_appendinvariantinputprimitivecoordinates))) -> (exists jt_factor_appendinvariantinputprimitivecoordinatesdivides. (jt_value_appendinvariantinputprimitivecoordinates)=(jt_divisor_appendinvariantinputprimitive)*jt_factor_appendinvariantinputprimitivecoordinatesdivides)) -> jt_divisor_appendinvariantinputprimitive=1) -> (exists jt_index_appendinvariantlisted jt_code_appendinvariantlisted jt_scale_appendinvariantlisted. ((exists jt_gap_appendinvariantlistedindex. jt_gap_appendinvariantlistedindex+S (jt_index_appendinvariantlisted)=(j)) /\\ (((((((exists fs_h_jt_appendinvariantlistedcode. fs_h_jt_appendinvariantlistedcode + S (jt_code_appendinvariantlisted) = S ((S (jt_index_appendinvariantlisted)) * C)) /\\ exists fs_q_jt_appendinvariantlistedcode. B = fs_q_jt_appendinvariantlistedcode * S ((S (jt_index_appendinvariantlisted)) * C) + (jt_code_appendinvariantlisted))) /\\ (((exists fs_h_jt_appendinvariantlistedscale. fs_h_jt_appendinvariantlistedscale + S (jt_scale_appendinvariantlisted) = S ((S (jt_index_appendinvariantlisted)) * E)) /\\ exists fs_q_jt_appendinvariantlistedscale. D = fs_q_jt_appendinvariantlistedscale * S ((S (jt_index_appendinvariantlisted)) * E) + (jt_scale_appendinvariantlisted))))) /\\ (forall jt_index_appendinvariantlistedequal jt_left_appendinvariantlistedequal jt_right_appendinvariantlistedequal. (exists jt_gap_appendinvariantlistedequalindex. jt_gap_appendinvariantlistedequalindex+S (jt_index_appendinvariantlistedequal)=(k)) -> (((exists fs_h_jt_appendinvariantlistedequalleft. fs_h_jt_appendinvariantlistedequalleft + S (jt_left_appendinvariantlistedequal) = S ((S (jt_index_appendinvariantlistedequal)) * c)) /\\ exists fs_q_jt_appendinvariantlistedequalleft. jt_z_appendinvariant = fs_q_jt_appendinvariantlistedequalleft * S ((S (jt_index_appendinvariantlistedequal)) * c) + (jt_left_appendinvariantlistedequal))) -> (((exists fs_h_jt_appendinvariantlistedequalright. fs_h_jt_appendinvariantlistedequalright + S (jt_right_appendinvariantlistedequal) = S ((S (jt_index_appendinvariantlistedequal)) * jt_scale_appendinvariantlisted)) /\\ exists fs_q_jt_appendinvariantlistedequalright. jt_code_appendinvariantlisted = fs_q_jt_appendinvariantlistedequalright * S ((S (jt_index_appendinvariantlistedequal)) * jt_scale_appendinvariantlisted) + (jt_right_appendinvariantlistedequal))) -> jt_left_appendinvariantlistedequal=jt_right_appendinvariantlistedequal)))))))))) -> (forall jt_index_appendbound. (exists jt_gap_appendboundindex. jt_gap_appendboundindex+S (jt_index_appendbound)=(k)) -> exists jt_value_appendbound. ((((exists fs_h_jt_appendboundat. fs_h_jt_appendboundat + S (jt_value_appendbound) = S ((S (jt_index_appendbound)) * c)) /\\ exists fs_q_jt_appendboundat. t = fs_q_jt_appendboundat * S ((S (jt_index_appendbound)) * c) + (jt_value_appendbound))) /\\ (exists jt_gap_appendboundvalue. jt_gap_appendboundvalue+S (jt_value_appendbound)=(n)))) -> (forall jt_divisor_appendprimitive. (exists jt_factor_appendprimitivemodulus. (n)=(jt_divisor_appendprimitive)*jt_factor_appendprimitivemodulus) -> (forall jt_index_appendprimitivecoordinates jt_value_appendprimitivecoordinates. (exists jt_gap_appendprimitivecoordinatesindex. jt_gap_appendprimitivecoordinatesindex+S (jt_index_appendprimitivecoordinates)=(k)) -> (((exists fs_h_jt_appendprimitivecoordinatesat. fs_h_jt_appendprimitivecoordinatesat + S (jt_value_appendprimitivecoordinates) = S ((S (jt_index_appendprimitivecoordinates)) * c)) /\\ exists fs_q_jt_appendprimitivecoordinatesat. t = fs_q_jt_appendprimitivecoordinatesat * S ((S (jt_index_appendprimitivecoordinates)) * c) + (jt_value_appendprimitivecoordinates))) -> (exists jt_factor_appendprimitivecoordinatesdivides. (jt_value_appendprimitivecoordinates)=(jt_divisor_appendprimitive)*jt_factor_appendprimitivecoordinatesdivides)) -> jt_divisor_appendprimitive=1) -> ~(exists jt_index_appendfresh jt_code_appendfresh jt_scale_appendfresh. ((exists jt_gap_appendfreshindex. jt_gap_appendfreshindex+S (jt_index_appendfresh)=(j)) /\\ (((((((exists fs_h_jt_appendfreshcode. fs_h_jt_appendfreshcode + S (jt_code_appendfresh) = S ((S (jt_index_appendfresh)) * C)) /\\ exists fs_q_jt_appendfreshcode. B = fs_q_jt_appendfreshcode * S ((S (jt_index_appendfresh)) * C) + (jt_code_appendfresh))) /\\ (((exists fs_h_jt_appendfreshscale. fs_h_jt_appendfreshscale + S (jt_scale_appendfresh) = S ((S (jt_index_appendfresh)) * E)) /\\ exists fs_q_jt_appendfreshscale. D = fs_q_jt_appendfreshscale * S ((S (jt_index_appendfresh)) * E) + (jt_scale_appendfresh))))) /\\ (forall jt_index_appendfreshequal jt_left_appendfreshequal jt_right_appendfreshequal. (exists jt_gap_appendfreshequalindex. jt_gap_appendfreshequalindex+S (jt_index_appendfreshequal)=(k)) -> (((exists fs_h_jt_appendfreshequalleft. fs_h_jt_appendfreshequalleft + S (jt_left_appendfreshequal) = S ((S (jt_index_appendfreshequal)) * c)) /\\ exists fs_q_jt_appendfreshequalleft. t = fs_q_jt_appendfreshequalleft * S ((S (jt_index_appendfreshequal)) * c) + (jt_left_appendfreshequal))) -> (((exists fs_h_jt_appendfreshequalright. fs_h_jt_appendfreshequalright + S (jt_right_appendfreshequal) = S ((S (jt_index_appendfreshequal)) * jt_scale_appendfresh)) /\\ exists fs_q_jt_appendfreshequalright. jt_code_appendfresh = fs_q_jt_appendfreshequalright * S ((S (jt_index_appendfreshequal)) * jt_scale_appendfresh) + (jt_right_appendfreshequal))) -> jt_left_appendfreshequal=jt_right_appendfreshequal))))) -> exists U V W X. ((forall jt_i_appendresult. (exists jt_gap_appendresultsoundindex. jt_gap_appendresultsoundindex+S (jt_i_appendresult)=(S j)) -> exists jt_b_appendresult jt_e_appendresult. ((((((exists fs_h_jt_appendresultsoundcode. fs_h_jt_appendresultsoundcode + S (jt_b_appendresult) = S ((S (jt_i_appendresult)) * V)) /\\ exists fs_q_jt_appendresultsoundcode. U = fs_q_jt_appendresultsoundcode * S ((S (jt_i_appendresult)) * V) + (jt_b_appendresult))) /\\ (((exists fs_h_jt_appendresultsoundscale. fs_h_jt_appendresultsoundscale + S (jt_e_appendresult) = S ((S (jt_i_appendresult)) * X)) /\\ exists fs_q_jt_appendresultsoundscale. W = fs_q_jt_appendresultsoundscale * S ((S (jt_i_appendresult)) * X) + (jt_e_appendresult))))) /\\ (((forall jt_index_appendresultbound. (exists jt_gap_appendresultboundindex. jt_gap_appendresultboundindex+S (jt_index_appendresultbound)=(k)) -> exists jt_value_appendresultbound. ((((exists fs_h_jt_appendresultboundat. fs_h_jt_appendresultboundat + S (jt_value_appendresultbound) = S ((S (jt_index_appendresultbound)) * jt_e_appendresult)) /\\ exists fs_q_jt_appendresultboundat. jt_b_appendresult = fs_q_jt_appendresultboundat * S ((S (jt_index_appendresultbound)) * jt_e_appendresult) + (jt_value_appendresultbound))) /\\ (exists jt_gap_appendresultboundvalue. jt_gap_appendresultboundvalue+S (jt_value_appendresultbound)=(n)))) /\\ (forall jt_divisor_appendresultprimitive. (exists jt_factor_appendresultprimitivemodulus. (n)=(jt_divisor_appendresultprimitive)*jt_factor_appendresultprimitivemodulus) -> (forall jt_index_appendresultprimitivecoordinates jt_value_appendresultprimitivecoordinates. (exists jt_gap_appendresultprimitivecoordinatesindex. jt_gap_appendresultprimitivecoordinatesindex+S (jt_index_appendresultprimitivecoordinates)=(k)) -> (((exists fs_h_jt_appendresultprimitivecoordinatesat. fs_h_jt_appendresultprimitivecoordinatesat + S (jt_value_appendresultprimitivecoordinates) = S ((S (jt_index_appendresultprimitivecoordinates)) * jt_e_appendresult)) /\\ exists fs_q_jt_appendresultprimitivecoordinatesat. jt_b_appendresult = fs_q_jt_appendresultprimitivecoordinatesat * S ((S (jt_index_appendresultprimitivecoordinates)) * jt_e_appendresult) + (jt_value_appendresultprimitivecoordinates))) -> (exists jt_factor_appendresultprimitivecoordinatesdivides. (jt_value_appendresultprimitivecoordinates)=(jt_divisor_appendresultprimitive)*jt_factor_appendresultprimitivecoordinatesdivides)) -> jt_divisor_appendresultprimitive=1))))) /\\ (((forall jt_i_appendresult jt_h_appendresult jt_b_appendresult jt_e_appendresult jt_d_appendresult jt_f_appendresult. (exists jt_gap_appendresultfirstindex. jt_gap_appendresultfirstindex+S (jt_i_appendresult)=(S j)) -> (exists jt_gap_appendresultsecondindex. jt_gap_appendresultsecondindex+S (jt_h_appendresult)=(S j)) -> (((((exists fs_h_jt_appendresultfirstcode. fs_h_jt_appendresultfirstcode + S (jt_b_appendresult) = S ((S (jt_i_appendresult)) * V)) /\\ exists fs_q_jt_appendresultfirstcode. U = fs_q_jt_appendresultfirstcode * S ((S (jt_i_appendresult)) * V) + (jt_b_appendresult))) /\\ (((exists fs_h_jt_appendresultfirstscale. fs_h_jt_appendresultfirstscale + S (jt_e_appendresult) = S ((S (jt_i_appendresult)) * X)) /\\ exists fs_q_jt_appendresultfirstscale. W = fs_q_jt_appendresultfirstscale * S ((S (jt_i_appendresult)) * X) + (jt_e_appendresult))))) -> (((((exists fs_h_jt_appendresultsecondcode. fs_h_jt_appendresultsecondcode + S (jt_d_appendresult) = S ((S (jt_h_appendresult)) * V)) /\\ exists fs_q_jt_appendresultsecondcode. U = fs_q_jt_appendresultsecondcode * S ((S (jt_h_appendresult)) * V) + (jt_d_appendresult))) /\\ (((exists fs_h_jt_appendresultsecondscale. fs_h_jt_appendresultsecondscale + S (jt_f_appendresult) = S ((S (jt_h_appendresult)) * X)) /\\ exists fs_q_jt_appendresultsecondscale. W = fs_q_jt_appendresultsecondscale * S ((S (jt_h_appendresult)) * X) + (jt_f_appendresult))))) -> (forall jt_index_appendresultsame jt_left_appendresultsame jt_right_appendresultsame. (exists jt_gap_appendresultsameindex. jt_gap_appendresultsameindex+S (jt_index_appendresultsame)=(k)) -> (((exists fs_h_jt_appendresultsameleft. fs_h_jt_appendresultsameleft + S (jt_left_appendresultsame) = S ((S (jt_index_appendresultsame)) * jt_e_appendresult)) /\\ exists fs_q_jt_appendresultsameleft. jt_b_appendresult = fs_q_jt_appendresultsameleft * S ((S (jt_index_appendresultsame)) * jt_e_appendresult) + (jt_left_appendresultsame))) -> (((exists fs_h_jt_appendresultsameright. fs_h_jt_appendresultsameright + S (jt_right_appendresultsame) = S ((S (jt_index_appendresultsame)) * jt_f_appendresult)) /\\ exists fs_q_jt_appendresultsameright. jt_d_appendresult = fs_q_jt_appendresultsameright * S ((S (jt_index_appendresultsame)) * jt_f_appendresult) + (jt_right_appendresultsame))) -> jt_left_appendresultsame=jt_right_appendresultsame) -> jt_i_appendresult=jt_h_appendresult) /\\ (forall jt_z_appendresult. (exists jt_gap_appendresultcodeindex. jt_gap_appendresultcodeindex+S (jt_z_appendresult)=(S t)) -> (forall jt_index_appendresultinputbound. (exists jt_gap_appendresultinputboundindex. jt_gap_appendresultinputboundindex+S (jt_index_appendresultinputbound)=(k)) -> exists jt_value_appendresultinputbound. ((((exists fs_h_jt_appendresultinputboundat. fs_h_jt_appendresultinputboundat + S (jt_value_appendresultinputbound) = S ((S (jt_index_appendresultinputbound)) * c)) /\\ exists fs_q_jt_appendresultinputboundat. jt_z_appendresult = fs_q_jt_appendresultinputboundat * S ((S (jt_index_appendresultinputbound)) * c) + (jt_value_appendresultinputbound))) /\\ (exists jt_gap_appendresultinputboundvalue. jt_gap_appendresultinputboundvalue+S (jt_value_appendresultinputbound)=(n)))) -> (forall jt_divisor_appendresultinputprimitive. (exists jt_factor_appendresultinputprimitivemodulus. (n)=(jt_divisor_appendresultinputprimitive)*jt_factor_appendresultinputprimitivemodulus) -> (forall jt_index_appendresultinputprimitivecoordinates jt_value_appendresultinputprimitivecoordinates. (exists jt_gap_appendresultinputprimitivecoordinatesindex. jt_gap_appendresultinputprimitivecoordinatesindex+S (jt_index_appendresultinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_appendresultinputprimitivecoordinatesat. fs_h_jt_appendresultinputprimitivecoordinatesat + S (jt_value_appendresultinputprimitivecoordinates) = S ((S (jt_index_appendresultinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_appendresultinputprimitivecoordinatesat. jt_z_appendresult = fs_q_jt_appendresultinputprimitivecoordinatesat * S ((S (jt_index_appendresultinputprimitivecoordinates)) * c) + (jt_value_appendresultinputprimitivecoordinates))) -> (exists jt_factor_appendresultinputprimitivecoordinatesdivides. (jt_value_appendresultinputprimitivecoordinates)=(jt_divisor_appendresultinputprimitive)*jt_factor_appendresultinputprimitivecoordinatesdivides)) -> jt_divisor_appendresultinputprimitive=1) -> (exists jt_index_appendresultlisted jt_code_appendresultlisted jt_scale_appendresultlisted. ((exists jt_gap_appendresultlistedindex. jt_gap_appendresultlistedindex+S (jt_index_appendresultlisted)=(S j)) /\\ (((((((exists fs_h_jt_appendresultlistedcode. fs_h_jt_appendresultlistedcode + S (jt_code_appendresultlisted) = S ((S (jt_index_appendresultlisted)) * V)) /\\ exists fs_q_jt_appendresultlistedcode. U = fs_q_jt_appendresultlistedcode * S ((S (jt_index_appendresultlisted)) * V) + (jt_code_appendresultlisted))) /\\ (((exists fs_h_jt_appendresultlistedscale. fs_h_jt_appendresultlistedscale + S (jt_scale_appendresultlisted) = S ((S (jt_index_appendresultlisted)) * X)) /\\ exists fs_q_jt_appendresultlistedscale. W = fs_q_jt_appendresultlistedscale * S ((S (jt_index_appendresultlisted)) * X) + (jt_scale_appendresultlisted))))) /\\ (forall jt_index_appendresultlistedequal jt_left_appendresultlistedequal jt_right_appendresultlistedequal. (exists jt_gap_appendresultlistedequalindex. jt_gap_appendresultlistedequalindex+S (jt_index_appendresultlistedequal)=(k)) -> (((exists fs_h_jt_appendresultlistedequalleft. fs_h_jt_appendresultlistedequalleft + S (jt_left_appendresultlistedequal) = S ((S (jt_index_appendresultlistedequal)) * c)) /\\ exists fs_q_jt_appendresultlistedequalleft. jt_z_appendresult = fs_q_jt_appendresultlistedequalleft * S ((S (jt_index_appendresultlistedequal)) * c) + (jt_left_appendresultlistedequal))) -> (((exists fs_h_jt_appendresultlistedequalright. fs_h_jt_appendresultlistedequalright + S (jt_right_appendresultlistedequal) = S ((S (jt_index_appendresultlistedequal)) * jt_scale_appendresultlisted)) /\\ exists fs_q_jt_appendresultlistedequalright. jt_code_appendresultlisted = fs_q_jt_appendresultlistedequalright * S ((S (jt_index_appendresultlistedequal)) * jt_scale_appendresultlisted) + (jt_right_appendresultlistedequal))) -> jt_left_appendresultlistedequal=jt_right_appendresultlistedequal)))))))))", "statement_sha256": "ac5d2d8b85e501803ff4d2130e971b02aa950449ffbdc6444027712912c72a0b", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Append an actually absent primitive tuple and prove full soundness, distinctness and prefix coverage." }, { "admission_dependencies": [ "jordan_tuple_scan_empty", "matrix_rank_bounded_prefix_decidable", "jordan_primitive_tuple_decidable", "jordan_tuple_listed_decidable", "jordan_tuple_scan_skip", "jordan_tuple_scan_append" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 38, "body_proof_nodes": 160, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro c", "intro hn", "induction t", "exists 0", "exists 0", "exists 0", "exists 0", "exists 0", "specialize jordan_tuple_scan_empty (k)", "specialize jordan_tuple_scan_empty (n)", "specialize jordan_tuple_scan_empty (c)", "specialize jordan_tuple_scan_empty (0)", "specialize jordan_tuple_scan_empty (0)", "specialize jordan_tuple_scan_empty (0)", "specialize jordan_tuple_scan_empty (0)", "apply jordan_tuple_scan_empty", "cases IH", "cases IH_witness", "cases IH_witness_witness", "cases IH_witness_witness_witness", "cases IH_witness_witness_witness_witness", "have hb : BetaPrefixInto(t,c,k,n) ∨ ¬BetaPrefixInto(t,c,k,n)", "specialize matrix_rank_bounded_prefix_decidable (t)", "specialize matrix_rank_bounded_prefix_decidable (c)", "specialize matrix_rank_bounded_prefix_decidable (k)", "specialize matrix_rank_bounded_prefix_decidable (n)", "apply matrix_rank_bounded_prefix_decidable", "cases hb", "have hp : JordanPrimitiveTuple(n,t,c,k) ∨ ¬JordanPrimitiveTuple(n,t,c,k)", "specialize jordan_primitive_tuple_decidable (n)", "specialize jordan_primitive_tuple_decidable (t)", "specialize jordan_primitive_tuple_decidable (c)", "specialize jordan_primitive_tuple_decidable (k)", "apply jordan_primitive_tuple_decidable", "exact hn", "cases hp", "have hl : JordanTupleListed(t,c,k,x,x1,x2,x3,x4) ∨ ¬JordanTupleListed(t,c,k,x,x1,x2,x3,x4)", "specialize jordan_tuple_listed_decidable (t)", "specialize jordan_tuple_listed_decidable (c)", "specialize jordan_tuple_listed_decidable (k)", "specialize jordan_tuple_listed_decidable (x)", "specialize jordan_tuple_listed_decidable (x1)", "specialize jordan_tuple_listed_decidable (x2)", "specialize jordan_tuple_listed_decidable (x3)", "specialize jordan_tuple_listed_decidable (x4)", "apply jordan_tuple_listed_decidable", "cases hl", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "specialize jordan_tuple_scan_skip (k)", "specialize jordan_tuple_scan_skip (n)", "specialize jordan_tuple_scan_skip (c)", "specialize jordan_tuple_scan_skip (t)", "specialize jordan_tuple_scan_skip (x)", "specialize jordan_tuple_scan_skip (x1)", "specialize jordan_tuple_scan_skip (x2)", "specialize jordan_tuple_scan_skip (x3)", "specialize jordan_tuple_scan_skip (x4)", "apply jordan_tuple_scan_skip", "exact IH_witness_witness_witness_witness_witness", "intro hbound", "intro hprimitive", "exact hl_left", "have hnew : ∃ U. ∃ V. ∃ W. ∃ X. JordanTupleScan(k,n,c,S t,U,V,W,X,S x4)", "specialize jordan_tuple_scan_append (k)", "specialize jordan_tuple_scan_append (n)", "specialize jordan_tuple_scan_append (c)", "specialize jordan_tuple_scan_append (t)", "specialize jordan_tuple_scan_append (x)", "specialize jordan_tuple_scan_append (x1)", "specialize jordan_tuple_scan_append (x2)", "specialize jordan_tuple_scan_append (x3)", "specialize jordan_tuple_scan_append (x4)", "apply jordan_tuple_scan_append", "exact IH_witness_witness_witness_witness_witness", "exact hb_left", "exact hp_left", "exact hl_right", "cases hnew", "cases hnew_witness", "cases hnew_witness_witness", "cases hnew_witness_witness_witness", "exists x5", "exists x6", "exists x7", "exists x8", "exists S x4", "exact hnew_witness_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "specialize jordan_tuple_scan_skip (k)", "specialize jordan_tuple_scan_skip (n)", "specialize jordan_tuple_scan_skip (c)", "specialize jordan_tuple_scan_skip (t)", "specialize jordan_tuple_scan_skip (x)", "specialize jordan_tuple_scan_skip (x1)", "specialize jordan_tuple_scan_skip (x2)", "specialize jordan_tuple_scan_skip (x3)", "specialize jordan_tuple_scan_skip (x4)", "apply jordan_tuple_scan_skip", "exact IH_witness_witness_witness_witness_witness", "intro hbound", "intro hprimitive", "exfalso", "apply hp_right", "exact hprimitive", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "specialize jordan_tuple_scan_skip (k)", "specialize jordan_tuple_scan_skip (n)", "specialize jordan_tuple_scan_skip (c)", "specialize jordan_tuple_scan_skip (t)", "specialize jordan_tuple_scan_skip (x)", "specialize jordan_tuple_scan_skip (x1)", "specialize jordan_tuple_scan_skip (x2)", "specialize jordan_tuple_scan_skip (x3)", "specialize jordan_tuple_scan_skip (x4)", "apply jordan_tuple_scan_skip", "exact IH_witness_witness_witness_witness_witness", "intro hbound", "intro hprimitive", "exfalso", "apply hb_right", "exact hbound" ], "defined_statement": "∀ k. ∀ n. ∀ c. ¬n = 0 → ∀ x. ∃ y. ∃ z. ∃ m. ∃ i. ∃ j. JordanTupleScan(k,n,c,x,y,z,m,i,j)", "defined_statement_sha256": "87d08e5f60e9e59e70e74d2641ae8a799ae37a039973637bf026ef187cd9bde7", "definition_uses": { "ND0262": 2, "ND0372": 2, "ND0376": 2, "ND0377": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "79137d417866db17e71e8fa85e60555919383d0873a6df6640c84cd8f4e93bf6", "free_names": [], "script_definition_uses": { "ND0262": 2, "ND0372": 2, "ND0376": 2, "ND0377": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "induction t" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (0)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (0)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (0)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_empty (0)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_empty" } ], [ { "kind": "text", "text": "cases IH" } ], [ { "kind": "text", "text": "cases IH_witness" } ], [ { "kind": "text", "text": "cases IH_witness_witness" } ], [ { "kind": "text", "text": "cases IH_witness_witness_witness" } ], [ { "kind": "text", "text": "cases IH_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "have hb : " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(t,c,k,n)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(t,c,k,n)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_decidable (t)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_decidable (c)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_decidable (k)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_decidable (n)" } ], [ { "kind": "text", "text": "apply matrix_rank_bounded_prefix_decidable" } ], [ { "kind": "text", "text": "cases hb" } ], [ { "kind": "text", "text": "have hp : " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,t,c,k)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,t,c,k)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_decidable (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_decidable (t)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_decidable (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_decidable (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_decidable" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "cases hp" } ], [ { "kind": "text", "text": "have hl : " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(t,c,k,x,x1,x2,x3,x4)" }, { "kind": "text", "text": " ∨ ¬" }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(t,c,k,x,x1,x2,x3,x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_listed_decidable (x4)" } ], [ { "kind": "text", "text": "apply jordan_tuple_listed_decidable" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x4)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_skip" } ], [ { "kind": "text", "text": "exact IH_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "intro hbound" } ], [ { "kind": "text", "text": "intro hprimitive" } ], [ { "kind": "text", "text": "exact hl_left" } ], [ { "kind": "text", "text": "have hnew : " }, { "kind": "text", "text": "∃ U. ∃ V. ∃ W. ∃ X. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,S t,U,V,W,X,S x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_append (x4)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_append" } ], [ { "kind": "text", "text": "exact IH_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "exact hb_left" } ], [ { "kind": "text", "text": "exact hp_left" } ], [ { "kind": "text", "text": "exact hl_right" } ], [ { "kind": "text", "text": "cases hnew" } ], [ { "kind": "text", "text": "cases hnew_witness" } ], [ { "kind": "text", "text": "cases hnew_witness_witness" } ], [ { "kind": "text", "text": "cases hnew_witness_witness_witness" } ], [ { "kind": "text", "text": "exists x5" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "exists x7" } ], [ { "kind": "text", "text": "exists x8" } ], [ { "kind": "text", "text": "exists S x4" } ], [ { "kind": "text", "text": "exact hnew_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x4)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_skip" } ], [ { "kind": "text", "text": "exact IH_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "intro hbound" } ], [ { "kind": "text", "text": "intro hprimitive" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hp_right" } ], [ { "kind": "text", "text": "exact hprimitive" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (t)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_skip (x4)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_skip" } ], [ { "kind": "text", "text": "exact IH_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "intro hbound" } ], [ { "kind": "text", "text": "intro hprimitive" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hb_right" } ], [ { "kind": "text", "text": "exact hbound" } ] ], "statement_definition_uses": { "ND0377": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ c. ¬n = 0 → ∀ x. ∃ y. ∃ z. ∃ m. ∃ i. ∃ j. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,c,x,y,z,m,i,j)" } ] }, "dependencies": [ "jordan_tuple_scan_empty", "matrix_rank_bounded_prefix_decidable", "jordan_primitive_tuple_decidable", "jordan_tuple_listed_decidable", "jordan_tuple_scan_skip", "jordan_tuple_scan_append" ], "enrolled_in_alpha": true, "factory": "make_jordan_scan_candidate_theorems", "first_admitted_version": "v35", "id": "JT0027", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_scan_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 296, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro c", "intro hn", "induction t", "exists 0", "exists 0", "exists 0", "exists 0", "exists 0", "specialize jordan_tuple_scan_empty (k)", "specialize jordan_tuple_scan_empty (n)", "specialize jordan_tuple_scan_empty (c)", "specialize jordan_tuple_scan_empty (0)", "specialize jordan_tuple_scan_empty (0)", "specialize jordan_tuple_scan_empty (0)", "specialize jordan_tuple_scan_empty (0)", "apply jordan_tuple_scan_empty", "cases IH", "cases IH_witness", "cases IH_witness_witness", "cases IH_witness_witness_witness", "cases IH_witness_witness_witness_witness", "have hb : (forall jt_index_scantotalyes. (exists jt_gap_scantotalyesindex. jt_gap_scantotalyesindex+S (jt_index_scantotalyes)=(k)) -> exists jt_value_scantotalyes. ((((exists fs_h_jt_scantotalyesat. fs_h_jt_scantotalyesat + S (jt_value_scantotalyes) = S ((S (jt_index_scantotalyes)) * c)) /\\ exists fs_q_jt_scantotalyesat. t = fs_q_jt_scantotalyesat * S ((S (jt_index_scantotalyes)) * c) + (jt_value_scantotalyes))) /\\ (exists jt_gap_scantotalyesvalue. jt_gap_scantotalyesvalue+S (jt_value_scantotalyes)=(n)))) \\/ ~(forall jt_index_scantotalno. (exists jt_gap_scantotalnoindex. jt_gap_scantotalnoindex+S (jt_index_scantotalno)=(k)) -> exists jt_value_scantotalno. ((((exists fs_h_jt_scantotalnoat. fs_h_jt_scantotalnoat + S (jt_value_scantotalno) = S ((S (jt_index_scantotalno)) * c)) /\\ exists fs_q_jt_scantotalnoat. t = fs_q_jt_scantotalnoat * S ((S (jt_index_scantotalno)) * c) + (jt_value_scantotalno))) /\\ (exists jt_gap_scantotalnovalue. jt_gap_scantotalnovalue+S (jt_value_scantotalno)=(n))))", "specialize matrix_rank_bounded_prefix_decidable (t)", "specialize matrix_rank_bounded_prefix_decidable (c)", "specialize matrix_rank_bounded_prefix_decidable (k)", "specialize matrix_rank_bounded_prefix_decidable (n)", "apply matrix_rank_bounded_prefix_decidable", "cases hb", "have hp : (forall jt_divisor_scanprimyes. (exists jt_factor_scanprimyesmodulus. (n)=(jt_divisor_scanprimyes)*jt_factor_scanprimyesmodulus) -> (forall jt_index_scanprimyescoordinates jt_value_scanprimyescoordinates. (exists jt_gap_scanprimyescoordinatesindex. jt_gap_scanprimyescoordinatesindex+S (jt_index_scanprimyescoordinates)=(k)) -> (((exists fs_h_jt_scanprimyescoordinatesat. fs_h_jt_scanprimyescoordinatesat + S (jt_value_scanprimyescoordinates) = S ((S (jt_index_scanprimyescoordinates)) * c)) /\\ exists fs_q_jt_scanprimyescoordinatesat. t = fs_q_jt_scanprimyescoordinatesat * S ((S (jt_index_scanprimyescoordinates)) * c) + (jt_value_scanprimyescoordinates))) -> (exists jt_factor_scanprimyescoordinatesdivides. (jt_value_scanprimyescoordinates)=(jt_divisor_scanprimyes)*jt_factor_scanprimyescoordinatesdivides)) -> jt_divisor_scanprimyes=1) \\/ ~(forall jt_divisor_scanprimno. (exists jt_factor_scanprimnomodulus. (n)=(jt_divisor_scanprimno)*jt_factor_scanprimnomodulus) -> (forall jt_index_scanprimnocoordinates jt_value_scanprimnocoordinates. (exists jt_gap_scanprimnocoordinatesindex. jt_gap_scanprimnocoordinatesindex+S (jt_index_scanprimnocoordinates)=(k)) -> (((exists fs_h_jt_scanprimnocoordinatesat. fs_h_jt_scanprimnocoordinatesat + S (jt_value_scanprimnocoordinates) = S ((S (jt_index_scanprimnocoordinates)) * c)) /\\ exists fs_q_jt_scanprimnocoordinatesat. t = fs_q_jt_scanprimnocoordinatesat * S ((S (jt_index_scanprimnocoordinates)) * c) + (jt_value_scanprimnocoordinates))) -> (exists jt_factor_scanprimnocoordinatesdivides. (jt_value_scanprimnocoordinates)=(jt_divisor_scanprimno)*jt_factor_scanprimnocoordinatesdivides)) -> jt_divisor_scanprimno=1)", "specialize jordan_primitive_tuple_decidable (n)", "specialize jordan_primitive_tuple_decidable (t)", "specialize jordan_primitive_tuple_decidable (c)", "specialize jordan_primitive_tuple_decidable (k)", "apply jordan_primitive_tuple_decidable", "exact hn", "cases hp", "have hl : (exists jt_index_scanlistedyes jt_code_scanlistedyes jt_scale_scanlistedyes. ((exists jt_gap_scanlistedyesindex. jt_gap_scanlistedyesindex+S (jt_index_scanlistedyes)=(x4)) /\\ (((((((exists fs_h_jt_scanlistedyescode. fs_h_jt_scanlistedyescode + S (jt_code_scanlistedyes) = S ((S (jt_index_scanlistedyes)) * x1)) /\\ exists fs_q_jt_scanlistedyescode. x = fs_q_jt_scanlistedyescode * S ((S (jt_index_scanlistedyes)) * x1) + (jt_code_scanlistedyes))) /\\ (((exists fs_h_jt_scanlistedyesscale. fs_h_jt_scanlistedyesscale + S (jt_scale_scanlistedyes) = S ((S (jt_index_scanlistedyes)) * x3)) /\\ exists fs_q_jt_scanlistedyesscale. x2 = fs_q_jt_scanlistedyesscale * S ((S (jt_index_scanlistedyes)) * x3) + (jt_scale_scanlistedyes))))) /\\ (forall jt_index_scanlistedyesequal jt_left_scanlistedyesequal jt_right_scanlistedyesequal. (exists jt_gap_scanlistedyesequalindex. jt_gap_scanlistedyesequalindex+S (jt_index_scanlistedyesequal)=(k)) -> (((exists fs_h_jt_scanlistedyesequalleft. fs_h_jt_scanlistedyesequalleft + S (jt_left_scanlistedyesequal) = S ((S (jt_index_scanlistedyesequal)) * c)) /\\ exists fs_q_jt_scanlistedyesequalleft. t = fs_q_jt_scanlistedyesequalleft * S ((S (jt_index_scanlistedyesequal)) * c) + (jt_left_scanlistedyesequal))) -> (((exists fs_h_jt_scanlistedyesequalright. fs_h_jt_scanlistedyesequalright + S (jt_right_scanlistedyesequal) = S ((S (jt_index_scanlistedyesequal)) * jt_scale_scanlistedyes)) /\\ exists fs_q_jt_scanlistedyesequalright. jt_code_scanlistedyes = fs_q_jt_scanlistedyesequalright * S ((S (jt_index_scanlistedyesequal)) * jt_scale_scanlistedyes) + (jt_right_scanlistedyesequal))) -> jt_left_scanlistedyesequal=jt_right_scanlistedyesequal))))) \\/ ~(exists jt_index_scanlistedno jt_code_scanlistedno jt_scale_scanlistedno. ((exists jt_gap_scanlistednoindex. jt_gap_scanlistednoindex+S (jt_index_scanlistedno)=(x4)) /\\ (((((((exists fs_h_jt_scanlistednocode. fs_h_jt_scanlistednocode + S (jt_code_scanlistedno) = S ((S (jt_index_scanlistedno)) * x1)) /\\ exists fs_q_jt_scanlistednocode. x = fs_q_jt_scanlistednocode * S ((S (jt_index_scanlistedno)) * x1) + (jt_code_scanlistedno))) /\\ (((exists fs_h_jt_scanlistednoscale. fs_h_jt_scanlistednoscale + S (jt_scale_scanlistedno) = S ((S (jt_index_scanlistedno)) * x3)) /\\ exists fs_q_jt_scanlistednoscale. x2 = fs_q_jt_scanlistednoscale * S ((S (jt_index_scanlistedno)) * x3) + (jt_scale_scanlistedno))))) /\\ (forall jt_index_scanlistednoequal jt_left_scanlistednoequal jt_right_scanlistednoequal. (exists jt_gap_scanlistednoequalindex. jt_gap_scanlistednoequalindex+S (jt_index_scanlistednoequal)=(k)) -> (((exists fs_h_jt_scanlistednoequalleft. fs_h_jt_scanlistednoequalleft + S (jt_left_scanlistednoequal) = S ((S (jt_index_scanlistednoequal)) * c)) /\\ exists fs_q_jt_scanlistednoequalleft. t = fs_q_jt_scanlistednoequalleft * S ((S (jt_index_scanlistednoequal)) * c) + (jt_left_scanlistednoequal))) -> (((exists fs_h_jt_scanlistednoequalright. fs_h_jt_scanlistednoequalright + S (jt_right_scanlistednoequal) = S ((S (jt_index_scanlistednoequal)) * jt_scale_scanlistedno)) /\\ exists fs_q_jt_scanlistednoequalright. jt_code_scanlistedno = fs_q_jt_scanlistednoequalright * S ((S (jt_index_scanlistednoequal)) * jt_scale_scanlistedno) + (jt_right_scanlistednoequal))) -> jt_left_scanlistednoequal=jt_right_scanlistednoequal)))))", "specialize jordan_tuple_listed_decidable (t)", "specialize jordan_tuple_listed_decidable (c)", "specialize jordan_tuple_listed_decidable (k)", "specialize jordan_tuple_listed_decidable (x)", "specialize jordan_tuple_listed_decidable (x1)", "specialize jordan_tuple_listed_decidable (x2)", "specialize jordan_tuple_listed_decidable (x3)", "specialize jordan_tuple_listed_decidable (x4)", "apply jordan_tuple_listed_decidable", "cases hl", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "specialize jordan_tuple_scan_skip (k)", "specialize jordan_tuple_scan_skip (n)", "specialize jordan_tuple_scan_skip (c)", "specialize jordan_tuple_scan_skip (t)", "specialize jordan_tuple_scan_skip (x)", "specialize jordan_tuple_scan_skip (x1)", "specialize jordan_tuple_scan_skip (x2)", "specialize jordan_tuple_scan_skip (x3)", "specialize jordan_tuple_scan_skip (x4)", "apply jordan_tuple_scan_skip", "exact IH_witness_witness_witness_witness_witness", "intro hbound", "intro hprimitive", "exact hl_left", "have hnew : exists U V W X. ((forall jt_i_scantotalnew. (exists jt_gap_scantotalnewsoundindex. jt_gap_scantotalnewsoundindex+S (jt_i_scantotalnew)=(S x4)) -> exists jt_b_scantotalnew jt_e_scantotalnew. ((((((exists fs_h_jt_scantotalnewsoundcode. fs_h_jt_scantotalnewsoundcode + S (jt_b_scantotalnew) = S ((S (jt_i_scantotalnew)) * V)) /\\ exists fs_q_jt_scantotalnewsoundcode. U = fs_q_jt_scantotalnewsoundcode * S ((S (jt_i_scantotalnew)) * V) + (jt_b_scantotalnew))) /\\ (((exists fs_h_jt_scantotalnewsoundscale. fs_h_jt_scantotalnewsoundscale + S (jt_e_scantotalnew) = S ((S (jt_i_scantotalnew)) * X)) /\\ exists fs_q_jt_scantotalnewsoundscale. W = fs_q_jt_scantotalnewsoundscale * S ((S (jt_i_scantotalnew)) * X) + (jt_e_scantotalnew))))) /\\ (((forall jt_index_scantotalnewbound. (exists jt_gap_scantotalnewboundindex. jt_gap_scantotalnewboundindex+S (jt_index_scantotalnewbound)=(k)) -> exists jt_value_scantotalnewbound. ((((exists fs_h_jt_scantotalnewboundat. fs_h_jt_scantotalnewboundat + S (jt_value_scantotalnewbound) = S ((S (jt_index_scantotalnewbound)) * jt_e_scantotalnew)) /\\ exists fs_q_jt_scantotalnewboundat. jt_b_scantotalnew = fs_q_jt_scantotalnewboundat * S ((S (jt_index_scantotalnewbound)) * jt_e_scantotalnew) + (jt_value_scantotalnewbound))) /\\ (exists jt_gap_scantotalnewboundvalue. jt_gap_scantotalnewboundvalue+S (jt_value_scantotalnewbound)=(n)))) /\\ (forall jt_divisor_scantotalnewprimitive. (exists jt_factor_scantotalnewprimitivemodulus. (n)=(jt_divisor_scantotalnewprimitive)*jt_factor_scantotalnewprimitivemodulus) -> (forall jt_index_scantotalnewprimitivecoordinates jt_value_scantotalnewprimitivecoordinates. (exists jt_gap_scantotalnewprimitivecoordinatesindex. jt_gap_scantotalnewprimitivecoordinatesindex+S (jt_index_scantotalnewprimitivecoordinates)=(k)) -> (((exists fs_h_jt_scantotalnewprimitivecoordinatesat. fs_h_jt_scantotalnewprimitivecoordinatesat + S (jt_value_scantotalnewprimitivecoordinates) = S ((S (jt_index_scantotalnewprimitivecoordinates)) * jt_e_scantotalnew)) /\\ exists fs_q_jt_scantotalnewprimitivecoordinatesat. jt_b_scantotalnew = fs_q_jt_scantotalnewprimitivecoordinatesat * S ((S (jt_index_scantotalnewprimitivecoordinates)) * jt_e_scantotalnew) + (jt_value_scantotalnewprimitivecoordinates))) -> (exists jt_factor_scantotalnewprimitivecoordinatesdivides. (jt_value_scantotalnewprimitivecoordinates)=(jt_divisor_scantotalnewprimitive)*jt_factor_scantotalnewprimitivecoordinatesdivides)) -> jt_divisor_scantotalnewprimitive=1))))) /\\ (((forall jt_i_scantotalnew jt_h_scantotalnew jt_b_scantotalnew jt_e_scantotalnew jt_d_scantotalnew jt_f_scantotalnew. (exists jt_gap_scantotalnewfirstindex. jt_gap_scantotalnewfirstindex+S (jt_i_scantotalnew)=(S x4)) -> (exists jt_gap_scantotalnewsecondindex. jt_gap_scantotalnewsecondindex+S (jt_h_scantotalnew)=(S x4)) -> (((((exists fs_h_jt_scantotalnewfirstcode. fs_h_jt_scantotalnewfirstcode + S (jt_b_scantotalnew) = S ((S (jt_i_scantotalnew)) * V)) /\\ exists fs_q_jt_scantotalnewfirstcode. U = fs_q_jt_scantotalnewfirstcode * S ((S (jt_i_scantotalnew)) * V) + (jt_b_scantotalnew))) /\\ (((exists fs_h_jt_scantotalnewfirstscale. fs_h_jt_scantotalnewfirstscale + S (jt_e_scantotalnew) = S ((S (jt_i_scantotalnew)) * X)) /\\ exists fs_q_jt_scantotalnewfirstscale. W = fs_q_jt_scantotalnewfirstscale * S ((S (jt_i_scantotalnew)) * X) + (jt_e_scantotalnew))))) -> (((((exists fs_h_jt_scantotalnewsecondcode. fs_h_jt_scantotalnewsecondcode + S (jt_d_scantotalnew) = S ((S (jt_h_scantotalnew)) * V)) /\\ exists fs_q_jt_scantotalnewsecondcode. U = fs_q_jt_scantotalnewsecondcode * S ((S (jt_h_scantotalnew)) * V) + (jt_d_scantotalnew))) /\\ (((exists fs_h_jt_scantotalnewsecondscale. fs_h_jt_scantotalnewsecondscale + S (jt_f_scantotalnew) = S ((S (jt_h_scantotalnew)) * X)) /\\ exists fs_q_jt_scantotalnewsecondscale. W = fs_q_jt_scantotalnewsecondscale * S ((S (jt_h_scantotalnew)) * X) + (jt_f_scantotalnew))))) -> (forall jt_index_scantotalnewsame jt_left_scantotalnewsame jt_right_scantotalnewsame. (exists jt_gap_scantotalnewsameindex. jt_gap_scantotalnewsameindex+S (jt_index_scantotalnewsame)=(k)) -> (((exists fs_h_jt_scantotalnewsameleft. fs_h_jt_scantotalnewsameleft + S (jt_left_scantotalnewsame) = S ((S (jt_index_scantotalnewsame)) * jt_e_scantotalnew)) /\\ exists fs_q_jt_scantotalnewsameleft. jt_b_scantotalnew = fs_q_jt_scantotalnewsameleft * S ((S (jt_index_scantotalnewsame)) * jt_e_scantotalnew) + (jt_left_scantotalnewsame))) -> (((exists fs_h_jt_scantotalnewsameright. fs_h_jt_scantotalnewsameright + S (jt_right_scantotalnewsame) = S ((S (jt_index_scantotalnewsame)) * jt_f_scantotalnew)) /\\ exists fs_q_jt_scantotalnewsameright. jt_d_scantotalnew = fs_q_jt_scantotalnewsameright * S ((S (jt_index_scantotalnewsame)) * jt_f_scantotalnew) + (jt_right_scantotalnewsame))) -> jt_left_scantotalnewsame=jt_right_scantotalnewsame) -> jt_i_scantotalnew=jt_h_scantotalnew) /\\ (forall jt_z_scantotalnew. (exists jt_gap_scantotalnewcodeindex. jt_gap_scantotalnewcodeindex+S (jt_z_scantotalnew)=(S t)) -> (forall jt_index_scantotalnewinputbound. (exists jt_gap_scantotalnewinputboundindex. jt_gap_scantotalnewinputboundindex+S (jt_index_scantotalnewinputbound)=(k)) -> exists jt_value_scantotalnewinputbound. ((((exists fs_h_jt_scantotalnewinputboundat. fs_h_jt_scantotalnewinputboundat + S (jt_value_scantotalnewinputbound) = S ((S (jt_index_scantotalnewinputbound)) * c)) /\\ exists fs_q_jt_scantotalnewinputboundat. jt_z_scantotalnew = fs_q_jt_scantotalnewinputboundat * S ((S (jt_index_scantotalnewinputbound)) * c) + (jt_value_scantotalnewinputbound))) /\\ (exists jt_gap_scantotalnewinputboundvalue. jt_gap_scantotalnewinputboundvalue+S (jt_value_scantotalnewinputbound)=(n)))) -> (forall jt_divisor_scantotalnewinputprimitive. (exists jt_factor_scantotalnewinputprimitivemodulus. (n)=(jt_divisor_scantotalnewinputprimitive)*jt_factor_scantotalnewinputprimitivemodulus) -> (forall jt_index_scantotalnewinputprimitivecoordinates jt_value_scantotalnewinputprimitivecoordinates. (exists jt_gap_scantotalnewinputprimitivecoordinatesindex. jt_gap_scantotalnewinputprimitivecoordinatesindex+S (jt_index_scantotalnewinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_scantotalnewinputprimitivecoordinatesat. fs_h_jt_scantotalnewinputprimitivecoordinatesat + S (jt_value_scantotalnewinputprimitivecoordinates) = S ((S (jt_index_scantotalnewinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_scantotalnewinputprimitivecoordinatesat. jt_z_scantotalnew = fs_q_jt_scantotalnewinputprimitivecoordinatesat * S ((S (jt_index_scantotalnewinputprimitivecoordinates)) * c) + (jt_value_scantotalnewinputprimitivecoordinates))) -> (exists jt_factor_scantotalnewinputprimitivecoordinatesdivides. (jt_value_scantotalnewinputprimitivecoordinates)=(jt_divisor_scantotalnewinputprimitive)*jt_factor_scantotalnewinputprimitivecoordinatesdivides)) -> jt_divisor_scantotalnewinputprimitive=1) -> (exists jt_index_scantotalnewlisted jt_code_scantotalnewlisted jt_scale_scantotalnewlisted. ((exists jt_gap_scantotalnewlistedindex. jt_gap_scantotalnewlistedindex+S (jt_index_scantotalnewlisted)=(S x4)) /\\ (((((((exists fs_h_jt_scantotalnewlistedcode. fs_h_jt_scantotalnewlistedcode + S (jt_code_scantotalnewlisted) = S ((S (jt_index_scantotalnewlisted)) * V)) /\\ exists fs_q_jt_scantotalnewlistedcode. U = fs_q_jt_scantotalnewlistedcode * S ((S (jt_index_scantotalnewlisted)) * V) + (jt_code_scantotalnewlisted))) /\\ (((exists fs_h_jt_scantotalnewlistedscale. fs_h_jt_scantotalnewlistedscale + S (jt_scale_scantotalnewlisted) = S ((S (jt_index_scantotalnewlisted)) * X)) /\\ exists fs_q_jt_scantotalnewlistedscale. W = fs_q_jt_scantotalnewlistedscale * S ((S (jt_index_scantotalnewlisted)) * X) + (jt_scale_scantotalnewlisted))))) /\\ (forall jt_index_scantotalnewlistedequal jt_left_scantotalnewlistedequal jt_right_scantotalnewlistedequal. (exists jt_gap_scantotalnewlistedequalindex. jt_gap_scantotalnewlistedequalindex+S (jt_index_scantotalnewlistedequal)=(k)) -> (((exists fs_h_jt_scantotalnewlistedequalleft. fs_h_jt_scantotalnewlistedequalleft + S (jt_left_scantotalnewlistedequal) = S ((S (jt_index_scantotalnewlistedequal)) * c)) /\\ exists fs_q_jt_scantotalnewlistedequalleft. jt_z_scantotalnew = fs_q_jt_scantotalnewlistedequalleft * S ((S (jt_index_scantotalnewlistedequal)) * c) + (jt_left_scantotalnewlistedequal))) -> (((exists fs_h_jt_scantotalnewlistedequalright. fs_h_jt_scantotalnewlistedequalright + S (jt_right_scantotalnewlistedequal) = S ((S (jt_index_scantotalnewlistedequal)) * jt_scale_scantotalnewlisted)) /\\ exists fs_q_jt_scantotalnewlistedequalright. jt_code_scantotalnewlisted = fs_q_jt_scantotalnewlistedequalright * S ((S (jt_index_scantotalnewlistedequal)) * jt_scale_scantotalnewlisted) + (jt_right_scantotalnewlistedequal))) -> jt_left_scantotalnewlistedequal=jt_right_scantotalnewlistedequal)))))))))", "specialize jordan_tuple_scan_append (k)", "specialize jordan_tuple_scan_append (n)", "specialize jordan_tuple_scan_append (c)", "specialize jordan_tuple_scan_append (t)", "specialize jordan_tuple_scan_append (x)", "specialize jordan_tuple_scan_append (x1)", "specialize jordan_tuple_scan_append (x2)", "specialize jordan_tuple_scan_append (x3)", "specialize jordan_tuple_scan_append (x4)", "apply jordan_tuple_scan_append", "exact IH_witness_witness_witness_witness_witness", "exact hb_left", "exact hp_left", "exact hl_right", "cases hnew", "cases hnew_witness", "cases hnew_witness_witness", "cases hnew_witness_witness_witness", "exists x5", "exists x6", "exists x7", "exists x8", "exists S x4", "exact hnew_witness_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "specialize jordan_tuple_scan_skip (k)", "specialize jordan_tuple_scan_skip (n)", "specialize jordan_tuple_scan_skip (c)", "specialize jordan_tuple_scan_skip (t)", "specialize jordan_tuple_scan_skip (x)", "specialize jordan_tuple_scan_skip (x1)", "specialize jordan_tuple_scan_skip (x2)", "specialize jordan_tuple_scan_skip (x3)", "specialize jordan_tuple_scan_skip (x4)", "apply jordan_tuple_scan_skip", "exact IH_witness_witness_witness_witness_witness", "intro hbound", "intro hprimitive", "exfalso", "apply hp_right", "exact hprimitive", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "specialize jordan_tuple_scan_skip (k)", "specialize jordan_tuple_scan_skip (n)", "specialize jordan_tuple_scan_skip (c)", "specialize jordan_tuple_scan_skip (t)", "specialize jordan_tuple_scan_skip (x)", "specialize jordan_tuple_scan_skip (x1)", "specialize jordan_tuple_scan_skip (x2)", "specialize jordan_tuple_scan_skip (x3)", "specialize jordan_tuple_scan_skip (x4)", "apply jordan_tuple_scan_skip", "exact IH_witness_witness_witness_witness_witness", "intro hbound", "intro hprimitive", "exfalso", "apply hb_right", "exact hbound" ], "script_sha256": "d85897a74b92bacd7a22eba63de1e10a82179be0fd040895c0e6ab6885542c23", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_scan_candidate_theorems", "script_sha256": "d85897a74b92bacd7a22eba63de1e10a82179be0fd040895c0e6ab6885542c23", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "79137d417866db17e71e8fa85e60555919383d0873a6df6640c84cd8f4e93bf6" } ], "stable_member": false, "statement": "forall k n c. ~(n=0) -> forall t. exists B C D E j. ((forall jt_i_scanexists. (exists jt_gap_scanexistssoundindex. jt_gap_scanexistssoundindex+S (jt_i_scanexists)=(j)) -> exists jt_b_scanexists jt_e_scanexists. ((((((exists fs_h_jt_scanexistssoundcode. fs_h_jt_scanexistssoundcode + S (jt_b_scanexists) = S ((S (jt_i_scanexists)) * C)) /\\ exists fs_q_jt_scanexistssoundcode. B = fs_q_jt_scanexistssoundcode * S ((S (jt_i_scanexists)) * C) + (jt_b_scanexists))) /\\ (((exists fs_h_jt_scanexistssoundscale. fs_h_jt_scanexistssoundscale + S (jt_e_scanexists) = S ((S (jt_i_scanexists)) * E)) /\\ exists fs_q_jt_scanexistssoundscale. D = fs_q_jt_scanexistssoundscale * S ((S (jt_i_scanexists)) * E) + (jt_e_scanexists))))) /\\ (((forall jt_index_scanexistsbound. (exists jt_gap_scanexistsboundindex. jt_gap_scanexistsboundindex+S (jt_index_scanexistsbound)=(k)) -> exists jt_value_scanexistsbound. ((((exists fs_h_jt_scanexistsboundat. fs_h_jt_scanexistsboundat + S (jt_value_scanexistsbound) = S ((S (jt_index_scanexistsbound)) * jt_e_scanexists)) /\\ exists fs_q_jt_scanexistsboundat. jt_b_scanexists = fs_q_jt_scanexistsboundat * S ((S (jt_index_scanexistsbound)) * jt_e_scanexists) + (jt_value_scanexistsbound))) /\\ (exists jt_gap_scanexistsboundvalue. jt_gap_scanexistsboundvalue+S (jt_value_scanexistsbound)=(n)))) /\\ (forall jt_divisor_scanexistsprimitive. (exists jt_factor_scanexistsprimitivemodulus. (n)=(jt_divisor_scanexistsprimitive)*jt_factor_scanexistsprimitivemodulus) -> (forall jt_index_scanexistsprimitivecoordinates jt_value_scanexistsprimitivecoordinates. (exists jt_gap_scanexistsprimitivecoordinatesindex. jt_gap_scanexistsprimitivecoordinatesindex+S (jt_index_scanexistsprimitivecoordinates)=(k)) -> (((exists fs_h_jt_scanexistsprimitivecoordinatesat. fs_h_jt_scanexistsprimitivecoordinatesat + S (jt_value_scanexistsprimitivecoordinates) = S ((S (jt_index_scanexistsprimitivecoordinates)) * jt_e_scanexists)) /\\ exists fs_q_jt_scanexistsprimitivecoordinatesat. jt_b_scanexists = fs_q_jt_scanexistsprimitivecoordinatesat * S ((S (jt_index_scanexistsprimitivecoordinates)) * jt_e_scanexists) + (jt_value_scanexistsprimitivecoordinates))) -> (exists jt_factor_scanexistsprimitivecoordinatesdivides. (jt_value_scanexistsprimitivecoordinates)=(jt_divisor_scanexistsprimitive)*jt_factor_scanexistsprimitivecoordinatesdivides)) -> jt_divisor_scanexistsprimitive=1))))) /\\ (((forall jt_i_scanexists jt_h_scanexists jt_b_scanexists jt_e_scanexists jt_d_scanexists jt_f_scanexists. (exists jt_gap_scanexistsfirstindex. jt_gap_scanexistsfirstindex+S (jt_i_scanexists)=(j)) -> (exists jt_gap_scanexistssecondindex. jt_gap_scanexistssecondindex+S (jt_h_scanexists)=(j)) -> (((((exists fs_h_jt_scanexistsfirstcode. fs_h_jt_scanexistsfirstcode + S (jt_b_scanexists) = S ((S (jt_i_scanexists)) * C)) /\\ exists fs_q_jt_scanexistsfirstcode. B = fs_q_jt_scanexistsfirstcode * S ((S (jt_i_scanexists)) * C) + (jt_b_scanexists))) /\\ (((exists fs_h_jt_scanexistsfirstscale. fs_h_jt_scanexistsfirstscale + S (jt_e_scanexists) = S ((S (jt_i_scanexists)) * E)) /\\ exists fs_q_jt_scanexistsfirstscale. D = fs_q_jt_scanexistsfirstscale * S ((S (jt_i_scanexists)) * E) + (jt_e_scanexists))))) -> (((((exists fs_h_jt_scanexistssecondcode. fs_h_jt_scanexistssecondcode + S (jt_d_scanexists) = S ((S (jt_h_scanexists)) * C)) /\\ exists fs_q_jt_scanexistssecondcode. B = fs_q_jt_scanexistssecondcode * S ((S (jt_h_scanexists)) * C) + (jt_d_scanexists))) /\\ (((exists fs_h_jt_scanexistssecondscale. fs_h_jt_scanexistssecondscale + S (jt_f_scanexists) = S ((S (jt_h_scanexists)) * E)) /\\ exists fs_q_jt_scanexistssecondscale. D = fs_q_jt_scanexistssecondscale * S ((S (jt_h_scanexists)) * E) + (jt_f_scanexists))))) -> (forall jt_index_scanexistssame jt_left_scanexistssame jt_right_scanexistssame. (exists jt_gap_scanexistssameindex. jt_gap_scanexistssameindex+S (jt_index_scanexistssame)=(k)) -> (((exists fs_h_jt_scanexistssameleft. fs_h_jt_scanexistssameleft + S (jt_left_scanexistssame) = S ((S (jt_index_scanexistssame)) * jt_e_scanexists)) /\\ exists fs_q_jt_scanexistssameleft. jt_b_scanexists = fs_q_jt_scanexistssameleft * S ((S (jt_index_scanexistssame)) * jt_e_scanexists) + (jt_left_scanexistssame))) -> (((exists fs_h_jt_scanexistssameright. fs_h_jt_scanexistssameright + S (jt_right_scanexistssame) = S ((S (jt_index_scanexistssame)) * jt_f_scanexists)) /\\ exists fs_q_jt_scanexistssameright. jt_d_scanexists = fs_q_jt_scanexistssameright * S ((S (jt_index_scanexistssame)) * jt_f_scanexists) + (jt_right_scanexistssame))) -> jt_left_scanexistssame=jt_right_scanexistssame) -> jt_i_scanexists=jt_h_scanexists) /\\ (forall jt_z_scanexists. (exists jt_gap_scanexistscodeindex. jt_gap_scanexistscodeindex+S (jt_z_scanexists)=(t)) -> (forall jt_index_scanexistsinputbound. (exists jt_gap_scanexistsinputboundindex. jt_gap_scanexistsinputboundindex+S (jt_index_scanexistsinputbound)=(k)) -> exists jt_value_scanexistsinputbound. ((((exists fs_h_jt_scanexistsinputboundat. fs_h_jt_scanexistsinputboundat + S (jt_value_scanexistsinputbound) = S ((S (jt_index_scanexistsinputbound)) * c)) /\\ exists fs_q_jt_scanexistsinputboundat. jt_z_scanexists = fs_q_jt_scanexistsinputboundat * S ((S (jt_index_scanexistsinputbound)) * c) + (jt_value_scanexistsinputbound))) /\\ (exists jt_gap_scanexistsinputboundvalue. jt_gap_scanexistsinputboundvalue+S (jt_value_scanexistsinputbound)=(n)))) -> (forall jt_divisor_scanexistsinputprimitive. (exists jt_factor_scanexistsinputprimitivemodulus. (n)=(jt_divisor_scanexistsinputprimitive)*jt_factor_scanexistsinputprimitivemodulus) -> (forall jt_index_scanexistsinputprimitivecoordinates jt_value_scanexistsinputprimitivecoordinates. (exists jt_gap_scanexistsinputprimitivecoordinatesindex. jt_gap_scanexistsinputprimitivecoordinatesindex+S (jt_index_scanexistsinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_scanexistsinputprimitivecoordinatesat. fs_h_jt_scanexistsinputprimitivecoordinatesat + S (jt_value_scanexistsinputprimitivecoordinates) = S ((S (jt_index_scanexistsinputprimitivecoordinates)) * c)) /\\ exists fs_q_jt_scanexistsinputprimitivecoordinatesat. jt_z_scanexists = fs_q_jt_scanexistsinputprimitivecoordinatesat * S ((S (jt_index_scanexistsinputprimitivecoordinates)) * c) + (jt_value_scanexistsinputprimitivecoordinates))) -> (exists jt_factor_scanexistsinputprimitivecoordinatesdivides. (jt_value_scanexistsinputprimitivecoordinates)=(jt_divisor_scanexistsinputprimitive)*jt_factor_scanexistsinputprimitivecoordinatesdivides)) -> jt_divisor_scanexistsinputprimitive=1) -> (exists jt_index_scanexistslisted jt_code_scanexistslisted jt_scale_scanexistslisted. ((exists jt_gap_scanexistslistedindex. jt_gap_scanexistslistedindex+S (jt_index_scanexistslisted)=(j)) /\\ (((((((exists fs_h_jt_scanexistslistedcode. fs_h_jt_scanexistslistedcode + S (jt_code_scanexistslisted) = S ((S (jt_index_scanexistslisted)) * C)) /\\ exists fs_q_jt_scanexistslistedcode. B = fs_q_jt_scanexistslistedcode * S ((S (jt_index_scanexistslisted)) * C) + (jt_code_scanexistslisted))) /\\ (((exists fs_h_jt_scanexistslistedscale. fs_h_jt_scanexistslistedscale + S (jt_scale_scanexistslisted) = S ((S (jt_index_scanexistslisted)) * E)) /\\ exists fs_q_jt_scanexistslistedscale. D = fs_q_jt_scanexistslistedscale * S ((S (jt_index_scanexistslisted)) * E) + (jt_scale_scanexistslisted))))) /\\ (forall jt_index_scanexistslistedequal jt_left_scanexistslistedequal jt_right_scanexistslistedequal. (exists jt_gap_scanexistslistedequalindex. jt_gap_scanexistslistedequalindex+S (jt_index_scanexistslistedequal)=(k)) -> (((exists fs_h_jt_scanexistslistedequalleft. fs_h_jt_scanexistslistedequalleft + S (jt_left_scanexistslistedequal) = S ((S (jt_index_scanexistslistedequal)) * c)) /\\ exists fs_q_jt_scanexistslistedequalleft. jt_z_scanexists = fs_q_jt_scanexistslistedequalleft * S ((S (jt_index_scanexistslistedequal)) * c) + (jt_left_scanexistslistedequal))) -> (((exists fs_h_jt_scanexistslistedequalright. fs_h_jt_scanexistslistedequalright + S (jt_right_scanexistslistedequal) = S ((S (jt_index_scanexistslistedequal)) * jt_scale_scanexistslisted)) /\\ exists fs_q_jt_scanexistslistedequalright. jt_code_scanexistslisted = fs_q_jt_scanexistslistedequalright * S ((S (jt_index_scanexistslistedequal)) * jt_scale_scanexistslisted) + (jt_right_scanexistslistedequal))) -> jt_left_scanexistslistedequal=jt_right_scanexistslistedequal)))))))))", "statement_sha256": "79137d417866db17e71e8fa85e60555919383d0873a6df6640c84cd8f4e93bf6", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Construct the duplicate-free finite scan by HA induction and three genuine finite decisions." }, { "admission_dependencies": [ "matrix_rank_uniform_beta_prefix_box_exists", "jordan_tuple_prefix_equal" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 21, "body_proof_nodes": 35, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "have hbox : ∃ c. ∃ T. UniformBetaPrefixBox(c,T,k,n)", "specialize matrix_rank_uniform_beta_prefix_box_exists (k)", "specialize matrix_rank_uniform_beta_prefix_box_exists (n)", "apply matrix_rank_uniform_beta_prefix_box_exists", "cases hbox", "cases hbox_witness", "cases hbox_witness_witness", "exists x", "exists x1", "intro b", "intro e", "intro hb", "have hz : ∃ z. Lt(z,x1) ∧ BetaPrefixEqual(b,e,z,x,k)", "specialize hbox_witness_witness_right (b)", "specialize hbox_witness_witness_right (e)", "apply hbox_witness_witness_right", "exact hb", "cases hz", "cases hz_witness", "exists x2", "split", "exact hz_witness_left", "specialize jordan_tuple_prefix_equal (b)", "specialize jordan_tuple_prefix_equal (e)", "specialize jordan_tuple_prefix_equal (x2)", "specialize jordan_tuple_prefix_equal (x)", "specialize jordan_tuple_prefix_equal (k)", "apply jordan_tuple_prefix_equal", "exact hz_witness_right" ], "defined_statement": "∀ k. ∀ n. ∃ c. ∃ T. JordanTupleRepresentatives(k,n,c,T)", "defined_statement_sha256": "a9fbd6b26a5ceafdea2b9ba02d672fac5076a7d112fac1812f41b2f877daf091", "definition_uses": { "ND0112": 1, "ND0263": 1, "ND0378": 1, "PD0002": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "048ae78b803a00e057d5e3699765ce661a0b628020f4afd6aaeba83f197c6c2f", "free_names": [], "script_definition_uses": { "ND0112": 1, "ND0263": 1, "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "have hbox : " }, { "kind": "text", "text": "∃ c. ∃ T. " }, { "definition": "ND0112", "kind": "definition", "text": "UniformBetaPrefixBox(c,T,k,n)" } ], [ { "kind": "text", "text": "specialize matrix_rank_uniform_beta_prefix_box_exists (k)" } ], [ { "kind": "text", "text": "specialize matrix_rank_uniform_beta_prefix_box_exists (n)" } ], [ { "kind": "text", "text": "apply matrix_rank_uniform_beta_prefix_box_exists" } ], [ { "kind": "text", "text": "cases hbox" } ], [ { "kind": "text", "text": "cases hbox_witness" } ], [ { "kind": "text", "text": "cases hbox_witness_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "have hz : " }, { "kind": "text", "text": "∃ z. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,x1)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0263", "kind": "definition", "text": "BetaPrefixEqual(b,e,z,x,k)" } ], [ { "kind": "text", "text": "specialize hbox_witness_witness_right (b)" } ], [ { "kind": "text", "text": "specialize hbox_witness_witness_right (e)" } ], [ { "kind": "text", "text": "apply hbox_witness_witness_right" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "cases hz" } ], [ { "kind": "text", "text": "cases hz_witness" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hz_witness_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_prefix_equal (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_prefix_equal" } ], [ { "kind": "text", "text": "exact hz_witness_right" } ] ], "statement_definition_uses": { "ND0378": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∃ c. ∃ T. " }, { "definition": "ND0378", "kind": "definition", "text": "JordanTupleRepresentatives(k,n,c,T)" } ] }, "dependencies": [ "matrix_rank_uniform_beta_prefix_box_exists", "jordan_tuple_prefix_equal" ], "enrolled_in_alpha": true, "factory": "make_jordan_scan_candidate_theorems", "first_admitted_version": "v35", "id": "JT0028", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_representatives_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 297, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "have hbox : exists c T. ((~(T=0)) /\\ (forall b e. (forall jt_index_boxinput. (exists jt_gap_boxinputindex. jt_gap_boxinputindex+S (jt_index_boxinput)=(k)) -> exists jt_value_boxinput. ((((exists fs_h_jt_boxinputat. fs_h_jt_boxinputat + S (jt_value_boxinput) = S ((S (jt_index_boxinput)) * e)) /\\ exists fs_q_jt_boxinputat. b = fs_q_jt_boxinputat * S ((S (jt_index_boxinput)) * e) + (jt_value_boxinput))) /\\ (exists jt_gap_boxinputvalue. jt_gap_boxinputvalue+S (jt_value_boxinput)=(n)))) -> exists z. ((exists jt_gap_boxindex. jt_gap_boxindex+S (z)=(T)) /\\ (forall jt_index_boxprefix jt_value_boxprefix. (exists jt_gap_boxprefixindex. jt_gap_boxprefixindex+S (jt_index_boxprefix)=(k)) -> (((exists fs_h_jt_boxprefixold. fs_h_jt_boxprefixold + S (jt_value_boxprefix) = S ((S (jt_index_boxprefix)) * e)) /\\ exists fs_q_jt_boxprefixold. b = fs_q_jt_boxprefixold * S ((S (jt_index_boxprefix)) * e) + (jt_value_boxprefix))) -> (((exists fs_h_jt_boxprefixnew. fs_h_jt_boxprefixnew + S (jt_value_boxprefix) = S ((S (jt_index_boxprefix)) * c)) /\\ exists fs_q_jt_boxprefixnew. z = fs_q_jt_boxprefixnew * S ((S (jt_index_boxprefix)) * c) + (jt_value_boxprefix)))))))", "specialize matrix_rank_uniform_beta_prefix_box_exists (k)", "specialize matrix_rank_uniform_beta_prefix_box_exists (n)", "apply matrix_rank_uniform_beta_prefix_box_exists", "cases hbox", "cases hbox_witness", "cases hbox_witness_witness", "exists x", "exists x1", "intro b", "intro e", "intro hb", "have hz : exists z. ((exists jt_gap_recodeindex. jt_gap_recodeindex+S (z)=(x1)) /\\ (forall jt_index_recodeprefix jt_value_recodeprefix. (exists jt_gap_recodeprefixindex. jt_gap_recodeprefixindex+S (jt_index_recodeprefix)=(k)) -> (((exists fs_h_jt_recodeprefixold. fs_h_jt_recodeprefixold + S (jt_value_recodeprefix) = S ((S (jt_index_recodeprefix)) * e)) /\\ exists fs_q_jt_recodeprefixold. b = fs_q_jt_recodeprefixold * S ((S (jt_index_recodeprefix)) * e) + (jt_value_recodeprefix))) -> (((exists fs_h_jt_recodeprefixnew. fs_h_jt_recodeprefixnew + S (jt_value_recodeprefix) = S ((S (jt_index_recodeprefix)) * x)) /\\ exists fs_q_jt_recodeprefixnew. z = fs_q_jt_recodeprefixnew * S ((S (jt_index_recodeprefix)) * x) + (jt_value_recodeprefix)))))", "specialize hbox_witness_witness_right (b)", "specialize hbox_witness_witness_right (e)", "apply hbox_witness_witness_right", "exact hb", "cases hz", "cases hz_witness", "exists x2", "split", "exact hz_witness_left", "specialize jordan_tuple_prefix_equal (b)", "specialize jordan_tuple_prefix_equal (e)", "specialize jordan_tuple_prefix_equal (x2)", "specialize jordan_tuple_prefix_equal (x)", "specialize jordan_tuple_prefix_equal (k)", "apply jordan_tuple_prefix_equal", "exact hz_witness_right" ], "script_sha256": "c64005670f40705332b4ccca8ff0c5c2b3511a3a0752d8d72378b879024a18a0", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_scan_candidate_theorems", "script_sha256": "c64005670f40705332b4ccca8ff0c5c2b3511a3a0752d8d72378b879024a18a0", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "048ae78b803a00e057d5e3699765ce661a0b628020f4afd6aaeba83f197c6c2f" } ], "stable_member": false, "statement": "forall k n. exists c T. forall jt_code_representativesexists jt_scale_representativesexists. (forall jt_index_representativesexistsbound. (exists jt_gap_representativesexistsboundindex. jt_gap_representativesexistsboundindex+S (jt_index_representativesexistsbound)=(k)) -> exists jt_value_representativesexistsbound. ((((exists fs_h_jt_representativesexistsboundat. fs_h_jt_representativesexistsboundat + S (jt_value_representativesexistsbound) = S ((S (jt_index_representativesexistsbound)) * jt_scale_representativesexists)) /\\ exists fs_q_jt_representativesexistsboundat. jt_code_representativesexists = fs_q_jt_representativesexistsboundat * S ((S (jt_index_representativesexistsbound)) * jt_scale_representativesexists) + (jt_value_representativesexistsbound))) /\\ (exists jt_gap_representativesexistsboundvalue. jt_gap_representativesexistsboundvalue+S (jt_value_representativesexistsbound)=(n)))) -> exists jt_representative_representativesexists. ((exists jt_gap_representativesexistsindex. jt_gap_representativesexistsindex+S (jt_representative_representativesexists)=(T)) /\\ (forall jt_index_representativesexistsequal jt_left_representativesexistsequal jt_right_representativesexistsequal. (exists jt_gap_representativesexistsequalindex. jt_gap_representativesexistsequalindex+S (jt_index_representativesexistsequal)=(k)) -> (((exists fs_h_jt_representativesexistsequalleft. fs_h_jt_representativesexistsequalleft + S (jt_left_representativesexistsequal) = S ((S (jt_index_representativesexistsequal)) * jt_scale_representativesexists)) /\\ exists fs_q_jt_representativesexistsequalleft. jt_code_representativesexists = fs_q_jt_representativesexistsequalleft * S ((S (jt_index_representativesexistsequal)) * jt_scale_representativesexists) + (jt_left_representativesexistsequal))) -> (((exists fs_h_jt_representativesexistsequalright. fs_h_jt_representativesexistsequalright + S (jt_right_representativesexistsequal) = S ((S (jt_index_representativesexistsequal)) * c)) /\\ exists fs_q_jt_representativesexistsequalright. jt_representative_representativesexists = fs_q_jt_representativesexistsequalright * S ((S (jt_index_representativesexistsequal)) * c) + (jt_right_representativesexistsequal))) -> jt_left_representativesexistsequal=jt_right_representativesexistsequal))", "statement_sha256": "048ae78b803a00e057d5e3699765ce661a0b628020f4afd6aaeba83f197c6c2f", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Extract an actual finite coordinate-representative box from the existing beta recoding theorem." }, { "admission_dependencies": [ "jordan_tuple_representatives_exists", "jordan_tuple_scan_exists", "jordan_totient_from_complete_scan" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 29, "body_proof_nodes": 48, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro hk", "intro hn", "have hbox : ∃ c. ∃ T. JordanTupleRepresentatives(k,n,c,T)", "specialize jordan_tuple_representatives_exists (k)", "specialize jordan_tuple_representatives_exists (n)", "apply jordan_tuple_representatives_exists", "cases hbox", "cases hbox_witness", "have hfamily : ∀ t. ∃ B. ∃ C. ∃ D. ∃ E. ∃ j. JordanTupleScan(k,n,x,t,B,C,D,E,j)", "specialize jordan_tuple_scan_exists (k)", "specialize jordan_tuple_scan_exists (n)", "specialize jordan_tuple_scan_exists (x)", "apply jordan_tuple_scan_exists", "exact hn", "have hs : ∃ B. ∃ C. ∃ D. ∃ E. ∃ j. JordanTupleScan(k,n,x,x1,B,C,D,E,j)", "specialize hfamily (x1)", "apply hfamily", "cases hs", "cases hs_witness", "cases hs_witness_witness", "cases hs_witness_witness_witness", "cases hs_witness_witness_witness_witness", "exists x6", "specialize jordan_totient_from_complete_scan (k)", "specialize jordan_totient_from_complete_scan (n)", "specialize jordan_totient_from_complete_scan (x)", "specialize jordan_totient_from_complete_scan (x1)", "specialize jordan_totient_from_complete_scan (x2)", "specialize jordan_totient_from_complete_scan (x3)", "specialize jordan_totient_from_complete_scan (x4)", "specialize jordan_totient_from_complete_scan (x5)", "specialize jordan_totient_from_complete_scan (x6)", "apply jordan_totient_from_complete_scan", "exact hk", "exact hn", "exact hbox_witness_witness", "exact hs_witness_witness_witness_witness_witness" ], "defined_statement": "∀ k. ∀ n. ¬k = 0 → ¬n = 0 → ∃ x. JordanTotient(k,n,x)", "defined_statement_sha256": "fa5e3a210e164c356d0a6297909fe3b775f81a5dc615341a627cf1a3b9cec093", "definition_uses": { "ND0375": 1, "ND0377": 2, "ND0378": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "713cee634bc57758c78bd9766fb66b5c233cb302149291bfe5cc85ac43cc3eea", "free_names": [], "script_definition_uses": { "ND0377": 2, "ND0378": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro hk" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "have hbox : " }, { "kind": "text", "text": "∃ c. ∃ T. " }, { "definition": "ND0378", "kind": "definition", "text": "JordanTupleRepresentatives(k,n,c,T)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_representatives_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_representatives_exists (n)" } ], [ { "kind": "text", "text": "apply jordan_tuple_representatives_exists" } ], [ { "kind": "text", "text": "cases hbox" } ], [ { "kind": "text", "text": "cases hbox_witness" } ], [ { "kind": "text", "text": "have hfamily : " }, { "kind": "text", "text": "∀ t. ∃ B. ∃ C. ∃ D. ∃ E. ∃ j. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,x,t,B,C,D,E,j)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_scan_exists (x)" } ], [ { "kind": "text", "text": "apply jordan_tuple_scan_exists" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "have hs : " }, { "kind": "text", "text": "∃ B. ∃ C. ∃ D. ∃ E. ∃ j. " }, { "definition": "ND0377", "kind": "definition", "text": "JordanTupleScan(k,n,x,x1,B,C,D,E,j)" } ], [ { "kind": "text", "text": "specialize hfamily (x1)" } ], [ { "kind": "text", "text": "apply hfamily" } ], [ { "kind": "text", "text": "cases hs" } ], [ { "kind": "text", "text": "cases hs_witness" } ], [ { "kind": "text", "text": "cases hs_witness_witness" } ], [ { "kind": "text", "text": "cases hs_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hs_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (n)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x1)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x2)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x3)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x4)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x5)" } ], [ { "kind": "text", "text": "specialize jordan_totient_from_complete_scan (x6)" } ], [ { "kind": "text", "text": "apply jordan_totient_from_complete_scan" } ], [ { "kind": "text", "text": "exact hk" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hbox_witness_witness" } ], [ { "kind": "text", "text": "exact hs_witness_witness_witness_witness_witness" } ] ], "statement_definition_uses": { "ND0375": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ¬k = 0 → ¬n = 0 → ∃ x. " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,n,x)" } ] }, "dependencies": [ "jordan_tuple_representatives_exists", "jordan_tuple_scan_exists", "jordan_totient_from_complete_scan" ], "enrolled_in_alpha": true, "factory": "make_jordan_scan_candidate_theorems", "first_admitted_version": "v35", "id": "JT0029", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 298, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro hk", "intro hn", "have hbox : exists c T. forall jt_code_totalbox jt_scale_totalbox. (forall jt_index_totalboxbound. (exists jt_gap_totalboxboundindex. jt_gap_totalboxboundindex+S (jt_index_totalboxbound)=(k)) -> exists jt_value_totalboxbound. ((((exists fs_h_jt_totalboxboundat. fs_h_jt_totalboxboundat + S (jt_value_totalboxbound) = S ((S (jt_index_totalboxbound)) * jt_scale_totalbox)) /\\ exists fs_q_jt_totalboxboundat. jt_code_totalbox = fs_q_jt_totalboxboundat * S ((S (jt_index_totalboxbound)) * jt_scale_totalbox) + (jt_value_totalboxbound))) /\\ (exists jt_gap_totalboxboundvalue. jt_gap_totalboxboundvalue+S (jt_value_totalboxbound)=(n)))) -> exists jt_representative_totalbox. ((exists jt_gap_totalboxindex. jt_gap_totalboxindex+S (jt_representative_totalbox)=(T)) /\\ (forall jt_index_totalboxequal jt_left_totalboxequal jt_right_totalboxequal. (exists jt_gap_totalboxequalindex. jt_gap_totalboxequalindex+S (jt_index_totalboxequal)=(k)) -> (((exists fs_h_jt_totalboxequalleft. fs_h_jt_totalboxequalleft + S (jt_left_totalboxequal) = S ((S (jt_index_totalboxequal)) * jt_scale_totalbox)) /\\ exists fs_q_jt_totalboxequalleft. jt_code_totalbox = fs_q_jt_totalboxequalleft * S ((S (jt_index_totalboxequal)) * jt_scale_totalbox) + (jt_left_totalboxequal))) -> (((exists fs_h_jt_totalboxequalright. fs_h_jt_totalboxequalright + S (jt_right_totalboxequal) = S ((S (jt_index_totalboxequal)) * c)) /\\ exists fs_q_jt_totalboxequalright. jt_representative_totalbox = fs_q_jt_totalboxequalright * S ((S (jt_index_totalboxequal)) * c) + (jt_right_totalboxequal))) -> jt_left_totalboxequal=jt_right_totalboxequal))", "specialize jordan_tuple_representatives_exists (k)", "specialize jordan_tuple_representatives_exists (n)", "apply jordan_tuple_representatives_exists", "cases hbox", "cases hbox_witness", "have hfamily : forall t. exists B C D E j. ((forall jt_i_totalfamily. (exists jt_gap_totalfamilysoundindex. jt_gap_totalfamilysoundindex+S (jt_i_totalfamily)=(j)) -> exists jt_b_totalfamily jt_e_totalfamily. ((((((exists fs_h_jt_totalfamilysoundcode. fs_h_jt_totalfamilysoundcode + S (jt_b_totalfamily) = S ((S (jt_i_totalfamily)) * C)) /\\ exists fs_q_jt_totalfamilysoundcode. B = fs_q_jt_totalfamilysoundcode * S ((S (jt_i_totalfamily)) * C) + (jt_b_totalfamily))) /\\ (((exists fs_h_jt_totalfamilysoundscale. fs_h_jt_totalfamilysoundscale + S (jt_e_totalfamily) = S ((S (jt_i_totalfamily)) * E)) /\\ exists fs_q_jt_totalfamilysoundscale. D = fs_q_jt_totalfamilysoundscale * S ((S (jt_i_totalfamily)) * E) + (jt_e_totalfamily))))) /\\ (((forall jt_index_totalfamilybound. (exists jt_gap_totalfamilyboundindex. jt_gap_totalfamilyboundindex+S (jt_index_totalfamilybound)=(k)) -> exists jt_value_totalfamilybound. ((((exists fs_h_jt_totalfamilyboundat. fs_h_jt_totalfamilyboundat + S (jt_value_totalfamilybound) = S ((S (jt_index_totalfamilybound)) * jt_e_totalfamily)) /\\ exists fs_q_jt_totalfamilyboundat. jt_b_totalfamily = fs_q_jt_totalfamilyboundat * S ((S (jt_index_totalfamilybound)) * jt_e_totalfamily) + (jt_value_totalfamilybound))) /\\ (exists jt_gap_totalfamilyboundvalue. jt_gap_totalfamilyboundvalue+S (jt_value_totalfamilybound)=(n)))) /\\ (forall jt_divisor_totalfamilyprimitive. (exists jt_factor_totalfamilyprimitivemodulus. (n)=(jt_divisor_totalfamilyprimitive)*jt_factor_totalfamilyprimitivemodulus) -> (forall jt_index_totalfamilyprimitivecoordinates jt_value_totalfamilyprimitivecoordinates. (exists jt_gap_totalfamilyprimitivecoordinatesindex. jt_gap_totalfamilyprimitivecoordinatesindex+S (jt_index_totalfamilyprimitivecoordinates)=(k)) -> (((exists fs_h_jt_totalfamilyprimitivecoordinatesat. fs_h_jt_totalfamilyprimitivecoordinatesat + S (jt_value_totalfamilyprimitivecoordinates) = S ((S (jt_index_totalfamilyprimitivecoordinates)) * jt_e_totalfamily)) /\\ exists fs_q_jt_totalfamilyprimitivecoordinatesat. jt_b_totalfamily = fs_q_jt_totalfamilyprimitivecoordinatesat * S ((S (jt_index_totalfamilyprimitivecoordinates)) * jt_e_totalfamily) + (jt_value_totalfamilyprimitivecoordinates))) -> (exists jt_factor_totalfamilyprimitivecoordinatesdivides. (jt_value_totalfamilyprimitivecoordinates)=(jt_divisor_totalfamilyprimitive)*jt_factor_totalfamilyprimitivecoordinatesdivides)) -> jt_divisor_totalfamilyprimitive=1))))) /\\ (((forall jt_i_totalfamily jt_h_totalfamily jt_b_totalfamily jt_e_totalfamily jt_d_totalfamily jt_f_totalfamily. (exists jt_gap_totalfamilyfirstindex. jt_gap_totalfamilyfirstindex+S (jt_i_totalfamily)=(j)) -> (exists jt_gap_totalfamilysecondindex. jt_gap_totalfamilysecondindex+S (jt_h_totalfamily)=(j)) -> (((((exists fs_h_jt_totalfamilyfirstcode. fs_h_jt_totalfamilyfirstcode + S (jt_b_totalfamily) = S ((S (jt_i_totalfamily)) * C)) /\\ exists fs_q_jt_totalfamilyfirstcode. B = fs_q_jt_totalfamilyfirstcode * S ((S (jt_i_totalfamily)) * C) + (jt_b_totalfamily))) /\\ (((exists fs_h_jt_totalfamilyfirstscale. fs_h_jt_totalfamilyfirstscale + S (jt_e_totalfamily) = S ((S (jt_i_totalfamily)) * E)) /\\ exists fs_q_jt_totalfamilyfirstscale. D = fs_q_jt_totalfamilyfirstscale * S ((S (jt_i_totalfamily)) * E) + (jt_e_totalfamily))))) -> (((((exists fs_h_jt_totalfamilysecondcode. fs_h_jt_totalfamilysecondcode + S (jt_d_totalfamily) = S ((S (jt_h_totalfamily)) * C)) /\\ exists fs_q_jt_totalfamilysecondcode. B = fs_q_jt_totalfamilysecondcode * S ((S (jt_h_totalfamily)) * C) + (jt_d_totalfamily))) /\\ (((exists fs_h_jt_totalfamilysecondscale. fs_h_jt_totalfamilysecondscale + S (jt_f_totalfamily) = S ((S (jt_h_totalfamily)) * E)) /\\ exists fs_q_jt_totalfamilysecondscale. D = fs_q_jt_totalfamilysecondscale * S ((S (jt_h_totalfamily)) * E) + (jt_f_totalfamily))))) -> (forall jt_index_totalfamilysame jt_left_totalfamilysame jt_right_totalfamilysame. (exists jt_gap_totalfamilysameindex. jt_gap_totalfamilysameindex+S (jt_index_totalfamilysame)=(k)) -> (((exists fs_h_jt_totalfamilysameleft. fs_h_jt_totalfamilysameleft + S (jt_left_totalfamilysame) = S ((S (jt_index_totalfamilysame)) * jt_e_totalfamily)) /\\ exists fs_q_jt_totalfamilysameleft. jt_b_totalfamily = fs_q_jt_totalfamilysameleft * S ((S (jt_index_totalfamilysame)) * jt_e_totalfamily) + (jt_left_totalfamilysame))) -> (((exists fs_h_jt_totalfamilysameright. fs_h_jt_totalfamilysameright + S (jt_right_totalfamilysame) = S ((S (jt_index_totalfamilysame)) * jt_f_totalfamily)) /\\ exists fs_q_jt_totalfamilysameright. jt_d_totalfamily = fs_q_jt_totalfamilysameright * S ((S (jt_index_totalfamilysame)) * jt_f_totalfamily) + (jt_right_totalfamilysame))) -> jt_left_totalfamilysame=jt_right_totalfamilysame) -> jt_i_totalfamily=jt_h_totalfamily) /\\ (forall jt_z_totalfamily. (exists jt_gap_totalfamilycodeindex. jt_gap_totalfamilycodeindex+S (jt_z_totalfamily)=(t)) -> (forall jt_index_totalfamilyinputbound. (exists jt_gap_totalfamilyinputboundindex. jt_gap_totalfamilyinputboundindex+S (jt_index_totalfamilyinputbound)=(k)) -> exists jt_value_totalfamilyinputbound. ((((exists fs_h_jt_totalfamilyinputboundat. fs_h_jt_totalfamilyinputboundat + S (jt_value_totalfamilyinputbound) = S ((S (jt_index_totalfamilyinputbound)) * x)) /\\ exists fs_q_jt_totalfamilyinputboundat. jt_z_totalfamily = fs_q_jt_totalfamilyinputboundat * S ((S (jt_index_totalfamilyinputbound)) * x) + (jt_value_totalfamilyinputbound))) /\\ (exists jt_gap_totalfamilyinputboundvalue. jt_gap_totalfamilyinputboundvalue+S (jt_value_totalfamilyinputbound)=(n)))) -> (forall jt_divisor_totalfamilyinputprimitive. (exists jt_factor_totalfamilyinputprimitivemodulus. (n)=(jt_divisor_totalfamilyinputprimitive)*jt_factor_totalfamilyinputprimitivemodulus) -> (forall jt_index_totalfamilyinputprimitivecoordinates jt_value_totalfamilyinputprimitivecoordinates. (exists jt_gap_totalfamilyinputprimitivecoordinatesindex. jt_gap_totalfamilyinputprimitivecoordinatesindex+S (jt_index_totalfamilyinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_totalfamilyinputprimitivecoordinatesat. fs_h_jt_totalfamilyinputprimitivecoordinatesat + S (jt_value_totalfamilyinputprimitivecoordinates) = S ((S (jt_index_totalfamilyinputprimitivecoordinates)) * x)) /\\ exists fs_q_jt_totalfamilyinputprimitivecoordinatesat. jt_z_totalfamily = fs_q_jt_totalfamilyinputprimitivecoordinatesat * S ((S (jt_index_totalfamilyinputprimitivecoordinates)) * x) + (jt_value_totalfamilyinputprimitivecoordinates))) -> (exists jt_factor_totalfamilyinputprimitivecoordinatesdivides. (jt_value_totalfamilyinputprimitivecoordinates)=(jt_divisor_totalfamilyinputprimitive)*jt_factor_totalfamilyinputprimitivecoordinatesdivides)) -> jt_divisor_totalfamilyinputprimitive=1) -> (exists jt_index_totalfamilylisted jt_code_totalfamilylisted jt_scale_totalfamilylisted. ((exists jt_gap_totalfamilylistedindex. jt_gap_totalfamilylistedindex+S (jt_index_totalfamilylisted)=(j)) /\\ (((((((exists fs_h_jt_totalfamilylistedcode. fs_h_jt_totalfamilylistedcode + S (jt_code_totalfamilylisted) = S ((S (jt_index_totalfamilylisted)) * C)) /\\ exists fs_q_jt_totalfamilylistedcode. B = fs_q_jt_totalfamilylistedcode * S ((S (jt_index_totalfamilylisted)) * C) + (jt_code_totalfamilylisted))) /\\ (((exists fs_h_jt_totalfamilylistedscale. fs_h_jt_totalfamilylistedscale + S (jt_scale_totalfamilylisted) = S ((S (jt_index_totalfamilylisted)) * E)) /\\ exists fs_q_jt_totalfamilylistedscale. D = fs_q_jt_totalfamilylistedscale * S ((S (jt_index_totalfamilylisted)) * E) + (jt_scale_totalfamilylisted))))) /\\ (forall jt_index_totalfamilylistedequal jt_left_totalfamilylistedequal jt_right_totalfamilylistedequal. (exists jt_gap_totalfamilylistedequalindex. jt_gap_totalfamilylistedequalindex+S (jt_index_totalfamilylistedequal)=(k)) -> (((exists fs_h_jt_totalfamilylistedequalleft. fs_h_jt_totalfamilylistedequalleft + S (jt_left_totalfamilylistedequal) = S ((S (jt_index_totalfamilylistedequal)) * x)) /\\ exists fs_q_jt_totalfamilylistedequalleft. jt_z_totalfamily = fs_q_jt_totalfamilylistedequalleft * S ((S (jt_index_totalfamilylistedequal)) * x) + (jt_left_totalfamilylistedequal))) -> (((exists fs_h_jt_totalfamilylistedequalright. fs_h_jt_totalfamilylistedequalright + S (jt_right_totalfamilylistedequal) = S ((S (jt_index_totalfamilylistedequal)) * jt_scale_totalfamilylisted)) /\\ exists fs_q_jt_totalfamilylistedequalright. jt_code_totalfamilylisted = fs_q_jt_totalfamilylistedequalright * S ((S (jt_index_totalfamilylistedequal)) * jt_scale_totalfamilylisted) + (jt_right_totalfamilylistedequal))) -> jt_left_totalfamilylistedequal=jt_right_totalfamilylistedequal)))))))))", "specialize jordan_tuple_scan_exists (k)", "specialize jordan_tuple_scan_exists (n)", "specialize jordan_tuple_scan_exists (x)", "apply jordan_tuple_scan_exists", "exact hn", "have hs : exists B C D E j. ((forall jt_i_totalscan. (exists jt_gap_totalscansoundindex. jt_gap_totalscansoundindex+S (jt_i_totalscan)=(j)) -> exists jt_b_totalscan jt_e_totalscan. ((((((exists fs_h_jt_totalscansoundcode. fs_h_jt_totalscansoundcode + S (jt_b_totalscan) = S ((S (jt_i_totalscan)) * C)) /\\ exists fs_q_jt_totalscansoundcode. B = fs_q_jt_totalscansoundcode * S ((S (jt_i_totalscan)) * C) + (jt_b_totalscan))) /\\ (((exists fs_h_jt_totalscansoundscale. fs_h_jt_totalscansoundscale + S (jt_e_totalscan) = S ((S (jt_i_totalscan)) * E)) /\\ exists fs_q_jt_totalscansoundscale. D = fs_q_jt_totalscansoundscale * S ((S (jt_i_totalscan)) * E) + (jt_e_totalscan))))) /\\ (((forall jt_index_totalscanbound. (exists jt_gap_totalscanboundindex. jt_gap_totalscanboundindex+S (jt_index_totalscanbound)=(k)) -> exists jt_value_totalscanbound. ((((exists fs_h_jt_totalscanboundat. fs_h_jt_totalscanboundat + S (jt_value_totalscanbound) = S ((S (jt_index_totalscanbound)) * jt_e_totalscan)) /\\ exists fs_q_jt_totalscanboundat. jt_b_totalscan = fs_q_jt_totalscanboundat * S ((S (jt_index_totalscanbound)) * jt_e_totalscan) + (jt_value_totalscanbound))) /\\ (exists jt_gap_totalscanboundvalue. jt_gap_totalscanboundvalue+S (jt_value_totalscanbound)=(n)))) /\\ (forall jt_divisor_totalscanprimitive. (exists jt_factor_totalscanprimitivemodulus. (n)=(jt_divisor_totalscanprimitive)*jt_factor_totalscanprimitivemodulus) -> (forall jt_index_totalscanprimitivecoordinates jt_value_totalscanprimitivecoordinates. (exists jt_gap_totalscanprimitivecoordinatesindex. jt_gap_totalscanprimitivecoordinatesindex+S (jt_index_totalscanprimitivecoordinates)=(k)) -> (((exists fs_h_jt_totalscanprimitivecoordinatesat. fs_h_jt_totalscanprimitivecoordinatesat + S (jt_value_totalscanprimitivecoordinates) = S ((S (jt_index_totalscanprimitivecoordinates)) * jt_e_totalscan)) /\\ exists fs_q_jt_totalscanprimitivecoordinatesat. jt_b_totalscan = fs_q_jt_totalscanprimitivecoordinatesat * S ((S (jt_index_totalscanprimitivecoordinates)) * jt_e_totalscan) + (jt_value_totalscanprimitivecoordinates))) -> (exists jt_factor_totalscanprimitivecoordinatesdivides. (jt_value_totalscanprimitivecoordinates)=(jt_divisor_totalscanprimitive)*jt_factor_totalscanprimitivecoordinatesdivides)) -> jt_divisor_totalscanprimitive=1))))) /\\ (((forall jt_i_totalscan jt_h_totalscan jt_b_totalscan jt_e_totalscan jt_d_totalscan jt_f_totalscan. (exists jt_gap_totalscanfirstindex. jt_gap_totalscanfirstindex+S (jt_i_totalscan)=(j)) -> (exists jt_gap_totalscansecondindex. jt_gap_totalscansecondindex+S (jt_h_totalscan)=(j)) -> (((((exists fs_h_jt_totalscanfirstcode. fs_h_jt_totalscanfirstcode + S (jt_b_totalscan) = S ((S (jt_i_totalscan)) * C)) /\\ exists fs_q_jt_totalscanfirstcode. B = fs_q_jt_totalscanfirstcode * S ((S (jt_i_totalscan)) * C) + (jt_b_totalscan))) /\\ (((exists fs_h_jt_totalscanfirstscale. fs_h_jt_totalscanfirstscale + S (jt_e_totalscan) = S ((S (jt_i_totalscan)) * E)) /\\ exists fs_q_jt_totalscanfirstscale. D = fs_q_jt_totalscanfirstscale * S ((S (jt_i_totalscan)) * E) + (jt_e_totalscan))))) -> (((((exists fs_h_jt_totalscansecondcode. fs_h_jt_totalscansecondcode + S (jt_d_totalscan) = S ((S (jt_h_totalscan)) * C)) /\\ exists fs_q_jt_totalscansecondcode. B = fs_q_jt_totalscansecondcode * S ((S (jt_h_totalscan)) * C) + (jt_d_totalscan))) /\\ (((exists fs_h_jt_totalscansecondscale. fs_h_jt_totalscansecondscale + S (jt_f_totalscan) = S ((S (jt_h_totalscan)) * E)) /\\ exists fs_q_jt_totalscansecondscale. D = fs_q_jt_totalscansecondscale * S ((S (jt_h_totalscan)) * E) + (jt_f_totalscan))))) -> (forall jt_index_totalscansame jt_left_totalscansame jt_right_totalscansame. (exists jt_gap_totalscansameindex. jt_gap_totalscansameindex+S (jt_index_totalscansame)=(k)) -> (((exists fs_h_jt_totalscansameleft. fs_h_jt_totalscansameleft + S (jt_left_totalscansame) = S ((S (jt_index_totalscansame)) * jt_e_totalscan)) /\\ exists fs_q_jt_totalscansameleft. jt_b_totalscan = fs_q_jt_totalscansameleft * S ((S (jt_index_totalscansame)) * jt_e_totalscan) + (jt_left_totalscansame))) -> (((exists fs_h_jt_totalscansameright. fs_h_jt_totalscansameright + S (jt_right_totalscansame) = S ((S (jt_index_totalscansame)) * jt_f_totalscan)) /\\ exists fs_q_jt_totalscansameright. jt_d_totalscan = fs_q_jt_totalscansameright * S ((S (jt_index_totalscansame)) * jt_f_totalscan) + (jt_right_totalscansame))) -> jt_left_totalscansame=jt_right_totalscansame) -> jt_i_totalscan=jt_h_totalscan) /\\ (forall jt_z_totalscan. (exists jt_gap_totalscancodeindex. jt_gap_totalscancodeindex+S (jt_z_totalscan)=(x1)) -> (forall jt_index_totalscaninputbound. (exists jt_gap_totalscaninputboundindex. jt_gap_totalscaninputboundindex+S (jt_index_totalscaninputbound)=(k)) -> exists jt_value_totalscaninputbound. ((((exists fs_h_jt_totalscaninputboundat. fs_h_jt_totalscaninputboundat + S (jt_value_totalscaninputbound) = S ((S (jt_index_totalscaninputbound)) * x)) /\\ exists fs_q_jt_totalscaninputboundat. jt_z_totalscan = fs_q_jt_totalscaninputboundat * S ((S (jt_index_totalscaninputbound)) * x) + (jt_value_totalscaninputbound))) /\\ (exists jt_gap_totalscaninputboundvalue. jt_gap_totalscaninputboundvalue+S (jt_value_totalscaninputbound)=(n)))) -> (forall jt_divisor_totalscaninputprimitive. (exists jt_factor_totalscaninputprimitivemodulus. (n)=(jt_divisor_totalscaninputprimitive)*jt_factor_totalscaninputprimitivemodulus) -> (forall jt_index_totalscaninputprimitivecoordinates jt_value_totalscaninputprimitivecoordinates. (exists jt_gap_totalscaninputprimitivecoordinatesindex. jt_gap_totalscaninputprimitivecoordinatesindex+S (jt_index_totalscaninputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_totalscaninputprimitivecoordinatesat. fs_h_jt_totalscaninputprimitivecoordinatesat + S (jt_value_totalscaninputprimitivecoordinates) = S ((S (jt_index_totalscaninputprimitivecoordinates)) * x)) /\\ exists fs_q_jt_totalscaninputprimitivecoordinatesat. jt_z_totalscan = fs_q_jt_totalscaninputprimitivecoordinatesat * S ((S (jt_index_totalscaninputprimitivecoordinates)) * x) + (jt_value_totalscaninputprimitivecoordinates))) -> (exists jt_factor_totalscaninputprimitivecoordinatesdivides. (jt_value_totalscaninputprimitivecoordinates)=(jt_divisor_totalscaninputprimitive)*jt_factor_totalscaninputprimitivecoordinatesdivides)) -> jt_divisor_totalscaninputprimitive=1) -> (exists jt_index_totalscanlisted jt_code_totalscanlisted jt_scale_totalscanlisted. ((exists jt_gap_totalscanlistedindex. jt_gap_totalscanlistedindex+S (jt_index_totalscanlisted)=(j)) /\\ (((((((exists fs_h_jt_totalscanlistedcode. fs_h_jt_totalscanlistedcode + S (jt_code_totalscanlisted) = S ((S (jt_index_totalscanlisted)) * C)) /\\ exists fs_q_jt_totalscanlistedcode. B = fs_q_jt_totalscanlistedcode * S ((S (jt_index_totalscanlisted)) * C) + (jt_code_totalscanlisted))) /\\ (((exists fs_h_jt_totalscanlistedscale. fs_h_jt_totalscanlistedscale + S (jt_scale_totalscanlisted) = S ((S (jt_index_totalscanlisted)) * E)) /\\ exists fs_q_jt_totalscanlistedscale. D = fs_q_jt_totalscanlistedscale * S ((S (jt_index_totalscanlisted)) * E) + (jt_scale_totalscanlisted))))) /\\ (forall jt_index_totalscanlistedequal jt_left_totalscanlistedequal jt_right_totalscanlistedequal. (exists jt_gap_totalscanlistedequalindex. jt_gap_totalscanlistedequalindex+S (jt_index_totalscanlistedequal)=(k)) -> (((exists fs_h_jt_totalscanlistedequalleft. fs_h_jt_totalscanlistedequalleft + S (jt_left_totalscanlistedequal) = S ((S (jt_index_totalscanlistedequal)) * x)) /\\ exists fs_q_jt_totalscanlistedequalleft. jt_z_totalscan = fs_q_jt_totalscanlistedequalleft * S ((S (jt_index_totalscanlistedequal)) * x) + (jt_left_totalscanlistedequal))) -> (((exists fs_h_jt_totalscanlistedequalright. fs_h_jt_totalscanlistedequalright + S (jt_right_totalscanlistedequal) = S ((S (jt_index_totalscanlistedequal)) * jt_scale_totalscanlisted)) /\\ exists fs_q_jt_totalscanlistedequalright. jt_code_totalscanlisted = fs_q_jt_totalscanlistedequalright * S ((S (jt_index_totalscanlistedequal)) * jt_scale_totalscanlisted) + (jt_right_totalscanlistedequal))) -> jt_left_totalscanlistedequal=jt_right_totalscanlistedequal)))))))))", "specialize hfamily (x1)", "apply hfamily", "cases hs", "cases hs_witness", "cases hs_witness_witness", "cases hs_witness_witness_witness", "cases hs_witness_witness_witness_witness", "exists x6", "specialize jordan_totient_from_complete_scan (k)", "specialize jordan_totient_from_complete_scan (n)", "specialize jordan_totient_from_complete_scan (x)", "specialize jordan_totient_from_complete_scan (x1)", "specialize jordan_totient_from_complete_scan (x2)", "specialize jordan_totient_from_complete_scan (x3)", "specialize jordan_totient_from_complete_scan (x4)", "specialize jordan_totient_from_complete_scan (x5)", "specialize jordan_totient_from_complete_scan (x6)", "apply jordan_totient_from_complete_scan", "exact hk", "exact hn", "exact hbox_witness_witness", "exact hs_witness_witness_witness_witness_witness" ], "script_sha256": "2ac47a3523480bbff3c5f5f02fd7d38f4b023a791faa97517c289e80b714f7dc", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_scan_candidate_theorems", "script_sha256": "2ac47a3523480bbff3c5f5f02fd7d38f4b023a791faa97517c289e80b714f7dc", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "713cee634bc57758c78bd9766fb66b5c233cb302149291bfe5cc85ac43cc3eea" } ], "stable_member": false, "statement": "forall k n. ~(k=0) -> ~(n=0) -> exists j. ((~((k)=0)) /\\ (((~((n)=0)) /\\ (exists jt_codes_jordanexists jt_code_scale_jordanexists jt_scales_jordanexists jt_scale_scale_jordanexists. ((forall jt_i_jordanexistsenum. (exists jt_gap_jordanexistsenumsoundindex. jt_gap_jordanexistsenumsoundindex+S (jt_i_jordanexistsenum)=(j)) -> exists jt_b_jordanexistsenum jt_c_jordanexistsenum. ((((((exists fs_h_jt_jordanexistsenumsoundcode. fs_h_jt_jordanexistsenumsoundcode + S (jt_b_jordanexistsenum) = S ((S (jt_i_jordanexistsenum)) * jt_code_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumsoundcode. jt_codes_jordanexists = fs_q_jt_jordanexistsenumsoundcode * S ((S (jt_i_jordanexistsenum)) * jt_code_scale_jordanexists) + (jt_b_jordanexistsenum))) /\\ (((exists fs_h_jt_jordanexistsenumsoundscale. fs_h_jt_jordanexistsenumsoundscale + S (jt_c_jordanexistsenum) = S ((S (jt_i_jordanexistsenum)) * jt_scale_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumsoundscale. jt_scales_jordanexists = fs_q_jt_jordanexistsenumsoundscale * S ((S (jt_i_jordanexistsenum)) * jt_scale_scale_jordanexists) + (jt_c_jordanexistsenum))))) /\\ (((forall jt_index_jordanexistsenumbound. (exists jt_gap_jordanexistsenumboundindex. jt_gap_jordanexistsenumboundindex+S (jt_index_jordanexistsenumbound)=(k)) -> exists jt_value_jordanexistsenumbound. ((((exists fs_h_jt_jordanexistsenumboundat. fs_h_jt_jordanexistsenumboundat + S (jt_value_jordanexistsenumbound) = S ((S (jt_index_jordanexistsenumbound)) * jt_c_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenumboundat. jt_b_jordanexistsenum = fs_q_jt_jordanexistsenumboundat * S ((S (jt_index_jordanexistsenumbound)) * jt_c_jordanexistsenum) + (jt_value_jordanexistsenumbound))) /\\ (exists jt_gap_jordanexistsenumboundvalue. jt_gap_jordanexistsenumboundvalue+S (jt_value_jordanexistsenumbound)=(n)))) /\\ (forall jt_divisor_jordanexistsenumprimitive. (exists jt_factor_jordanexistsenumprimitivemodulus. (n)=(jt_divisor_jordanexistsenumprimitive)*jt_factor_jordanexistsenumprimitivemodulus) -> (forall jt_index_jordanexistsenumprimitivecoordinates jt_value_jordanexistsenumprimitivecoordinates. (exists jt_gap_jordanexistsenumprimitivecoordinatesindex. jt_gap_jordanexistsenumprimitivecoordinatesindex+S (jt_index_jordanexistsenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jordanexistsenumprimitivecoordinatesat. fs_h_jt_jordanexistsenumprimitivecoordinatesat + S (jt_value_jordanexistsenumprimitivecoordinates) = S ((S (jt_index_jordanexistsenumprimitivecoordinates)) * jt_c_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenumprimitivecoordinatesat. jt_b_jordanexistsenum = fs_q_jt_jordanexistsenumprimitivecoordinatesat * S ((S (jt_index_jordanexistsenumprimitivecoordinates)) * jt_c_jordanexistsenum) + (jt_value_jordanexistsenumprimitivecoordinates))) -> (exists jt_factor_jordanexistsenumprimitivecoordinatesdivides. (jt_value_jordanexistsenumprimitivecoordinates)=(jt_divisor_jordanexistsenumprimitive)*jt_factor_jordanexistsenumprimitivecoordinatesdivides)) -> jt_divisor_jordanexistsenumprimitive=1))))) /\\ (((forall jt_b_jordanexistsenum jt_c_jordanexistsenum. (forall jt_index_jordanexistsenuminputbound. (exists jt_gap_jordanexistsenuminputboundindex. jt_gap_jordanexistsenuminputboundindex+S (jt_index_jordanexistsenuminputbound)=(k)) -> exists jt_value_jordanexistsenuminputbound. ((((exists fs_h_jt_jordanexistsenuminputboundat. fs_h_jt_jordanexistsenuminputboundat + S (jt_value_jordanexistsenuminputbound) = S ((S (jt_index_jordanexistsenuminputbound)) * jt_c_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenuminputboundat. jt_b_jordanexistsenum = fs_q_jt_jordanexistsenuminputboundat * S ((S (jt_index_jordanexistsenuminputbound)) * jt_c_jordanexistsenum) + (jt_value_jordanexistsenuminputbound))) /\\ (exists jt_gap_jordanexistsenuminputboundvalue. jt_gap_jordanexistsenuminputboundvalue+S (jt_value_jordanexistsenuminputbound)=(n)))) -> (forall jt_divisor_jordanexistsenuminputprimitive. (exists jt_factor_jordanexistsenuminputprimitivemodulus. (n)=(jt_divisor_jordanexistsenuminputprimitive)*jt_factor_jordanexistsenuminputprimitivemodulus) -> (forall jt_index_jordanexistsenuminputprimitivecoordinates jt_value_jordanexistsenuminputprimitivecoordinates. (exists jt_gap_jordanexistsenuminputprimitivecoordinatesindex. jt_gap_jordanexistsenuminputprimitivecoordinatesindex+S (jt_index_jordanexistsenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jordanexistsenuminputprimitivecoordinatesat. fs_h_jt_jordanexistsenuminputprimitivecoordinatesat + S (jt_value_jordanexistsenuminputprimitivecoordinates) = S ((S (jt_index_jordanexistsenuminputprimitivecoordinates)) * jt_c_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenuminputprimitivecoordinatesat. jt_b_jordanexistsenum = fs_q_jt_jordanexistsenuminputprimitivecoordinatesat * S ((S (jt_index_jordanexistsenuminputprimitivecoordinates)) * jt_c_jordanexistsenum) + (jt_value_jordanexistsenuminputprimitivecoordinates))) -> (exists jt_factor_jordanexistsenuminputprimitivecoordinatesdivides. (jt_value_jordanexistsenuminputprimitivecoordinates)=(jt_divisor_jordanexistsenuminputprimitive)*jt_factor_jordanexistsenuminputprimitivecoordinatesdivides)) -> jt_divisor_jordanexistsenuminputprimitive=1) -> exists jt_i_jordanexistsenum jt_d_jordanexistsenum jt_e_jordanexistsenum. ((exists jt_gap_jordanexistsenumcompleteindex. jt_gap_jordanexistsenumcompleteindex+S (jt_i_jordanexistsenum)=(j)) /\\ (((((((exists fs_h_jt_jordanexistsenumcompletecode. fs_h_jt_jordanexistsenumcompletecode + S (jt_d_jordanexistsenum) = S ((S (jt_i_jordanexistsenum)) * jt_code_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumcompletecode. jt_codes_jordanexists = fs_q_jt_jordanexistsenumcompletecode * S ((S (jt_i_jordanexistsenum)) * jt_code_scale_jordanexists) + (jt_d_jordanexistsenum))) /\\ (((exists fs_h_jt_jordanexistsenumcompletescale. fs_h_jt_jordanexistsenumcompletescale + S (jt_e_jordanexistsenum) = S ((S (jt_i_jordanexistsenum)) * jt_scale_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumcompletescale. jt_scales_jordanexists = fs_q_jt_jordanexistsenumcompletescale * S ((S (jt_i_jordanexistsenum)) * jt_scale_scale_jordanexists) + (jt_e_jordanexistsenum))))) /\\ (forall jt_index_jordanexistsenumrepresented jt_left_jordanexistsenumrepresented jt_right_jordanexistsenumrepresented. (exists jt_gap_jordanexistsenumrepresentedindex. jt_gap_jordanexistsenumrepresentedindex+S (jt_index_jordanexistsenumrepresented)=(k)) -> (((exists fs_h_jt_jordanexistsenumrepresentedleft. fs_h_jt_jordanexistsenumrepresentedleft + S (jt_left_jordanexistsenumrepresented) = S ((S (jt_index_jordanexistsenumrepresented)) * jt_c_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenumrepresentedleft. jt_b_jordanexistsenum = fs_q_jt_jordanexistsenumrepresentedleft * S ((S (jt_index_jordanexistsenumrepresented)) * jt_c_jordanexistsenum) + (jt_left_jordanexistsenumrepresented))) -> (((exists fs_h_jt_jordanexistsenumrepresentedright. fs_h_jt_jordanexistsenumrepresentedright + S (jt_right_jordanexistsenumrepresented) = S ((S (jt_index_jordanexistsenumrepresented)) * jt_e_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenumrepresentedright. jt_d_jordanexistsenum = fs_q_jt_jordanexistsenumrepresentedright * S ((S (jt_index_jordanexistsenumrepresented)) * jt_e_jordanexistsenum) + (jt_right_jordanexistsenumrepresented))) -> jt_left_jordanexistsenumrepresented=jt_right_jordanexistsenumrepresented))))) /\\ (forall jt_i_jordanexistsenum jt_h_jordanexistsenum jt_b_jordanexistsenum jt_c_jordanexistsenum jt_d_jordanexistsenum jt_e_jordanexistsenum. (exists jt_gap_jordanexistsenumfirstindex. jt_gap_jordanexistsenumfirstindex+S (jt_i_jordanexistsenum)=(j)) -> (exists jt_gap_jordanexistsenumsecondindex. jt_gap_jordanexistsenumsecondindex+S (jt_h_jordanexistsenum)=(j)) -> (((((exists fs_h_jt_jordanexistsenumfirstcode. fs_h_jt_jordanexistsenumfirstcode + S (jt_b_jordanexistsenum) = S ((S (jt_i_jordanexistsenum)) * jt_code_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumfirstcode. jt_codes_jordanexists = fs_q_jt_jordanexistsenumfirstcode * S ((S (jt_i_jordanexistsenum)) * jt_code_scale_jordanexists) + (jt_b_jordanexistsenum))) /\\ (((exists fs_h_jt_jordanexistsenumfirstscale. fs_h_jt_jordanexistsenumfirstscale + S (jt_c_jordanexistsenum) = S ((S (jt_i_jordanexistsenum)) * jt_scale_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumfirstscale. jt_scales_jordanexists = fs_q_jt_jordanexistsenumfirstscale * S ((S (jt_i_jordanexistsenum)) * jt_scale_scale_jordanexists) + (jt_c_jordanexistsenum))))) -> (((((exists fs_h_jt_jordanexistsenumsecondcode. fs_h_jt_jordanexistsenumsecondcode + S (jt_d_jordanexistsenum) = S ((S (jt_h_jordanexistsenum)) * jt_code_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumsecondcode. jt_codes_jordanexists = fs_q_jt_jordanexistsenumsecondcode * S ((S (jt_h_jordanexistsenum)) * jt_code_scale_jordanexists) + (jt_d_jordanexistsenum))) /\\ (((exists fs_h_jt_jordanexistsenumsecondscale. fs_h_jt_jordanexistsenumsecondscale + S (jt_e_jordanexistsenum) = S ((S (jt_h_jordanexistsenum)) * jt_scale_scale_jordanexists)) /\\ exists fs_q_jt_jordanexistsenumsecondscale. jt_scales_jordanexists = fs_q_jt_jordanexistsenumsecondscale * S ((S (jt_h_jordanexistsenum)) * jt_scale_scale_jordanexists) + (jt_e_jordanexistsenum))))) -> (forall jt_index_jordanexistsenumsame jt_left_jordanexistsenumsame jt_right_jordanexistsenumsame. (exists jt_gap_jordanexistsenumsameindex. jt_gap_jordanexistsenumsameindex+S (jt_index_jordanexistsenumsame)=(k)) -> (((exists fs_h_jt_jordanexistsenumsameleft. fs_h_jt_jordanexistsenumsameleft + S (jt_left_jordanexistsenumsame) = S ((S (jt_index_jordanexistsenumsame)) * jt_c_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenumsameleft. jt_b_jordanexistsenum = fs_q_jt_jordanexistsenumsameleft * S ((S (jt_index_jordanexistsenumsame)) * jt_c_jordanexistsenum) + (jt_left_jordanexistsenumsame))) -> (((exists fs_h_jt_jordanexistsenumsameright. fs_h_jt_jordanexistsenumsameright + S (jt_right_jordanexistsenumsame) = S ((S (jt_index_jordanexistsenumsame)) * jt_e_jordanexistsenum)) /\\ exists fs_q_jt_jordanexistsenumsameright. jt_d_jordanexistsenum = fs_q_jt_jordanexistsenumsameright * S ((S (jt_index_jordanexistsenumsame)) * jt_e_jordanexistsenum) + (jt_right_jordanexistsenumsame))) -> jt_left_jordanexistsenumsame=jt_right_jordanexistsenumsame) -> jt_i_jordanexistsenum=jt_h_jordanexistsenum))))))))", "statement_sha256": "713cee634bc57758c78bd9766fb66b5c233cb302149291bfe5cc85ac43cc3eea", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Obtain an actual finite tuple cardinality from independently constructed duplicate-free enumeration." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 21, "body_proof_nodes": 30, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro i", "intro hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. JordanTupleCRT(m,n,b,c,d,e,f,g,0)", "defined_statement_sha256": "ffd37c4bf8c8b323b4c467438b1454bba54d6b78f33f8df6f7a0ee43836e851a", "definition_uses": { "ND0379": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "bb23802c2a55ed623b800faf965f88e3d1c8507e970103abbcdd8d4b9fc2d367", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (i)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "specialize zero_le (i)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "ND0379": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,f,g,0)" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_crt_tuple_candidate_theorems", "first_admitted_version": "v35", "id": "JT002A", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_crt_tuple_empty", "original_ha_bundle_verified": true, "proof_bundle_node_id": 299, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro i", "intro hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "script_sha256": "2b0c5becda2c3201c7b55d6f76b0ef4fae23459fc66f44e664c5ff5c6d9962a4", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_crt_tuple_candidate_theorems", "script_sha256": "2b0c5becda2c3201c7b55d6f76b0ef4fae23459fc66f44e664c5ff5c6d9962a4", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "bb23802c2a55ed623b800faf965f88e3d1c8507e970103abbcdd8d4b9fc2d367" } ], "stable_member": false, "statement": "forall m n b c d e f g. forall jt_index_crtempty. (exists jt_gap_crtemptyindex. jt_gap_crtemptyindex+S (jt_index_crtempty)=(0)) -> exists jt_left_crtempty jt_right_crtempty jt_output_crtempty. ((((exists fs_h_jt_crtemptyleft. fs_h_jt_crtemptyleft + S (jt_left_crtempty) = S ((S (jt_index_crtempty)) * c)) /\\ exists fs_q_jt_crtemptyleft. b = fs_q_jt_crtemptyleft * S ((S (jt_index_crtempty)) * c) + (jt_left_crtempty))) /\\ (((((exists fs_h_jt_crtemptyright. fs_h_jt_crtemptyright + S (jt_right_crtempty) = S ((S (jt_index_crtempty)) * e)) /\\ exists fs_q_jt_crtemptyright. d = fs_q_jt_crtemptyright * S ((S (jt_index_crtempty)) * e) + (jt_right_crtempty))) /\\ (((((exists fs_h_jt_crtemptyoutput. fs_h_jt_crtemptyoutput + S (jt_output_crtempty) = S ((S (jt_index_crtempty)) * g)) /\\ exists fs_q_jt_crtemptyoutput. f = fs_q_jt_crtemptyoutput * S ((S (jt_index_crtempty)) * g) + (jt_output_crtempty))) /\\ (((exists jt_left_crtemptymodleft jt_right_crtemptymodleft. (jt_output_crtempty)+(m)*jt_left_crtemptymodleft=(jt_left_crtempty)+(m)*jt_right_crtemptymodleft) /\\ (exists jt_left_crtemptymodright jt_right_crtemptymodright. (jt_output_crtempty)+(n)*jt_left_crtemptymodright=(jt_right_crtempty)+(n)*jt_right_crtemptymodright))))))))", "statement_sha256": "bb23802c2a55ed623b800faf965f88e3d1c8507e970103abbcdd8d4b9fc2d367", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The empty simultaneous coordinate congruence has no missing witness." }, { "admission_dependencies": [ "beta_prefix_extend", "finite_lt_succ_eq_or_lt" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 41, "body_proof_nodes": 110, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro a", "intro z", "intro w", "intro hprefix", "intro ha", "intro hz", "intro hm", "intro hn", "have hext : ∃ u. ∃ v. BetaAt(u,v,k,w) ∧ BetaPrefixEqual(f,g,u,v,k)", "specialize beta_prefix_extend (k)", "specialize beta_prefix_extend (f)", "specialize beta_prefix_extend (g)", "specialize beta_prefix_extend (w)", "apply beta_prefix_extend", "cases hext", "cases hext_witness", "cases hext_witness_witness", "exists x", "exists x1", "intro i", "intro hi", "have hc : i = k ∨ Lt(i,k)", "specialize finite_lt_succ_eq_or_lt (k)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hc", "exists a", "exists z", "exists w", "split", "rewrite hc_left", "rewrite hc_left", "exact ha", "split", "rewrite hc_left", "rewrite hc_left", "exact hz", "split", "rewrite hc_left", "rewrite hc_left", "exact hext_witness_witness_left", "split", "exact hm", "exact hn", "have hvalue : ∃ a. ∃ z. ∃ w. BetaAt(b,c,i,a) ∧ (BetaAt(d,e,i,z) ∧ (BetaAt(f,g,i,w) ∧ (ModEq(m,w,a) ∧ ModEq(n,w,z))))", "specialize hprefix (i)", "apply hprefix", "exact hc_right", "cases hvalue", "cases hvalue_witness", "cases hvalue_witness_witness", "cases hvalue_witness_witness_witness", "cases hvalue_witness_witness_witness_right", "cases hvalue_witness_witness_witness_right_right", "cases hvalue_witness_witness_witness_right_right_right", "exists x2", "exists x3", "exists x4", "split", "exact hvalue_witness_witness_witness_left", "split", "exact hvalue_witness_witness_witness_right_left", "split", "specialize hext_witness_witness_right (i)", "specialize hext_witness_witness_right (x4)", "apply hext_witness_witness_right", "exact hc_right", "exact hvalue_witness_witness_witness_right_right_left", "split", "exact hvalue_witness_witness_witness_right_right_right_left", "exact hvalue_witness_witness_witness_right_right_right_right" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. ∀ a. ∀ z. ∀ w. JordanTupleCRT(m,n,b,c,d,e,f,g,k) → BetaAt(b,c,k,a) → BetaAt(d,e,k,z) → ModEq(m,w,a) → ModEq(n,w,z) → ∃ x. ∃ y. JordanTupleCRT(m,n,b,c,d,e,x,y,S k)", "defined_statement_sha256": "17f96b9f569213558f756be9cf99cdad5e6a752572177cb13f332715301f7a81", "definition_uses": { "ND0263": 1, "ND0379": 2, "PD0002": 1, "PD0008": 4, "PD0013": 6 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "79fb8eec9564ba5f1ba16d551d9d4bb8ccfc6587ddce5b94db818cc63ff4bbde", "free_names": [], "script_definition_uses": { "ND0263": 1, "PD0002": 1, "PD0008": 2, "PD0013": 4 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro w" } ], [ { "kind": "text", "text": "intro hprefix" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "have hext : " }, { "kind": "text", "text": "∃ u. ∃ v. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(u,v,k,w)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0263", "kind": "definition", "text": "BetaPrefixEqual(f,g,u,v,k)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (k)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (f)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (g)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (w)" } ], [ { "kind": "text", "text": "apply beta_prefix_extend" } ], [ { "kind": "text", "text": "cases hext" } ], [ { "kind": "text", "text": "cases hext_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "i = k ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,k)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (k)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (i)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "exists a" } ], [ { "kind": "text", "text": "exists z" } ], [ { "kind": "text", "text": "exists w" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "have hvalue : " }, { "kind": "text", "text": "∃ a. ∃ z. ∃ w. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,a)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,z)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(f,g,i,w)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,w,a)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,w,z)" }, { "kind": "text", "text": ")))" } ], [ { "kind": "text", "text": "specialize hprefix (i)" } ], [ { "kind": "text", "text": "apply hprefix" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "cases hvalue" } ], [ { "kind": "text", "text": "cases hvalue_witness" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hvalue_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize hext_witness_witness_right (i)" } ], [ { "kind": "text", "text": "specialize hext_witness_witness_right (x4)" } ], [ { "kind": "text", "text": "apply hext_witness_witness_right" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "exact hvalue_witness_witness_witness_right_right_right_right" } ] ], "statement_definition_uses": { "ND0379": 2, "PD0008": 2, "PD0013": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. ∀ a. ∀ z. ∀ w. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,k,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,k,z)" }, { "kind": "text", "text": " → " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,w,a)" }, { "kind": "text", "text": " → " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,w,z)" }, { "kind": "text", "text": " → ∃ x. ∃ y. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,x,y,S k)" } ] }, "dependencies": [ "beta_prefix_extend", "finite_lt_succ_eq_or_lt" ], "enrolled_in_alpha": true, "factory": "make_jordan_crt_tuple_candidate_theorems", "first_admitted_version": "v35", "id": "JT002B", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_crt_tuple_extend", "original_ha_bundle_verified": true, "proof_bundle_node_id": 300, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro a", "intro z", "intro w", "intro hprefix", "intro ha", "intro hz", "intro hm", "intro hn", "have hext : exists u v. ((((exists fs_h_jt_crtextendlast. fs_h_jt_crtextendlast + S (w) = S ((S (k)) * v)) /\\ exists fs_q_jt_crtextendlast. u = fs_q_jt_crtextendlast * S ((S (k)) * v) + (w))) /\\ (forall jt_index_crtextendprefix jt_value_crtextendprefix. (exists jt_gap_crtextendprefixindex. jt_gap_crtextendprefixindex+S (jt_index_crtextendprefix)=(k)) -> (((exists fs_h_jt_crtextendprefixold. fs_h_jt_crtextendprefixold + S (jt_value_crtextendprefix) = S ((S (jt_index_crtextendprefix)) * g)) /\\ exists fs_q_jt_crtextendprefixold. f = fs_q_jt_crtextendprefixold * S ((S (jt_index_crtextendprefix)) * g) + (jt_value_crtextendprefix))) -> (((exists fs_h_jt_crtextendprefixnew. fs_h_jt_crtextendprefixnew + S (jt_value_crtextendprefix) = S ((S (jt_index_crtextendprefix)) * v)) /\\ exists fs_q_jt_crtextendprefixnew. u = fs_q_jt_crtextendprefixnew * S ((S (jt_index_crtextendprefix)) * v) + (jt_value_crtextendprefix)))))", "specialize beta_prefix_extend (k)", "specialize beta_prefix_extend (f)", "specialize beta_prefix_extend (g)", "specialize beta_prefix_extend (w)", "apply beta_prefix_extend", "cases hext", "cases hext_witness", "cases hext_witness_witness", "exists x", "exists x1", "intro i", "intro hi", "have hc : i=k \\/ (exists jt_gap_crtextendcase. jt_gap_crtextendcase+S (i)=(k))", "specialize finite_lt_succ_eq_or_lt (k)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hc", "exists a", "exists z", "exists w", "split", "rewrite hc_left", "rewrite hc_left", "exact ha", "split", "rewrite hc_left", "rewrite hc_left", "exact hz", "split", "rewrite hc_left", "rewrite hc_left", "exact hext_witness_witness_left", "split", "exact hm", "exact hn", "have hvalue : exists a z w. ((((exists fs_h_jt_crtoldleft. fs_h_jt_crtoldleft + S (a) = S ((S (i)) * c)) /\\ exists fs_q_jt_crtoldleft. b = fs_q_jt_crtoldleft * S ((S (i)) * c) + (a))) /\\ (((((exists fs_h_jt_crtoldright. fs_h_jt_crtoldright + S (z) = S ((S (i)) * e)) /\\ exists fs_q_jt_crtoldright. d = fs_q_jt_crtoldright * S ((S (i)) * e) + (z))) /\\ (((((exists fs_h_jt_crtoldoutput. fs_h_jt_crtoldoutput + S (w) = S ((S (i)) * g)) /\\ exists fs_q_jt_crtoldoutput. f = fs_q_jt_crtoldoutput * S ((S (i)) * g) + (w))) /\\ (((exists jt_left_crtoldmodleft jt_right_crtoldmodleft. (w)+(m)*jt_left_crtoldmodleft=(a)+(m)*jt_right_crtoldmodleft) /\\ (exists jt_left_crtoldmodright jt_right_crtoldmodright. (w)+(n)*jt_left_crtoldmodright=(z)+(n)*jt_right_crtoldmodright))))))))", "specialize hprefix (i)", "apply hprefix", "exact hc_right", "cases hvalue", "cases hvalue_witness", "cases hvalue_witness_witness", "cases hvalue_witness_witness_witness", "cases hvalue_witness_witness_witness_right", "cases hvalue_witness_witness_witness_right_right", "cases hvalue_witness_witness_witness_right_right_right", "exists x2", "exists x3", "exists x4", "split", "exact hvalue_witness_witness_witness_left", "split", "exact hvalue_witness_witness_witness_right_left", "split", "specialize hext_witness_witness_right (i)", "specialize hext_witness_witness_right (x4)", "apply hext_witness_witness_right", "exact hc_right", "exact hvalue_witness_witness_witness_right_right_left", "split", "exact hvalue_witness_witness_witness_right_right_right_left", "exact hvalue_witness_witness_witness_right_right_right_right" ], "script_sha256": "40676e164e7db4f537d823a072b8d6ab954454313bfca8572ee9351e65f509a7", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_crt_tuple_candidate_theorems", "script_sha256": "40676e164e7db4f537d823a072b8d6ab954454313bfca8572ee9351e65f509a7", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "79fb8eec9564ba5f1ba16d551d9d4bb8ccfc6587ddce5b94db818cc63ff4bbde" } ], "stable_member": false, "statement": "forall m n b c d e f g k a z w. (forall jt_index_crtprefix. (exists jt_gap_crtprefixindex. jt_gap_crtprefixindex+S (jt_index_crtprefix)=(k)) -> exists jt_left_crtprefix jt_right_crtprefix jt_output_crtprefix. ((((exists fs_h_jt_crtprefixleft. fs_h_jt_crtprefixleft + S (jt_left_crtprefix) = S ((S (jt_index_crtprefix)) * c)) /\\ exists fs_q_jt_crtprefixleft. b = fs_q_jt_crtprefixleft * S ((S (jt_index_crtprefix)) * c) + (jt_left_crtprefix))) /\\ (((((exists fs_h_jt_crtprefixright. fs_h_jt_crtprefixright + S (jt_right_crtprefix) = S ((S (jt_index_crtprefix)) * e)) /\\ exists fs_q_jt_crtprefixright. d = fs_q_jt_crtprefixright * S ((S (jt_index_crtprefix)) * e) + (jt_right_crtprefix))) /\\ (((((exists fs_h_jt_crtprefixoutput. fs_h_jt_crtprefixoutput + S (jt_output_crtprefix) = S ((S (jt_index_crtprefix)) * g)) /\\ exists fs_q_jt_crtprefixoutput. f = fs_q_jt_crtprefixoutput * S ((S (jt_index_crtprefix)) * g) + (jt_output_crtprefix))) /\\ (((exists jt_left_crtprefixmodleft jt_right_crtprefixmodleft. (jt_output_crtprefix)+(m)*jt_left_crtprefixmodleft=(jt_left_crtprefix)+(m)*jt_right_crtprefixmodleft) /\\ (exists jt_left_crtprefixmodright jt_right_crtprefixmodright. (jt_output_crtprefix)+(n)*jt_left_crtprefixmodright=(jt_right_crtprefix)+(n)*jt_right_crtprefixmodright))))))))) -> (((exists fs_h_jt_crtlastleft. fs_h_jt_crtlastleft + S (a) = S ((S (k)) * c)) /\\ exists fs_q_jt_crtlastleft. b = fs_q_jt_crtlastleft * S ((S (k)) * c) + (a))) -> (((exists fs_h_jt_crtlastright. fs_h_jt_crtlastright + S (z) = S ((S (k)) * e)) /\\ exists fs_q_jt_crtlastright. d = fs_q_jt_crtlastright * S ((S (k)) * e) + (z))) -> (exists jt_left_crtlastmodleft jt_right_crtlastmodleft. (w)+(m)*jt_left_crtlastmodleft=(a)+(m)*jt_right_crtlastmodleft) -> (exists jt_left_crtlastmodright jt_right_crtlastmodright. (w)+(n)*jt_left_crtlastmodright=(z)+(n)*jt_right_crtlastmodright) -> exists u v. forall jt_index_crtextended. (exists jt_gap_crtextendedindex. jt_gap_crtextendedindex+S (jt_index_crtextended)=(S k)) -> exists jt_left_crtextended jt_right_crtextended jt_output_crtextended. ((((exists fs_h_jt_crtextendedleft. fs_h_jt_crtextendedleft + S (jt_left_crtextended) = S ((S (jt_index_crtextended)) * c)) /\\ exists fs_q_jt_crtextendedleft. b = fs_q_jt_crtextendedleft * S ((S (jt_index_crtextended)) * c) + (jt_left_crtextended))) /\\ (((((exists fs_h_jt_crtextendedright. fs_h_jt_crtextendedright + S (jt_right_crtextended) = S ((S (jt_index_crtextended)) * e)) /\\ exists fs_q_jt_crtextendedright. d = fs_q_jt_crtextendedright * S ((S (jt_index_crtextended)) * e) + (jt_right_crtextended))) /\\ (((((exists fs_h_jt_crtextendedoutput. fs_h_jt_crtextendedoutput + S (jt_output_crtextended) = S ((S (jt_index_crtextended)) * v)) /\\ exists fs_q_jt_crtextendedoutput. u = fs_q_jt_crtextendedoutput * S ((S (jt_index_crtextended)) * v) + (jt_output_crtextended))) /\\ (((exists jt_left_crtextendedmodleft jt_right_crtextendedmodleft. (jt_output_crtextended)+(m)*jt_left_crtextendedmodleft=(jt_left_crtextended)+(m)*jt_right_crtextendedmodleft) /\\ (exists jt_left_crtextendedmodright jt_right_crtextendedmodright. (jt_output_crtextended)+(n)*jt_left_crtextendedmodright=(jt_right_crtextended)+(n)*jt_right_crtextendedmodright))))))))", "statement_sha256": "79fb8eec9564ba5f1ba16d551d9d4bb8ccfc6587ddce5b94db818cc63ff4bbde", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Append one genuine scalar CRT solution using an actual beta prefix extension." }, { "admission_dependencies": [ "jordan_crt_tuple_empty", "beta_at_exists", "binary_crt", "jordan_crt_tuple_extend" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 39, "body_proof_nodes": 78, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro hm", "intro hn", "intro hcop", "induction k", "exists 0", "exists 0", "specialize jordan_crt_tuple_empty (m)", "specialize jordan_crt_tuple_empty (n)", "specialize jordan_crt_tuple_empty (b)", "specialize jordan_crt_tuple_empty (c)", "specialize jordan_crt_tuple_empty (d)", "specialize jordan_crt_tuple_empty (e)", "specialize jordan_crt_tuple_empty (0)", "specialize jordan_crt_tuple_empty (0)", "apply jordan_crt_tuple_empty", "cases IH", "cases IH_witness", "have ha : ∃ a. BetaAt(b,c,k,a)", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases ha", "have hz : ∃ z. BetaAt(d,e,k,z)", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases hz", "have hw : ∃ w. ModEq(m,w,x2) ∧ ModEq(n,w,x3)", "specialize binary_crt (m)", "specialize binary_crt (n)", "specialize binary_crt (x2)", "specialize binary_crt (x3)", "apply binary_crt", "exact hm", "exact hn", "exact hcop", "cases hw", "cases hw_witness", "specialize jordan_crt_tuple_extend (m)", "specialize jordan_crt_tuple_extend (n)", "specialize jordan_crt_tuple_extend (b)", "specialize jordan_crt_tuple_extend (c)", "specialize jordan_crt_tuple_extend (d)", "specialize jordan_crt_tuple_extend (e)", "specialize jordan_crt_tuple_extend (x)", "specialize jordan_crt_tuple_extend (x1)", "specialize jordan_crt_tuple_extend (k)", "specialize jordan_crt_tuple_extend (x2)", "specialize jordan_crt_tuple_extend (x3)", "specialize jordan_crt_tuple_extend (x4)", "apply jordan_crt_tuple_extend", "exact IH_witness_witness", "exact ha_witness", "exact hz_witness", "exact hw_witness_left", "exact hw_witness_right" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ¬m = 0 → ¬n = 0 → Coprime(m,n) → ∀ x. ∃ y. ∃ z. JordanTupleCRT(m,n,b,c,d,e,y,z,x)", "defined_statement_sha256": "49901da35bcfe019db9132d7689b1c44ab5eb5d5d455621351d99ec7b2024d82", "definition_uses": { "ND0379": 1, "PD0005": 1, "PD0008": 2, "PD0013": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "a8d4b35910586ec93d2486eb0bb1ef4feb2a8be0567d4863fcda263dedfe35fb", "free_names": [], "script_definition_uses": { "PD0008": 2, "PD0013": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "induction k" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (m)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (n)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (b)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (c)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (d)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (e)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (0)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_empty (0)" } ], [ { "kind": "text", "text": "apply jordan_crt_tuple_empty" } ], [ { "kind": "text", "text": "cases IH" } ], [ { "kind": "text", "text": "cases IH_witness" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ a. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,k,a)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (b)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (c)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (k)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "have hz : " }, { "kind": "text", "text": "∃ z. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,k,z)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (k)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hz" } ], [ { "kind": "text", "text": "have hw : " }, { "kind": "text", "text": "∃ w. " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,w,x2)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,w,x3)" } ], [ { "kind": "text", "text": "specialize binary_crt (m)" } ], [ { "kind": "text", "text": "specialize binary_crt (n)" } ], [ { "kind": "text", "text": "specialize binary_crt (x2)" } ], [ { "kind": "text", "text": "specialize binary_crt (x3)" } ], [ { "kind": "text", "text": "apply binary_crt" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "cases hw" } ], [ { "kind": "text", "text": "cases hw_witness" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (m)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (n)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (b)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (c)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (d)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (e)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (x)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (x1)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (k)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (x2)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (x3)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_extend (x4)" } ], [ { "kind": "text", "text": "apply jordan_crt_tuple_extend" } ], [ { "kind": "text", "text": "exact IH_witness_witness" } ], [ { "kind": "text", "text": "exact ha_witness" } ], [ { "kind": "text", "text": "exact hz_witness" } ], [ { "kind": "text", "text": "exact hw_witness_left" } ], [ { "kind": "text", "text": "exact hw_witness_right" } ] ], "statement_definition_uses": { "ND0379": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → ∀ x. ∃ y. ∃ z. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,y,z,x)" } ] }, "dependencies": [ "jordan_crt_tuple_empty", "beta_at_exists", "binary_crt", "jordan_crt_tuple_extend" ], "enrolled_in_alpha": true, "factory": "make_jordan_crt_tuple_candidate_theorems", "first_admitted_version": "v35", "id": "JT002C", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_crt_tuple_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 301, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro hm", "intro hn", "intro hcop", "induction k", "exists 0", "exists 0", "specialize jordan_crt_tuple_empty (m)", "specialize jordan_crt_tuple_empty (n)", "specialize jordan_crt_tuple_empty (b)", "specialize jordan_crt_tuple_empty (c)", "specialize jordan_crt_tuple_empty (d)", "specialize jordan_crt_tuple_empty (e)", "specialize jordan_crt_tuple_empty (0)", "specialize jordan_crt_tuple_empty (0)", "apply jordan_crt_tuple_empty", "cases IH", "cases IH_witness", "have ha : exists a. ((exists fs_h_jt_crttotalleft. fs_h_jt_crttotalleft + S (a) = S ((S (k)) * c)) /\\ exists fs_q_jt_crttotalleft. b = fs_q_jt_crttotalleft * S ((S (k)) * c) + (a))", "specialize beta_at_exists (b)", "specialize beta_at_exists (c)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases ha", "have hz : exists z. ((exists fs_h_jt_crttotalright. fs_h_jt_crttotalright + S (z) = S ((S (k)) * e)) /\\ exists fs_q_jt_crttotalright. d = fs_q_jt_crttotalright * S ((S (k)) * e) + (z))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (k)", "apply beta_at_exists", "cases hz", "have hw : exists w. ((exists jt_left_crttotalmodleft jt_right_crttotalmodleft. (w)+(m)*jt_left_crttotalmodleft=(x2)+(m)*jt_right_crttotalmodleft) /\\ (exists jt_left_crttotalmodright jt_right_crttotalmodright. (w)+(n)*jt_left_crttotalmodright=(x3)+(n)*jt_right_crttotalmodright))", "specialize binary_crt (m)", "specialize binary_crt (n)", "specialize binary_crt (x2)", "specialize binary_crt (x3)", "apply binary_crt", "exact hm", "exact hn", "exact hcop", "cases hw", "cases hw_witness", "specialize jordan_crt_tuple_extend (m)", "specialize jordan_crt_tuple_extend (n)", "specialize jordan_crt_tuple_extend (b)", "specialize jordan_crt_tuple_extend (c)", "specialize jordan_crt_tuple_extend (d)", "specialize jordan_crt_tuple_extend (e)", "specialize jordan_crt_tuple_extend (x)", "specialize jordan_crt_tuple_extend (x1)", "specialize jordan_crt_tuple_extend (k)", "specialize jordan_crt_tuple_extend (x2)", "specialize jordan_crt_tuple_extend (x3)", "specialize jordan_crt_tuple_extend (x4)", "apply jordan_crt_tuple_extend", "exact IH_witness_witness", "exact ha_witness", "exact hz_witness", "exact hw_witness_left", "exact hw_witness_right" ], "script_sha256": "708e806b1a145c18099a91951c2ec2833a5e10694d3e8bd25bfabf9938ebb5a2", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_crt_tuple_candidate_theorems", "script_sha256": "708e806b1a145c18099a91951c2ec2833a5e10694d3e8bd25bfabf9938ebb5a2", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "a8d4b35910586ec93d2486eb0bb1ef4feb2a8be0567d4863fcda263dedfe35fb" } ], "stable_member": false, "statement": "forall m n b c d e. ~(m=0) -> ~(n=0) -> (forall jt_divisor_crtcoprime. (exists jt_factor_crtcoprimea. (m)=(jt_divisor_crtcoprime)*jt_factor_crtcoprimea) -> (exists jt_factor_crtcoprimeb. (n)=(jt_divisor_crtcoprime)*jt_factor_crtcoprimeb) -> jt_divisor_crtcoprime=1) -> forall k. exists f g. forall jt_index_crtexists. (exists jt_gap_crtexistsindex. jt_gap_crtexistsindex+S (jt_index_crtexists)=(k)) -> exists jt_left_crtexists jt_right_crtexists jt_output_crtexists. ((((exists fs_h_jt_crtexistsleft. fs_h_jt_crtexistsleft + S (jt_left_crtexists) = S ((S (jt_index_crtexists)) * c)) /\\ exists fs_q_jt_crtexistsleft. b = fs_q_jt_crtexistsleft * S ((S (jt_index_crtexists)) * c) + (jt_left_crtexists))) /\\ (((((exists fs_h_jt_crtexistsright. fs_h_jt_crtexistsright + S (jt_right_crtexists) = S ((S (jt_index_crtexists)) * e)) /\\ exists fs_q_jt_crtexistsright. d = fs_q_jt_crtexistsright * S ((S (jt_index_crtexists)) * e) + (jt_right_crtexists))) /\\ (((((exists fs_h_jt_crtexistsoutput. fs_h_jt_crtexistsoutput + S (jt_output_crtexists) = S ((S (jt_index_crtexists)) * g)) /\\ exists fs_q_jt_crtexistsoutput. f = fs_q_jt_crtexistsoutput * S ((S (jt_index_crtexists)) * g) + (jt_output_crtexists))) /\\ (((exists jt_left_crtexistsmodleft jt_right_crtexistsmodleft. (jt_output_crtexists)+(m)*jt_left_crtexistsmodleft=(jt_left_crtexists)+(m)*jt_right_crtexistsmodleft) /\\ (exists jt_left_crtexistsmodright jt_right_crtexistsmodright. (jt_output_crtexists)+(n)*jt_left_crtexistsmodright=(jt_right_crtexists)+(n)*jt_right_crtexistsmodright))))))))", "statement_sha256": "a8d4b35910586ec93d2486eb0bb1ef4feb2a8be0567d4863fcda263dedfe35fb", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Construct simultaneous residue representatives coordinate by coordinate, without any tuple totality premise." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 30, "body_proof_nodes": 57, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro h", "intro i", "intro w", "intro a", "intro hi", "intro hw", "intro ha", "have ht : ∃ u. ∃ v. ∃ q. BetaAt(b,c,i,u) ∧ (BetaAt(d,e,i,v) ∧ (BetaAt(f,g,i,q) ∧ (ModEq(m,q,u) ∧ ModEq(n,q,v))))", "specialize h (i)", "apply h", "exact hi", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "cases ht_witness_witness_witness_right_right_right", "have hout : x2=w", "specialize beta_at_unique (f)", "specialize beta_at_unique (g)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x2)", "specialize beta_at_unique (w)", "apply beta_at_unique", "exact ht_witness_witness_witness_right_right_left", "exact hw", "have hin : x=a", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (a)", "apply beta_at_unique", "exact ht_witness_witness_witness_left", "exact ha", "rewrite hout at ht_witness_witness_witness_right_right_right_left", "rewrite hin at ht_witness_witness_witness_right_right_right_left", "exact ht_witness_witness_witness_right_right_right_left" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. JordanTupleCRT(m,n,b,c,d,e,f,g,k) → JordanTupleCongruence(m,f,g,b,c,k)", "defined_statement_sha256": "d93c1e3a70e275f83554609912db43e27d992c42c25118a5538ee78b2a6c9670", "definition_uses": { "ND0373": 1, "ND0379": 1, "PD0008": 2, "PD0013": 3 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "2d3ed54d409e347385d053f08bc783ac402f8de2f69c33d617544d94d148ec93", "free_names": [], "script_definition_uses": { "PD0008": 2, "PD0013": 3 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro w" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hw" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "have ht : " }, { "kind": "text", "text": "∃ u. ∃ v. ∃ q. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,v)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(f,g,i,q)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,q,u)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,q,v)" }, { "kind": "text", "text": ")))" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases ht" } ], [ { "kind": "text", "text": "cases ht_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "have hout : x2=w" } ], [ { "kind": "text", "text": "specialize beta_at_unique (f)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (g)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x2)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (w)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact ht_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "exact hw" } ], [ { "kind": "text", "text": "have hin : x=a" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact ht_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "rewrite hout at ht_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hin at ht_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "exact ht_witness_witness_witness_right_right_right_left" } ] ], "statement_definition_uses": { "ND0373": 1, "ND0379": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m,f,g,b,c,k)" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_crt_tuple_candidate_theorems", "first_admitted_version": "v35", "id": "JT002D", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_crt_tuple_left", "original_ha_bundle_verified": true, "proof_bundle_node_id": 302, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro h", "intro i", "intro w", "intro a", "intro hi", "intro hw", "intro ha", "have ht : exists u v q. ((((exists fs_h_jt_crtpointleft. fs_h_jt_crtpointleft + S (u) = S ((S (i)) * c)) /\\ exists fs_q_jt_crtpointleft. b = fs_q_jt_crtpointleft * S ((S (i)) * c) + (u))) /\\ (((((exists fs_h_jt_crtpointright. fs_h_jt_crtpointright + S (v) = S ((S (i)) * e)) /\\ exists fs_q_jt_crtpointright. d = fs_q_jt_crtpointright * S ((S (i)) * e) + (v))) /\\ (((((exists fs_h_jt_crtpointoutput. fs_h_jt_crtpointoutput + S (q) = S ((S (i)) * g)) /\\ exists fs_q_jt_crtpointoutput. f = fs_q_jt_crtpointoutput * S ((S (i)) * g) + (q))) /\\ (((exists jt_left_crtpointmodleft jt_right_crtpointmodleft. (q)+(m)*jt_left_crtpointmodleft=(u)+(m)*jt_right_crtpointmodleft) /\\ (exists jt_left_crtpointmodright jt_right_crtpointmodright. (q)+(n)*jt_left_crtpointmodright=(v)+(n)*jt_right_crtpointmodright))))))))", "specialize h (i)", "apply h", "exact hi", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "cases ht_witness_witness_witness_right_right_right", "have hout : x2=w", "specialize beta_at_unique (f)", "specialize beta_at_unique (g)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x2)", "specialize beta_at_unique (w)", "apply beta_at_unique", "exact ht_witness_witness_witness_right_right_left", "exact hw", "have hin : x=a", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (a)", "apply beta_at_unique", "exact ht_witness_witness_witness_left", "exact ha", "rewrite hout at ht_witness_witness_witness_right_right_right_left", "rewrite hin at ht_witness_witness_witness_right_right_right_left", "exact ht_witness_witness_witness_right_right_right_left" ], "script_sha256": "29a5f0d2ff5632a7c550a3f0ebfe701798d44ce4b15fc7f90bb255335cd422b5", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_crt_tuple_candidate_theorems", "script_sha256": "29a5f0d2ff5632a7c550a3f0ebfe701798d44ce4b15fc7f90bb255335cd422b5", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "2d3ed54d409e347385d053f08bc783ac402f8de2f69c33d617544d94d148ec93" } ], "stable_member": false, "statement": "forall m n b c d e f g k. (forall jt_index_crtleft. (exists jt_gap_crtleftindex. jt_gap_crtleftindex+S (jt_index_crtleft)=(k)) -> exists jt_left_crtleft jt_right_crtleft jt_output_crtleft. ((((exists fs_h_jt_crtleftleft. fs_h_jt_crtleftleft + S (jt_left_crtleft) = S ((S (jt_index_crtleft)) * c)) /\\ exists fs_q_jt_crtleftleft. b = fs_q_jt_crtleftleft * S ((S (jt_index_crtleft)) * c) + (jt_left_crtleft))) /\\ (((((exists fs_h_jt_crtleftright. fs_h_jt_crtleftright + S (jt_right_crtleft) = S ((S (jt_index_crtleft)) * e)) /\\ exists fs_q_jt_crtleftright. d = fs_q_jt_crtleftright * S ((S (jt_index_crtleft)) * e) + (jt_right_crtleft))) /\\ (((((exists fs_h_jt_crtleftoutput. fs_h_jt_crtleftoutput + S (jt_output_crtleft) = S ((S (jt_index_crtleft)) * g)) /\\ exists fs_q_jt_crtleftoutput. f = fs_q_jt_crtleftoutput * S ((S (jt_index_crtleft)) * g) + (jt_output_crtleft))) /\\ (((exists jt_left_crtleftmodleft jt_right_crtleftmodleft. (jt_output_crtleft)+(m)*jt_left_crtleftmodleft=(jt_left_crtleft)+(m)*jt_right_crtleftmodleft) /\\ (exists jt_left_crtleftmodright jt_right_crtleftmodright. (jt_output_crtleft)+(n)*jt_left_crtleftmodright=(jt_right_crtleft)+(n)*jt_right_crtleftmodright))))))))) -> (forall jt_index_crtprojectionleft jt_left_crtprojectionleft jt_right_crtprojectionleft. (exists jt_gap_crtprojectionleftindex. jt_gap_crtprojectionleftindex+S (jt_index_crtprojectionleft)=(k)) -> (((exists fs_h_jt_crtprojectionleftleft. fs_h_jt_crtprojectionleftleft + S (jt_left_crtprojectionleft) = S ((S (jt_index_crtprojectionleft)) * g)) /\\ exists fs_q_jt_crtprojectionleftleft. f = fs_q_jt_crtprojectionleftleft * S ((S (jt_index_crtprojectionleft)) * g) + (jt_left_crtprojectionleft))) -> (((exists fs_h_jt_crtprojectionleftright. fs_h_jt_crtprojectionleftright + S (jt_right_crtprojectionleft) = S ((S (jt_index_crtprojectionleft)) * c)) /\\ exists fs_q_jt_crtprojectionleftright. b = fs_q_jt_crtprojectionleftright * S ((S (jt_index_crtprojectionleft)) * c) + (jt_right_crtprojectionleft))) -> (exists jt_left_crtprojectionleftmod jt_right_crtprojectionleftmod. (jt_left_crtprojectionleft)+(m)*jt_left_crtprojectionleftmod=(jt_right_crtprojectionleft)+(m)*jt_right_crtprojectionleftmod))", "statement_sha256": "2d3ed54d409e347385d053f08bc783ac402f8de2f69c33d617544d94d148ec93", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every actual output coordinate has the required left congruence." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 30, "body_proof_nodes": 58, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro h", "intro i", "intro w", "intro a", "intro hi", "intro hw", "intro ha", "have ht : ∃ u. ∃ v. ∃ q. BetaAt(b,c,i,u) ∧ (BetaAt(d,e,i,v) ∧ (BetaAt(f,g,i,q) ∧ (ModEq(m,q,u) ∧ ModEq(n,q,v))))", "specialize h (i)", "apply h", "exact hi", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "cases ht_witness_witness_witness_right_right_right", "have hout : x2=w", "specialize beta_at_unique (f)", "specialize beta_at_unique (g)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x2)", "specialize beta_at_unique (w)", "apply beta_at_unique", "exact ht_witness_witness_witness_right_right_left", "exact hw", "have hin : x1=a", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (a)", "apply beta_at_unique", "exact ht_witness_witness_witness_right_left", "exact ha", "rewrite hout at ht_witness_witness_witness_right_right_right_right", "rewrite hin at ht_witness_witness_witness_right_right_right_right", "exact ht_witness_witness_witness_right_right_right_right" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. JordanTupleCRT(m,n,b,c,d,e,f,g,k) → JordanTupleCongruence(n,f,g,d,e,k)", "defined_statement_sha256": "a60432756e4cec6225b8fa1fdbca6b1f7921f3c8ebe7c41ed7793a38d8d46690", "definition_uses": { "ND0373": 1, "ND0379": 1, "PD0008": 2, "PD0013": 3 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "eeab9da250778c4c08d8bb3f3d8fa1c3d37f30f0c3a2c3a24a0eb0d2405fa466", "free_names": [], "script_definition_uses": { "PD0008": 2, "PD0013": 3 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro w" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hw" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "have ht : " }, { "kind": "text", "text": "∃ u. ∃ v. ∃ q. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,v)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(f,g,i,q)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(m,q,u)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0008", "kind": "definition", "text": "ModEq(n,q,v)" }, { "kind": "text", "text": ")))" } ], [ { "kind": "text", "text": "specialize h (i)" } ], [ { "kind": "text", "text": "apply h" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases ht" } ], [ { "kind": "text", "text": "cases ht_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases ht_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "have hout : x2=w" } ], [ { "kind": "text", "text": "specialize beta_at_unique (f)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (g)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x2)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (w)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact ht_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "exact hw" } ], [ { "kind": "text", "text": "have hin : x1=a" } ], [ { "kind": "text", "text": "specialize beta_at_unique (d)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (e)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact ht_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "rewrite hout at ht_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "rewrite hin at ht_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "exact ht_witness_witness_witness_right_right_right_right" } ] ], "statement_definition_uses": { "ND0373": 1, "ND0379": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,f,g,d,e,k)" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_crt_tuple_candidate_theorems", "first_admitted_version": "v35", "id": "JT002E", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_crt_tuple_right", "original_ha_bundle_verified": true, "proof_bundle_node_id": 303, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro h", "intro i", "intro w", "intro a", "intro hi", "intro hw", "intro ha", "have ht : exists u v q. ((((exists fs_h_jt_crtpointleft. fs_h_jt_crtpointleft + S (u) = S ((S (i)) * c)) /\\ exists fs_q_jt_crtpointleft. b = fs_q_jt_crtpointleft * S ((S (i)) * c) + (u))) /\\ (((((exists fs_h_jt_crtpointright. fs_h_jt_crtpointright + S (v) = S ((S (i)) * e)) /\\ exists fs_q_jt_crtpointright. d = fs_q_jt_crtpointright * S ((S (i)) * e) + (v))) /\\ (((((exists fs_h_jt_crtpointoutput. fs_h_jt_crtpointoutput + S (q) = S ((S (i)) * g)) /\\ exists fs_q_jt_crtpointoutput. f = fs_q_jt_crtpointoutput * S ((S (i)) * g) + (q))) /\\ (((exists jt_left_crtpointmodleft jt_right_crtpointmodleft. (q)+(m)*jt_left_crtpointmodleft=(u)+(m)*jt_right_crtpointmodleft) /\\ (exists jt_left_crtpointmodright jt_right_crtpointmodright. (q)+(n)*jt_left_crtpointmodright=(v)+(n)*jt_right_crtpointmodright))))))))", "specialize h (i)", "apply h", "exact hi", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_witness", "cases ht_witness_witness_witness_right", "cases ht_witness_witness_witness_right_right", "cases ht_witness_witness_witness_right_right_right", "have hout : x2=w", "specialize beta_at_unique (f)", "specialize beta_at_unique (g)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x2)", "specialize beta_at_unique (w)", "apply beta_at_unique", "exact ht_witness_witness_witness_right_right_left", "exact hw", "have hin : x1=a", "specialize beta_at_unique (d)", "specialize beta_at_unique (e)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (a)", "apply beta_at_unique", "exact ht_witness_witness_witness_right_left", "exact ha", "rewrite hout at ht_witness_witness_witness_right_right_right_right", "rewrite hin at ht_witness_witness_witness_right_right_right_right", "exact ht_witness_witness_witness_right_right_right_right" ], "script_sha256": "053c62a9db3de910c5fa71bd3480ced993790eed52f77caf426ef5c7db0b4ad5", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_crt_tuple_candidate_theorems", "script_sha256": "053c62a9db3de910c5fa71bd3480ced993790eed52f77caf426ef5c7db0b4ad5", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "eeab9da250778c4c08d8bb3f3d8fa1c3d37f30f0c3a2c3a24a0eb0d2405fa466" } ], "stable_member": false, "statement": "forall m n b c d e f g k. (forall jt_index_crtright. (exists jt_gap_crtrightindex. jt_gap_crtrightindex+S (jt_index_crtright)=(k)) -> exists jt_left_crtright jt_right_crtright jt_output_crtright. ((((exists fs_h_jt_crtrightleft. fs_h_jt_crtrightleft + S (jt_left_crtright) = S ((S (jt_index_crtright)) * c)) /\\ exists fs_q_jt_crtrightleft. b = fs_q_jt_crtrightleft * S ((S (jt_index_crtright)) * c) + (jt_left_crtright))) /\\ (((((exists fs_h_jt_crtrightright. fs_h_jt_crtrightright + S (jt_right_crtright) = S ((S (jt_index_crtright)) * e)) /\\ exists fs_q_jt_crtrightright. d = fs_q_jt_crtrightright * S ((S (jt_index_crtright)) * e) + (jt_right_crtright))) /\\ (((((exists fs_h_jt_crtrightoutput. fs_h_jt_crtrightoutput + S (jt_output_crtright) = S ((S (jt_index_crtright)) * g)) /\\ exists fs_q_jt_crtrightoutput. f = fs_q_jt_crtrightoutput * S ((S (jt_index_crtright)) * g) + (jt_output_crtright))) /\\ (((exists jt_left_crtrightmodleft jt_right_crtrightmodleft. (jt_output_crtright)+(m)*jt_left_crtrightmodleft=(jt_left_crtright)+(m)*jt_right_crtrightmodleft) /\\ (exists jt_left_crtrightmodright jt_right_crtrightmodright. (jt_output_crtright)+(n)*jt_left_crtrightmodright=(jt_right_crtright)+(n)*jt_right_crtrightmodright))))))))) -> (forall jt_index_crtprojectionright jt_left_crtprojectionright jt_right_crtprojectionright. (exists jt_gap_crtprojectionrightindex. jt_gap_crtprojectionrightindex+S (jt_index_crtprojectionright)=(k)) -> (((exists fs_h_jt_crtprojectionrightleft. fs_h_jt_crtprojectionrightleft + S (jt_left_crtprojectionright) = S ((S (jt_index_crtprojectionright)) * g)) /\\ exists fs_q_jt_crtprojectionrightleft. f = fs_q_jt_crtprojectionrightleft * S ((S (jt_index_crtprojectionright)) * g) + (jt_left_crtprojectionright))) -> (((exists fs_h_jt_crtprojectionrightright. fs_h_jt_crtprojectionrightright + S (jt_right_crtprojectionright) = S ((S (jt_index_crtprojectionright)) * e)) /\\ exists fs_q_jt_crtprojectionrightright. d = fs_q_jt_crtprojectionrightright * S ((S (jt_index_crtprojectionright)) * e) + (jt_right_crtprojectionright))) -> (exists jt_left_crtprojectionrightmod jt_right_crtprojectionrightmod. (jt_left_crtprojectionright)+(n)*jt_left_crtprojectionrightmod=(jt_right_crtprojectionright)+(n)*jt_right_crtprojectionrightmod))", "statement_sha256": "eeab9da250778c4c08d8bb3f3d8fa1c3d37f30f0c3a2c3a24a0eb0d2405fa466", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every actual output coordinate has the required right congruence." }, { "admission_dependencies": [ "prime_field_polynomial_normalization_exists", "prime_field_polynomial_normalization_bounded", "prime_field_polynomial_normalization_entry" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 34, "body_proof_nodes": 55, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro k", "intro hn", "have hnorm : ∃ d. ∃ e. FpCoefficientReduction(n,b,c,d,e,k)", "specialize prime_field_polynomial_normalization_exists (n)", "specialize prime_field_polynomial_normalization_exists (b)", "specialize prime_field_polynomial_normalization_exists (c)", "specialize prime_field_polynomial_normalization_exists (k)", "apply prime_field_polynomial_normalization_exists", "exact hn", "cases hnorm", "cases hnorm_witness", "exists x", "exists x1", "split", "specialize prime_field_polynomial_normalization_bounded (n)", "specialize prime_field_polynomial_normalization_bounded (b)", "specialize prime_field_polynomial_normalization_bounded (c)", "specialize prime_field_polynomial_normalization_bounded (x)", "specialize prime_field_polynomial_normalization_bounded (x1)", "specialize prime_field_polynomial_normalization_bounded (k)", "apply prime_field_polynomial_normalization_bounded", "exact hnorm_witness_witness", "intro i", "intro a", "intro r", "intro hi", "intro ha", "intro hr", "have hvalue : CanonicalModularResidue(n,a,r)", "specialize prime_field_polynomial_normalization_entry (n)", "specialize prime_field_polynomial_normalization_entry (b)", "specialize prime_field_polynomial_normalization_entry (c)", "specialize prime_field_polynomial_normalization_entry (x)", "specialize prime_field_polynomial_normalization_entry (x1)", "specialize prime_field_polynomial_normalization_entry (k)", "specialize prime_field_polynomial_normalization_entry (i)", "specialize prime_field_polynomial_normalization_entry (a)", "specialize prime_field_polynomial_normalization_entry (r)", "apply prime_field_polynomial_normalization_entry", "exact hnorm_witness_witness", "exact hi", "exact ha", "exact hr", "cases hvalue", "exact hvalue_right" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ k. ¬n = 0 → ∃ x. ∃ y. BetaPrefixInto(x,y,k,n) ∧ JordanTupleCongruence(n,b,c,x,y,k)", "defined_statement_sha256": "2c2ace5712d7da7ff595332b231ae7768c97ae44fddf1e014eb64f25d56a65a3", "definition_uses": { "ND0023": 1, "ND0262": 1, "ND0269": 1, "ND0373": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "a6659077a2de6cac97641a55abf2da0c8ec52e48e7ecb73a4950f2ed859e70ae", "free_names": [], "script_definition_uses": { "ND0023": 1, "ND0269": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "have hnorm : " }, { "kind": "text", "text": "∃ d. ∃ e. " }, { "definition": "ND0269", "kind": "definition", "text": "FpCoefficientReduction(n,b,c,d,e,k)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_exists (n)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_exists (b)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_exists (c)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_exists (k)" } ], [ { "kind": "text", "text": "apply prime_field_polynomial_normalization_exists" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "cases hnorm" } ], [ { "kind": "text", "text": "cases hnorm_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_bounded (n)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_bounded (b)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_bounded (c)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_bounded (x)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_bounded (x1)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_bounded (k)" } ], [ { "kind": "text", "text": "apply prime_field_polynomial_normalization_bounded" } ], [ { "kind": "text", "text": "exact hnorm_witness_witness" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro r" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "have hvalue : " }, { "definition": "ND0023", "kind": "definition", "text": "CanonicalModularResidue(n,a,r)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (n)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (b)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (c)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (x)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (x1)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (k)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (i)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (a)" } ], [ { "kind": "text", "text": "specialize prime_field_polynomial_normalization_entry (r)" } ], [ { "kind": "text", "text": "apply prime_field_polynomial_normalization_entry" } ], [ { "kind": "text", "text": "exact hnorm_witness_witness" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "cases hvalue" } ], [ { "kind": "text", "text": "exact hvalue_right" } ] ], "statement_definition_uses": { "ND0262": 1, "ND0373": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ k. ¬n = 0 → ∃ x. ∃ y. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x,y,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,x,y,k)" } ] }, "dependencies": [ "prime_field_polynomial_normalization_exists", "prime_field_polynomial_normalization_bounded", "prime_field_polynomial_normalization_entry" ], "enrolled_in_alpha": true, "factory": "make_jordan_canonical_crt_candidate_theorems", "first_admitted_version": "v35", "id": "JT002F", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_normalize_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 308, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro k", "intro hn", "have hnorm : exists d e. forall jt_index_normalizeactual. (exists jt_gap_normalizeactualindex. jt_gap_normalizeactualindex+S (jt_index_normalizeactual)=(k)) -> exists jt_input_normalizeactual jt_output_normalizeactual. ((((exists fs_h_jt_normalizeactualinput. fs_h_jt_normalizeactualinput + S (jt_input_normalizeactual) = S ((S (jt_index_normalizeactual)) * c)) /\\ exists fs_q_jt_normalizeactualinput. b = fs_q_jt_normalizeactualinput * S ((S (jt_index_normalizeactual)) * c) + (jt_input_normalizeactual))) /\\ (((((exists fs_h_jt_normalizeactualoutput. fs_h_jt_normalizeactualoutput + S (jt_output_normalizeactual) = S ((S (jt_index_normalizeactual)) * e)) /\\ exists fs_q_jt_normalizeactualoutput. d = fs_q_jt_normalizeactualoutput * S ((S (jt_index_normalizeactual)) * e) + (jt_output_normalizeactual))) /\\ (((exists jt_gap_normalizeactualbound. jt_gap_normalizeactualbound+S (jt_output_normalizeactual)=(n)) /\\ (exists jt_left_normalizeactualmod jt_right_normalizeactualmod. (jt_input_normalizeactual)+(n)*jt_left_normalizeactualmod=(jt_output_normalizeactual)+(n)*jt_right_normalizeactualmod))))))", "specialize prime_field_polynomial_normalization_exists (n)", "specialize prime_field_polynomial_normalization_exists (b)", "specialize prime_field_polynomial_normalization_exists (c)", "specialize prime_field_polynomial_normalization_exists (k)", "apply prime_field_polynomial_normalization_exists", "exact hn", "cases hnorm", "cases hnorm_witness", "exists x", "exists x1", "split", "specialize prime_field_polynomial_normalization_bounded (n)", "specialize prime_field_polynomial_normalization_bounded (b)", "specialize prime_field_polynomial_normalization_bounded (c)", "specialize prime_field_polynomial_normalization_bounded (x)", "specialize prime_field_polynomial_normalization_bounded (x1)", "specialize prime_field_polynomial_normalization_bounded (k)", "apply prime_field_polynomial_normalization_bounded", "exact hnorm_witness_witness", "intro i", "intro a", "intro r", "intro hi", "intro ha", "intro hr", "have hvalue : ((exists jt_gap_normalizeresiduebound. jt_gap_normalizeresiduebound+S (r)=(n)) /\\ (exists jt_left_normalizeresiduemod jt_right_normalizeresiduemod. (a)+(n)*jt_left_normalizeresiduemod=(r)+(n)*jt_right_normalizeresiduemod))", "specialize prime_field_polynomial_normalization_entry (n)", "specialize prime_field_polynomial_normalization_entry (b)", "specialize prime_field_polynomial_normalization_entry (c)", "specialize prime_field_polynomial_normalization_entry (x)", "specialize prime_field_polynomial_normalization_entry (x1)", "specialize prime_field_polynomial_normalization_entry (k)", "specialize prime_field_polynomial_normalization_entry (i)", "specialize prime_field_polynomial_normalization_entry (a)", "specialize prime_field_polynomial_normalization_entry (r)", "apply prime_field_polynomial_normalization_entry", "exact hnorm_witness_witness", "exact hi", "exact ha", "exact hr", "cases hvalue", "exact hvalue_right" ], "script_sha256": "6fccc5e702d1f6edd0e38054f386f216de1a6d18e6aa86f8410f7aeb55a6bfcb", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_canonical_crt_candidate_theorems", "script_sha256": "6fccc5e702d1f6edd0e38054f386f216de1a6d18e6aa86f8410f7aeb55a6bfcb", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "a6659077a2de6cac97641a55abf2da0c8ec52e48e7ecb73a4950f2ed859e70ae" } ], "stable_member": false, "statement": "forall n b c k. ~(n=0) -> exists d e. ((forall jt_index_normalizebound. (exists jt_gap_normalizeboundindex. jt_gap_normalizeboundindex+S (jt_index_normalizebound)=(k)) -> exists jt_value_normalizebound. ((((exists fs_h_jt_normalizeboundat. fs_h_jt_normalizeboundat + S (jt_value_normalizebound) = S ((S (jt_index_normalizebound)) * e)) /\\ exists fs_q_jt_normalizeboundat. d = fs_q_jt_normalizeboundat * S ((S (jt_index_normalizebound)) * e) + (jt_value_normalizebound))) /\\ (exists jt_gap_normalizeboundvalue. jt_gap_normalizeboundvalue+S (jt_value_normalizebound)=(n)))) /\\ (forall jt_index_normalizemod jt_left_normalizemod jt_right_normalizemod. (exists jt_gap_normalizemodindex. jt_gap_normalizemodindex+S (jt_index_normalizemod)=(k)) -> (((exists fs_h_jt_normalizemodleft. fs_h_jt_normalizemodleft + S (jt_left_normalizemod) = S ((S (jt_index_normalizemod)) * c)) /\\ exists fs_q_jt_normalizemodleft. b = fs_q_jt_normalizemodleft * S ((S (jt_index_normalizemod)) * c) + (jt_left_normalizemod))) -> (((exists fs_h_jt_normalizemodright. fs_h_jt_normalizemodright + S (jt_right_normalizemod) = S ((S (jt_index_normalizemod)) * e)) /\\ exists fs_q_jt_normalizemodright. d = fs_q_jt_normalizemodright * S ((S (jt_index_normalizemod)) * e) + (jt_right_normalizemod))) -> (exists jt_left_normalizemodmod jt_right_normalizemodmod. (jt_left_normalizemod)+(n)*jt_left_normalizemodmod=(jt_right_normalizemod)+(n)*jt_right_normalizemodmod)))", "statement_sha256": "a6659077a2de6cac97641a55abf2da0c8ec52e48e7ecb73a4950f2ed859e70ae", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Canonical coordinate reduction works for every nonzero modulus, not just fields." }, { "admission_dependencies": [ "beta_at_exists", "mod_eq_trans" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 28, "body_proof_nodes": 50, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro hleft", "intro hright", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "have hx : ∃ x. BetaAt(d,e,i,x)", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hx", "specialize mod_eq_trans (n)", "specialize mod_eq_trans (a)", "specialize mod_eq_trans (x)", "specialize mod_eq_trans (z)", "apply mod_eq_trans", "specialize hleft (i)", "specialize hleft (a)", "specialize hleft (x)", "apply hleft", "exact hi", "exact ha", "exact hx_witness", "specialize hright (i)", "specialize hright (x)", "specialize hright (z)", "apply hright", "exact hi", "exact hx_witness", "exact hz" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. JordanTupleCongruence(n,b,c,d,e,k) → JordanTupleCongruence(n,d,e,f,g,k) → JordanTupleCongruence(n,b,c,f,g,k)", "defined_statement_sha256": "690d8288fb5bacc5f8f2d3f7638e0b00ead84714595c82716c929c64c93f3f8f", "definition_uses": { "ND0373": 3, "PD0013": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "5f04fde14711923c8a06a647f654240381012e625a4b48c7d956ea3c7791c817", "free_names": [], "script_definition_uses": { "PD0013": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hleft" } ], [ { "kind": "text", "text": "intro hright" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "have hx : " }, { "kind": "text", "text": "∃ x. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(d,e,i,x)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (d)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (e)" } ], [ { "kind": "text", "text": "specialize beta_at_exists (i)" } ], [ { "kind": "text", "text": "apply beta_at_exists" } ], [ { "kind": "text", "text": "cases hx" } ], [ { "kind": "text", "text": "specialize mod_eq_trans (n)" } ], [ { "kind": "text", "text": "specialize mod_eq_trans (a)" } ], [ { "kind": "text", "text": "specialize mod_eq_trans (x)" } ], [ { "kind": "text", "text": "specialize mod_eq_trans (z)" } ], [ { "kind": "text", "text": "apply mod_eq_trans" } ], [ { "kind": "text", "text": "specialize hleft (i)" } ], [ { "kind": "text", "text": "specialize hleft (a)" } ], [ { "kind": "text", "text": "specialize hleft (x)" } ], [ { "kind": "text", "text": "apply hleft" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hx_witness" } ], [ { "kind": "text", "text": "specialize hright (i)" } ], [ { "kind": "text", "text": "specialize hright (x)" } ], [ { "kind": "text", "text": "specialize hright (z)" } ], [ { "kind": "text", "text": "apply hright" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hx_witness" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0373": 3 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ k. " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,d,e,f,g,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,f,g,k)" } ] }, "dependencies": [ "beta_at_exists", "mod_eq_trans" ], "enrolled_in_alpha": true, "factory": "make_jordan_canonical_crt_candidate_theorems", "first_admitted_version": "v35", "id": "JT0030", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_congruence_trans", "original_ha_bundle_verified": true, "proof_bundle_node_id": 309, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro k", "intro hleft", "intro hright", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "have hx : exists x. ((exists fs_h_jt_modtransmiddle. fs_h_jt_modtransmiddle + S (x) = S ((S (i)) * e)) /\\ exists fs_q_jt_modtransmiddle. d = fs_q_jt_modtransmiddle * S ((S (i)) * e) + (x))", "specialize beta_at_exists (d)", "specialize beta_at_exists (e)", "specialize beta_at_exists (i)", "apply beta_at_exists", "cases hx", "specialize mod_eq_trans (n)", "specialize mod_eq_trans (a)", "specialize mod_eq_trans (x)", "specialize mod_eq_trans (z)", "apply mod_eq_trans", "specialize hleft (i)", "specialize hleft (a)", "specialize hleft (x)", "apply hleft", "exact hi", "exact ha", "exact hx_witness", "specialize hright (i)", "specialize hright (x)", "specialize hright (z)", "apply hright", "exact hi", "exact hx_witness", "exact hz" ], "script_sha256": "6f1f510e6fb1169e7db38c4e33596aef3381264121202923941d80836693cb53", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_canonical_crt_candidate_theorems", "script_sha256": "6f1f510e6fb1169e7db38c4e33596aef3381264121202923941d80836693cb53", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "5f04fde14711923c8a06a647f654240381012e625a4b48c7d956ea3c7791c817" } ], "stable_member": false, "statement": "forall n b c d e f g k. (forall jt_index_modtransfirst jt_left_modtransfirst jt_right_modtransfirst. (exists jt_gap_modtransfirstindex. jt_gap_modtransfirstindex+S (jt_index_modtransfirst)=(k)) -> (((exists fs_h_jt_modtransfirstleft. fs_h_jt_modtransfirstleft + S (jt_left_modtransfirst) = S ((S (jt_index_modtransfirst)) * c)) /\\ exists fs_q_jt_modtransfirstleft. b = fs_q_jt_modtransfirstleft * S ((S (jt_index_modtransfirst)) * c) + (jt_left_modtransfirst))) -> (((exists fs_h_jt_modtransfirstright. fs_h_jt_modtransfirstright + S (jt_right_modtransfirst) = S ((S (jt_index_modtransfirst)) * e)) /\\ exists fs_q_jt_modtransfirstright. d = fs_q_jt_modtransfirstright * S ((S (jt_index_modtransfirst)) * e) + (jt_right_modtransfirst))) -> (exists jt_left_modtransfirstmod jt_right_modtransfirstmod. (jt_left_modtransfirst)+(n)*jt_left_modtransfirstmod=(jt_right_modtransfirst)+(n)*jt_right_modtransfirstmod)) -> (forall jt_index_modtranssecond jt_left_modtranssecond jt_right_modtranssecond. (exists jt_gap_modtranssecondindex. jt_gap_modtranssecondindex+S (jt_index_modtranssecond)=(k)) -> (((exists fs_h_jt_modtranssecondleft. fs_h_jt_modtranssecondleft + S (jt_left_modtranssecond) = S ((S (jt_index_modtranssecond)) * e)) /\\ exists fs_q_jt_modtranssecondleft. d = fs_q_jt_modtranssecondleft * S ((S (jt_index_modtranssecond)) * e) + (jt_left_modtranssecond))) -> (((exists fs_h_jt_modtranssecondright. fs_h_jt_modtranssecondright + S (jt_right_modtranssecond) = S ((S (jt_index_modtranssecond)) * g)) /\\ exists fs_q_jt_modtranssecondright. f = fs_q_jt_modtranssecondright * S ((S (jt_index_modtranssecond)) * g) + (jt_right_modtranssecond))) -> (exists jt_left_modtranssecondmod jt_right_modtranssecondmod. (jt_left_modtranssecond)+(n)*jt_left_modtranssecondmod=(jt_right_modtranssecond)+(n)*jt_right_modtranssecondmod)) -> (forall jt_index_modtransresult jt_left_modtransresult jt_right_modtransresult. (exists jt_gap_modtransresultindex. jt_gap_modtransresultindex+S (jt_index_modtransresult)=(k)) -> (((exists fs_h_jt_modtransresultleft. fs_h_jt_modtransresultleft + S (jt_left_modtransresult) = S ((S (jt_index_modtransresult)) * c)) /\\ exists fs_q_jt_modtransresultleft. b = fs_q_jt_modtransresultleft * S ((S (jt_index_modtransresult)) * c) + (jt_left_modtransresult))) -> (((exists fs_h_jt_modtransresultright. fs_h_jt_modtransresultright + S (jt_right_modtransresult) = S ((S (jt_index_modtransresult)) * g)) /\\ exists fs_q_jt_modtransresultright. f = fs_q_jt_modtransresultright * S ((S (jt_index_modtransresult)) * g) + (jt_right_modtransresult))) -> (exists jt_left_modtransresultmod jt_right_modtransresultmod. (jt_left_modtransresult)+(n)*jt_left_modtransresultmod=(jt_right_modtransresult)+(n)*jt_right_modtransresultmod))", "statement_sha256": "5f04fde14711923c8a06a647f654240381012e625a4b48c7d956ea3c7791c817", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Actual decoded middle entries witness transitivity of coordinate congruence." }, { "admission_dependencies": [ "mod_eq_of_mod_eq_multiple" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 35, "body_proof_nodes": 55, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hdiv", "intro hmod", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_of_mod_eq_multiple (m)", "specialize mod_eq_of_mod_eq_multiple (n)", "specialize mod_eq_of_mod_eq_multiple (a)", "specialize mod_eq_of_mod_eq_multiple (z)", "apply mod_eq_of_mod_eq_multiple", "exact hdiv", "specialize hmod (i)", "specialize hmod (a)", "specialize hmod (z)", "apply hmod", "exact hi", "exact ha", "exact hz" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. Dvd(m,n) → JordanTupleCongruence(n,b,c,d,e,k) → JordanTupleCongruence(m,b,c,d,e,k)", "defined_statement_sha256": "d70fde4f982a811246d8d5d3d03cb2e399c734d8547a5c769dce90417aa4fb52", "definition_uses": { "ND0373": 2, "PD0003": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "6e8fff0689002c8a3fd7b4317a116860afefcfa9ab93ed0b933ebd69347ff2db", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hdiv" } ], [ { "kind": "text", "text": "intro hmod" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize mod_eq_of_mod_eq_multiple (m)" } ], [ { "kind": "text", "text": "specialize mod_eq_of_mod_eq_multiple (n)" } ], [ { "kind": "text", "text": "specialize mod_eq_of_mod_eq_multiple (a)" } ], [ { "kind": "text", "text": "specialize mod_eq_of_mod_eq_multiple (z)" } ], [ { "kind": "text", "text": "apply mod_eq_of_mod_eq_multiple" } ], [ { "kind": "text", "text": "exact hdiv" } ], [ { "kind": "text", "text": "specialize hmod (i)" } ], [ { "kind": "text", "text": "specialize hmod (a)" } ], [ { "kind": "text", "text": "specialize hmod (z)" } ], [ { "kind": "text", "text": "apply hmod" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0373": 2, "PD0003": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m,b,c,d,e,k)" } ] }, "dependencies": [ "mod_eq_of_mod_eq_multiple" ], "enrolled_in_alpha": true, "factory": "make_jordan_canonical_crt_candidate_theorems", "first_admitted_version": "v35", "id": "JT0031", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_congruence_divisor", "original_ha_bundle_verified": true, "proof_bundle_node_id": 310, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hdiv", "intro hmod", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_of_mod_eq_multiple (m)", "specialize mod_eq_of_mod_eq_multiple (n)", "specialize mod_eq_of_mod_eq_multiple (a)", "specialize mod_eq_of_mod_eq_multiple (z)", "apply mod_eq_of_mod_eq_multiple", "exact hdiv", "specialize hmod (i)", "specialize hmod (a)", "specialize hmod (z)", "apply hmod", "exact hi", "exact ha", "exact hz" ], "script_sha256": "3f42967da8e189031dea38e036e78139b427b121effb4690a7a042f046f0551a", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_canonical_crt_candidate_theorems", "script_sha256": "3f42967da8e189031dea38e036e78139b427b121effb4690a7a042f046f0551a", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "6e8fff0689002c8a3fd7b4317a116860afefcfa9ab93ed0b933ebd69347ff2db" } ], "stable_member": false, "statement": "forall m n b c d e k. (exists jt_factor_moddivisor. (n)=(m)*jt_factor_moddivisor) -> (forall jt_index_modlarger jt_left_modlarger jt_right_modlarger. (exists jt_gap_modlargerindex. jt_gap_modlargerindex+S (jt_index_modlarger)=(k)) -> (((exists fs_h_jt_modlargerleft. fs_h_jt_modlargerleft + S (jt_left_modlarger) = S ((S (jt_index_modlarger)) * c)) /\\ exists fs_q_jt_modlargerleft. b = fs_q_jt_modlargerleft * S ((S (jt_index_modlarger)) * c) + (jt_left_modlarger))) -> (((exists fs_h_jt_modlargerright. fs_h_jt_modlargerright + S (jt_right_modlarger) = S ((S (jt_index_modlarger)) * e)) /\\ exists fs_q_jt_modlargerright. d = fs_q_jt_modlargerright * S ((S (jt_index_modlarger)) * e) + (jt_right_modlarger))) -> (exists jt_left_modlargermod jt_right_modlargermod. (jt_left_modlarger)+(n)*jt_left_modlargermod=(jt_right_modlarger)+(n)*jt_right_modlargermod)) -> (forall jt_index_modsmaller jt_left_modsmaller jt_right_modsmaller. (exists jt_gap_modsmallerindex. jt_gap_modsmallerindex+S (jt_index_modsmaller)=(k)) -> (((exists fs_h_jt_modsmallerleft. fs_h_jt_modsmallerleft + S (jt_left_modsmaller) = S ((S (jt_index_modsmaller)) * c)) /\\ exists fs_q_jt_modsmallerleft. b = fs_q_jt_modsmallerleft * S ((S (jt_index_modsmaller)) * c) + (jt_left_modsmaller))) -> (((exists fs_h_jt_modsmallerright. fs_h_jt_modsmallerright + S (jt_right_modsmaller) = S ((S (jt_index_modsmaller)) * e)) /\\ exists fs_q_jt_modsmallerright. d = fs_q_jt_modsmallerright * S ((S (jt_index_modsmaller)) * e) + (jt_right_modsmaller))) -> (exists jt_left_modsmallermod jt_right_modsmallermod. (jt_left_modsmaller)+(m)*jt_left_modsmallermod=(jt_right_modsmaller)+(m)*jt_right_modsmallermod))", "statement_sha256": "6e8fff0689002c8a3fd7b4317a116860afefcfa9ab93ed0b933ebd69347ff2db", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Coordinate congruence descends along actual divisibility of moduli." }, { "admission_dependencies": [ "jordan_crt_tuple_exists", "mul_ne_zero", "jordan_tuple_normalize_exists", "jordan_tuple_congruence_divisor", "mul_comm", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm", "jordan_crt_tuple_left", "jordan_crt_tuple_right" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 40, "body_proof_nodes": 153, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hm", "intro hn", "intro hcop", "have hfamily : ∀ L. ∃ f. ∃ g. JordanTupleCRT(m,n,b,c,d,e,f,g,L)", "specialize jordan_crt_tuple_exists (m)", "specialize jordan_crt_tuple_exists (n)", "specialize jordan_crt_tuple_exists (b)", "specialize jordan_crt_tuple_exists (c)", "specialize jordan_crt_tuple_exists (d)", "specialize jordan_crt_tuple_exists (e)", "apply jordan_crt_tuple_exists", "exact hm", "exact hn", "exact hcop", "have hraw : ∃ f. ∃ g. JordanTupleCRT(m,n,b,c,d,e,f,g,k)", "specialize hfamily (k)", "apply hfamily", "cases hraw", "cases hraw_witness", "have hproduct : ~(m*n=0)", "intro hz", "specialize mul_ne_zero (m)", "specialize mul_ne_zero (n)", "apply mul_ne_zero", "exact hm", "exact hn", "exact hz", "have hnorm : ∃ u. ∃ v. BetaPrefixInto(u,v,k,m · n) ∧ JordanTupleCongruence(m · n,x,x1,u,v,k)", "specialize jordan_tuple_normalize_exists (m*n)", "specialize jordan_tuple_normalize_exists (x)", "specialize jordan_tuple_normalize_exists (x1)", "specialize jordan_tuple_normalize_exists (k)", "apply jordan_tuple_normalize_exists", "exact hproduct", "cases hnorm", "cases hnorm_witness", "cases hnorm_witness_witness", "exists x2", "exists x3", "split", "exact hnorm_witness_witness_left", "split", "have hsmall : JordanTupleCongruence(m,x,x1,x2,x3,k)", "specialize jordan_tuple_congruence_divisor (m)", "specialize jordan_tuple_congruence_divisor (m*n)", "specialize jordan_tuple_congruence_divisor (x)", "specialize jordan_tuple_congruence_divisor (x1)", "specialize jordan_tuple_congruence_divisor (x2)", "specialize jordan_tuple_congruence_divisor (x3)", "specialize jordan_tuple_congruence_divisor (k)", "apply jordan_tuple_congruence_divisor", "exists n", "refl", "exact hnorm_witness_witness_right", "specialize jordan_tuple_congruence_trans (m)", "specialize jordan_tuple_congruence_trans (x2)", "specialize jordan_tuple_congruence_trans (x3)", "specialize jordan_tuple_congruence_trans (x)", "specialize jordan_tuple_congruence_trans (x1)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_congruence_symm (m)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (x2)", "specialize jordan_tuple_congruence_symm (x3)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hsmall", "specialize jordan_crt_tuple_left (m)", "specialize jordan_crt_tuple_left (n)", "specialize jordan_crt_tuple_left (b)", "specialize jordan_crt_tuple_left (c)", "specialize jordan_crt_tuple_left (d)", "specialize jordan_crt_tuple_left (e)", "specialize jordan_crt_tuple_left (x)", "specialize jordan_crt_tuple_left (x1)", "specialize jordan_crt_tuple_left (k)", "apply jordan_crt_tuple_left", "exact hraw_witness_witness", "have hsmall : JordanTupleCongruence(n,x,x1,x2,x3,k)", "specialize jordan_tuple_congruence_divisor (n)", "specialize jordan_tuple_congruence_divisor (m*n)", "specialize jordan_tuple_congruence_divisor (x)", "specialize jordan_tuple_congruence_divisor (x1)", "specialize jordan_tuple_congruence_divisor (x2)", "specialize jordan_tuple_congruence_divisor (x3)", "specialize jordan_tuple_congruence_divisor (k)", "apply jordan_tuple_congruence_divisor", "exists m", "specialize mul_comm (m)", "specialize mul_comm (n)", "apply mul_comm", "exact hnorm_witness_witness_right", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (x2)", "specialize jordan_tuple_congruence_trans (x3)", "specialize jordan_tuple_congruence_trans (x)", "specialize jordan_tuple_congruence_trans (x1)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (x2)", "specialize jordan_tuple_congruence_symm (x3)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hsmall", "specialize jordan_crt_tuple_right (m)", "specialize jordan_crt_tuple_right (n)", "specialize jordan_crt_tuple_right (b)", "specialize jordan_crt_tuple_right (c)", "specialize jordan_crt_tuple_right (d)", "specialize jordan_crt_tuple_right (e)", "specialize jordan_crt_tuple_right (x)", "specialize jordan_crt_tuple_right (x1)", "specialize jordan_crt_tuple_right (k)", "apply jordan_crt_tuple_right", "exact hraw_witness_witness" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ¬m = 0 → ¬n = 0 → Coprime(m,n) → ∃ x. ∃ y. JordanCanonicalTupleCRT(m,n,b,c,d,e,x,y,k)", "defined_statement_sha256": "190b38fa5ed8b3784f1986db6af31c08f086b565a7388ad718ecf352753b4879", "definition_uses": { "ND0262": 1, "ND0373": 3, "ND0379": 2, "ND0380": 1, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "600b2631d0969f307fd951d8bf78ee88877422ce635eb0bf8b94021a37b8c7f3", "free_names": [], "script_definition_uses": { "ND0262": 1, "ND0373": 3, "ND0379": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "have hfamily : " }, { "kind": "text", "text": "∀ L. ∃ f. ∃ g. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,f,g,L)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_exists (m)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_exists (b)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_exists (c)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_exists (d)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_exists (e)" } ], [ { "kind": "text", "text": "apply jordan_crt_tuple_exists" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "have hraw : " }, { "kind": "text", "text": "∃ f. ∃ g. " }, { "definition": "ND0379", "kind": "definition", "text": "JordanTupleCRT(m,n,b,c,d,e,f,g,k)" } ], [ { "kind": "text", "text": "specialize hfamily (k)" } ], [ { "kind": "text", "text": "apply hfamily" } ], [ { "kind": "text", "text": "cases hraw" } ], [ { "kind": "text", "text": "cases hraw_witness" } ], [ { "kind": "text", "text": "have hproduct : ~(m*n=0)" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize mul_ne_zero (m)" } ], [ { "kind": "text", "text": "specialize mul_ne_zero (n)" } ], [ { "kind": "text", "text": "apply mul_ne_zero" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "have hnorm : " }, { "kind": "text", "text": "∃ u. ∃ v. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(u,v,k,m · n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m · n,x,x1,u,v,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (m*n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_normalize_exists" } ], [ { "kind": "text", "text": "exact hproduct" } ], [ { "kind": "text", "text": "cases hnorm" } ], [ { "kind": "text", "text": "cases hnorm_witness" } ], [ { "kind": "text", "text": "cases hnorm_witness_witness" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hnorm_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "have hsmall : " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m,x,x1,x2,x3,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (m*n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_divisor" } ], [ { "kind": "text", "text": "exists n" } ], [ { "kind": "text", "text": "refl" } ], [ { "kind": "text", "text": "exact hnorm_witness_witness_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hsmall" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (m)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (n)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (b)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (c)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (d)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (e)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (x)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (x1)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_left (k)" } ], [ { "kind": "text", "text": "apply jordan_crt_tuple_left" } ], [ { "kind": "text", "text": "exact hraw_witness_witness" } ], [ { "kind": "text", "text": "have hsmall : " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,x,x1,x2,x3,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (m*n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_divisor (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_divisor" } ], [ { "kind": "text", "text": "exists m" } ], [ { "kind": "text", "text": "specialize mul_comm (m)" } ], [ { "kind": "text", "text": "specialize mul_comm (n)" } ], [ { "kind": "text", "text": "apply mul_comm" } ], [ { "kind": "text", "text": "exact hnorm_witness_witness_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hsmall" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (m)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (n)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (b)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (c)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (d)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (e)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (x)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (x1)" } ], [ { "kind": "text", "text": "specialize jordan_crt_tuple_right (k)" } ], [ { "kind": "text", "text": "apply jordan_crt_tuple_right" } ], [ { "kind": "text", "text": "exact hraw_witness_witness" } ] ], "statement_definition_uses": { "ND0380": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → ∃ x. ∃ y. " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,x,y,k)" } ] }, "dependencies": [ "jordan_crt_tuple_exists", "mul_ne_zero", "jordan_tuple_normalize_exists", "jordan_tuple_congruence_divisor", "mul_comm", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm", "jordan_crt_tuple_left", "jordan_crt_tuple_right" ], "enrolled_in_alpha": true, "factory": "make_jordan_canonical_crt_candidate_theorems", "first_admitted_version": "v35", "id": "JT0032", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_canonical_crt_tuple_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 311, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hm", "intro hn", "intro hcop", "have hfamily : forall L. exists f g. forall jt_index_canonicalfamily. (exists jt_gap_canonicalfamilyindex. jt_gap_canonicalfamilyindex+S (jt_index_canonicalfamily)=(L)) -> exists jt_left_canonicalfamily jt_right_canonicalfamily jt_output_canonicalfamily. ((((exists fs_h_jt_canonicalfamilyleft. fs_h_jt_canonicalfamilyleft + S (jt_left_canonicalfamily) = S ((S (jt_index_canonicalfamily)) * c)) /\\ exists fs_q_jt_canonicalfamilyleft. b = fs_q_jt_canonicalfamilyleft * S ((S (jt_index_canonicalfamily)) * c) + (jt_left_canonicalfamily))) /\\ (((((exists fs_h_jt_canonicalfamilyright. fs_h_jt_canonicalfamilyright + S (jt_right_canonicalfamily) = S ((S (jt_index_canonicalfamily)) * e)) /\\ exists fs_q_jt_canonicalfamilyright. d = fs_q_jt_canonicalfamilyright * S ((S (jt_index_canonicalfamily)) * e) + (jt_right_canonicalfamily))) /\\ (((((exists fs_h_jt_canonicalfamilyoutput. fs_h_jt_canonicalfamilyoutput + S (jt_output_canonicalfamily) = S ((S (jt_index_canonicalfamily)) * g)) /\\ exists fs_q_jt_canonicalfamilyoutput. f = fs_q_jt_canonicalfamilyoutput * S ((S (jt_index_canonicalfamily)) * g) + (jt_output_canonicalfamily))) /\\ (((exists jt_left_canonicalfamilymodleft jt_right_canonicalfamilymodleft. (jt_output_canonicalfamily)+(m)*jt_left_canonicalfamilymodleft=(jt_left_canonicalfamily)+(m)*jt_right_canonicalfamilymodleft) /\\ (exists jt_left_canonicalfamilymodright jt_right_canonicalfamilymodright. (jt_output_canonicalfamily)+(n)*jt_left_canonicalfamilymodright=(jt_right_canonicalfamily)+(n)*jt_right_canonicalfamilymodright))))))))", "specialize jordan_crt_tuple_exists (m)", "specialize jordan_crt_tuple_exists (n)", "specialize jordan_crt_tuple_exists (b)", "specialize jordan_crt_tuple_exists (c)", "specialize jordan_crt_tuple_exists (d)", "specialize jordan_crt_tuple_exists (e)", "apply jordan_crt_tuple_exists", "exact hm", "exact hn", "exact hcop", "have hraw : exists f g. forall jt_index_canonicalraw. (exists jt_gap_canonicalrawindex. jt_gap_canonicalrawindex+S (jt_index_canonicalraw)=(k)) -> exists jt_left_canonicalraw jt_right_canonicalraw jt_output_canonicalraw. ((((exists fs_h_jt_canonicalrawleft. fs_h_jt_canonicalrawleft + S (jt_left_canonicalraw) = S ((S (jt_index_canonicalraw)) * c)) /\\ exists fs_q_jt_canonicalrawleft. b = fs_q_jt_canonicalrawleft * S ((S (jt_index_canonicalraw)) * c) + (jt_left_canonicalraw))) /\\ (((((exists fs_h_jt_canonicalrawright. fs_h_jt_canonicalrawright + S (jt_right_canonicalraw) = S ((S (jt_index_canonicalraw)) * e)) /\\ exists fs_q_jt_canonicalrawright. d = fs_q_jt_canonicalrawright * S ((S (jt_index_canonicalraw)) * e) + (jt_right_canonicalraw))) /\\ (((((exists fs_h_jt_canonicalrawoutput. fs_h_jt_canonicalrawoutput + S (jt_output_canonicalraw) = S ((S (jt_index_canonicalraw)) * g)) /\\ exists fs_q_jt_canonicalrawoutput. f = fs_q_jt_canonicalrawoutput * S ((S (jt_index_canonicalraw)) * g) + (jt_output_canonicalraw))) /\\ (((exists jt_left_canonicalrawmodleft jt_right_canonicalrawmodleft. (jt_output_canonicalraw)+(m)*jt_left_canonicalrawmodleft=(jt_left_canonicalraw)+(m)*jt_right_canonicalrawmodleft) /\\ (exists jt_left_canonicalrawmodright jt_right_canonicalrawmodright. (jt_output_canonicalraw)+(n)*jt_left_canonicalrawmodright=(jt_right_canonicalraw)+(n)*jt_right_canonicalrawmodright))))))))", "specialize hfamily (k)", "apply hfamily", "cases hraw", "cases hraw_witness", "have hproduct : ~(m*n=0)", "intro hz", "specialize mul_ne_zero (m)", "specialize mul_ne_zero (n)", "apply mul_ne_zero", "exact hm", "exact hn", "exact hz", "have hnorm : exists u v. ((forall jt_index_canonicalbound. (exists jt_gap_canonicalboundindex. jt_gap_canonicalboundindex+S (jt_index_canonicalbound)=(k)) -> exists jt_value_canonicalbound. ((((exists fs_h_jt_canonicalboundat. fs_h_jt_canonicalboundat + S (jt_value_canonicalbound) = S ((S (jt_index_canonicalbound)) * v)) /\\ exists fs_q_jt_canonicalboundat. u = fs_q_jt_canonicalboundat * S ((S (jt_index_canonicalbound)) * v) + (jt_value_canonicalbound))) /\\ (exists jt_gap_canonicalboundvalue. jt_gap_canonicalboundvalue+S (jt_value_canonicalbound)=(m*n)))) /\\ (forall jt_index_canonicalmod jt_left_canonicalmod jt_right_canonicalmod. (exists jt_gap_canonicalmodindex. jt_gap_canonicalmodindex+S (jt_index_canonicalmod)=(k)) -> (((exists fs_h_jt_canonicalmodleft. fs_h_jt_canonicalmodleft + S (jt_left_canonicalmod) = S ((S (jt_index_canonicalmod)) * x1)) /\\ exists fs_q_jt_canonicalmodleft. x = fs_q_jt_canonicalmodleft * S ((S (jt_index_canonicalmod)) * x1) + (jt_left_canonicalmod))) -> (((exists fs_h_jt_canonicalmodright. fs_h_jt_canonicalmodright + S (jt_right_canonicalmod) = S ((S (jt_index_canonicalmod)) * v)) /\\ exists fs_q_jt_canonicalmodright. u = fs_q_jt_canonicalmodright * S ((S (jt_index_canonicalmod)) * v) + (jt_right_canonicalmod))) -> (exists jt_left_canonicalmodmod jt_right_canonicalmodmod. (jt_left_canonicalmod)+(m*n)*jt_left_canonicalmodmod=(jt_right_canonicalmod)+(m*n)*jt_right_canonicalmodmod)))", "specialize jordan_tuple_normalize_exists (m*n)", "specialize jordan_tuple_normalize_exists (x)", "specialize jordan_tuple_normalize_exists (x1)", "specialize jordan_tuple_normalize_exists (k)", "apply jordan_tuple_normalize_exists", "exact hproduct", "cases hnorm", "cases hnorm_witness", "cases hnorm_witness_witness", "exists x2", "exists x3", "split", "exact hnorm_witness_witness_left", "split", "have hsmall : forall jt_index_canonicalsmallleft jt_left_canonicalsmallleft jt_right_canonicalsmallleft. (exists jt_gap_canonicalsmallleftindex. jt_gap_canonicalsmallleftindex+S (jt_index_canonicalsmallleft)=(k)) -> (((exists fs_h_jt_canonicalsmallleftleft. fs_h_jt_canonicalsmallleftleft + S (jt_left_canonicalsmallleft) = S ((S (jt_index_canonicalsmallleft)) * x1)) /\\ exists fs_q_jt_canonicalsmallleftleft. x = fs_q_jt_canonicalsmallleftleft * S ((S (jt_index_canonicalsmallleft)) * x1) + (jt_left_canonicalsmallleft))) -> (((exists fs_h_jt_canonicalsmallleftright. fs_h_jt_canonicalsmallleftright + S (jt_right_canonicalsmallleft) = S ((S (jt_index_canonicalsmallleft)) * x3)) /\\ exists fs_q_jt_canonicalsmallleftright. x2 = fs_q_jt_canonicalsmallleftright * S ((S (jt_index_canonicalsmallleft)) * x3) + (jt_right_canonicalsmallleft))) -> (exists jt_left_canonicalsmallleftmod jt_right_canonicalsmallleftmod. (jt_left_canonicalsmallleft)+(m)*jt_left_canonicalsmallleftmod=(jt_right_canonicalsmallleft)+(m)*jt_right_canonicalsmallleftmod)", "specialize jordan_tuple_congruence_divisor (m)", "specialize jordan_tuple_congruence_divisor (m*n)", "specialize jordan_tuple_congruence_divisor (x)", "specialize jordan_tuple_congruence_divisor (x1)", "specialize jordan_tuple_congruence_divisor (x2)", "specialize jordan_tuple_congruence_divisor (x3)", "specialize jordan_tuple_congruence_divisor (k)", "apply jordan_tuple_congruence_divisor", "exists n", "refl", "exact hnorm_witness_witness_right", "specialize jordan_tuple_congruence_trans (m)", "specialize jordan_tuple_congruence_trans (x2)", "specialize jordan_tuple_congruence_trans (x3)", "specialize jordan_tuple_congruence_trans (x)", "specialize jordan_tuple_congruence_trans (x1)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_congruence_symm (m)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (x2)", "specialize jordan_tuple_congruence_symm (x3)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hsmall", "specialize jordan_crt_tuple_left (m)", "specialize jordan_crt_tuple_left (n)", "specialize jordan_crt_tuple_left (b)", "specialize jordan_crt_tuple_left (c)", "specialize jordan_crt_tuple_left (d)", "specialize jordan_crt_tuple_left (e)", "specialize jordan_crt_tuple_left (x)", "specialize jordan_crt_tuple_left (x1)", "specialize jordan_crt_tuple_left (k)", "apply jordan_crt_tuple_left", "exact hraw_witness_witness", "have hsmall : forall jt_index_canonicalsmallright jt_left_canonicalsmallright jt_right_canonicalsmallright. (exists jt_gap_canonicalsmallrightindex. jt_gap_canonicalsmallrightindex+S (jt_index_canonicalsmallright)=(k)) -> (((exists fs_h_jt_canonicalsmallrightleft. fs_h_jt_canonicalsmallrightleft + S (jt_left_canonicalsmallright) = S ((S (jt_index_canonicalsmallright)) * x1)) /\\ exists fs_q_jt_canonicalsmallrightleft. x = fs_q_jt_canonicalsmallrightleft * S ((S (jt_index_canonicalsmallright)) * x1) + (jt_left_canonicalsmallright))) -> (((exists fs_h_jt_canonicalsmallrightright. fs_h_jt_canonicalsmallrightright + S (jt_right_canonicalsmallright) = S ((S (jt_index_canonicalsmallright)) * x3)) /\\ exists fs_q_jt_canonicalsmallrightright. x2 = fs_q_jt_canonicalsmallrightright * S ((S (jt_index_canonicalsmallright)) * x3) + (jt_right_canonicalsmallright))) -> (exists jt_left_canonicalsmallrightmod jt_right_canonicalsmallrightmod. (jt_left_canonicalsmallright)+(n)*jt_left_canonicalsmallrightmod=(jt_right_canonicalsmallright)+(n)*jt_right_canonicalsmallrightmod)", "specialize jordan_tuple_congruence_divisor (n)", "specialize jordan_tuple_congruence_divisor (m*n)", "specialize jordan_tuple_congruence_divisor (x)", "specialize jordan_tuple_congruence_divisor (x1)", "specialize jordan_tuple_congruence_divisor (x2)", "specialize jordan_tuple_congruence_divisor (x3)", "specialize jordan_tuple_congruence_divisor (k)", "apply jordan_tuple_congruence_divisor", "exists m", "specialize mul_comm (m)", "specialize mul_comm (n)", "apply mul_comm", "exact hnorm_witness_witness_right", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (x2)", "specialize jordan_tuple_congruence_trans (x3)", "specialize jordan_tuple_congruence_trans (x)", "specialize jordan_tuple_congruence_trans (x1)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (x2)", "specialize jordan_tuple_congruence_symm (x3)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hsmall", "specialize jordan_crt_tuple_right (m)", "specialize jordan_crt_tuple_right (n)", "specialize jordan_crt_tuple_right (b)", "specialize jordan_crt_tuple_right (c)", "specialize jordan_crt_tuple_right (d)", "specialize jordan_crt_tuple_right (e)", "specialize jordan_crt_tuple_right (x)", "specialize jordan_crt_tuple_right (x1)", "specialize jordan_crt_tuple_right (k)", "apply jordan_crt_tuple_right", "exact hraw_witness_witness" ], "script_sha256": "02a2291cbfadd9d3520cf3c54eeaa7505e36f73d346cf3d5969909ba29ecf10b", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_canonical_crt_candidate_theorems", "script_sha256": "02a2291cbfadd9d3520cf3c54eeaa7505e36f73d346cf3d5969909ba29ecf10b", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "600b2631d0969f307fd951d8bf78ee88877422ce635eb0bf8b94021a37b8c7f3" } ], "stable_member": false, "statement": "forall m n b c d e k. ~(m=0) -> ~(n=0) -> (forall jt_divisor_canonicalcrtcoprime. (exists jt_factor_canonicalcrtcoprimea. (m)=(jt_divisor_canonicalcrtcoprime)*jt_factor_canonicalcrtcoprimea) -> (exists jt_factor_canonicalcrtcoprimeb. (n)=(jt_divisor_canonicalcrtcoprime)*jt_factor_canonicalcrtcoprimeb) -> jt_divisor_canonicalcrtcoprime=1) -> exists f g. ((forall jt_index_canonicalcrtexistsbound. (exists jt_gap_canonicalcrtexistsboundindex. jt_gap_canonicalcrtexistsboundindex+S (jt_index_canonicalcrtexistsbound)=(k)) -> exists jt_value_canonicalcrtexistsbound. ((((exists fs_h_jt_canonicalcrtexistsboundat. fs_h_jt_canonicalcrtexistsboundat + S (jt_value_canonicalcrtexistsbound) = S ((S (jt_index_canonicalcrtexistsbound)) * g)) /\\ exists fs_q_jt_canonicalcrtexistsboundat. f = fs_q_jt_canonicalcrtexistsboundat * S ((S (jt_index_canonicalcrtexistsbound)) * g) + (jt_value_canonicalcrtexistsbound))) /\\ (exists jt_gap_canonicalcrtexistsboundvalue. jt_gap_canonicalcrtexistsboundvalue+S (jt_value_canonicalcrtexistsbound)=(m*n)))) /\\ (((forall jt_index_canonicalcrtexistsleft jt_left_canonicalcrtexistsleft jt_right_canonicalcrtexistsleft. (exists jt_gap_canonicalcrtexistsleftindex. jt_gap_canonicalcrtexistsleftindex+S (jt_index_canonicalcrtexistsleft)=(k)) -> (((exists fs_h_jt_canonicalcrtexistsleftleft. fs_h_jt_canonicalcrtexistsleftleft + S (jt_left_canonicalcrtexistsleft) = S ((S (jt_index_canonicalcrtexistsleft)) * g)) /\\ exists fs_q_jt_canonicalcrtexistsleftleft. f = fs_q_jt_canonicalcrtexistsleftleft * S ((S (jt_index_canonicalcrtexistsleft)) * g) + (jt_left_canonicalcrtexistsleft))) -> (((exists fs_h_jt_canonicalcrtexistsleftright. fs_h_jt_canonicalcrtexistsleftright + S (jt_right_canonicalcrtexistsleft) = S ((S (jt_index_canonicalcrtexistsleft)) * c)) /\\ exists fs_q_jt_canonicalcrtexistsleftright. b = fs_q_jt_canonicalcrtexistsleftright * S ((S (jt_index_canonicalcrtexistsleft)) * c) + (jt_right_canonicalcrtexistsleft))) -> (exists jt_left_canonicalcrtexistsleftmod jt_right_canonicalcrtexistsleftmod. (jt_left_canonicalcrtexistsleft)+(m)*jt_left_canonicalcrtexistsleftmod=(jt_right_canonicalcrtexistsleft)+(m)*jt_right_canonicalcrtexistsleftmod)) /\\ (forall jt_index_canonicalcrtexistsright jt_left_canonicalcrtexistsright jt_right_canonicalcrtexistsright. (exists jt_gap_canonicalcrtexistsrightindex. jt_gap_canonicalcrtexistsrightindex+S (jt_index_canonicalcrtexistsright)=(k)) -> (((exists fs_h_jt_canonicalcrtexistsrightleft. fs_h_jt_canonicalcrtexistsrightleft + S (jt_left_canonicalcrtexistsright) = S ((S (jt_index_canonicalcrtexistsright)) * g)) /\\ exists fs_q_jt_canonicalcrtexistsrightleft. f = fs_q_jt_canonicalcrtexistsrightleft * S ((S (jt_index_canonicalcrtexistsright)) * g) + (jt_left_canonicalcrtexistsright))) -> (((exists fs_h_jt_canonicalcrtexistsrightright. fs_h_jt_canonicalcrtexistsrightright + S (jt_right_canonicalcrtexistsright) = S ((S (jt_index_canonicalcrtexistsright)) * e)) /\\ exists fs_q_jt_canonicalcrtexistsrightright. d = fs_q_jt_canonicalcrtexistsrightright * S ((S (jt_index_canonicalcrtexistsright)) * e) + (jt_right_canonicalcrtexistsright))) -> (exists jt_left_canonicalcrtexistsrightmod jt_right_canonicalcrtexistsrightmod. (jt_left_canonicalcrtexistsright)+(n)*jt_left_canonicalcrtexistsrightmod=(jt_right_canonicalcrtexistsright)+(n)*jt_right_canonicalcrtexistsrightmod)))))", "statement_sha256": "600b2631d0969f307fd951d8bf78ee88877422ce635eb0bf8b94021a37b8c7f3", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Construct an actual tuple bounded by the product modulus with both prescribed residue tuples." }, { "admission_dependencies": [ "jordan_canonical_crt_tuple_exists", "jordan_primitive_tuple_coprime_product", "jordan_primitive_tuple_congruence_transport", "jordan_tuple_congruence_symm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 33, "body_proof_nodes": 102, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "have hcrt : ∃ f. ∃ g. JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)", "specialize jordan_canonical_crt_tuple_exists (m)", "specialize jordan_canonical_crt_tuple_exists (n)", "specialize jordan_canonical_crt_tuple_exists (b)", "specialize jordan_canonical_crt_tuple_exists (c)", "specialize jordan_canonical_crt_tuple_exists (d)", "specialize jordan_canonical_crt_tuple_exists (e)", "specialize jordan_canonical_crt_tuple_exists (k)", "apply jordan_canonical_crt_tuple_exists", "exact hm", "exact hn", "exact hcop", "cases hcrt", "cases hcrt_witness", "cases hcrt_witness_witness", "cases hcrt_witness_witness_right", "exists x", "exists x1", "split", "split", "exact hcrt_witness_witness_left", "split", "exact hcrt_witness_witness_right_left", "exact hcrt_witness_witness_right_right", "specialize jordan_primitive_tuple_coprime_product (m)", "specialize jordan_primitive_tuple_coprime_product (n)", "specialize jordan_primitive_tuple_coprime_product (x)", "specialize jordan_primitive_tuple_coprime_product (x1)", "specialize jordan_primitive_tuple_coprime_product (k)", "apply jordan_primitive_tuple_coprime_product", "exact hm", "exact hn", "exact hcop", "specialize jordan_primitive_tuple_congruence_transport (m)", "specialize jordan_primitive_tuple_congruence_transport (b)", "specialize jordan_primitive_tuple_congruence_transport (c)", "specialize jordan_primitive_tuple_congruence_transport (x)", "specialize jordan_primitive_tuple_congruence_transport (x1)", "specialize jordan_primitive_tuple_congruence_transport (k)", "apply jordan_primitive_tuple_congruence_transport", "specialize jordan_tuple_congruence_symm (m)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (b)", "specialize jordan_tuple_congruence_symm (c)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hcrt_witness_witness_right_left", "exact hleft", "specialize jordan_primitive_tuple_congruence_transport (n)", "specialize jordan_primitive_tuple_congruence_transport (d)", "specialize jordan_primitive_tuple_congruence_transport (e)", "specialize jordan_primitive_tuple_congruence_transport (x)", "specialize jordan_primitive_tuple_congruence_transport (x1)", "specialize jordan_primitive_tuple_congruence_transport (k)", "apply jordan_primitive_tuple_congruence_transport", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (d)", "specialize jordan_tuple_congruence_symm (e)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hcrt_witness_witness_right_right", "exact hright" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanPrimitiveTuple(m,b,c,k) → JordanPrimitiveTuple(n,d,e,k) → ∃ x. ∃ y. JordanCanonicalTupleCRT(m,n,b,c,d,e,x,y,k) ∧ JordanPrimitiveTuple(m · n,x,y,k)", "defined_statement_sha256": "197b033691ac087bf409139321269fa870f37a76777478166e18cf0e5cbaf51a", "definition_uses": { "ND0372": 3, "ND0380": 2, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "49ee99529794f38d85e5097131bb9b82a09d960901449727d8b03d99d826fec1", "free_names": [], "script_definition_uses": { "ND0380": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hleft" } ], [ { "kind": "text", "text": "intro hright" } ], [ { "kind": "text", "text": "have hcrt : " }, { "kind": "text", "text": "∃ f. ∃ g. " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (m)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (b)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (c)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (d)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (e)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_exists (k)" } ], [ { "kind": "text", "text": "apply jordan_canonical_crt_tuple_exists" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "cases hcrt" } ], [ { "kind": "text", "text": "cases hcrt_witness" } ], [ { "kind": "text", "text": "cases hcrt_witness_witness" } ], [ { "kind": "text", "text": "cases hcrt_witness_witness_right" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hcrt_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hcrt_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hcrt_witness_witness_right_right" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_coprime_product (m)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_coprime_product (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_coprime_product (x)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_coprime_product (x1)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_coprime_product (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_coprime_product" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (m)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (x1)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_congruence_transport" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hcrt_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hleft" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (d)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (e)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (x1)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_congruence_transport" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hcrt_witness_witness_right_right" } ], [ { "kind": "text", "text": "exact hright" } ] ], "statement_definition_uses": { "ND0372": 3, "ND0380": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,d,e,k)" }, { "kind": "text", "text": " → ∃ x. ∃ y. " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,x,y,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,x,y,k)" } ] }, "dependencies": [ "jordan_canonical_crt_tuple_exists", "jordan_primitive_tuple_coprime_product", "jordan_primitive_tuple_congruence_transport", "jordan_tuple_congruence_symm" ], "enrolled_in_alpha": true, "factory": "make_jordan_canonical_crt_candidate_theorems", "first_admitted_version": "v35", "id": "JT0033", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_crt_tuple_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 312, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "have hcrt : exists f g. ((forall jt_index_primitivecrtactualbound. (exists jt_gap_primitivecrtactualboundindex. jt_gap_primitivecrtactualboundindex+S (jt_index_primitivecrtactualbound)=(k)) -> exists jt_value_primitivecrtactualbound. ((((exists fs_h_jt_primitivecrtactualboundat. fs_h_jt_primitivecrtactualboundat + S (jt_value_primitivecrtactualbound) = S ((S (jt_index_primitivecrtactualbound)) * g)) /\\ exists fs_q_jt_primitivecrtactualboundat. f = fs_q_jt_primitivecrtactualboundat * S ((S (jt_index_primitivecrtactualbound)) * g) + (jt_value_primitivecrtactualbound))) /\\ (exists jt_gap_primitivecrtactualboundvalue. jt_gap_primitivecrtactualboundvalue+S (jt_value_primitivecrtactualbound)=(m*n)))) /\\ (((forall jt_index_primitivecrtactualleft jt_left_primitivecrtactualleft jt_right_primitivecrtactualleft. (exists jt_gap_primitivecrtactualleftindex. jt_gap_primitivecrtactualleftindex+S (jt_index_primitivecrtactualleft)=(k)) -> (((exists fs_h_jt_primitivecrtactualleftleft. fs_h_jt_primitivecrtactualleftleft + S (jt_left_primitivecrtactualleft) = S ((S (jt_index_primitivecrtactualleft)) * g)) /\\ exists fs_q_jt_primitivecrtactualleftleft. f = fs_q_jt_primitivecrtactualleftleft * S ((S (jt_index_primitivecrtactualleft)) * g) + (jt_left_primitivecrtactualleft))) -> (((exists fs_h_jt_primitivecrtactualleftright. fs_h_jt_primitivecrtactualleftright + S (jt_right_primitivecrtactualleft) = S ((S (jt_index_primitivecrtactualleft)) * c)) /\\ exists fs_q_jt_primitivecrtactualleftright. b = fs_q_jt_primitivecrtactualleftright * S ((S (jt_index_primitivecrtactualleft)) * c) + (jt_right_primitivecrtactualleft))) -> (exists jt_left_primitivecrtactualleftmod jt_right_primitivecrtactualleftmod. (jt_left_primitivecrtactualleft)+(m)*jt_left_primitivecrtactualleftmod=(jt_right_primitivecrtactualleft)+(m)*jt_right_primitivecrtactualleftmod)) /\\ (forall jt_index_primitivecrtactualright jt_left_primitivecrtactualright jt_right_primitivecrtactualright. (exists jt_gap_primitivecrtactualrightindex. jt_gap_primitivecrtactualrightindex+S (jt_index_primitivecrtactualright)=(k)) -> (((exists fs_h_jt_primitivecrtactualrightleft. fs_h_jt_primitivecrtactualrightleft + S (jt_left_primitivecrtactualright) = S ((S (jt_index_primitivecrtactualright)) * g)) /\\ exists fs_q_jt_primitivecrtactualrightleft. f = fs_q_jt_primitivecrtactualrightleft * S ((S (jt_index_primitivecrtactualright)) * g) + (jt_left_primitivecrtactualright))) -> (((exists fs_h_jt_primitivecrtactualrightright. fs_h_jt_primitivecrtactualrightright + S (jt_right_primitivecrtactualright) = S ((S (jt_index_primitivecrtactualright)) * e)) /\\ exists fs_q_jt_primitivecrtactualrightright. d = fs_q_jt_primitivecrtactualrightright * S ((S (jt_index_primitivecrtactualright)) * e) + (jt_right_primitivecrtactualright))) -> (exists jt_left_primitivecrtactualrightmod jt_right_primitivecrtactualrightmod. (jt_left_primitivecrtactualright)+(n)*jt_left_primitivecrtactualrightmod=(jt_right_primitivecrtactualright)+(n)*jt_right_primitivecrtactualrightmod)))))", "specialize jordan_canonical_crt_tuple_exists (m)", "specialize jordan_canonical_crt_tuple_exists (n)", "specialize jordan_canonical_crt_tuple_exists (b)", "specialize jordan_canonical_crt_tuple_exists (c)", "specialize jordan_canonical_crt_tuple_exists (d)", "specialize jordan_canonical_crt_tuple_exists (e)", "specialize jordan_canonical_crt_tuple_exists (k)", "apply jordan_canonical_crt_tuple_exists", "exact hm", "exact hn", "exact hcop", "cases hcrt", "cases hcrt_witness", "cases hcrt_witness_witness", "cases hcrt_witness_witness_right", "exists x", "exists x1", "split", "split", "exact hcrt_witness_witness_left", "split", "exact hcrt_witness_witness_right_left", "exact hcrt_witness_witness_right_right", "specialize jordan_primitive_tuple_coprime_product (m)", "specialize jordan_primitive_tuple_coprime_product (n)", "specialize jordan_primitive_tuple_coprime_product (x)", "specialize jordan_primitive_tuple_coprime_product (x1)", "specialize jordan_primitive_tuple_coprime_product (k)", "apply jordan_primitive_tuple_coprime_product", "exact hm", "exact hn", "exact hcop", "specialize jordan_primitive_tuple_congruence_transport (m)", "specialize jordan_primitive_tuple_congruence_transport (b)", "specialize jordan_primitive_tuple_congruence_transport (c)", "specialize jordan_primitive_tuple_congruence_transport (x)", "specialize jordan_primitive_tuple_congruence_transport (x1)", "specialize jordan_primitive_tuple_congruence_transport (k)", "apply jordan_primitive_tuple_congruence_transport", "specialize jordan_tuple_congruence_symm (m)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (b)", "specialize jordan_tuple_congruence_symm (c)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hcrt_witness_witness_right_left", "exact hleft", "specialize jordan_primitive_tuple_congruence_transport (n)", "specialize jordan_primitive_tuple_congruence_transport (d)", "specialize jordan_primitive_tuple_congruence_transport (e)", "specialize jordan_primitive_tuple_congruence_transport (x)", "specialize jordan_primitive_tuple_congruence_transport (x1)", "specialize jordan_primitive_tuple_congruence_transport (k)", "apply jordan_primitive_tuple_congruence_transport", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (x)", "specialize jordan_tuple_congruence_symm (x1)", "specialize jordan_tuple_congruence_symm (d)", "specialize jordan_tuple_congruence_symm (e)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hcrt_witness_witness_right_right", "exact hright" ], "script_sha256": "d685a845aafb2c7528139a79ce20d009f6d88e6bce466051b119634750590ba6", "source_filename": "jordan_totient_candidate.py", "source_module": "peano_lab.library.jordan_totient_candidate", "sources": [ { "factory": "make_jordan_canonical_crt_candidate_theorems", "script_sha256": "d685a845aafb2c7528139a79ce20d009f6d88e6bce466051b119634750590ba6", "selected": true, "source_module": "peano_lab.library.jordan_totient_candidate", "source_sha256": "ec2f9c368b4d30dfb8ffe0a2c89dca6e82966d3c8819ce10d29123189fe7052c", "statement_sha256": "49ee99529794f38d85e5097131bb9b82a09d960901449727d8b03d99d826fec1" } ], "stable_member": false, "statement": "forall m n b c d e k. ~(m=0) -> ~(n=0) -> (forall jt_divisor_primitivecrtcoprime. (exists jt_factor_primitivecrtcoprimea. (m)=(jt_divisor_primitivecrtcoprime)*jt_factor_primitivecrtcoprimea) -> (exists jt_factor_primitivecrtcoprimeb. (n)=(jt_divisor_primitivecrtcoprime)*jt_factor_primitivecrtcoprimeb) -> jt_divisor_primitivecrtcoprime=1) -> (forall jt_divisor_primitivecrtleft. (exists jt_factor_primitivecrtleftmodulus. (m)=(jt_divisor_primitivecrtleft)*jt_factor_primitivecrtleftmodulus) -> (forall jt_index_primitivecrtleftcoordinates jt_value_primitivecrtleftcoordinates. (exists jt_gap_primitivecrtleftcoordinatesindex. jt_gap_primitivecrtleftcoordinatesindex+S (jt_index_primitivecrtleftcoordinates)=(k)) -> (((exists fs_h_jt_primitivecrtleftcoordinatesat. fs_h_jt_primitivecrtleftcoordinatesat + S (jt_value_primitivecrtleftcoordinates) = S ((S (jt_index_primitivecrtleftcoordinates)) * c)) /\\ exists fs_q_jt_primitivecrtleftcoordinatesat. b = fs_q_jt_primitivecrtleftcoordinatesat * S ((S (jt_index_primitivecrtleftcoordinates)) * c) + (jt_value_primitivecrtleftcoordinates))) -> (exists jt_factor_primitivecrtleftcoordinatesdivides. (jt_value_primitivecrtleftcoordinates)=(jt_divisor_primitivecrtleft)*jt_factor_primitivecrtleftcoordinatesdivides)) -> jt_divisor_primitivecrtleft=1) -> (forall jt_divisor_primitivecrtright. (exists jt_factor_primitivecrtrightmodulus. (n)=(jt_divisor_primitivecrtright)*jt_factor_primitivecrtrightmodulus) -> (forall jt_index_primitivecrtrightcoordinates jt_value_primitivecrtrightcoordinates. (exists jt_gap_primitivecrtrightcoordinatesindex. jt_gap_primitivecrtrightcoordinatesindex+S (jt_index_primitivecrtrightcoordinates)=(k)) -> (((exists fs_h_jt_primitivecrtrightcoordinatesat. fs_h_jt_primitivecrtrightcoordinatesat + S (jt_value_primitivecrtrightcoordinates) = S ((S (jt_index_primitivecrtrightcoordinates)) * e)) /\\ exists fs_q_jt_primitivecrtrightcoordinatesat. d = fs_q_jt_primitivecrtrightcoordinatesat * S ((S (jt_index_primitivecrtrightcoordinates)) * e) + (jt_value_primitivecrtrightcoordinates))) -> (exists jt_factor_primitivecrtrightcoordinatesdivides. (jt_value_primitivecrtrightcoordinates)=(jt_divisor_primitivecrtright)*jt_factor_primitivecrtrightcoordinatesdivides)) -> jt_divisor_primitivecrtright=1) -> exists f g. ((((forall jt_index_primitivecrtresultbound. (exists jt_gap_primitivecrtresultboundindex. jt_gap_primitivecrtresultboundindex+S (jt_index_primitivecrtresultbound)=(k)) -> exists jt_value_primitivecrtresultbound. ((((exists fs_h_jt_primitivecrtresultboundat. fs_h_jt_primitivecrtresultboundat + S (jt_value_primitivecrtresultbound) = S ((S (jt_index_primitivecrtresultbound)) * g)) /\\ exists fs_q_jt_primitivecrtresultboundat. f = fs_q_jt_primitivecrtresultboundat * S ((S (jt_index_primitivecrtresultbound)) * g) + (jt_value_primitivecrtresultbound))) /\\ (exists jt_gap_primitivecrtresultboundvalue. jt_gap_primitivecrtresultboundvalue+S (jt_value_primitivecrtresultbound)=(m*n)))) /\\ (((forall jt_index_primitivecrtresultleft jt_left_primitivecrtresultleft jt_right_primitivecrtresultleft. (exists jt_gap_primitivecrtresultleftindex. jt_gap_primitivecrtresultleftindex+S (jt_index_primitivecrtresultleft)=(k)) -> (((exists fs_h_jt_primitivecrtresultleftleft. fs_h_jt_primitivecrtresultleftleft + S (jt_left_primitivecrtresultleft) = S ((S (jt_index_primitivecrtresultleft)) * g)) /\\ exists fs_q_jt_primitivecrtresultleftleft. f = fs_q_jt_primitivecrtresultleftleft * S ((S (jt_index_primitivecrtresultleft)) * g) + (jt_left_primitivecrtresultleft))) -> (((exists fs_h_jt_primitivecrtresultleftright. fs_h_jt_primitivecrtresultleftright + S (jt_right_primitivecrtresultleft) = S ((S (jt_index_primitivecrtresultleft)) * c)) /\\ exists fs_q_jt_primitivecrtresultleftright. b = fs_q_jt_primitivecrtresultleftright * S ((S (jt_index_primitivecrtresultleft)) * c) + (jt_right_primitivecrtresultleft))) -> (exists jt_left_primitivecrtresultleftmod jt_right_primitivecrtresultleftmod. (jt_left_primitivecrtresultleft)+(m)*jt_left_primitivecrtresultleftmod=(jt_right_primitivecrtresultleft)+(m)*jt_right_primitivecrtresultleftmod)) /\\ (forall jt_index_primitivecrtresultright jt_left_primitivecrtresultright jt_right_primitivecrtresultright. (exists jt_gap_primitivecrtresultrightindex. jt_gap_primitivecrtresultrightindex+S (jt_index_primitivecrtresultright)=(k)) -> (((exists fs_h_jt_primitivecrtresultrightleft. fs_h_jt_primitivecrtresultrightleft + S (jt_left_primitivecrtresultright) = S ((S (jt_index_primitivecrtresultright)) * g)) /\\ exists fs_q_jt_primitivecrtresultrightleft. f = fs_q_jt_primitivecrtresultrightleft * S ((S (jt_index_primitivecrtresultright)) * g) + (jt_left_primitivecrtresultright))) -> (((exists fs_h_jt_primitivecrtresultrightright. fs_h_jt_primitivecrtresultrightright + S (jt_right_primitivecrtresultright) = S ((S (jt_index_primitivecrtresultright)) * e)) /\\ exists fs_q_jt_primitivecrtresultrightright. d = fs_q_jt_primitivecrtresultrightright * S ((S (jt_index_primitivecrtresultright)) * e) + (jt_right_primitivecrtresultright))) -> (exists jt_left_primitivecrtresultrightmod jt_right_primitivecrtresultrightmod. (jt_left_primitivecrtresultright)+(n)*jt_left_primitivecrtresultrightmod=(jt_right_primitivecrtresultright)+(n)*jt_right_primitivecrtresultrightmod)))))) /\\ (forall jt_divisor_primitivecrtproduct. (exists jt_factor_primitivecrtproductmodulus. (m*n)=(jt_divisor_primitivecrtproduct)*jt_factor_primitivecrtproductmodulus) -> (forall jt_index_primitivecrtproductcoordinates jt_value_primitivecrtproductcoordinates. (exists jt_gap_primitivecrtproductcoordinatesindex. jt_gap_primitivecrtproductcoordinatesindex+S (jt_index_primitivecrtproductcoordinates)=(k)) -> (((exists fs_h_jt_primitivecrtproductcoordinatesat. fs_h_jt_primitivecrtproductcoordinatesat + S (jt_value_primitivecrtproductcoordinates) = S ((S (jt_index_primitivecrtproductcoordinates)) * g)) /\\ exists fs_q_jt_primitivecrtproductcoordinatesat. f = fs_q_jt_primitivecrtproductcoordinatesat * S ((S (jt_index_primitivecrtproductcoordinates)) * g) + (jt_value_primitivecrtproductcoordinates))) -> (exists jt_factor_primitivecrtproductcoordinatesdivides. (jt_value_primitivecrtproductcoordinates)=(jt_divisor_primitivecrtproduct)*jt_factor_primitivecrtproductcoordinatesdivides)) -> jt_divisor_primitivecrtproduct=1))", "statement_sha256": "49ee99529794f38d85e5097131bb9b82a09d960901449727d8b03d99d826fec1", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The constructed canonical CRT tuple is primitive collectively, by coprime common-divisor decomposition." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 15, "body_proof_nodes": 24, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro u", "intro v", "intro p", "intro hp", "intro hv", "have hprod : u*v=0", "rewrite hv", "rewrite PA5", "refl", "rewrite hprod at hp", "specialize lt_not_le (p)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hp", "specialize zero_le (p)", "apply zero_le" ], "defined_statement": "∀ u. ∀ v. ∀ p. Lt(p,u · v) → ¬v = 0", "defined_statement_sha256": "2b47bce928cf06535466d27ccea607e254e09694474dcc1b95a676098c48ff41", "definition_uses": { "PD0002": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "194340a8d480ffe6dcb2390254e95bc5605f872df949aa24d276ae3056780117", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hv" } ], [ { "kind": "text", "text": "have hprod : u*v=0" } ], [ { "kind": "text", "text": "rewrite hv" } ], [ { "kind": "text", "text": "rewrite PA5" } ], [ { "kind": "text", "text": "refl" } ], [ { "kind": "text", "text": "rewrite hprod at hp" } ], [ { "kind": "text", "text": "specialize lt_not_le (p)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "specialize zero_le (p)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "PD0002": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ u. ∀ v. ∀ p. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(p,u · v)" }, { "kind": "text", "text": " → ¬v = 0" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0034", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_width_nonzero", "original_ha_bundle_verified": true, "proof_bundle_node_id": 313, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro u", "intro v", "intro p", "intro hp", "intro hv", "have hprod : u*v=0", "rewrite hv", "rewrite PA5", "refl", "rewrite hprod at hp", "specialize lt_not_le (p)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hp", "specialize zero_le (p)", "apply zero_le" ], "script_sha256": "7ede5d8b5cb444ed64bc7d9f84dacde410129f6aafa8cb5eaad57189a28c84fc", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "7ede5d8b5cb444ed64bc7d9f84dacde410129f6aafa8cb5eaad57189a28c84fc", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "194340a8d480ffe6dcb2390254e95bc5605f872df949aa24d276ae3056780117" } ], "stable_member": false, "statement": "forall u v p. (exists jt_gap_widthbound. jt_gap_widthbound+S (p)=(u*v)) -> ~(v=0)", "statement_sha256": "194340a8d480ffe6dcb2390254e95bc5605f872df949aa24d276ae3056780117", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A genuine index below u*v forces the rectangle width to be positive." }, { "admission_dependencies": [ "le_or_lt", "lt_not_le", "le_trans", "mul_le_mul_left", "mul_comm", "le_add_right" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 25, "body_proof_nodes": 48, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro u", "intro v", "intro p", "intro i", "intro j", "intro hp", "intro heq", "intro hj", "have hc : Le(u,i) ∨ Lt(i,u)", "specialize le_or_lt (u)", "specialize le_or_lt (i)", "apply le_or_lt", "cases hc", "exfalso", "specialize lt_not_le (p)", "specialize lt_not_le (u*v)", "apply lt_not_le", "exact hp", "specialize le_trans (u*v)", "specialize le_trans (v*i)", "specialize le_trans (p)", "apply le_trans", "have hm : Le(v · u,v · i)", "specialize mul_le_mul_left (u)", "specialize mul_le_mul_left (i)", "specialize mul_le_mul_left (v)", "apply mul_le_mul_left", "exact hc_left", "have hcomm : v*u=u*v", "specialize mul_comm (v)", "specialize mul_comm (u)", "apply mul_comm", "rewrite hcomm at hm", "exact hm", "rewrite heq", "specialize le_add_right (v*i)", "specialize le_add_right (j)", "apply le_add_right", "exact hc_right" ], "defined_statement": "∀ u. ∀ v. ∀ p. ∀ i. ∀ j. Lt(p,u · v) → p = v · i + j → Lt(j,v) → Lt(i,u)", "defined_statement_sha256": "b9f7707f54c4d017eef2e2f34ea1db3f9a43b1ff1e011cf79dc32b59993b319a", "definition_uses": { "PD0001": 2, "PD0002": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "b195a0ce32325ce6d68d55fa78dd7c12c0d61556e09da899da066807e6700356", "free_names": [], "script_definition_uses": { "PD0001": 2, "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "have hc : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(u,i)" }, { "kind": "text", "text": " ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" } ], [ { "kind": "text", "text": "specialize le_or_lt (u)" } ], [ { "kind": "text", "text": "specialize le_or_lt (i)" } ], [ { "kind": "text", "text": "apply le_or_lt" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (p)" } ], [ { "kind": "text", "text": "specialize lt_not_le (u*v)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "specialize le_trans (u*v)" } ], [ { "kind": "text", "text": "specialize le_trans (v*i)" } ], [ { "kind": "text", "text": "specialize le_trans (p)" } ], [ { "kind": "text", "text": "apply le_trans" } ], [ { "kind": "text", "text": "have hm : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(v · u,v · i)" } ], [ { "kind": "text", "text": "specialize mul_le_mul_left (u)" } ], [ { "kind": "text", "text": "specialize mul_le_mul_left (i)" } ], [ { "kind": "text", "text": "specialize mul_le_mul_left (v)" } ], [ { "kind": "text", "text": "apply mul_le_mul_left" } ], [ { "kind": "text", "text": "exact hc_left" } ], [ { "kind": "text", "text": "have hcomm : v*u=u*v" } ], [ { "kind": "text", "text": "specialize mul_comm (v)" } ], [ { "kind": "text", "text": "specialize mul_comm (u)" } ], [ { "kind": "text", "text": "apply mul_comm" } ], [ { "kind": "text", "text": "rewrite hcomm at hm" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "rewrite heq" } ], [ { "kind": "text", "text": "specialize le_add_right (v*i)" } ], [ { "kind": "text", "text": "specialize le_add_right (j)" } ], [ { "kind": "text", "text": "apply le_add_right" } ], [ { "kind": "text", "text": "exact hc_right" } ] ], "statement_definition_uses": { "PD0002": 3 }, "statement_parts": [ { "kind": "text", "text": "∀ u. ∀ v. ∀ p. ∀ i. ∀ j. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(p,u · v)" }, { "kind": "text", "text": " → p = v · i + j → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" } ] }, "dependencies": [ "le_or_lt", "lt_not_le", "le_trans", "mul_le_mul_left", "mul_comm", "le_add_right" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0035", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_quotient_bound", "original_ha_bundle_verified": true, "proof_bundle_node_id": 314, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro u", "intro v", "intro p", "intro i", "intro j", "intro hp", "intro heq", "intro hj", "have hc : (exists jt_gap_quotle. jt_gap_quotle+(u)=(i)) \\/ (exists jt_gap_quotlt. jt_gap_quotlt+S (i)=(u))", "specialize le_or_lt (u)", "specialize le_or_lt (i)", "apply le_or_lt", "cases hc", "exfalso", "specialize lt_not_le (p)", "specialize lt_not_le (u*v)", "apply lt_not_le", "exact hp", "specialize le_trans (u*v)", "specialize le_trans (v*i)", "specialize le_trans (p)", "apply le_trans", "have hm : exists jt_gap_quotmult. jt_gap_quotmult+(v*u)=(v*i)", "specialize mul_le_mul_left (u)", "specialize mul_le_mul_left (i)", "specialize mul_le_mul_left (v)", "apply mul_le_mul_left", "exact hc_left", "have hcomm : v*u=u*v", "specialize mul_comm (v)", "specialize mul_comm (u)", "apply mul_comm", "rewrite hcomm at hm", "exact hm", "rewrite heq", "specialize le_add_right (v*i)", "specialize le_add_right (j)", "apply le_add_right", "exact hc_right" ], "script_sha256": "487c088c7248dd4826018f687cab344bc7b64f08249fb6f12a57048b23bec7d0", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "487c088c7248dd4826018f687cab344bc7b64f08249fb6f12a57048b23bec7d0", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "b195a0ce32325ce6d68d55fa78dd7c12c0d61556e09da899da066807e6700356" } ], "stable_member": false, "statement": "forall u v p i j. (exists jt_gap_quotbound. jt_gap_quotbound+S (p)=(u*v)) -> p=v*i+j -> (exists jt_gap_quotremainder. jt_gap_quotremainder+S (j)=(v)) -> (exists jt_gap_quotresult. jt_gap_quotresult+S (i)=(u))", "statement_sha256": "b195a0ce32325ce6d68d55fa78dd7c12c0d61556e09da899da066807e6700356", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "An actual bounded row-major index has a row index below u; no division oracle is assumed." }, { "admission_dependencies": [ "finite_add_lt_of_le_of_lt", "le_refl", "lt_of_lt_of_le", "mul_le_mul_right", "mul_comm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 20, "body_proof_nodes": 41, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro u", "intro v", "intro i", "intro j", "intro hi", "intro hj", "have hs : Lt(v · i + j,v · i + v)", "specialize finite_add_lt_of_le_of_lt (v*i)", "specialize finite_add_lt_of_le_of_lt (v*i)", "specialize finite_add_lt_of_le_of_lt (j)", "specialize finite_add_lt_of_le_of_lt (v)", "apply finite_add_lt_of_le_of_lt", "specialize le_refl (v*i)", "apply le_refl", "exact hj", "specialize lt_of_lt_of_le (v*i+j)", "specialize lt_of_lt_of_le (v*i+v)", "specialize lt_of_lt_of_le (u*v)", "apply lt_of_lt_of_le", "exact hs", "have hm : Le(S i · v,u · v)", "specialize mul_le_mul_right (S i)", "specialize mul_le_mul_right (u)", "specialize mul_le_mul_right (v)", "apply mul_le_mul_right", "exact hi", "have hcomm : S i*v=v*S i", "specialize mul_comm (S i)", "specialize mul_comm (v)", "apply mul_comm", "rewrite hcomm at hm", "have hstep : v*S i=v*i+v", "apply PA6", "rewrite hstep at hm", "exact hm" ], "defined_statement": "∀ u. ∀ v. ∀ i. ∀ j. Lt(i,u) → Lt(j,v) → Lt(v · i + j,u · v)", "defined_statement_sha256": "dcbb4a818f6618df0b7781986eaa11d0058e1bf8c0ab848616605095e26f0949", "definition_uses": { "PD0001": 1, "PD0002": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "b42447fc0d36cde0ac306699e6526ecfa02ab25baa88a9810fd557e5a3a2ce50", "free_names": [], "script_definition_uses": { "PD0001": 1, "PD0002": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "have hs : " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(v · i + j,v · i + v)" } ], [ { "kind": "text", "text": "specialize finite_add_lt_of_le_of_lt (v*i)" } ], [ { "kind": "text", "text": "specialize finite_add_lt_of_le_of_lt (v*i)" } ], [ { "kind": "text", "text": "specialize finite_add_lt_of_le_of_lt (j)" } ], [ { "kind": "text", "text": "specialize finite_add_lt_of_le_of_lt (v)" } ], [ { "kind": "text", "text": "apply finite_add_lt_of_le_of_lt" } ], [ { "kind": "text", "text": "specialize le_refl (v*i)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "specialize lt_of_lt_of_le (v*i+j)" } ], [ { "kind": "text", "text": "specialize lt_of_lt_of_le (v*i+v)" } ], [ { "kind": "text", "text": "specialize lt_of_lt_of_le (u*v)" } ], [ { "kind": "text", "text": "apply lt_of_lt_of_le" } ], [ { "kind": "text", "text": "exact hs" } ], [ { "kind": "text", "text": "have hm : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(S i · v,u · v)" } ], [ { "kind": "text", "text": "specialize mul_le_mul_right (S i)" } ], [ { "kind": "text", "text": "specialize mul_le_mul_right (u)" } ], [ { "kind": "text", "text": "specialize mul_le_mul_right (v)" } ], [ { "kind": "text", "text": "apply mul_le_mul_right" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "have hcomm : S i*v=v*S i" } ], [ { "kind": "text", "text": "specialize mul_comm (S i)" } ], [ { "kind": "text", "text": "specialize mul_comm (v)" } ], [ { "kind": "text", "text": "apply mul_comm" } ], [ { "kind": "text", "text": "rewrite hcomm at hm" } ], [ { "kind": "text", "text": "have hstep : v*S i=v*i+v" } ], [ { "kind": "text", "text": "apply PA6" } ], [ { "kind": "text", "text": "rewrite hstep at hm" } ], [ { "kind": "text", "text": "exact hm" } ] ], "statement_definition_uses": { "PD0002": 3 }, "statement_parts": [ { "kind": "text", "text": "∀ u. ∀ v. ∀ i. ∀ j. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(v · i + j,u · v)" } ] }, "dependencies": [ "finite_add_lt_of_le_of_lt", "le_refl", "lt_of_lt_of_le", "mul_le_mul_right", "mul_comm" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0036", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_flat_bound", "original_ha_bundle_verified": true, "proof_bundle_node_id": 315, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro u", "intro v", "intro i", "intro j", "intro hi", "intro hj", "have hs : exists jt_gap_flatadd. jt_gap_flatadd+S (v*i+j)=(v*i+v)", "specialize finite_add_lt_of_le_of_lt (v*i)", "specialize finite_add_lt_of_le_of_lt (v*i)", "specialize finite_add_lt_of_le_of_lt (j)", "specialize finite_add_lt_of_le_of_lt (v)", "apply finite_add_lt_of_le_of_lt", "specialize le_refl (v*i)", "apply le_refl", "exact hj", "specialize lt_of_lt_of_le (v*i+j)", "specialize lt_of_lt_of_le (v*i+v)", "specialize lt_of_lt_of_le (u*v)", "apply lt_of_lt_of_le", "exact hs", "have hm : exists jt_gap_flatmult. jt_gap_flatmult+(S i*v)=(u*v)", "specialize mul_le_mul_right (S i)", "specialize mul_le_mul_right (u)", "specialize mul_le_mul_right (v)", "apply mul_le_mul_right", "exact hi", "have hcomm : S i*v=v*S i", "specialize mul_comm (S i)", "specialize mul_comm (v)", "apply mul_comm", "rewrite hcomm at hm", "have hstep : v*S i=v*i+v", "apply PA6", "rewrite hstep at hm", "exact hm" ], "script_sha256": "ea7dde23404b65f731b7186c7177ee0d92b3a807c20a3d3acb9c77313df22f1e", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "ea7dde23404b65f731b7186c7177ee0d92b3a807c20a3d3acb9c77313df22f1e", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "b42447fc0d36cde0ac306699e6526ecfa02ab25baa88a9810fd557e5a3a2ce50" } ], "stable_member": false, "statement": "forall u v i j. (exists jt_gap_flatrow. jt_gap_flatrow+S (i)=(u)) -> (exists jt_gap_flatcolumn. jt_gap_flatcolumn+S (j)=(v)) -> (exists jt_gap_flatresult. jt_gap_flatresult+S (v*i+j)=(u*v))", "statement_sha256": "b42447fc0d36cde0ac306699e6526ecfa02ab25baa88a9810fd557e5a3a2ce50", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every actual pair of bounded indices has a row-major index below the literal product." }, { "admission_dependencies": [ "division_remainder_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 26, "body_proof_nodes": 42, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro v", "intro i", "intro j", "intro r", "intro s", "intro hj", "intro hs", "intro heq", "specialize division_remainder_unique (v)", "specialize division_remainder_unique (v*i+j)", "specialize division_remainder_unique (i)", "specialize division_remainder_unique (j)", "specialize division_remainder_unique (r)", "specialize division_remainder_unique (s)", "apply division_remainder_unique", "refl", "exact hj", "exact heq", "exact hs" ], "defined_statement": "∀ v. ∀ i. ∀ j. ∀ r. ∀ s. Lt(j,v) → Lt(s,v) → v · i + j = v · r + s → i = r ∧ j = s", "defined_statement_sha256": "4125ca2aa934756cd20007444d31dc02b9a346915405d43b1af2efc6bb5a06f5", "definition_uses": { "PD0002": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "7b6a6161af780e9a73710a2cba1f8a512e1a823e96f9fa55acd5b16c80564fd4", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro r" } ], [ { "kind": "text", "text": "intro s" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "intro hs" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "specialize division_remainder_unique (v)" } ], [ { "kind": "text", "text": "specialize division_remainder_unique (v*i+j)" } ], [ { "kind": "text", "text": "specialize division_remainder_unique (i)" } ], [ { "kind": "text", "text": "specialize division_remainder_unique (j)" } ], [ { "kind": "text", "text": "specialize division_remainder_unique (r)" } ], [ { "kind": "text", "text": "specialize division_remainder_unique (s)" } ], [ { "kind": "text", "text": "apply division_remainder_unique" } ], [ { "kind": "text", "text": "refl" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "exact hs" } ] ], "statement_definition_uses": { "PD0002": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ v. ∀ i. ∀ j. ∀ r. ∀ s. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(s,v)" }, { "kind": "text", "text": " → v · i + j = v · r + s → i = r ∧ j = s" } ] }, "dependencies": [ "division_remainder_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0037", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_pair_unique", "original_ha_bundle_verified": true, "proof_bundle_node_id": 316, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro v", "intro i", "intro j", "intro r", "intro s", "intro hj", "intro hs", "intro heq", "specialize division_remainder_unique (v)", "specialize division_remainder_unique (v*i+j)", "specialize division_remainder_unique (i)", "specialize division_remainder_unique (j)", "specialize division_remainder_unique (r)", "specialize division_remainder_unique (s)", "apply division_remainder_unique", "refl", "exact hj", "exact heq", "exact hs" ], "script_sha256": "28c9ce823d9564858e62349706232c79bea1c7413145ce36d7c1a05975f49b7b", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "28c9ce823d9564858e62349706232c79bea1c7413145ce36d7c1a05975f49b7b", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "7b6a6161af780e9a73710a2cba1f8a512e1a823e96f9fa55acd5b16c80564fd4" } ], "stable_member": false, "statement": "forall v i j r s. (exists jt_gap_uniquecolumn. jt_gap_uniquecolumn+S (j)=(v)) -> (exists jt_gap_uniqueother. jt_gap_uniqueother+S (s)=(v)) -> v*i+j=v*r+s -> (i=r /\\ j=s)", "statement_sha256": "7b6a6161af780e9a73710a2cba1f8a512e1a823e96f9fa55acd5b16c80564fd4", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Actual bounded quotient/remainder decomposition uniquely recovers both indices." }, { "admission_dependencies": [ "mod_eq_refl" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 23, "body_proof_nodes": 29, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro heq", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "have hval : a=z", "specialize heq (i)", "specialize heq (a)", "specialize heq (z)", "apply heq", "exact hi", "exact ha", "exact hz", "rewrite hval", "specialize mod_eq_refl (n)", "specialize mod_eq_refl (z)", "apply mod_eq_refl" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. IntegerVectorZero(b,c,d,e,k) → JordanTupleCongruence(n,b,c,d,e,k)", "defined_statement_sha256": "812bc850bdf037d20d941a101dd1babff0613d73a6ed2214ec6907fc0b9fdb7e", "definition_uses": { "ND0121": 1, "ND0373": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "019277c060ec67805f891d6adecd42dcb44b9192c7dbfe9f6427615499bf8df3", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "have hval : a=z" } ], [ { "kind": "text", "text": "specialize heq (i)" } ], [ { "kind": "text", "text": "specialize heq (a)" } ], [ { "kind": "text", "text": "specialize heq (z)" } ], [ { "kind": "text", "text": "apply heq" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "rewrite hval" } ], [ { "kind": "text", "text": "specialize mod_eq_refl (n)" } ], [ { "kind": "text", "text": "specialize mod_eq_refl (z)" } ], [ { "kind": "text", "text": "apply mod_eq_refl" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0373": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" } ] }, "dependencies": [ "mod_eq_refl" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0038", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_equal_congruence", "original_ha_bundle_verified": true, "proof_bundle_node_id": 317, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro heq", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "have hval : a=z", "specialize heq (i)", "specialize heq (a)", "specialize heq (z)", "apply heq", "exact hi", "exact ha", "exact hz", "rewrite hval", "specialize mod_eq_refl (n)", "specialize mod_eq_refl (z)", "apply mod_eq_refl" ], "script_sha256": "03685165793d3e6d2679c074a6add3dbb16324c1940fdb1dfb7d8ca0779e0eb1", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "03685165793d3e6d2679c074a6add3dbb16324c1940fdb1dfb7d8ca0779e0eb1", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "019277c060ec67805f891d6adecd42dcb44b9192c7dbfe9f6427615499bf8df3" } ], "stable_member": false, "statement": "forall n b c d e k. (forall jt_index_equalcong jt_left_equalcong jt_right_equalcong. (exists jt_gap_equalcongindex. jt_gap_equalcongindex+S (jt_index_equalcong)=(k)) -> (((exists fs_h_jt_equalcongleft. fs_h_jt_equalcongleft + S (jt_left_equalcong) = S ((S (jt_index_equalcong)) * c)) /\\ exists fs_q_jt_equalcongleft. b = fs_q_jt_equalcongleft * S ((S (jt_index_equalcong)) * c) + (jt_left_equalcong))) -> (((exists fs_h_jt_equalcongright. fs_h_jt_equalcongright + S (jt_right_equalcong) = S ((S (jt_index_equalcong)) * e)) /\\ exists fs_q_jt_equalcongright. d = fs_q_jt_equalcongright * S ((S (jt_index_equalcong)) * e) + (jt_right_equalcong))) -> jt_left_equalcong=jt_right_equalcong) -> (forall jt_index_equalcongresult jt_left_equalcongresult jt_right_equalcongresult. (exists jt_gap_equalcongresultindex. jt_gap_equalcongresultindex+S (jt_index_equalcongresult)=(k)) -> (((exists fs_h_jt_equalcongresultleft. fs_h_jt_equalcongresultleft + S (jt_left_equalcongresult) = S ((S (jt_index_equalcongresult)) * c)) /\\ exists fs_q_jt_equalcongresultleft. b = fs_q_jt_equalcongresultleft * S ((S (jt_index_equalcongresult)) * c) + (jt_left_equalcongresult))) -> (((exists fs_h_jt_equalcongresultright. fs_h_jt_equalcongresultright + S (jt_right_equalcongresult) = S ((S (jt_index_equalcongresult)) * e)) /\\ exists fs_q_jt_equalcongresultright. d = fs_q_jt_equalcongresultright * S ((S (jt_index_equalcongresult)) * e) + (jt_right_equalcongresult))) -> (exists jt_left_equalcongresultmod jt_right_equalcongresultmod. (jt_left_equalcongresult)+(n)*jt_left_equalcongresultmod=(jt_right_equalcongresult)+(n)*jt_right_equalcongresultmod))", "statement_sha256": "019277c060ec67805f891d6adecd42dcb44b9192c7dbfe9f6427615499bf8df3", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Equal decoded coordinates are congruent at every modulus, including zero." }, { "admission_dependencies": [ "mod_eq_bounded_unique", "matrix_rank_bounded_prefix_value" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 42, "body_proof_nodes": 114, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hb", "intro hd", "intro hm", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_bounded_unique (n)", "specialize mod_eq_bounded_unique (a)", "specialize mod_eq_bounded_unique (z)", "apply mod_eq_bounded_unique", "specialize matrix_rank_bounded_prefix_value (b)", "specialize matrix_rank_bounded_prefix_value (c)", "specialize matrix_rank_bounded_prefix_value (k)", "specialize matrix_rank_bounded_prefix_value (n)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (a)", "apply matrix_rank_bounded_prefix_value", "exact hb", "exact hi", "exact ha", "specialize matrix_rank_bounded_prefix_value (d)", "specialize matrix_rank_bounded_prefix_value (e)", "specialize matrix_rank_bounded_prefix_value (k)", "specialize matrix_rank_bounded_prefix_value (n)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (z)", "apply matrix_rank_bounded_prefix_value", "exact hd", "exact hi", "exact hz", "specialize hm (i)", "specialize hm (a)", "specialize hm (z)", "apply hm", "exact hi", "exact ha", "exact hz" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. BetaPrefixInto(b,c,k,n) → BetaPrefixInto(d,e,k,n) → JordanTupleCongruence(n,b,c,d,e,k) → IntegerVectorZero(b,c,d,e,k)", "defined_statement_sha256": "a5ca37325c6edd9566363b7ceb76c685f250b4bfb033ac02034d0e5068584736", "definition_uses": { "ND0121": 1, "ND0262": 2, "ND0373": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "dac2f0f8154a68c50443764a472b2b535738e971f64e9ee0ee70d2405881393b", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize mod_eq_bounded_unique (n)" } ], [ { "kind": "text", "text": "specialize mod_eq_bounded_unique (a)" } ], [ { "kind": "text", "text": "specialize mod_eq_bounded_unique (z)" } ], [ { "kind": "text", "text": "apply mod_eq_bounded_unique" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (b)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (c)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (k)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (n)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (i)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (a)" } ], [ { "kind": "text", "text": "apply matrix_rank_bounded_prefix_value" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (d)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (e)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (k)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (n)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (i)" } ], [ { "kind": "text", "text": "specialize matrix_rank_bounded_prefix_value (z)" } ], [ { "kind": "text", "text": "apply matrix_rank_bounded_prefix_value" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "specialize hm (i)" } ], [ { "kind": "text", "text": "specialize hm (a)" } ], [ { "kind": "text", "text": "specialize hm (z)" } ], [ { "kind": "text", "text": "apply hm" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0262": 2, "ND0373": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(d,e,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" } ] }, "dependencies": [ "mod_eq_bounded_unique", "matrix_rank_bounded_prefix_value" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0039", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_bounded_congruence_equal", "original_ha_bundle_verified": true, "proof_bundle_node_id": 318, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hb", "intro hd", "intro hm", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_bounded_unique (n)", "specialize mod_eq_bounded_unique (a)", "specialize mod_eq_bounded_unique (z)", "apply mod_eq_bounded_unique", "specialize matrix_rank_bounded_prefix_value (b)", "specialize matrix_rank_bounded_prefix_value (c)", "specialize matrix_rank_bounded_prefix_value (k)", "specialize matrix_rank_bounded_prefix_value (n)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (a)", "apply matrix_rank_bounded_prefix_value", "exact hb", "exact hi", "exact ha", "specialize matrix_rank_bounded_prefix_value (d)", "specialize matrix_rank_bounded_prefix_value (e)", "specialize matrix_rank_bounded_prefix_value (k)", "specialize matrix_rank_bounded_prefix_value (n)", "specialize matrix_rank_bounded_prefix_value (i)", "specialize matrix_rank_bounded_prefix_value (z)", "apply matrix_rank_bounded_prefix_value", "exact hd", "exact hi", "exact hz", "specialize hm (i)", "specialize hm (a)", "specialize hm (z)", "apply hm", "exact hi", "exact ha", "exact hz" ], "script_sha256": "c3ac07a036669493a1493bcf69645b9bd9ce2d450543aacf385563fe4e73f182", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "c3ac07a036669493a1493bcf69645b9bd9ce2d450543aacf385563fe4e73f182", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "dac2f0f8154a68c50443764a472b2b535738e971f64e9ee0ee70d2405881393b" } ], "stable_member": false, "statement": "forall n b c d e k. (forall jt_index_congboundleft. (exists jt_gap_congboundleftindex. jt_gap_congboundleftindex+S (jt_index_congboundleft)=(k)) -> exists jt_value_congboundleft. ((((exists fs_h_jt_congboundleftat. fs_h_jt_congboundleftat + S (jt_value_congboundleft) = S ((S (jt_index_congboundleft)) * c)) /\\ exists fs_q_jt_congboundleftat. b = fs_q_jt_congboundleftat * S ((S (jt_index_congboundleft)) * c) + (jt_value_congboundleft))) /\\ (exists jt_gap_congboundleftvalue. jt_gap_congboundleftvalue+S (jt_value_congboundleft)=(n)))) -> (forall jt_index_congboundright. (exists jt_gap_congboundrightindex. jt_gap_congboundrightindex+S (jt_index_congboundright)=(k)) -> exists jt_value_congboundright. ((((exists fs_h_jt_congboundrightat. fs_h_jt_congboundrightat + S (jt_value_congboundright) = S ((S (jt_index_congboundright)) * e)) /\\ exists fs_q_jt_congboundrightat. d = fs_q_jt_congboundrightat * S ((S (jt_index_congboundright)) * e) + (jt_value_congboundright))) /\\ (exists jt_gap_congboundrightvalue. jt_gap_congboundrightvalue+S (jt_value_congboundright)=(n)))) -> (forall jt_index_congboth jt_left_congboth jt_right_congboth. (exists jt_gap_congbothindex. jt_gap_congbothindex+S (jt_index_congboth)=(k)) -> (((exists fs_h_jt_congbothleft. fs_h_jt_congbothleft + S (jt_left_congboth) = S ((S (jt_index_congboth)) * c)) /\\ exists fs_q_jt_congbothleft. b = fs_q_jt_congbothleft * S ((S (jt_index_congboth)) * c) + (jt_left_congboth))) -> (((exists fs_h_jt_congbothright. fs_h_jt_congbothright + S (jt_right_congboth) = S ((S (jt_index_congboth)) * e)) /\\ exists fs_q_jt_congbothright. d = fs_q_jt_congbothright * S ((S (jt_index_congboth)) * e) + (jt_right_congboth))) -> (exists jt_left_congbothmod jt_right_congbothmod. (jt_left_congboth)+(n)*jt_left_congbothmod=(jt_right_congboth)+(n)*jt_right_congbothmod)) -> (forall jt_index_congequal jt_left_congequal jt_right_congequal. (exists jt_gap_congequalindex. jt_gap_congequalindex+S (jt_index_congequal)=(k)) -> (((exists fs_h_jt_congequalleft. fs_h_jt_congequalleft + S (jt_left_congequal) = S ((S (jt_index_congequal)) * c)) /\\ exists fs_q_jt_congequalleft. b = fs_q_jt_congequalleft * S ((S (jt_index_congequal)) * c) + (jt_left_congequal))) -> (((exists fs_h_jt_congequalright. fs_h_jt_congequalright + S (jt_right_congequal) = S ((S (jt_index_congequal)) * e)) /\\ exists fs_q_jt_congequalright. d = fs_q_jt_congequalright * S ((S (jt_index_congequal)) * e) + (jt_right_congequal))) -> jt_left_congequal=jt_right_congequal)", "statement_sha256": "dac2f0f8154a68c50443764a472b2b535738e971f64e9ee0ee70d2405881393b", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Actual canonical residues with pointwise congruence are equal as coordinate functions." }, { "admission_dependencies": [ "mod_eq_lcm_merge", "coprime_product_is_lcm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 40, "body_proof_nodes": 91, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hcop", "intro hm", "intro hn", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_lcm_merge (m*n)", "specialize mod_eq_lcm_merge (m)", "specialize mod_eq_lcm_merge (n)", "specialize mod_eq_lcm_merge (a)", "specialize mod_eq_lcm_merge (z)", "apply mod_eq_lcm_merge", "specialize coprime_product_is_lcm (m)", "specialize coprime_product_is_lcm (n)", "apply coprime_product_is_lcm", "exact hcop", "specialize hm (i)", "specialize hm (a)", "specialize hm (z)", "apply hm", "exact hi", "exact ha", "exact hz", "specialize hn (i)", "specialize hn (a)", "specialize hn (z)", "apply hn", "exact hi", "exact ha", "exact hz" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. Coprime(m,n) → JordanTupleCongruence(m,b,c,d,e,k) → JordanTupleCongruence(n,b,c,d,e,k) → JordanTupleCongruence(m · n,b,c,d,e,k)", "defined_statement_sha256": "d3f0f3b5632b1ac4365a12249e5be86051a1b658f10cb9cd3cacd1d097c7cfbe", "definition_uses": { "ND0373": 3, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "c69d0a6156b2e951ea01f60211fe18305a8308c3851b486c295eac033923d914", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize mod_eq_lcm_merge (m*n)" } ], [ { "kind": "text", "text": "specialize mod_eq_lcm_merge (m)" } ], [ { "kind": "text", "text": "specialize mod_eq_lcm_merge (n)" } ], [ { "kind": "text", "text": "specialize mod_eq_lcm_merge (a)" } ], [ { "kind": "text", "text": "specialize mod_eq_lcm_merge (z)" } ], [ { "kind": "text", "text": "apply mod_eq_lcm_merge" } ], [ { "kind": "text", "text": "specialize coprime_product_is_lcm (m)" } ], [ { "kind": "text", "text": "specialize coprime_product_is_lcm (n)" } ], [ { "kind": "text", "text": "apply coprime_product_is_lcm" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "specialize hm (i)" } ], [ { "kind": "text", "text": "specialize hm (a)" } ], [ { "kind": "text", "text": "specialize hm (z)" } ], [ { "kind": "text", "text": "apply hm" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "specialize hn (i)" } ], [ { "kind": "text", "text": "specialize hn (a)" } ], [ { "kind": "text", "text": "specialize hn (z)" } ], [ { "kind": "text", "text": "apply hn" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hz" } ] ], "statement_definition_uses": { "ND0373": 3, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ k. " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m · n,b,c,d,e,k)" } ] }, "dependencies": [ "mod_eq_lcm_merge", "coprime_product_is_lcm" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT003A", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_congruence_coprime_product", "original_ha_bundle_verified": true, "proof_bundle_node_id": 319, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro k", "intro hcop", "intro hm", "intro hn", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "specialize mod_eq_lcm_merge (m*n)", "specialize mod_eq_lcm_merge (m)", "specialize mod_eq_lcm_merge (n)", "specialize mod_eq_lcm_merge (a)", "specialize mod_eq_lcm_merge (z)", "apply mod_eq_lcm_merge", "specialize coprime_product_is_lcm (m)", "specialize coprime_product_is_lcm (n)", "apply coprime_product_is_lcm", "exact hcop", "specialize hm (i)", "specialize hm (a)", "specialize hm (z)", "apply hm", "exact hi", "exact ha", "exact hz", "specialize hn (i)", "specialize hn (a)", "specialize hn (z)", "apply hn", "exact hi", "exact ha", "exact hz" ], "script_sha256": "6bc122e7587afb2578679c69bd006149a3a65b8bd19afac117d882a4b90f3216", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "6bc122e7587afb2578679c69bd006149a3a65b8bd19afac117d882a4b90f3216", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "c69d0a6156b2e951ea01f60211fe18305a8308c3851b486c295eac033923d914" } ], "stable_member": false, "statement": "forall m n b c d e k. (forall jt_divisor_combinecop. (exists jt_factor_combinecopa. (m)=(jt_divisor_combinecop)*jt_factor_combinecopa) -> (exists jt_factor_combinecopb. (n)=(jt_divisor_combinecop)*jt_factor_combinecopb) -> jt_divisor_combinecop=1) -> (forall jt_index_combinem jt_left_combinem jt_right_combinem. (exists jt_gap_combinemindex. jt_gap_combinemindex+S (jt_index_combinem)=(k)) -> (((exists fs_h_jt_combinemleft. fs_h_jt_combinemleft + S (jt_left_combinem) = S ((S (jt_index_combinem)) * c)) /\\ exists fs_q_jt_combinemleft. b = fs_q_jt_combinemleft * S ((S (jt_index_combinem)) * c) + (jt_left_combinem))) -> (((exists fs_h_jt_combinemright. fs_h_jt_combinemright + S (jt_right_combinem) = S ((S (jt_index_combinem)) * e)) /\\ exists fs_q_jt_combinemright. d = fs_q_jt_combinemright * S ((S (jt_index_combinem)) * e) + (jt_right_combinem))) -> (exists jt_left_combinemmod jt_right_combinemmod. (jt_left_combinem)+(m)*jt_left_combinemmod=(jt_right_combinem)+(m)*jt_right_combinemmod)) -> (forall jt_index_combinen jt_left_combinen jt_right_combinen. (exists jt_gap_combinenindex. jt_gap_combinenindex+S (jt_index_combinen)=(k)) -> (((exists fs_h_jt_combinenleft. fs_h_jt_combinenleft + S (jt_left_combinen) = S ((S (jt_index_combinen)) * c)) /\\ exists fs_q_jt_combinenleft. b = fs_q_jt_combinenleft * S ((S (jt_index_combinen)) * c) + (jt_left_combinen))) -> (((exists fs_h_jt_combinenright. fs_h_jt_combinenright + S (jt_right_combinen) = S ((S (jt_index_combinen)) * e)) /\\ exists fs_q_jt_combinenright. d = fs_q_jt_combinenright * S ((S (jt_index_combinen)) * e) + (jt_right_combinen))) -> (exists jt_left_combinenmod jt_right_combinenmod. (jt_left_combinen)+(n)*jt_left_combinenmod=(jt_right_combinen)+(n)*jt_right_combinenmod)) -> (forall jt_index_combineproduct jt_left_combineproduct jt_right_combineproduct. (exists jt_gap_combineproductindex. jt_gap_combineproductindex+S (jt_index_combineproduct)=(k)) -> (((exists fs_h_jt_combineproductleft. fs_h_jt_combineproductleft + S (jt_left_combineproduct) = S ((S (jt_index_combineproduct)) * c)) /\\ exists fs_q_jt_combineproductleft. b = fs_q_jt_combineproductleft * S ((S (jt_index_combineproduct)) * c) + (jt_left_combineproduct))) -> (((exists fs_h_jt_combineproductright. fs_h_jt_combineproductright + S (jt_right_combineproduct) = S ((S (jt_index_combineproduct)) * e)) /\\ exists fs_q_jt_combineproductright. d = fs_q_jt_combineproductright * S ((S (jt_index_combineproduct)) * e) + (jt_right_combineproduct))) -> (exists jt_left_combineproductmod jt_right_combineproductmod. (jt_left_combineproduct)+(m*n)*jt_left_combineproductmod=(jt_right_combineproduct)+(m*n)*jt_right_combineproductmod))", "statement_sha256": "c69d0a6156b2e951ea01f60211fe18305a8308c3851b486c295eac033923d914", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The actual universal-property lcm merges both coordinate congruences." }, { "admission_dependencies": [ "jordan_tuple_bounded_congruence_equal", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm", "jordan_tuple_equal_congruence" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 32, "body_proof_nodes": 72, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro h", "intro s", "intro k", "intro hb", "intro hd", "intro hleft", "intro hright", "intro heq", "have hmiddle : JordanTupleCongruence(n,f,g,d,e,k)", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (h)", "specialize jordan_tuple_congruence_trans (s)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_equal_congruence (n)", "specialize jordan_tuple_equal_congruence (f)", "specialize jordan_tuple_equal_congruence (g)", "specialize jordan_tuple_equal_congruence (h)", "specialize jordan_tuple_equal_congruence (s)", "specialize jordan_tuple_equal_congruence (k)", "apply jordan_tuple_equal_congruence", "exact heq", "exact hright", "specialize jordan_tuple_bounded_congruence_equal (n)", "specialize jordan_tuple_bounded_congruence_equal (b)", "specialize jordan_tuple_bounded_congruence_equal (c)", "specialize jordan_tuple_bounded_congruence_equal (d)", "specialize jordan_tuple_bounded_congruence_equal (e)", "specialize jordan_tuple_bounded_congruence_equal (k)", "apply jordan_tuple_bounded_congruence_equal", "exact hb", "exact hd", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (f)", "specialize jordan_tuple_congruence_symm (g)", "specialize jordan_tuple_congruence_symm (b)", "specialize jordan_tuple_congruence_symm (c)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hleft", "exact hmiddle" ], "defined_statement": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ s. ∀ k. BetaPrefixInto(b,c,k,n) → BetaPrefixInto(d,e,k,n) → JordanTupleCongruence(n,f,g,b,c,k) → JordanTupleCongruence(n,h,s,d,e,k) → IntegerVectorZero(f,g,h,s,k) → IntegerVectorZero(b,c,d,e,k)", "defined_statement_sha256": "67ecd9b332b1cb7225352768317585b15fc437fce3e59adcbdfa978729bdf46f", "definition_uses": { "ND0121": 2, "ND0262": 2, "ND0373": 3 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "40d63417a1fe92b830db3c8aeb41b49d834cf1777a03689da6cd0906e0f72341", "free_names": [], "script_definition_uses": { "ND0373": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro s" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hleft" } ], [ { "kind": "text", "text": "intro hright" } ], [ { "kind": "text", "text": "intro heq" } ], [ { "kind": "text", "text": "have hmiddle : " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,f,g,d,e,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_congruence" } ], [ { "kind": "text", "text": "exact heq" } ], [ { "kind": "text", "text": "exact hright" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_bounded_congruence_equal" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hleft" } ], [ { "kind": "text", "text": "exact hmiddle" } ] ], "statement_definition_uses": { "ND0121": 2, "ND0262": 2, "ND0373": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ s. ∀ k. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(d,e,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,f,g,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,h,s,d,e,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(f,g,h,s,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" } ] }, "dependencies": [ "jordan_tuple_bounded_congruence_equal", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm", "jordan_tuple_equal_congruence" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT003B", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_crt_component_recovery", "original_ha_bundle_verified": true, "proof_bundle_node_id": 320, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro h", "intro s", "intro k", "intro hb", "intro hd", "intro hleft", "intro hright", "intro heq", "have hmiddle : forall jt_index_recovermiddle jt_left_recovermiddle jt_right_recovermiddle. (exists jt_gap_recovermiddleindex. jt_gap_recovermiddleindex+S (jt_index_recovermiddle)=(k)) -> (((exists fs_h_jt_recovermiddleleft. fs_h_jt_recovermiddleleft + S (jt_left_recovermiddle) = S ((S (jt_index_recovermiddle)) * g)) /\\ exists fs_q_jt_recovermiddleleft. f = fs_q_jt_recovermiddleleft * S ((S (jt_index_recovermiddle)) * g) + (jt_left_recovermiddle))) -> (((exists fs_h_jt_recovermiddleright. fs_h_jt_recovermiddleright + S (jt_right_recovermiddle) = S ((S (jt_index_recovermiddle)) * e)) /\\ exists fs_q_jt_recovermiddleright. d = fs_q_jt_recovermiddleright * S ((S (jt_index_recovermiddle)) * e) + (jt_right_recovermiddle))) -> (exists jt_left_recovermiddlemod jt_right_recovermiddlemod. (jt_left_recovermiddle)+(n)*jt_left_recovermiddlemod=(jt_right_recovermiddle)+(n)*jt_right_recovermiddlemod)", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (h)", "specialize jordan_tuple_congruence_trans (s)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_equal_congruence (n)", "specialize jordan_tuple_equal_congruence (f)", "specialize jordan_tuple_equal_congruence (g)", "specialize jordan_tuple_equal_congruence (h)", "specialize jordan_tuple_equal_congruence (s)", "specialize jordan_tuple_equal_congruence (k)", "apply jordan_tuple_equal_congruence", "exact heq", "exact hright", "specialize jordan_tuple_bounded_congruence_equal (n)", "specialize jordan_tuple_bounded_congruence_equal (b)", "specialize jordan_tuple_bounded_congruence_equal (c)", "specialize jordan_tuple_bounded_congruence_equal (d)", "specialize jordan_tuple_bounded_congruence_equal (e)", "specialize jordan_tuple_bounded_congruence_equal (k)", "apply jordan_tuple_bounded_congruence_equal", "exact hb", "exact hd", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (f)", "specialize jordan_tuple_congruence_symm (g)", "specialize jordan_tuple_congruence_symm (b)", "specialize jordan_tuple_congruence_symm (c)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hleft", "exact hmiddle" ], "script_sha256": "875f9bb452a62c4f62a2c3b62e7ce4f685b2a1722a7029f909d5c81d845ac582", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "875f9bb452a62c4f62a2c3b62e7ce4f685b2a1722a7029f909d5c81d845ac582", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "40d63417a1fe92b830db3c8aeb41b49d834cf1777a03689da6cd0906e0f72341" } ], "stable_member": false, "statement": "forall n b c d e f g h s k. (forall jt_index_recoverleft. (exists jt_gap_recoverleftindex. jt_gap_recoverleftindex+S (jt_index_recoverleft)=(k)) -> exists jt_value_recoverleft. ((((exists fs_h_jt_recoverleftat. fs_h_jt_recoverleftat + S (jt_value_recoverleft) = S ((S (jt_index_recoverleft)) * c)) /\\ exists fs_q_jt_recoverleftat. b = fs_q_jt_recoverleftat * S ((S (jt_index_recoverleft)) * c) + (jt_value_recoverleft))) /\\ (exists jt_gap_recoverleftvalue. jt_gap_recoverleftvalue+S (jt_value_recoverleft)=(n)))) -> (forall jt_index_recoverright. (exists jt_gap_recoverrightindex. jt_gap_recoverrightindex+S (jt_index_recoverright)=(k)) -> exists jt_value_recoverright. ((((exists fs_h_jt_recoverrightat. fs_h_jt_recoverrightat + S (jt_value_recoverright) = S ((S (jt_index_recoverright)) * e)) /\\ exists fs_q_jt_recoverrightat. d = fs_q_jt_recoverrightat * S ((S (jt_index_recoverright)) * e) + (jt_value_recoverright))) /\\ (exists jt_gap_recoverrightvalue. jt_gap_recoverrightvalue+S (jt_value_recoverright)=(n)))) -> (forall jt_index_recoverfirst jt_left_recoverfirst jt_right_recoverfirst. (exists jt_gap_recoverfirstindex. jt_gap_recoverfirstindex+S (jt_index_recoverfirst)=(k)) -> (((exists fs_h_jt_recoverfirstleft. fs_h_jt_recoverfirstleft + S (jt_left_recoverfirst) = S ((S (jt_index_recoverfirst)) * g)) /\\ exists fs_q_jt_recoverfirstleft. f = fs_q_jt_recoverfirstleft * S ((S (jt_index_recoverfirst)) * g) + (jt_left_recoverfirst))) -> (((exists fs_h_jt_recoverfirstright. fs_h_jt_recoverfirstright + S (jt_right_recoverfirst) = S ((S (jt_index_recoverfirst)) * c)) /\\ exists fs_q_jt_recoverfirstright. b = fs_q_jt_recoverfirstright * S ((S (jt_index_recoverfirst)) * c) + (jt_right_recoverfirst))) -> (exists jt_left_recoverfirstmod jt_right_recoverfirstmod. (jt_left_recoverfirst)+(n)*jt_left_recoverfirstmod=(jt_right_recoverfirst)+(n)*jt_right_recoverfirstmod)) -> (forall jt_index_recoversecond jt_left_recoversecond jt_right_recoversecond. (exists jt_gap_recoversecondindex. jt_gap_recoversecondindex+S (jt_index_recoversecond)=(k)) -> (((exists fs_h_jt_recoversecondleft. fs_h_jt_recoversecondleft + S (jt_left_recoversecond) = S ((S (jt_index_recoversecond)) * s)) /\\ exists fs_q_jt_recoversecondleft. h = fs_q_jt_recoversecondleft * S ((S (jt_index_recoversecond)) * s) + (jt_left_recoversecond))) -> (((exists fs_h_jt_recoversecondright. fs_h_jt_recoversecondright + S (jt_right_recoversecond) = S ((S (jt_index_recoversecond)) * e)) /\\ exists fs_q_jt_recoversecondright. d = fs_q_jt_recoversecondright * S ((S (jt_index_recoversecond)) * e) + (jt_right_recoversecond))) -> (exists jt_left_recoversecondmod jt_right_recoversecondmod. (jt_left_recoversecond)+(n)*jt_left_recoversecondmod=(jt_right_recoversecond)+(n)*jt_right_recoversecondmod)) -> (forall jt_index_recoveroutput jt_left_recoveroutput jt_right_recoveroutput. (exists jt_gap_recoveroutputindex. jt_gap_recoveroutputindex+S (jt_index_recoveroutput)=(k)) -> (((exists fs_h_jt_recoveroutputleft. fs_h_jt_recoveroutputleft + S (jt_left_recoveroutput) = S ((S (jt_index_recoveroutput)) * g)) /\\ exists fs_q_jt_recoveroutputleft. f = fs_q_jt_recoveroutputleft * S ((S (jt_index_recoveroutput)) * g) + (jt_left_recoveroutput))) -> (((exists fs_h_jt_recoveroutputright. fs_h_jt_recoveroutputright + S (jt_right_recoveroutput) = S ((S (jt_index_recoveroutput)) * s)) /\\ exists fs_q_jt_recoveroutputright. h = fs_q_jt_recoveroutputright * S ((S (jt_index_recoveroutput)) * s) + (jt_right_recoveroutput))) -> jt_left_recoveroutput=jt_right_recoveroutput) -> (forall jt_index_recoverinputs jt_left_recoverinputs jt_right_recoverinputs. (exists jt_gap_recoverinputsindex. jt_gap_recoverinputsindex+S (jt_index_recoverinputs)=(k)) -> (((exists fs_h_jt_recoverinputsleft. fs_h_jt_recoverinputsleft + S (jt_left_recoverinputs) = S ((S (jt_index_recoverinputs)) * c)) /\\ exists fs_q_jt_recoverinputsleft. b = fs_q_jt_recoverinputsleft * S ((S (jt_index_recoverinputs)) * c) + (jt_left_recoverinputs))) -> (((exists fs_h_jt_recoverinputsright. fs_h_jt_recoverinputsright + S (jt_right_recoverinputs) = S ((S (jt_index_recoverinputs)) * e)) /\\ exists fs_q_jt_recoverinputsright. d = fs_q_jt_recoverinputsright * S ((S (jt_index_recoverinputs)) * e) + (jt_right_recoverinputs))) -> jt_left_recoverinputs=jt_right_recoverinputs)", "statement_sha256": "40d63417a1fe92b830db3c8aeb41b49d834cf1777a03689da6cd0906e0f72341", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Equal output tuples recover equal canonical input components, not equal raw beta codes." }, { "admission_dependencies": [ "jordan_tuple_bounded_congruence_equal", "jordan_tuple_congruence_coprime_product", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 94, "body_proof_nodes": 239, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro h", "intro s", "intro k", "intro hcop", "intro hf", "intro hh", "cases hf", "cases hf_right", "cases hh", "cases hh_right", "specialize jordan_tuple_bounded_congruence_equal (m*n)", "specialize jordan_tuple_bounded_congruence_equal (f)", "specialize jordan_tuple_bounded_congruence_equal (g)", "specialize jordan_tuple_bounded_congruence_equal (h)", "specialize jordan_tuple_bounded_congruence_equal (s)", "specialize jordan_tuple_bounded_congruence_equal (k)", "apply jordan_tuple_bounded_congruence_equal", "exact hf_left", "exact hh_left", "specialize jordan_tuple_congruence_coprime_product (m)", "specialize jordan_tuple_congruence_coprime_product (n)", "specialize jordan_tuple_congruence_coprime_product (f)", "specialize jordan_tuple_congruence_coprime_product (g)", "specialize jordan_tuple_congruence_coprime_product (h)", "specialize jordan_tuple_congruence_coprime_product (s)", "specialize jordan_tuple_congruence_coprime_product (k)", "apply jordan_tuple_congruence_coprime_product", "exact hcop", "specialize jordan_tuple_congruence_trans (m)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (h)", "specialize jordan_tuple_congruence_trans (s)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "exact hf_right_left", "specialize jordan_tuple_congruence_symm (m)", "specialize jordan_tuple_congruence_symm (h)", "specialize jordan_tuple_congruence_symm (s)", "specialize jordan_tuple_congruence_symm (b)", "specialize jordan_tuple_congruence_symm (c)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hh_right_left", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (h)", "specialize jordan_tuple_congruence_trans (s)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "exact hf_right_right", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (h)", "specialize jordan_tuple_congruence_symm (s)", "specialize jordan_tuple_congruence_symm (d)", "specialize jordan_tuple_congruence_symm (e)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hh_right_right" ], "defined_statement": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ s. ∀ k. Coprime(m,n) → JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k) → JordanCanonicalTupleCRT(m,n,b,c,d,e,h,s,k) → IntegerVectorZero(f,g,h,s,k)", "defined_statement_sha256": "0fa4af292bb906de1982dbce46b8b49226d57dab05c36893c813d964be9f1582", "definition_uses": { "ND0121": 1, "ND0380": 2, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "4e160147ddf158d2332c82424e11fabe02462d71d43bd149a8c31f86f6d5be30", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro s" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hf" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "cases hf" } ], [ { "kind": "text", "text": "cases hf_right" } ], [ { "kind": "text", "text": "cases hh" } ], [ { "kind": "text", "text": "cases hh_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (m*n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_bounded_congruence_equal (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_bounded_congruence_equal" } ], [ { "kind": "text", "text": "exact hf_left" } ], [ { "kind": "text", "text": "exact hh_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_coprime_product (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_coprime_product" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "exact hf_right_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (m)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hh_right_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (f)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (g)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "exact hf_right_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (h)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (s)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (e)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_symm" } ], [ { "kind": "text", "text": "exact hh_right_right" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0380": 2, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ b. ∀ c. ∀ d. ∀ e. ∀ f. ∀ g. ∀ h. ∀ s. ∀ k. " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,h,s,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(f,g,h,s,k)" } ] }, "dependencies": [ "jordan_tuple_bounded_congruence_equal", "jordan_tuple_congruence_coprime_product", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT003C", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_canonical_crt_tuple_unique", "original_ha_bundle_verified": true, "proof_bundle_node_id": 321, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro b", "intro c", "intro d", "intro e", "intro f", "intro g", "intro h", "intro s", "intro k", "intro hcop", "intro hf", "intro hh", "cases hf", "cases hf_right", "cases hh", "cases hh_right", "specialize jordan_tuple_bounded_congruence_equal (m*n)", "specialize jordan_tuple_bounded_congruence_equal (f)", "specialize jordan_tuple_bounded_congruence_equal (g)", "specialize jordan_tuple_bounded_congruence_equal (h)", "specialize jordan_tuple_bounded_congruence_equal (s)", "specialize jordan_tuple_bounded_congruence_equal (k)", "apply jordan_tuple_bounded_congruence_equal", "exact hf_left", "exact hh_left", "specialize jordan_tuple_congruence_coprime_product (m)", "specialize jordan_tuple_congruence_coprime_product (n)", "specialize jordan_tuple_congruence_coprime_product (f)", "specialize jordan_tuple_congruence_coprime_product (g)", "specialize jordan_tuple_congruence_coprime_product (h)", "specialize jordan_tuple_congruence_coprime_product (s)", "specialize jordan_tuple_congruence_coprime_product (k)", "apply jordan_tuple_congruence_coprime_product", "exact hcop", "specialize jordan_tuple_congruence_trans (m)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (h)", "specialize jordan_tuple_congruence_trans (s)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "exact hf_right_left", "specialize jordan_tuple_congruence_symm (m)", "specialize jordan_tuple_congruence_symm (h)", "specialize jordan_tuple_congruence_symm (s)", "specialize jordan_tuple_congruence_symm (b)", "specialize jordan_tuple_congruence_symm (c)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hh_right_left", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (f)", "specialize jordan_tuple_congruence_trans (g)", "specialize jordan_tuple_congruence_trans (d)", "specialize jordan_tuple_congruence_trans (e)", "specialize jordan_tuple_congruence_trans (h)", "specialize jordan_tuple_congruence_trans (s)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "exact hf_right_right", "specialize jordan_tuple_congruence_symm (n)", "specialize jordan_tuple_congruence_symm (h)", "specialize jordan_tuple_congruence_symm (s)", "specialize jordan_tuple_congruence_symm (d)", "specialize jordan_tuple_congruence_symm (e)", "specialize jordan_tuple_congruence_symm (k)", "apply jordan_tuple_congruence_symm", "exact hh_right_right" ], "script_sha256": "cd1e062fb52ecad46f818d0d5cad43a989ea4aeedad92ce5306c8d44971916d9", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "cd1e062fb52ecad46f818d0d5cad43a989ea4aeedad92ce5306c8d44971916d9", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "4e160147ddf158d2332c82424e11fabe02462d71d43bd149a8c31f86f6d5be30" } ], "stable_member": false, "statement": "forall m n b c d e f g h s k. (forall jt_divisor_uniquecrtcop. (exists jt_factor_uniquecrtcopa. (m)=(jt_divisor_uniquecrtcop)*jt_factor_uniquecrtcopa) -> (exists jt_factor_uniquecrtcopb. (n)=(jt_divisor_uniquecrtcop)*jt_factor_uniquecrtcopb) -> jt_divisor_uniquecrtcop=1) -> (((forall jt_index_uniquecrtfirstbound. (exists jt_gap_uniquecrtfirstboundindex. jt_gap_uniquecrtfirstboundindex+S (jt_index_uniquecrtfirstbound)=(k)) -> exists jt_value_uniquecrtfirstbound. ((((exists fs_h_jt_uniquecrtfirstboundat. fs_h_jt_uniquecrtfirstboundat + S (jt_value_uniquecrtfirstbound) = S ((S (jt_index_uniquecrtfirstbound)) * g)) /\\ exists fs_q_jt_uniquecrtfirstboundat. f = fs_q_jt_uniquecrtfirstboundat * S ((S (jt_index_uniquecrtfirstbound)) * g) + (jt_value_uniquecrtfirstbound))) /\\ (exists jt_gap_uniquecrtfirstboundvalue. jt_gap_uniquecrtfirstboundvalue+S (jt_value_uniquecrtfirstbound)=(m*n)))) /\\ (((forall jt_index_uniquecrtfirstleft jt_left_uniquecrtfirstleft jt_right_uniquecrtfirstleft. (exists jt_gap_uniquecrtfirstleftindex. jt_gap_uniquecrtfirstleftindex+S (jt_index_uniquecrtfirstleft)=(k)) -> (((exists fs_h_jt_uniquecrtfirstleftleft. fs_h_jt_uniquecrtfirstleftleft + S (jt_left_uniquecrtfirstleft) = S ((S (jt_index_uniquecrtfirstleft)) * g)) /\\ exists fs_q_jt_uniquecrtfirstleftleft. f = fs_q_jt_uniquecrtfirstleftleft * S ((S (jt_index_uniquecrtfirstleft)) * g) + (jt_left_uniquecrtfirstleft))) -> (((exists fs_h_jt_uniquecrtfirstleftright. fs_h_jt_uniquecrtfirstleftright + S (jt_right_uniquecrtfirstleft) = S ((S (jt_index_uniquecrtfirstleft)) * c)) /\\ exists fs_q_jt_uniquecrtfirstleftright. b = fs_q_jt_uniquecrtfirstleftright * S ((S (jt_index_uniquecrtfirstleft)) * c) + (jt_right_uniquecrtfirstleft))) -> (exists jt_left_uniquecrtfirstleftmod jt_right_uniquecrtfirstleftmod. (jt_left_uniquecrtfirstleft)+(m)*jt_left_uniquecrtfirstleftmod=(jt_right_uniquecrtfirstleft)+(m)*jt_right_uniquecrtfirstleftmod)) /\\ (forall jt_index_uniquecrtfirstright jt_left_uniquecrtfirstright jt_right_uniquecrtfirstright. (exists jt_gap_uniquecrtfirstrightindex. jt_gap_uniquecrtfirstrightindex+S (jt_index_uniquecrtfirstright)=(k)) -> (((exists fs_h_jt_uniquecrtfirstrightleft. fs_h_jt_uniquecrtfirstrightleft + S (jt_left_uniquecrtfirstright) = S ((S (jt_index_uniquecrtfirstright)) * g)) /\\ exists fs_q_jt_uniquecrtfirstrightleft. f = fs_q_jt_uniquecrtfirstrightleft * S ((S (jt_index_uniquecrtfirstright)) * g) + (jt_left_uniquecrtfirstright))) -> (((exists fs_h_jt_uniquecrtfirstrightright. fs_h_jt_uniquecrtfirstrightright + S (jt_right_uniquecrtfirstright) = S ((S (jt_index_uniquecrtfirstright)) * e)) /\\ exists fs_q_jt_uniquecrtfirstrightright. d = fs_q_jt_uniquecrtfirstrightright * S ((S (jt_index_uniquecrtfirstright)) * e) + (jt_right_uniquecrtfirstright))) -> (exists jt_left_uniquecrtfirstrightmod jt_right_uniquecrtfirstrightmod. (jt_left_uniquecrtfirstright)+(n)*jt_left_uniquecrtfirstrightmod=(jt_right_uniquecrtfirstright)+(n)*jt_right_uniquecrtfirstrightmod)))))) -> (((forall jt_index_uniquecrtsecondbound. (exists jt_gap_uniquecrtsecondboundindex. jt_gap_uniquecrtsecondboundindex+S (jt_index_uniquecrtsecondbound)=(k)) -> exists jt_value_uniquecrtsecondbound. ((((exists fs_h_jt_uniquecrtsecondboundat. fs_h_jt_uniquecrtsecondboundat + S (jt_value_uniquecrtsecondbound) = S ((S (jt_index_uniquecrtsecondbound)) * s)) /\\ exists fs_q_jt_uniquecrtsecondboundat. h = fs_q_jt_uniquecrtsecondboundat * S ((S (jt_index_uniquecrtsecondbound)) * s) + (jt_value_uniquecrtsecondbound))) /\\ (exists jt_gap_uniquecrtsecondboundvalue. jt_gap_uniquecrtsecondboundvalue+S (jt_value_uniquecrtsecondbound)=(m*n)))) /\\ (((forall jt_index_uniquecrtsecondleft jt_left_uniquecrtsecondleft jt_right_uniquecrtsecondleft. (exists jt_gap_uniquecrtsecondleftindex. jt_gap_uniquecrtsecondleftindex+S (jt_index_uniquecrtsecondleft)=(k)) -> (((exists fs_h_jt_uniquecrtsecondleftleft. fs_h_jt_uniquecrtsecondleftleft + S (jt_left_uniquecrtsecondleft) = S ((S (jt_index_uniquecrtsecondleft)) * s)) /\\ exists fs_q_jt_uniquecrtsecondleftleft. h = fs_q_jt_uniquecrtsecondleftleft * S ((S (jt_index_uniquecrtsecondleft)) * s) + (jt_left_uniquecrtsecondleft))) -> (((exists fs_h_jt_uniquecrtsecondleftright. fs_h_jt_uniquecrtsecondleftright + S (jt_right_uniquecrtsecondleft) = S ((S (jt_index_uniquecrtsecondleft)) * c)) /\\ exists fs_q_jt_uniquecrtsecondleftright. b = fs_q_jt_uniquecrtsecondleftright * S ((S (jt_index_uniquecrtsecondleft)) * c) + (jt_right_uniquecrtsecondleft))) -> (exists jt_left_uniquecrtsecondleftmod jt_right_uniquecrtsecondleftmod. (jt_left_uniquecrtsecondleft)+(m)*jt_left_uniquecrtsecondleftmod=(jt_right_uniquecrtsecondleft)+(m)*jt_right_uniquecrtsecondleftmod)) /\\ (forall jt_index_uniquecrtsecondright jt_left_uniquecrtsecondright jt_right_uniquecrtsecondright. (exists jt_gap_uniquecrtsecondrightindex. jt_gap_uniquecrtsecondrightindex+S (jt_index_uniquecrtsecondright)=(k)) -> (((exists fs_h_jt_uniquecrtsecondrightleft. fs_h_jt_uniquecrtsecondrightleft + S (jt_left_uniquecrtsecondright) = S ((S (jt_index_uniquecrtsecondright)) * s)) /\\ exists fs_q_jt_uniquecrtsecondrightleft. h = fs_q_jt_uniquecrtsecondrightleft * S ((S (jt_index_uniquecrtsecondright)) * s) + (jt_left_uniquecrtsecondright))) -> (((exists fs_h_jt_uniquecrtsecondrightright. fs_h_jt_uniquecrtsecondrightright + S (jt_right_uniquecrtsecondright) = S ((S (jt_index_uniquecrtsecondright)) * e)) /\\ exists fs_q_jt_uniquecrtsecondrightright. d = fs_q_jt_uniquecrtsecondrightright * S ((S (jt_index_uniquecrtsecondright)) * e) + (jt_right_uniquecrtsecondright))) -> (exists jt_left_uniquecrtsecondrightmod jt_right_uniquecrtsecondrightmod. (jt_left_uniquecrtsecondright)+(n)*jt_left_uniquecrtsecondrightmod=(jt_right_uniquecrtsecondright)+(n)*jt_right_uniquecrtsecondrightmod)))))) -> (forall jt_index_uniquecrtoutputs jt_left_uniquecrtoutputs jt_right_uniquecrtoutputs. (exists jt_gap_uniquecrtoutputsindex. jt_gap_uniquecrtoutputsindex+S (jt_index_uniquecrtoutputs)=(k)) -> (((exists fs_h_jt_uniquecrtoutputsleft. fs_h_jt_uniquecrtoutputsleft + S (jt_left_uniquecrtoutputs) = S ((S (jt_index_uniquecrtoutputs)) * g)) /\\ exists fs_q_jt_uniquecrtoutputsleft. f = fs_q_jt_uniquecrtoutputsleft * S ((S (jt_index_uniquecrtoutputs)) * g) + (jt_left_uniquecrtoutputs))) -> (((exists fs_h_jt_uniquecrtoutputsright. fs_h_jt_uniquecrtoutputsright + S (jt_right_uniquecrtoutputs) = S ((S (jt_index_uniquecrtoutputs)) * s)) /\\ exists fs_q_jt_uniquecrtoutputsright. h = fs_q_jt_uniquecrtoutputsright * S ((S (jt_index_uniquecrtoutputs)) * s) + (jt_right_uniquecrtoutputs))) -> jt_left_uniquecrtoutputs=jt_right_uniquecrtoutputs)", "statement_sha256": "4e160147ddf158d2332c82424e11fabe02462d71d43bd149a8c31f86f6d5be30", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The actual two-coordinate CRT output is unique below the product modulus." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 28, "body_proof_nodes": 113, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro j", "intro i", "intro b", "intro c", "intro he", "intro hi", "intro hentry", "cases he", "cases he_right", "cases hentry", "have hv : ∃ d. ∃ e. BetaAt(A,B,i,d) ∧ BetaAt(C,D,i,e) ∧ (BetaPrefixInto(d,e,k,n) ∧ JordanPrimitiveTuple(n,d,e,k))", "specialize he_left (i)", "apply he_left", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_right", "cases hv_witness_witness_left", "have hb : x=b", "specialize beta_at_unique (A)", "specialize beta_at_unique (B)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (b)", "apply beta_at_unique", "exact hv_witness_witness_left_left", "exact hentry_left", "have hc : x1=c", "specialize beta_at_unique (C)", "specialize beta_at_unique (D)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (c)", "apply beta_at_unique", "exact hv_witness_witness_left_right", "exact hentry_right", "split", "rewrite hb at hv_witness_witness_right_left", "rewrite hc at hv_witness_witness_right_left", "rewrite hc at hv_witness_witness_right_left", "exact hv_witness_witness_right_left", "rewrite hb at hv_witness_witness_right_right", "rewrite hc at hv_witness_witness_right_right", "rewrite hc at hv_witness_witness_right_right", "exact hv_witness_witness_right_right" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ i. ∀ b. ∀ c. JordanTupleEnumeration(k,n,A,B,C,D,j) → Lt(i,j) → BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) → BetaPrefixInto(b,c,k,n) ∧ JordanPrimitiveTuple(n,b,c,k)", "defined_statement_sha256": "905516daa5bd9c8f4bcdb34eadddf34a03cd777fb5395e312083ea204191f951", "definition_uses": { "ND0262": 2, "ND0372": 2, "ND0374": 1, "PD0002": 1, "PD0013": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "aab729fff5720d895b51e7beb4f51350a575443433c26090d1dc0da941f416fa", "free_names": [], "script_definition_uses": { "ND0262": 1, "ND0372": 1, "PD0013": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hentry" } ], [ { "kind": "text", "text": "cases he" } ], [ { "kind": "text", "text": "cases he_right" } ], [ { "kind": "text", "text": "cases hentry" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ d. ∃ e. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(d,e,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,d,e,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize he_left (i)" } ], [ { "kind": "text", "text": "apply he_left" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_left" } ], [ { "kind": "text", "text": "have hb : x=b" } ], [ { "kind": "text", "text": "specialize beta_at_unique (A)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (B)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hv_witness_witness_left_left" } ], [ { "kind": "text", "text": "exact hentry_left" } ], [ { "kind": "text", "text": "have hc : x1=c" } ], [ { "kind": "text", "text": "specialize beta_at_unique (C)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (D)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hv_witness_witness_left_right" } ], [ { "kind": "text", "text": "exact hentry_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hb at hv_witness_witness_right_left" } ], [ { "kind": "text", "text": "rewrite hc at hv_witness_witness_right_left" } ], [ { "kind": "text", "text": "rewrite hc at hv_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_right_left" } ], [ { "kind": "text", "text": "rewrite hb at hv_witness_witness_right_right" } ], [ { "kind": "text", "text": "rewrite hc at hv_witness_witness_right_right" } ], [ { "kind": "text", "text": "rewrite hc at hv_witness_witness_right_right" } ], [ { "kind": "text", "text": "exact hv_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0374": 1, "PD0002": 1, "PD0013": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ i. ∀ b. ∀ c. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT003D", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_actual_value", "original_ha_bundle_verified": true, "proof_bundle_node_id": 322, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro j", "intro i", "intro b", "intro c", "intro he", "intro hi", "intro hentry", "cases he", "cases he_right", "cases hentry", "have hv : exists d e. ((((((exists fs_h_jt_enumvaluecode. fs_h_jt_enumvaluecode + S (d) = S ((S (i)) * B)) /\\ exists fs_q_jt_enumvaluecode. A = fs_q_jt_enumvaluecode * S ((S (i)) * B) + (d))) /\\ (((exists fs_h_jt_enumvaluescale. fs_h_jt_enumvaluescale + S (e) = S ((S (i)) * D)) /\\ exists fs_q_jt_enumvaluescale. C = fs_q_jt_enumvaluescale * S ((S (i)) * D) + (e))))) /\\ (((forall jt_index_enumvaluebound. (exists jt_gap_enumvalueboundindex. jt_gap_enumvalueboundindex+S (jt_index_enumvaluebound)=(k)) -> exists jt_value_enumvaluebound. ((((exists fs_h_jt_enumvalueboundat. fs_h_jt_enumvalueboundat + S (jt_value_enumvaluebound) = S ((S (jt_index_enumvaluebound)) * e)) /\\ exists fs_q_jt_enumvalueboundat. d = fs_q_jt_enumvalueboundat * S ((S (jt_index_enumvaluebound)) * e) + (jt_value_enumvaluebound))) /\\ (exists jt_gap_enumvalueboundvalue. jt_gap_enumvalueboundvalue+S (jt_value_enumvaluebound)=(n)))) /\\ (forall jt_divisor_enumvalueprimitive. (exists jt_factor_enumvalueprimitivemodulus. (n)=(jt_divisor_enumvalueprimitive)*jt_factor_enumvalueprimitivemodulus) -> (forall jt_index_enumvalueprimitivecoordinates jt_value_enumvalueprimitivecoordinates. (exists jt_gap_enumvalueprimitivecoordinatesindex. jt_gap_enumvalueprimitivecoordinatesindex+S (jt_index_enumvalueprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumvalueprimitivecoordinatesat. fs_h_jt_enumvalueprimitivecoordinatesat + S (jt_value_enumvalueprimitivecoordinates) = S ((S (jt_index_enumvalueprimitivecoordinates)) * e)) /\\ exists fs_q_jt_enumvalueprimitivecoordinatesat. d = fs_q_jt_enumvalueprimitivecoordinatesat * S ((S (jt_index_enumvalueprimitivecoordinates)) * e) + (jt_value_enumvalueprimitivecoordinates))) -> (exists jt_factor_enumvalueprimitivecoordinatesdivides. (jt_value_enumvalueprimitivecoordinates)=(jt_divisor_enumvalueprimitive)*jt_factor_enumvalueprimitivecoordinatesdivides)) -> jt_divisor_enumvalueprimitive=1))))", "specialize he_left (i)", "apply he_left", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_right", "cases hv_witness_witness_left", "have hb : x=b", "specialize beta_at_unique (A)", "specialize beta_at_unique (B)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (b)", "apply beta_at_unique", "exact hv_witness_witness_left_left", "exact hentry_left", "have hc : x1=c", "specialize beta_at_unique (C)", "specialize beta_at_unique (D)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (c)", "apply beta_at_unique", "exact hv_witness_witness_left_right", "exact hentry_right", "split", "rewrite hb at hv_witness_witness_right_left", "rewrite hc at hv_witness_witness_right_left", "rewrite hc at hv_witness_witness_right_left", "exact hv_witness_witness_right_left", "rewrite hb at hv_witness_witness_right_right", "rewrite hc at hv_witness_witness_right_right", "rewrite hc at hv_witness_witness_right_right", "exact hv_witness_witness_right_right" ], "script_sha256": "eed1aa1cc3fa51f9a812cc2cebc381776ad8c4c6a4b2661d469dd4ed2674e11a", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "eed1aa1cc3fa51f9a812cc2cebc381776ad8c4c6a4b2661d469dd4ed2674e11a", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "aab729fff5720d895b51e7beb4f51350a575443433c26090d1dc0da941f416fa" } ], "stable_member": false, "statement": "forall k n A B C D j i b c. (((forall jt_i_actualenum. (exists jt_gap_actualenumsoundindex. jt_gap_actualenumsoundindex+S (jt_i_actualenum)=(j)) -> exists jt_b_actualenum jt_c_actualenum. ((((((exists fs_h_jt_actualenumsoundcode. fs_h_jt_actualenumsoundcode + S (jt_b_actualenum) = S ((S (jt_i_actualenum)) * B)) /\\ exists fs_q_jt_actualenumsoundcode. A = fs_q_jt_actualenumsoundcode * S ((S (jt_i_actualenum)) * B) + (jt_b_actualenum))) /\\ (((exists fs_h_jt_actualenumsoundscale. fs_h_jt_actualenumsoundscale + S (jt_c_actualenum) = S ((S (jt_i_actualenum)) * D)) /\\ exists fs_q_jt_actualenumsoundscale. C = fs_q_jt_actualenumsoundscale * S ((S (jt_i_actualenum)) * D) + (jt_c_actualenum))))) /\\ (((forall jt_index_actualenumbound. (exists jt_gap_actualenumboundindex. jt_gap_actualenumboundindex+S (jt_index_actualenumbound)=(k)) -> exists jt_value_actualenumbound. ((((exists fs_h_jt_actualenumboundat. fs_h_jt_actualenumboundat + S (jt_value_actualenumbound) = S ((S (jt_index_actualenumbound)) * jt_c_actualenum)) /\\ exists fs_q_jt_actualenumboundat. jt_b_actualenum = fs_q_jt_actualenumboundat * S ((S (jt_index_actualenumbound)) * jt_c_actualenum) + (jt_value_actualenumbound))) /\\ (exists jt_gap_actualenumboundvalue. jt_gap_actualenumboundvalue+S (jt_value_actualenumbound)=(n)))) /\\ (forall jt_divisor_actualenumprimitive. (exists jt_factor_actualenumprimitivemodulus. (n)=(jt_divisor_actualenumprimitive)*jt_factor_actualenumprimitivemodulus) -> (forall jt_index_actualenumprimitivecoordinates jt_value_actualenumprimitivecoordinates. (exists jt_gap_actualenumprimitivecoordinatesindex. jt_gap_actualenumprimitivecoordinatesindex+S (jt_index_actualenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_actualenumprimitivecoordinatesat. fs_h_jt_actualenumprimitivecoordinatesat + S (jt_value_actualenumprimitivecoordinates) = S ((S (jt_index_actualenumprimitivecoordinates)) * jt_c_actualenum)) /\\ exists fs_q_jt_actualenumprimitivecoordinatesat. jt_b_actualenum = fs_q_jt_actualenumprimitivecoordinatesat * S ((S (jt_index_actualenumprimitivecoordinates)) * jt_c_actualenum) + (jt_value_actualenumprimitivecoordinates))) -> (exists jt_factor_actualenumprimitivecoordinatesdivides. (jt_value_actualenumprimitivecoordinates)=(jt_divisor_actualenumprimitive)*jt_factor_actualenumprimitivecoordinatesdivides)) -> jt_divisor_actualenumprimitive=1))))) /\\ (((forall jt_b_actualenum jt_c_actualenum. (forall jt_index_actualenuminputbound. (exists jt_gap_actualenuminputboundindex. jt_gap_actualenuminputboundindex+S (jt_index_actualenuminputbound)=(k)) -> exists jt_value_actualenuminputbound. ((((exists fs_h_jt_actualenuminputboundat. fs_h_jt_actualenuminputboundat + S (jt_value_actualenuminputbound) = S ((S (jt_index_actualenuminputbound)) * jt_c_actualenum)) /\\ exists fs_q_jt_actualenuminputboundat. jt_b_actualenum = fs_q_jt_actualenuminputboundat * S ((S (jt_index_actualenuminputbound)) * jt_c_actualenum) + (jt_value_actualenuminputbound))) /\\ (exists jt_gap_actualenuminputboundvalue. jt_gap_actualenuminputboundvalue+S (jt_value_actualenuminputbound)=(n)))) -> (forall jt_divisor_actualenuminputprimitive. (exists jt_factor_actualenuminputprimitivemodulus. (n)=(jt_divisor_actualenuminputprimitive)*jt_factor_actualenuminputprimitivemodulus) -> (forall jt_index_actualenuminputprimitivecoordinates jt_value_actualenuminputprimitivecoordinates. (exists jt_gap_actualenuminputprimitivecoordinatesindex. jt_gap_actualenuminputprimitivecoordinatesindex+S (jt_index_actualenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_actualenuminputprimitivecoordinatesat. fs_h_jt_actualenuminputprimitivecoordinatesat + S (jt_value_actualenuminputprimitivecoordinates) = S ((S (jt_index_actualenuminputprimitivecoordinates)) * jt_c_actualenum)) /\\ exists fs_q_jt_actualenuminputprimitivecoordinatesat. jt_b_actualenum = fs_q_jt_actualenuminputprimitivecoordinatesat * S ((S (jt_index_actualenuminputprimitivecoordinates)) * jt_c_actualenum) + (jt_value_actualenuminputprimitivecoordinates))) -> (exists jt_factor_actualenuminputprimitivecoordinatesdivides. (jt_value_actualenuminputprimitivecoordinates)=(jt_divisor_actualenuminputprimitive)*jt_factor_actualenuminputprimitivecoordinatesdivides)) -> jt_divisor_actualenuminputprimitive=1) -> exists jt_i_actualenum jt_d_actualenum jt_e_actualenum. ((exists jt_gap_actualenumcompleteindex. jt_gap_actualenumcompleteindex+S (jt_i_actualenum)=(j)) /\\ (((((((exists fs_h_jt_actualenumcompletecode. fs_h_jt_actualenumcompletecode + S (jt_d_actualenum) = S ((S (jt_i_actualenum)) * B)) /\\ exists fs_q_jt_actualenumcompletecode. A = fs_q_jt_actualenumcompletecode * S ((S (jt_i_actualenum)) * B) + (jt_d_actualenum))) /\\ (((exists fs_h_jt_actualenumcompletescale. fs_h_jt_actualenumcompletescale + S (jt_e_actualenum) = S ((S (jt_i_actualenum)) * D)) /\\ exists fs_q_jt_actualenumcompletescale. C = fs_q_jt_actualenumcompletescale * S ((S (jt_i_actualenum)) * D) + (jt_e_actualenum))))) /\\ (forall jt_index_actualenumrepresented jt_left_actualenumrepresented jt_right_actualenumrepresented. (exists jt_gap_actualenumrepresentedindex. jt_gap_actualenumrepresentedindex+S (jt_index_actualenumrepresented)=(k)) -> (((exists fs_h_jt_actualenumrepresentedleft. fs_h_jt_actualenumrepresentedleft + S (jt_left_actualenumrepresented) = S ((S (jt_index_actualenumrepresented)) * jt_c_actualenum)) /\\ exists fs_q_jt_actualenumrepresentedleft. jt_b_actualenum = fs_q_jt_actualenumrepresentedleft * S ((S (jt_index_actualenumrepresented)) * jt_c_actualenum) + (jt_left_actualenumrepresented))) -> (((exists fs_h_jt_actualenumrepresentedright. fs_h_jt_actualenumrepresentedright + S (jt_right_actualenumrepresented) = S ((S (jt_index_actualenumrepresented)) * jt_e_actualenum)) /\\ exists fs_q_jt_actualenumrepresentedright. jt_d_actualenum = fs_q_jt_actualenumrepresentedright * S ((S (jt_index_actualenumrepresented)) * jt_e_actualenum) + (jt_right_actualenumrepresented))) -> jt_left_actualenumrepresented=jt_right_actualenumrepresented))))) /\\ (forall jt_i_actualenum jt_h_actualenum jt_b_actualenum jt_c_actualenum jt_d_actualenum jt_e_actualenum. (exists jt_gap_actualenumfirstindex. jt_gap_actualenumfirstindex+S (jt_i_actualenum)=(j)) -> (exists jt_gap_actualenumsecondindex. jt_gap_actualenumsecondindex+S (jt_h_actualenum)=(j)) -> (((((exists fs_h_jt_actualenumfirstcode. fs_h_jt_actualenumfirstcode + S (jt_b_actualenum) = S ((S (jt_i_actualenum)) * B)) /\\ exists fs_q_jt_actualenumfirstcode. A = fs_q_jt_actualenumfirstcode * S ((S (jt_i_actualenum)) * B) + (jt_b_actualenum))) /\\ (((exists fs_h_jt_actualenumfirstscale. fs_h_jt_actualenumfirstscale + S (jt_c_actualenum) = S ((S (jt_i_actualenum)) * D)) /\\ exists fs_q_jt_actualenumfirstscale. C = fs_q_jt_actualenumfirstscale * S ((S (jt_i_actualenum)) * D) + (jt_c_actualenum))))) -> (((((exists fs_h_jt_actualenumsecondcode. fs_h_jt_actualenumsecondcode + S (jt_d_actualenum) = S ((S (jt_h_actualenum)) * B)) /\\ exists fs_q_jt_actualenumsecondcode. A = fs_q_jt_actualenumsecondcode * S ((S (jt_h_actualenum)) * B) + (jt_d_actualenum))) /\\ (((exists fs_h_jt_actualenumsecondscale. fs_h_jt_actualenumsecondscale + S (jt_e_actualenum) = S ((S (jt_h_actualenum)) * D)) /\\ exists fs_q_jt_actualenumsecondscale. C = fs_q_jt_actualenumsecondscale * S ((S (jt_h_actualenum)) * D) + (jt_e_actualenum))))) -> (forall jt_index_actualenumsame jt_left_actualenumsame jt_right_actualenumsame. (exists jt_gap_actualenumsameindex. jt_gap_actualenumsameindex+S (jt_index_actualenumsame)=(k)) -> (((exists fs_h_jt_actualenumsameleft. fs_h_jt_actualenumsameleft + S (jt_left_actualenumsame) = S ((S (jt_index_actualenumsame)) * jt_c_actualenum)) /\\ exists fs_q_jt_actualenumsameleft. jt_b_actualenum = fs_q_jt_actualenumsameleft * S ((S (jt_index_actualenumsame)) * jt_c_actualenum) + (jt_left_actualenumsame))) -> (((exists fs_h_jt_actualenumsameright. fs_h_jt_actualenumsameright + S (jt_right_actualenumsame) = S ((S (jt_index_actualenumsame)) * jt_e_actualenum)) /\\ exists fs_q_jt_actualenumsameright. jt_d_actualenum = fs_q_jt_actualenumsameright * S ((S (jt_index_actualenumsame)) * jt_e_actualenum) + (jt_right_actualenumsame))) -> jt_left_actualenumsame=jt_right_actualenumsame) -> jt_i_actualenum=jt_h_actualenum))))) -> (exists jt_gap_actualindex. jt_gap_actualindex+S (i)=(j)) -> (((((exists fs_h_jt_actualentrycode. fs_h_jt_actualentrycode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_actualentrycode. A = fs_q_jt_actualentrycode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_actualentryscale. fs_h_jt_actualentryscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_actualentryscale. C = fs_q_jt_actualentryscale * S ((S (i)) * D) + (c))))) -> ((forall jt_index_actualbound. (exists jt_gap_actualboundindex. jt_gap_actualboundindex+S (jt_index_actualbound)=(k)) -> exists jt_value_actualbound. ((((exists fs_h_jt_actualboundat. fs_h_jt_actualboundat + S (jt_value_actualbound) = S ((S (jt_index_actualbound)) * c)) /\\ exists fs_q_jt_actualboundat. b = fs_q_jt_actualboundat * S ((S (jt_index_actualbound)) * c) + (jt_value_actualbound))) /\\ (exists jt_gap_actualboundvalue. jt_gap_actualboundvalue+S (jt_value_actualbound)=(n)))) /\\ (forall jt_divisor_actualprimitive. (exists jt_factor_actualprimitivemodulus. (n)=(jt_divisor_actualprimitive)*jt_factor_actualprimitivemodulus) -> (forall jt_index_actualprimitivecoordinates jt_value_actualprimitivecoordinates. (exists jt_gap_actualprimitivecoordinatesindex. jt_gap_actualprimitivecoordinatesindex+S (jt_index_actualprimitivecoordinates)=(k)) -> (((exists fs_h_jt_actualprimitivecoordinatesat. fs_h_jt_actualprimitivecoordinatesat + S (jt_value_actualprimitivecoordinates) = S ((S (jt_index_actualprimitivecoordinates)) * c)) /\\ exists fs_q_jt_actualprimitivecoordinatesat. b = fs_q_jt_actualprimitivecoordinatesat * S ((S (jt_index_actualprimitivecoordinates)) * c) + (jt_value_actualprimitivecoordinates))) -> (exists jt_factor_actualprimitivecoordinatesdivides. (jt_value_actualprimitivecoordinates)=(jt_divisor_actualprimitive)*jt_factor_actualprimitivecoordinatesdivides)) -> jt_divisor_actualprimitive=1))", "statement_sha256": "aab729fff5720d895b51e7beb4f51350a575443433c26090d1dc0da941f416fa", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every actual decoded enumeration entry is bounded and primitive." }, { "admission_dependencies": [], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 27, "body_proof_nodes": 41, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro j", "intro b", "intro c", "intro he", "intro hb", "intro hp", "cases he", "cases he_right", "specialize he_right_left (b)", "specialize he_right_left (c)", "apply he_right_left", "exact hb", "exact hp" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ b. ∀ c. JordanTupleEnumeration(k,n,A,B,C,D,j) → BetaPrefixInto(b,c,k,n) → JordanPrimitiveTuple(n,b,c,k) → JordanTupleListed(b,c,k,A,B,C,D,j)", "defined_statement_sha256": "5e994b045423421f32e7378847c2feedfa29f8faf627fb7456fa7142fe85b734", "definition_uses": { "ND0262": 1, "ND0372": 1, "ND0374": 1, "ND0376": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "a8e0db5a87d119b2df49e06bc34fd6f587c77a530d28264073a9c684d7574442", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "cases he" } ], [ { "kind": "text", "text": "cases he_right" } ], [ { "kind": "text", "text": "specialize he_right_left (b)" } ], [ { "kind": "text", "text": "specialize he_right_left (c)" } ], [ { "kind": "text", "text": "apply he_right_left" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hp" } ] ], "statement_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0374": 1, "ND0376": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ b. ∀ c. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,A,B,C,D,j)" } ] }, "dependencies": [], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT003E", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_complete", "original_ha_bundle_verified": true, "proof_bundle_node_id": 323, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro j", "intro b", "intro c", "intro he", "intro hb", "intro hp", "cases he", "cases he_right", "specialize he_right_left (b)", "specialize he_right_left (c)", "apply he_right_left", "exact hb", "exact hp" ], "script_sha256": "72dc103e3019ea90cb55eaffb91de56b8dc605dcedea321a94af0e7d3bee84b1", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "72dc103e3019ea90cb55eaffb91de56b8dc605dcedea321a94af0e7d3bee84b1", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "a8e0db5a87d119b2df49e06bc34fd6f587c77a530d28264073a9c684d7574442" } ], "stable_member": false, "statement": "forall k n A B C D j b c. (((forall jt_i_completeenum. (exists jt_gap_completeenumsoundindex. jt_gap_completeenumsoundindex+S (jt_i_completeenum)=(j)) -> exists jt_b_completeenum jt_c_completeenum. ((((((exists fs_h_jt_completeenumsoundcode. fs_h_jt_completeenumsoundcode + S (jt_b_completeenum) = S ((S (jt_i_completeenum)) * B)) /\\ exists fs_q_jt_completeenumsoundcode. A = fs_q_jt_completeenumsoundcode * S ((S (jt_i_completeenum)) * B) + (jt_b_completeenum))) /\\ (((exists fs_h_jt_completeenumsoundscale. fs_h_jt_completeenumsoundscale + S (jt_c_completeenum) = S ((S (jt_i_completeenum)) * D)) /\\ exists fs_q_jt_completeenumsoundscale. C = fs_q_jt_completeenumsoundscale * S ((S (jt_i_completeenum)) * D) + (jt_c_completeenum))))) /\\ (((forall jt_index_completeenumbound. (exists jt_gap_completeenumboundindex. jt_gap_completeenumboundindex+S (jt_index_completeenumbound)=(k)) -> exists jt_value_completeenumbound. ((((exists fs_h_jt_completeenumboundat. fs_h_jt_completeenumboundat + S (jt_value_completeenumbound) = S ((S (jt_index_completeenumbound)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumboundat. jt_b_completeenum = fs_q_jt_completeenumboundat * S ((S (jt_index_completeenumbound)) * jt_c_completeenum) + (jt_value_completeenumbound))) /\\ (exists jt_gap_completeenumboundvalue. jt_gap_completeenumboundvalue+S (jt_value_completeenumbound)=(n)))) /\\ (forall jt_divisor_completeenumprimitive. (exists jt_factor_completeenumprimitivemodulus. (n)=(jt_divisor_completeenumprimitive)*jt_factor_completeenumprimitivemodulus) -> (forall jt_index_completeenumprimitivecoordinates jt_value_completeenumprimitivecoordinates. (exists jt_gap_completeenumprimitivecoordinatesindex. jt_gap_completeenumprimitivecoordinatesindex+S (jt_index_completeenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completeenumprimitivecoordinatesat. fs_h_jt_completeenumprimitivecoordinatesat + S (jt_value_completeenumprimitivecoordinates) = S ((S (jt_index_completeenumprimitivecoordinates)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumprimitivecoordinatesat. jt_b_completeenum = fs_q_jt_completeenumprimitivecoordinatesat * S ((S (jt_index_completeenumprimitivecoordinates)) * jt_c_completeenum) + (jt_value_completeenumprimitivecoordinates))) -> (exists jt_factor_completeenumprimitivecoordinatesdivides. (jt_value_completeenumprimitivecoordinates)=(jt_divisor_completeenumprimitive)*jt_factor_completeenumprimitivecoordinatesdivides)) -> jt_divisor_completeenumprimitive=1))))) /\\ (((forall jt_b_completeenum jt_c_completeenum. (forall jt_index_completeenuminputbound. (exists jt_gap_completeenuminputboundindex. jt_gap_completeenuminputboundindex+S (jt_index_completeenuminputbound)=(k)) -> exists jt_value_completeenuminputbound. ((((exists fs_h_jt_completeenuminputboundat. fs_h_jt_completeenuminputboundat + S (jt_value_completeenuminputbound) = S ((S (jt_index_completeenuminputbound)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenuminputboundat. jt_b_completeenum = fs_q_jt_completeenuminputboundat * S ((S (jt_index_completeenuminputbound)) * jt_c_completeenum) + (jt_value_completeenuminputbound))) /\\ (exists jt_gap_completeenuminputboundvalue. jt_gap_completeenuminputboundvalue+S (jt_value_completeenuminputbound)=(n)))) -> (forall jt_divisor_completeenuminputprimitive. (exists jt_factor_completeenuminputprimitivemodulus. (n)=(jt_divisor_completeenuminputprimitive)*jt_factor_completeenuminputprimitivemodulus) -> (forall jt_index_completeenuminputprimitivecoordinates jt_value_completeenuminputprimitivecoordinates. (exists jt_gap_completeenuminputprimitivecoordinatesindex. jt_gap_completeenuminputprimitivecoordinatesindex+S (jt_index_completeenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completeenuminputprimitivecoordinatesat. fs_h_jt_completeenuminputprimitivecoordinatesat + S (jt_value_completeenuminputprimitivecoordinates) = S ((S (jt_index_completeenuminputprimitivecoordinates)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenuminputprimitivecoordinatesat. jt_b_completeenum = fs_q_jt_completeenuminputprimitivecoordinatesat * S ((S (jt_index_completeenuminputprimitivecoordinates)) * jt_c_completeenum) + (jt_value_completeenuminputprimitivecoordinates))) -> (exists jt_factor_completeenuminputprimitivecoordinatesdivides. (jt_value_completeenuminputprimitivecoordinates)=(jt_divisor_completeenuminputprimitive)*jt_factor_completeenuminputprimitivecoordinatesdivides)) -> jt_divisor_completeenuminputprimitive=1) -> exists jt_i_completeenum jt_d_completeenum jt_e_completeenum. ((exists jt_gap_completeenumcompleteindex. jt_gap_completeenumcompleteindex+S (jt_i_completeenum)=(j)) /\\ (((((((exists fs_h_jt_completeenumcompletecode. fs_h_jt_completeenumcompletecode + S (jt_d_completeenum) = S ((S (jt_i_completeenum)) * B)) /\\ exists fs_q_jt_completeenumcompletecode. A = fs_q_jt_completeenumcompletecode * S ((S (jt_i_completeenum)) * B) + (jt_d_completeenum))) /\\ (((exists fs_h_jt_completeenumcompletescale. fs_h_jt_completeenumcompletescale + S (jt_e_completeenum) = S ((S (jt_i_completeenum)) * D)) /\\ exists fs_q_jt_completeenumcompletescale. C = fs_q_jt_completeenumcompletescale * S ((S (jt_i_completeenum)) * D) + (jt_e_completeenum))))) /\\ (forall jt_index_completeenumrepresented jt_left_completeenumrepresented jt_right_completeenumrepresented. (exists jt_gap_completeenumrepresentedindex. jt_gap_completeenumrepresentedindex+S (jt_index_completeenumrepresented)=(k)) -> (((exists fs_h_jt_completeenumrepresentedleft. fs_h_jt_completeenumrepresentedleft + S (jt_left_completeenumrepresented) = S ((S (jt_index_completeenumrepresented)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumrepresentedleft. jt_b_completeenum = fs_q_jt_completeenumrepresentedleft * S ((S (jt_index_completeenumrepresented)) * jt_c_completeenum) + (jt_left_completeenumrepresented))) -> (((exists fs_h_jt_completeenumrepresentedright. fs_h_jt_completeenumrepresentedright + S (jt_right_completeenumrepresented) = S ((S (jt_index_completeenumrepresented)) * jt_e_completeenum)) /\\ exists fs_q_jt_completeenumrepresentedright. jt_d_completeenum = fs_q_jt_completeenumrepresentedright * S ((S (jt_index_completeenumrepresented)) * jt_e_completeenum) + (jt_right_completeenumrepresented))) -> jt_left_completeenumrepresented=jt_right_completeenumrepresented))))) /\\ (forall jt_i_completeenum jt_h_completeenum jt_b_completeenum jt_c_completeenum jt_d_completeenum jt_e_completeenum. (exists jt_gap_completeenumfirstindex. jt_gap_completeenumfirstindex+S (jt_i_completeenum)=(j)) -> (exists jt_gap_completeenumsecondindex. jt_gap_completeenumsecondindex+S (jt_h_completeenum)=(j)) -> (((((exists fs_h_jt_completeenumfirstcode. fs_h_jt_completeenumfirstcode + S (jt_b_completeenum) = S ((S (jt_i_completeenum)) * B)) /\\ exists fs_q_jt_completeenumfirstcode. A = fs_q_jt_completeenumfirstcode * S ((S (jt_i_completeenum)) * B) + (jt_b_completeenum))) /\\ (((exists fs_h_jt_completeenumfirstscale. fs_h_jt_completeenumfirstscale + S (jt_c_completeenum) = S ((S (jt_i_completeenum)) * D)) /\\ exists fs_q_jt_completeenumfirstscale. C = fs_q_jt_completeenumfirstscale * S ((S (jt_i_completeenum)) * D) + (jt_c_completeenum))))) -> (((((exists fs_h_jt_completeenumsecondcode. fs_h_jt_completeenumsecondcode + S (jt_d_completeenum) = S ((S (jt_h_completeenum)) * B)) /\\ exists fs_q_jt_completeenumsecondcode. A = fs_q_jt_completeenumsecondcode * S ((S (jt_h_completeenum)) * B) + (jt_d_completeenum))) /\\ (((exists fs_h_jt_completeenumsecondscale. fs_h_jt_completeenumsecondscale + S (jt_e_completeenum) = S ((S (jt_h_completeenum)) * D)) /\\ exists fs_q_jt_completeenumsecondscale. C = fs_q_jt_completeenumsecondscale * S ((S (jt_h_completeenum)) * D) + (jt_e_completeenum))))) -> (forall jt_index_completeenumsame jt_left_completeenumsame jt_right_completeenumsame. (exists jt_gap_completeenumsameindex. jt_gap_completeenumsameindex+S (jt_index_completeenumsame)=(k)) -> (((exists fs_h_jt_completeenumsameleft. fs_h_jt_completeenumsameleft + S (jt_left_completeenumsame) = S ((S (jt_index_completeenumsame)) * jt_c_completeenum)) /\\ exists fs_q_jt_completeenumsameleft. jt_b_completeenum = fs_q_jt_completeenumsameleft * S ((S (jt_index_completeenumsame)) * jt_c_completeenum) + (jt_left_completeenumsame))) -> (((exists fs_h_jt_completeenumsameright. fs_h_jt_completeenumsameright + S (jt_right_completeenumsame) = S ((S (jt_index_completeenumsame)) * jt_e_completeenum)) /\\ exists fs_q_jt_completeenumsameright. jt_d_completeenum = fs_q_jt_completeenumsameright * S ((S (jt_index_completeenumsame)) * jt_e_completeenum) + (jt_right_completeenumsame))) -> jt_left_completeenumsame=jt_right_completeenumsame) -> jt_i_completeenum=jt_h_completeenum))))) -> (forall jt_index_completebound. (exists jt_gap_completeboundindex. jt_gap_completeboundindex+S (jt_index_completebound)=(k)) -> exists jt_value_completebound. ((((exists fs_h_jt_completeboundat. fs_h_jt_completeboundat + S (jt_value_completebound) = S ((S (jt_index_completebound)) * c)) /\\ exists fs_q_jt_completeboundat. b = fs_q_jt_completeboundat * S ((S (jt_index_completebound)) * c) + (jt_value_completebound))) /\\ (exists jt_gap_completeboundvalue. jt_gap_completeboundvalue+S (jt_value_completebound)=(n)))) -> (forall jt_divisor_completeprimitive. (exists jt_factor_completeprimitivemodulus. (n)=(jt_divisor_completeprimitive)*jt_factor_completeprimitivemodulus) -> (forall jt_index_completeprimitivecoordinates jt_value_completeprimitivecoordinates. (exists jt_gap_completeprimitivecoordinatesindex. jt_gap_completeprimitivecoordinatesindex+S (jt_index_completeprimitivecoordinates)=(k)) -> (((exists fs_h_jt_completeprimitivecoordinatesat. fs_h_jt_completeprimitivecoordinatesat + S (jt_value_completeprimitivecoordinates) = S ((S (jt_index_completeprimitivecoordinates)) * c)) /\\ exists fs_q_jt_completeprimitivecoordinatesat. b = fs_q_jt_completeprimitivecoordinatesat * S ((S (jt_index_completeprimitivecoordinates)) * c) + (jt_value_completeprimitivecoordinates))) -> (exists jt_factor_completeprimitivecoordinatesdivides. (jt_value_completeprimitivecoordinates)=(jt_divisor_completeprimitive)*jt_factor_completeprimitivecoordinatesdivides)) -> jt_divisor_completeprimitive=1) -> (exists jt_index_completelisted jt_code_completelisted jt_scale_completelisted. ((exists jt_gap_completelistedindex. jt_gap_completelistedindex+S (jt_index_completelisted)=(j)) /\\ (((((((exists fs_h_jt_completelistedcode. fs_h_jt_completelistedcode + S (jt_code_completelisted) = S ((S (jt_index_completelisted)) * B)) /\\ exists fs_q_jt_completelistedcode. A = fs_q_jt_completelistedcode * S ((S (jt_index_completelisted)) * B) + (jt_code_completelisted))) /\\ (((exists fs_h_jt_completelistedscale. fs_h_jt_completelistedscale + S (jt_scale_completelisted) = S ((S (jt_index_completelisted)) * D)) /\\ exists fs_q_jt_completelistedscale. C = fs_q_jt_completelistedscale * S ((S (jt_index_completelisted)) * D) + (jt_scale_completelisted))))) /\\ (forall jt_index_completelistedequal jt_left_completelistedequal jt_right_completelistedequal. (exists jt_gap_completelistedequalindex. jt_gap_completelistedequalindex+S (jt_index_completelistedequal)=(k)) -> (((exists fs_h_jt_completelistedequalleft. fs_h_jt_completelistedequalleft + S (jt_left_completelistedequal) = S ((S (jt_index_completelistedequal)) * c)) /\\ exists fs_q_jt_completelistedequalleft. b = fs_q_jt_completelistedequalleft * S ((S (jt_index_completelistedequal)) * c) + (jt_left_completelistedequal))) -> (((exists fs_h_jt_completelistedequalright. fs_h_jt_completelistedequalright + S (jt_right_completelistedequal) = S ((S (jt_index_completelistedequal)) * jt_scale_completelisted)) /\\ exists fs_q_jt_completelistedequalright. jt_code_completelisted = fs_q_jt_completelistedequalright * S ((S (jt_index_completelistedequal)) * jt_scale_completelisted) + (jt_right_completelistedequal))) -> jt_left_completelistedequal=jt_right_completelistedequal)))))", "statement_sha256": "a8e0db5a87d119b2df49e06bc34fd6f587c77a530d28264073a9c684d7574442", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Extract an actual list position for any primitive canonical tuple." }, { "admission_dependencies": [], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 45, "body_proof_nodes": 70, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro j", "intro i", "intro h", "intro b", "intro c", "intro d", "intro e", "intro he", "intro hi", "intro hh", "intro hfirst", "intro hsecond", "intro hsame", "cases he", "cases he_right", "specialize he_right_right (i)", "specialize he_right_right (h)", "specialize he_right_right (b)", "specialize he_right_right (c)", "specialize he_right_right (d)", "specialize he_right_right (e)", "apply he_right_right", "exact hi", "exact hh", "exact hfirst", "exact hsecond", "exact hsame" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ i. ∀ h. ∀ b. ∀ c. ∀ d. ∀ e. JordanTupleEnumeration(k,n,A,B,C,D,j) → Lt(i,j) → Lt(h,j) → BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) → BetaAt(A,B,h,d) ∧ BetaAt(C,D,h,e) → IntegerVectorZero(b,c,d,e,k) → i = h", "defined_statement_sha256": "5868cead8f359dd476dee947379cfe400c08a32ab25c6209089feef4d74bc511", "definition_uses": { "ND0121": 1, "ND0374": 1, "PD0002": 2, "PD0013": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "05a2da2b0fe3175a4b9f3e5ead105bd2b7e21057202e94cf2c620104e697fdf9", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "intro hfirst" } ], [ { "kind": "text", "text": "intro hsecond" } ], [ { "kind": "text", "text": "intro hsame" } ], [ { "kind": "text", "text": "cases he" } ], [ { "kind": "text", "text": "cases he_right" } ], [ { "kind": "text", "text": "specialize he_right_right (i)" } ], [ { "kind": "text", "text": "specialize he_right_right (h)" } ], [ { "kind": "text", "text": "specialize he_right_right (b)" } ], [ { "kind": "text", "text": "specialize he_right_right (c)" } ], [ { "kind": "text", "text": "specialize he_right_right (d)" } ], [ { "kind": "text", "text": "specialize he_right_right (e)" } ], [ { "kind": "text", "text": "apply he_right_right" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact hfirst" } ], [ { "kind": "text", "text": "exact hsecond" } ], [ { "kind": "text", "text": "exact hsame" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0374": 1, "PD0002": 2, "PD0013": 4 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ i. ∀ h. ∀ b. ∀ c. ∀ d. ∀ e. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(h,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,h,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,h,e)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → i = h" } ] }, "dependencies": [], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT003F", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_distinct", "original_ha_bundle_verified": true, "proof_bundle_node_id": 324, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro j", "intro i", "intro h", "intro b", "intro c", "intro d", "intro e", "intro he", "intro hi", "intro hh", "intro hfirst", "intro hsecond", "intro hsame", "cases he", "cases he_right", "specialize he_right_right (i)", "specialize he_right_right (h)", "specialize he_right_right (b)", "specialize he_right_right (c)", "specialize he_right_right (d)", "specialize he_right_right (e)", "apply he_right_right", "exact hi", "exact hh", "exact hfirst", "exact hsecond", "exact hsame" ], "script_sha256": "0cc93acd4c6b8dfe9fa41e5068d5f7bb219c6426f104050278c2599475243431", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "0cc93acd4c6b8dfe9fa41e5068d5f7bb219c6426f104050278c2599475243431", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "05a2da2b0fe3175a4b9f3e5ead105bd2b7e21057202e94cf2c620104e697fdf9" } ], "stable_member": false, "statement": "forall k n A B C D j i h b c d e. (((forall jt_i_distinctenum. (exists jt_gap_distinctenumsoundindex. jt_gap_distinctenumsoundindex+S (jt_i_distinctenum)=(j)) -> exists jt_b_distinctenum jt_c_distinctenum. ((((((exists fs_h_jt_distinctenumsoundcode. fs_h_jt_distinctenumsoundcode + S (jt_b_distinctenum) = S ((S (jt_i_distinctenum)) * B)) /\\ exists fs_q_jt_distinctenumsoundcode. A = fs_q_jt_distinctenumsoundcode * S ((S (jt_i_distinctenum)) * B) + (jt_b_distinctenum))) /\\ (((exists fs_h_jt_distinctenumsoundscale. fs_h_jt_distinctenumsoundscale + S (jt_c_distinctenum) = S ((S (jt_i_distinctenum)) * D)) /\\ exists fs_q_jt_distinctenumsoundscale. C = fs_q_jt_distinctenumsoundscale * S ((S (jt_i_distinctenum)) * D) + (jt_c_distinctenum))))) /\\ (((forall jt_index_distinctenumbound. (exists jt_gap_distinctenumboundindex. jt_gap_distinctenumboundindex+S (jt_index_distinctenumbound)=(k)) -> exists jt_value_distinctenumbound. ((((exists fs_h_jt_distinctenumboundat. fs_h_jt_distinctenumboundat + S (jt_value_distinctenumbound) = S ((S (jt_index_distinctenumbound)) * jt_c_distinctenum)) /\\ exists fs_q_jt_distinctenumboundat. jt_b_distinctenum = fs_q_jt_distinctenumboundat * S ((S (jt_index_distinctenumbound)) * jt_c_distinctenum) + (jt_value_distinctenumbound))) /\\ (exists jt_gap_distinctenumboundvalue. jt_gap_distinctenumboundvalue+S (jt_value_distinctenumbound)=(n)))) /\\ (forall jt_divisor_distinctenumprimitive. (exists jt_factor_distinctenumprimitivemodulus. (n)=(jt_divisor_distinctenumprimitive)*jt_factor_distinctenumprimitivemodulus) -> (forall jt_index_distinctenumprimitivecoordinates jt_value_distinctenumprimitivecoordinates. (exists jt_gap_distinctenumprimitivecoordinatesindex. jt_gap_distinctenumprimitivecoordinatesindex+S (jt_index_distinctenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_distinctenumprimitivecoordinatesat. fs_h_jt_distinctenumprimitivecoordinatesat + S (jt_value_distinctenumprimitivecoordinates) = S ((S (jt_index_distinctenumprimitivecoordinates)) * jt_c_distinctenum)) /\\ exists fs_q_jt_distinctenumprimitivecoordinatesat. jt_b_distinctenum = fs_q_jt_distinctenumprimitivecoordinatesat * S ((S (jt_index_distinctenumprimitivecoordinates)) * jt_c_distinctenum) + (jt_value_distinctenumprimitivecoordinates))) -> (exists jt_factor_distinctenumprimitivecoordinatesdivides. (jt_value_distinctenumprimitivecoordinates)=(jt_divisor_distinctenumprimitive)*jt_factor_distinctenumprimitivecoordinatesdivides)) -> jt_divisor_distinctenumprimitive=1))))) /\\ (((forall jt_b_distinctenum jt_c_distinctenum. (forall jt_index_distinctenuminputbound. (exists jt_gap_distinctenuminputboundindex. jt_gap_distinctenuminputboundindex+S (jt_index_distinctenuminputbound)=(k)) -> exists jt_value_distinctenuminputbound. ((((exists fs_h_jt_distinctenuminputboundat. fs_h_jt_distinctenuminputboundat + S (jt_value_distinctenuminputbound) = S ((S (jt_index_distinctenuminputbound)) * jt_c_distinctenum)) /\\ exists fs_q_jt_distinctenuminputboundat. jt_b_distinctenum = fs_q_jt_distinctenuminputboundat * S ((S (jt_index_distinctenuminputbound)) * jt_c_distinctenum) + (jt_value_distinctenuminputbound))) /\\ (exists jt_gap_distinctenuminputboundvalue. jt_gap_distinctenuminputboundvalue+S (jt_value_distinctenuminputbound)=(n)))) -> (forall jt_divisor_distinctenuminputprimitive. (exists jt_factor_distinctenuminputprimitivemodulus. (n)=(jt_divisor_distinctenuminputprimitive)*jt_factor_distinctenuminputprimitivemodulus) -> (forall jt_index_distinctenuminputprimitivecoordinates jt_value_distinctenuminputprimitivecoordinates. (exists jt_gap_distinctenuminputprimitivecoordinatesindex. jt_gap_distinctenuminputprimitivecoordinatesindex+S (jt_index_distinctenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_distinctenuminputprimitivecoordinatesat. fs_h_jt_distinctenuminputprimitivecoordinatesat + S (jt_value_distinctenuminputprimitivecoordinates) = S ((S (jt_index_distinctenuminputprimitivecoordinates)) * jt_c_distinctenum)) /\\ exists fs_q_jt_distinctenuminputprimitivecoordinatesat. jt_b_distinctenum = fs_q_jt_distinctenuminputprimitivecoordinatesat * S ((S (jt_index_distinctenuminputprimitivecoordinates)) * jt_c_distinctenum) + (jt_value_distinctenuminputprimitivecoordinates))) -> (exists jt_factor_distinctenuminputprimitivecoordinatesdivides. (jt_value_distinctenuminputprimitivecoordinates)=(jt_divisor_distinctenuminputprimitive)*jt_factor_distinctenuminputprimitivecoordinatesdivides)) -> jt_divisor_distinctenuminputprimitive=1) -> exists jt_i_distinctenum jt_d_distinctenum jt_e_distinctenum. ((exists jt_gap_distinctenumcompleteindex. jt_gap_distinctenumcompleteindex+S (jt_i_distinctenum)=(j)) /\\ (((((((exists fs_h_jt_distinctenumcompletecode. fs_h_jt_distinctenumcompletecode + S (jt_d_distinctenum) = S ((S (jt_i_distinctenum)) * B)) /\\ exists fs_q_jt_distinctenumcompletecode. A = fs_q_jt_distinctenumcompletecode * S ((S (jt_i_distinctenum)) * B) + (jt_d_distinctenum))) /\\ (((exists fs_h_jt_distinctenumcompletescale. fs_h_jt_distinctenumcompletescale + S (jt_e_distinctenum) = S ((S (jt_i_distinctenum)) * D)) /\\ exists fs_q_jt_distinctenumcompletescale. C = fs_q_jt_distinctenumcompletescale * S ((S (jt_i_distinctenum)) * D) + (jt_e_distinctenum))))) /\\ (forall jt_index_distinctenumrepresented jt_left_distinctenumrepresented jt_right_distinctenumrepresented. (exists jt_gap_distinctenumrepresentedindex. jt_gap_distinctenumrepresentedindex+S (jt_index_distinctenumrepresented)=(k)) -> (((exists fs_h_jt_distinctenumrepresentedleft. fs_h_jt_distinctenumrepresentedleft + S (jt_left_distinctenumrepresented) = S ((S (jt_index_distinctenumrepresented)) * jt_c_distinctenum)) /\\ exists fs_q_jt_distinctenumrepresentedleft. jt_b_distinctenum = fs_q_jt_distinctenumrepresentedleft * S ((S (jt_index_distinctenumrepresented)) * jt_c_distinctenum) + (jt_left_distinctenumrepresented))) -> (((exists fs_h_jt_distinctenumrepresentedright. fs_h_jt_distinctenumrepresentedright + S (jt_right_distinctenumrepresented) = S ((S (jt_index_distinctenumrepresented)) * jt_e_distinctenum)) /\\ exists fs_q_jt_distinctenumrepresentedright. jt_d_distinctenum = fs_q_jt_distinctenumrepresentedright * S ((S (jt_index_distinctenumrepresented)) * jt_e_distinctenum) + (jt_right_distinctenumrepresented))) -> jt_left_distinctenumrepresented=jt_right_distinctenumrepresented))))) /\\ (forall jt_i_distinctenum jt_h_distinctenum jt_b_distinctenum jt_c_distinctenum jt_d_distinctenum jt_e_distinctenum. (exists jt_gap_distinctenumfirstindex. jt_gap_distinctenumfirstindex+S (jt_i_distinctenum)=(j)) -> (exists jt_gap_distinctenumsecondindex. jt_gap_distinctenumsecondindex+S (jt_h_distinctenum)=(j)) -> (((((exists fs_h_jt_distinctenumfirstcode. fs_h_jt_distinctenumfirstcode + S (jt_b_distinctenum) = S ((S (jt_i_distinctenum)) * B)) /\\ exists fs_q_jt_distinctenumfirstcode. A = fs_q_jt_distinctenumfirstcode * S ((S (jt_i_distinctenum)) * B) + (jt_b_distinctenum))) /\\ (((exists fs_h_jt_distinctenumfirstscale. fs_h_jt_distinctenumfirstscale + S (jt_c_distinctenum) = S ((S (jt_i_distinctenum)) * D)) /\\ exists fs_q_jt_distinctenumfirstscale. C = fs_q_jt_distinctenumfirstscale * S ((S (jt_i_distinctenum)) * D) + (jt_c_distinctenum))))) -> (((((exists fs_h_jt_distinctenumsecondcode. fs_h_jt_distinctenumsecondcode + S (jt_d_distinctenum) = S ((S (jt_h_distinctenum)) * B)) /\\ exists fs_q_jt_distinctenumsecondcode. A = fs_q_jt_distinctenumsecondcode * S ((S (jt_h_distinctenum)) * B) + (jt_d_distinctenum))) /\\ (((exists fs_h_jt_distinctenumsecondscale. fs_h_jt_distinctenumsecondscale + S (jt_e_distinctenum) = S ((S (jt_h_distinctenum)) * D)) /\\ exists fs_q_jt_distinctenumsecondscale. C = fs_q_jt_distinctenumsecondscale * S ((S (jt_h_distinctenum)) * D) + (jt_e_distinctenum))))) -> (forall jt_index_distinctenumsame jt_left_distinctenumsame jt_right_distinctenumsame. (exists jt_gap_distinctenumsameindex. jt_gap_distinctenumsameindex+S (jt_index_distinctenumsame)=(k)) -> (((exists fs_h_jt_distinctenumsameleft. fs_h_jt_distinctenumsameleft + S (jt_left_distinctenumsame) = S ((S (jt_index_distinctenumsame)) * jt_c_distinctenum)) /\\ exists fs_q_jt_distinctenumsameleft. jt_b_distinctenum = fs_q_jt_distinctenumsameleft * S ((S (jt_index_distinctenumsame)) * jt_c_distinctenum) + (jt_left_distinctenumsame))) -> (((exists fs_h_jt_distinctenumsameright. fs_h_jt_distinctenumsameright + S (jt_right_distinctenumsame) = S ((S (jt_index_distinctenumsame)) * jt_e_distinctenum)) /\\ exists fs_q_jt_distinctenumsameright. jt_d_distinctenum = fs_q_jt_distinctenumsameright * S ((S (jt_index_distinctenumsame)) * jt_e_distinctenum) + (jt_right_distinctenumsame))) -> jt_left_distinctenumsame=jt_right_distinctenumsame) -> jt_i_distinctenum=jt_h_distinctenum))))) -> (exists jt_gap_distinctfirst. jt_gap_distinctfirst+S (i)=(j)) -> (exists jt_gap_distinctsecond. jt_gap_distinctsecond+S (h)=(j)) -> (((((exists fs_h_jt_distinctentryfirstcode. fs_h_jt_distinctentryfirstcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_distinctentryfirstcode. A = fs_q_jt_distinctentryfirstcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_distinctentryfirstscale. fs_h_jt_distinctentryfirstscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_distinctentryfirstscale. C = fs_q_jt_distinctentryfirstscale * S ((S (i)) * D) + (c))))) -> (((((exists fs_h_jt_distinctentrysecondcode. fs_h_jt_distinctentrysecondcode + S (d) = S ((S (h)) * B)) /\\ exists fs_q_jt_distinctentrysecondcode. A = fs_q_jt_distinctentrysecondcode * S ((S (h)) * B) + (d))) /\\ (((exists fs_h_jt_distinctentrysecondscale. fs_h_jt_distinctentrysecondscale + S (e) = S ((S (h)) * D)) /\\ exists fs_q_jt_distinctentrysecondscale. C = fs_q_jt_distinctentrysecondscale * S ((S (h)) * D) + (e))))) -> (forall jt_index_distinctequal jt_left_distinctequal jt_right_distinctequal. (exists jt_gap_distinctequalindex. jt_gap_distinctequalindex+S (jt_index_distinctequal)=(k)) -> (((exists fs_h_jt_distinctequalleft. fs_h_jt_distinctequalleft + S (jt_left_distinctequal) = S ((S (jt_index_distinctequal)) * c)) /\\ exists fs_q_jt_distinctequalleft. b = fs_q_jt_distinctequalleft * S ((S (jt_index_distinctequal)) * c) + (jt_left_distinctequal))) -> (((exists fs_h_jt_distinctequalright. fs_h_jt_distinctequalright + S (jt_right_distinctequal) = S ((S (jt_index_distinctequal)) * e)) /\\ exists fs_q_jt_distinctequalright. d = fs_q_jt_distinctequalright * S ((S (jt_index_distinctequal)) * e) + (jt_right_distinctequal))) -> jt_left_distinctequal=jt_right_distinctequal) -> i=h", "statement_sha256": "05a2da2b0fe3175a4b9f3e5ead105bd2b7e21057202e94cf2c620104e697fdf9", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The independent enumeration graph identifies equal coordinate tuples with equal positions." }, { "admission_dependencies": [ "jordan_rectangle_width_nonzero", "division_remainder_exists", "jordan_rectangle_quotient_bound", "jordan_primitive_crt_tuple_exists", "jordan_tuple_outer_append_exists", "finite_lt_succ_eq_or_lt", "jordan_tuple_equal_entry" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 85, "body_proof_nodes": 311, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro q", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "intro hold", "intro hq", "have hv : ~(v=0)", "intro hvzero", "specialize jordan_rectangle_width_nonzero (u)", "specialize jordan_rectangle_width_nonzero (v)", "specialize jordan_rectangle_width_nonzero (q)", "apply jordan_rectangle_width_nonzero", "exact hq", "exact hvzero", "have hcoords : ∃ i. ∃ j. DivRem(q,v,i,j)", "specialize division_remainder_exists (v)", "specialize division_remainder_exists (q)", "apply division_remainder_exists", "exact hv", "cases hcoords", "cases hcoords_witness", "cases hcoords_witness_witness", "have hrow : Lt(x,u)", "specialize jordan_rectangle_quotient_bound (u)", "specialize jordan_rectangle_quotient_bound (v)", "specialize jordan_rectangle_quotient_bound (q)", "specialize jordan_rectangle_quotient_bound (x)", "specialize jordan_rectangle_quotient_bound (x1)", "apply jordan_rectangle_quotient_bound", "exact hq", "exact hcoords_witness_witness_left", "exact hcoords_witness_witness_right", "have hlsound : ∀ i. Lt(i,u) → ∃ x. ∃ y. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) ∧ (BetaPrefixInto(x,y,k,m) ∧ JordanPrimitiveTuple(m,x,y,k))", "have hcopy : JordanTupleEnumeration(k,m,A,B,C,D,u)", "exact hleft", "cases hcopy", "exact hcopy_left", "have hl : ∃ b. ∃ c. BetaAt(A,B,x,b) ∧ BetaAt(C,D,x,c) ∧ (BetaPrefixInto(b,c,k,m) ∧ JordanPrimitiveTuple(m,b,c,k))", "specialize hlsound (x)", "apply hlsound", "exact hrow", "cases hl", "cases hl_witness", "cases hl_witness_witness", "cases hl_witness_witness_right", "have hrsound : ∀ i. Lt(i,v) → ∃ x. ∃ y. BetaAt(E,F,i,x) ∧ BetaAt(G,H,i,y) ∧ (BetaPrefixInto(x,y,k,n) ∧ JordanPrimitiveTuple(n,x,y,k))", "have hcopy : JordanTupleEnumeration(k,n,E,F,G,H,v)", "exact hright", "cases hcopy", "exact hcopy_left", "have hr : ∃ b. ∃ c. BetaAt(E,F,x1,b) ∧ BetaAt(G,H,x1,c) ∧ (BetaPrefixInto(b,c,k,n) ∧ JordanPrimitiveTuple(n,b,c,k))", "specialize hrsound (x1)", "apply hrsound", "exact hcoords_witness_witness_right", "cases hr", "cases hr_witness", "cases hr_witness_witness", "cases hr_witness_witness_right", "have hc : ∃ f. ∃ g. JordanCanonicalTupleCRT(m,n,x2,x3,x4,x5,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k)", "specialize jordan_primitive_crt_tuple_exists (m)", "specialize jordan_primitive_crt_tuple_exists (n)", "specialize jordan_primitive_crt_tuple_exists (x2)", "specialize jordan_primitive_crt_tuple_exists (x3)", "specialize jordan_primitive_crt_tuple_exists (x4)", "specialize jordan_primitive_crt_tuple_exists (x5)", "specialize jordan_primitive_crt_tuple_exists (k)", "apply jordan_primitive_crt_tuple_exists", "exact hm", "exact hn", "exact hcop", "exact hl_witness_witness_right_right", "exact hr_witness_witness_right_right", "cases hc", "cases hc_witness", "cases hc_witness_witness", "have hext : ∃ U. ∃ V. ∃ W. ∃ X. IntegerVectorZero(P,Q,U,V,q) ∧ (IntegerVectorZero(R,T,W,X,q) ∧ (BetaAt(U,V,q,x6) ∧ BetaAt(W,X,q,x7)))", "specialize jordan_tuple_outer_append_exists (P)", "specialize jordan_tuple_outer_append_exists (Q)", "specialize jordan_tuple_outer_append_exists (R)", "specialize jordan_tuple_outer_append_exists (T)", "specialize jordan_tuple_outer_append_exists (q)", "specialize jordan_tuple_outer_append_exists (x6)", "specialize jordan_tuple_outer_append_exists (x7)", "apply jordan_tuple_outer_append_exists", "cases hext", "cases hext_witness", "cases hext_witness_witness", "cases hext_witness_witness_witness", "cases hext_witness_witness_witness_witness", "cases hext_witness_witness_witness_witness_right", "cases hext_witness_witness_witness_witness_right_right", "exists x8", "exists x9", "exists x10", "exists x11", "intro p", "intro hp", "have hpc : p = q ∨ Lt(p,q)", "specialize finite_lt_succ_eq_or_lt (q)", "specialize finite_lt_succ_eq_or_lt (p)", "apply finite_lt_succ_eq_or_lt", "exact hp", "cases hpc", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "split", "exact hrow", "split", "exact hcoords_witness_witness_right", "split", "rewrite hpc_left", "exact hcoords_witness_witness_left", "split", "exact hl_witness_witness_left", "split", "exact hr_witness_witness_left", "split", "split", "rewrite hpc_left", "rewrite hpc_left", "exact hext_witness_witness_witness_witness_right_right_left", "rewrite hpc_left", "rewrite hpc_left", "exact hext_witness_witness_witness_witness_right_right_right", "split", "exact hc_witness_witness_left", "exact hc_witness_witness_right", "have hprev : ∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. ∃ f. ∃ g. Lt(i,u) ∧ (Lt(j,v) ∧ (p = v · i + j ∧ (BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) ∧ (BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) ∧ (JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k)))))))", "specialize hold (p)", "apply hold", "exact hpc_right", "cases hprev", "cases hprev_witness", "cases hprev_witness_witness", "cases hprev_witness_witness_witness", "cases hprev_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "exists x12", "exists x13", "exists x14", "exists x15", "exists x16", "exists x17", "exists x18", "exists x19", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "split", "split", "specialize jordan_tuple_equal_entry (P)", "specialize jordan_tuple_equal_entry (Q)", "specialize jordan_tuple_equal_entry (x8)", "specialize jordan_tuple_equal_entry (x9)", "specialize jordan_tuple_equal_entry (q)", "specialize jordan_tuple_equal_entry (p)", "specialize jordan_tuple_equal_entry (x18)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_left", "exact hpc_right", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_left", "specialize jordan_tuple_equal_entry (R)", "specialize jordan_tuple_equal_entry (T)", "specialize jordan_tuple_equal_entry (x10)", "specialize jordan_tuple_equal_entry (x11)", "specialize jordan_tuple_equal_entry (q)", "specialize jordan_tuple_equal_entry (p)", "specialize jordan_tuple_equal_entry (x19)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_right_left", "exact hpc_right", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_right", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ q. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q) → Lt(q,u · v) → ∃ x. ∃ y. ∃ z. ∃ i. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,S q)", "defined_statement_sha256": "75a80242538937b0ff260a170af2b60551c05030a2905cf7b3c2b55c425fc569", "definition_uses": { "ND0121": 2, "ND0262": 4, "ND0372": 6, "ND0374": 4, "ND0380": 2, "ND0381": 2, "PD0002": 7, "PD0005": 1, "PD0007": 1, "PD0013": 16 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "a41266d40b155ab1e76de188be6cd138158dc4478ba7a7cc8cbba1e85bd5215d", "free_names": [], "script_definition_uses": { "ND0121": 2, "ND0262": 4, "ND0372": 6, "ND0374": 2, "ND0380": 2, "PD0002": 6, "PD0007": 1, "PD0013": 16 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro P" } ], [ { "kind": "text", "text": "intro Q" } ], [ { "kind": "text", "text": "intro R" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hleft" } ], [ { "kind": "text", "text": "intro hright" } ], [ { "kind": "text", "text": "intro hold" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "have hv : ~(v=0)" } ], [ { "kind": "text", "text": "intro hvzero" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_width_nonzero (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_width_nonzero (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_width_nonzero (q)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_width_nonzero" } ], [ { "kind": "text", "text": "exact hq" } ], [ { "kind": "text", "text": "exact hvzero" } ], [ { "kind": "text", "text": "have hcoords : " }, { "kind": "text", "text": "∃ i. ∃ j. " }, { "definition": "PD0007", "kind": "definition", "text": "DivRem(q,v,i,j)" } ], [ { "kind": "text", "text": "specialize division_remainder_exists (v)" } ], [ { "kind": "text", "text": "specialize division_remainder_exists (q)" } ], [ { "kind": "text", "text": "apply division_remainder_exists" } ], [ { "kind": "text", "text": "exact hv" } ], [ { "kind": "text", "text": "cases hcoords" } ], [ { "kind": "text", "text": "cases hcoords_witness" } ], [ { "kind": "text", "text": "cases hcoords_witness_witness" } ], [ { "kind": "text", "text": "have hrow : " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_quotient_bound (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_quotient_bound (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_quotient_bound (q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_quotient_bound (x)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_quotient_bound (x1)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_quotient_bound" } ], [ { "kind": "text", "text": "exact hq" } ], [ { "kind": "text", "text": "exact hcoords_witness_witness_left" } ], [ { "kind": "text", "text": "exact hcoords_witness_witness_right" } ], [ { "kind": "text", "text": "have hlsound : " }, { "kind": "text", "text": "∀ i. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " → ∃ x. ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x,y,k,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,x,y,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "have hcopy : " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" } ], [ { "kind": "text", "text": "exact hleft" } ], [ { "kind": "text", "text": "cases hcopy" } ], [ { "kind": "text", "text": "exact hcopy_left" } ], [ { "kind": "text", "text": "have hl : " }, { "kind": "text", "text": "∃ b. ∃ c. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,b,c,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hlsound (x)" } ], [ { "kind": "text", "text": "apply hlsound" } ], [ { "kind": "text", "text": "exact hrow" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hl_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness_right" } ], [ { "kind": "text", "text": "have hrsound : " }, { "kind": "text", "text": "∀ i. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,v)" }, { "kind": "text", "text": " → ∃ x. ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,i,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x,y,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,x,y,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "have hcopy : " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" } ], [ { "kind": "text", "text": "exact hright" } ], [ { "kind": "text", "text": "cases hcopy" } ], [ { "kind": "text", "text": "exact hcopy_left" } ], [ { "kind": "text", "text": "have hr : " }, { "kind": "text", "text": "∃ b. ∃ c. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,x1,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,x1,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hrsound (x1)" } ], [ { "kind": "text", "text": "apply hrsound" } ], [ { "kind": "text", "text": "exact hcoords_witness_witness_right" } ], [ { "kind": "text", "text": "cases hr" } ], [ { "kind": "text", "text": "cases hr_witness" } ], [ { "kind": "text", "text": "cases hr_witness_witness" } ], [ { "kind": "text", "text": "cases hr_witness_witness_right" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "∃ f. ∃ g. " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,x2,x3,x4,x5,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (m)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (x2)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (x3)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (x4)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (x5)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_crt_tuple_exists (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_crt_tuple_exists" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hl_witness_witness_right_right" } ], [ { "kind": "text", "text": "exact hr_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "cases hc_witness" } ], [ { "kind": "text", "text": "cases hc_witness_witness" } ], [ { "kind": "text", "text": "have hext : " }, { "kind": "text", "text": "∃ U. ∃ V. ∃ W. ∃ X. " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(P,Q,U,V,q)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(R,T,W,X,q)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(U,V,q,x6)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(W,X,q,x7)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (P)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (Q)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (R)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (T)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (q)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (x6)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_outer_append_exists (x7)" } ], [ { "kind": "text", "text": "apply jordan_tuple_outer_append_exists" } ], [ { "kind": "text", "text": "cases hext" } ], [ { "kind": "text", "text": "cases hext_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hext_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "exists x8" } ], [ { "kind": "text", "text": "exists x9" } ], [ { "kind": "text", "text": "exists x10" } ], [ { "kind": "text", "text": "exists x11" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "have hpc : " }, { "kind": "text", "text": "p = q ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(p,q)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (q)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (p)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "cases hpc" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "exists x5" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "exists x7" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hrow" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hcoords_witness_witness_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hpc_left" } ], [ { "kind": "text", "text": "exact hcoords_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hl_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hr_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hpc_left" } ], [ { "kind": "text", "text": "rewrite hpc_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "rewrite hpc_left" } ], [ { "kind": "text", "text": "rewrite hpc_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hc_witness_witness_left" } ], [ { "kind": "text", "text": "exact hc_witness_witness_right" } ], [ { "kind": "text", "text": "have hprev : " }, { "kind": "text", "text": "∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. ∃ f. ∃ g. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (p = v · i + j ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" }, { "kind": "text", "text": "))))))" } ], [ { "kind": "text", "text": "specialize hold (p)" } ], [ { "kind": "text", "text": "apply hold" } ], [ { "kind": "text", "text": "exact hpc_right" } ], [ { "kind": "text", "text": "cases hprev" } ], [ { "kind": "text", "text": "cases hprev_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "exists x12" } ], [ { "kind": "text", "text": "exists x13" } ], [ { "kind": "text", "text": "exists x14" } ], [ { "kind": "text", "text": "exists x15" } ], [ { "kind": "text", "text": "exists x16" } ], [ { "kind": "text", "text": "exists x17" } ], [ { "kind": "text", "text": "exists x18" } ], [ { "kind": "text", "text": "exists x19" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (P)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (Q)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x8)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x9)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (q)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (p)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x18)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hpc_right" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (R)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (T)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x10)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x11)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (q)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (p)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_entry (x19)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_entry" } ], [ { "kind": "text", "text": "exact hext_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hpc_right" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ] ], "statement_definition_uses": { "ND0374": 2, "ND0381": 2, "PD0002": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ q. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(q,u · v)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ i. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,S q)" } ] }, "dependencies": [ "jordan_rectangle_width_nonzero", "division_remainder_exists", "jordan_rectangle_quotient_bound", "jordan_primitive_crt_tuple_exists", "jordan_tuple_outer_append_exists", "finite_lt_succ_eq_or_lt", "jordan_tuple_equal_entry" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0040", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_append", "original_ha_bundle_verified": true, "proof_bundle_node_id": 325, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro q", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "intro hold", "intro hq", "have hv : ~(v=0)", "intro hvzero", "specialize jordan_rectangle_width_nonzero (u)", "specialize jordan_rectangle_width_nonzero (v)", "specialize jordan_rectangle_width_nonzero (q)", "apply jordan_rectangle_width_nonzero", "exact hq", "exact hvzero", "have hcoords : exists i j. ((q=v*i+j) /\\ (exists jt_gap_rectcolumn. jt_gap_rectcolumn+S (j)=(v)))", "specialize division_remainder_exists (v)", "specialize division_remainder_exists (q)", "apply division_remainder_exists", "exact hv", "cases hcoords", "cases hcoords_witness", "cases hcoords_witness_witness", "have hrow : exists jt_gap_rectrow. jt_gap_rectrow+S (x)=(u)", "specialize jordan_rectangle_quotient_bound (u)", "specialize jordan_rectangle_quotient_bound (v)", "specialize jordan_rectangle_quotient_bound (q)", "specialize jordan_rectangle_quotient_bound (x)", "specialize jordan_rectangle_quotient_bound (x1)", "apply jordan_rectangle_quotient_bound", "exact hq", "exact hcoords_witness_witness_left", "exact hcoords_witness_witness_right", "have hlsound : forall i. (exists jt_gap_hlsoundindex. jt_gap_hlsoundindex+S (i)=(u)) -> exists b c. ((((((exists fs_h_jt_hlsoundentrycode. fs_h_jt_hlsoundentrycode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_hlsoundentrycode. A = fs_q_jt_hlsoundentrycode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_hlsoundentryscale. fs_h_jt_hlsoundentryscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_hlsoundentryscale. C = fs_q_jt_hlsoundentryscale * S ((S (i)) * D) + (c))))) /\\ (((forall jt_index_hlsoundbound. (exists jt_gap_hlsoundboundindex. jt_gap_hlsoundboundindex+S (jt_index_hlsoundbound)=(k)) -> exists jt_value_hlsoundbound. ((((exists fs_h_jt_hlsoundboundat. fs_h_jt_hlsoundboundat + S (jt_value_hlsoundbound) = S ((S (jt_index_hlsoundbound)) * c)) /\\ exists fs_q_jt_hlsoundboundat. b = fs_q_jt_hlsoundboundat * S ((S (jt_index_hlsoundbound)) * c) + (jt_value_hlsoundbound))) /\\ (exists jt_gap_hlsoundboundvalue. jt_gap_hlsoundboundvalue+S (jt_value_hlsoundbound)=(m)))) /\\ (forall jt_divisor_hlsoundprimitive. (exists jt_factor_hlsoundprimitivemodulus. (m)=(jt_divisor_hlsoundprimitive)*jt_factor_hlsoundprimitivemodulus) -> (forall jt_index_hlsoundprimitivecoordinates jt_value_hlsoundprimitivecoordinates. (exists jt_gap_hlsoundprimitivecoordinatesindex. jt_gap_hlsoundprimitivecoordinatesindex+S (jt_index_hlsoundprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hlsoundprimitivecoordinatesat. fs_h_jt_hlsoundprimitivecoordinatesat + S (jt_value_hlsoundprimitivecoordinates) = S ((S (jt_index_hlsoundprimitivecoordinates)) * c)) /\\ exists fs_q_jt_hlsoundprimitivecoordinatesat. b = fs_q_jt_hlsoundprimitivecoordinatesat * S ((S (jt_index_hlsoundprimitivecoordinates)) * c) + (jt_value_hlsoundprimitivecoordinates))) -> (exists jt_factor_hlsoundprimitivecoordinatesdivides. (jt_value_hlsoundprimitivecoordinates)=(jt_divisor_hlsoundprimitive)*jt_factor_hlsoundprimitivecoordinatesdivides)) -> jt_divisor_hlsoundprimitive=1))))", "have hcopy : ((forall jt_i_hlenumcopy. (exists jt_gap_hlenumcopysoundindex. jt_gap_hlenumcopysoundindex+S (jt_i_hlenumcopy)=(u)) -> exists jt_b_hlenumcopy jt_c_hlenumcopy. ((((((exists fs_h_jt_hlenumcopysoundcode. fs_h_jt_hlenumcopysoundcode + S (jt_b_hlenumcopy) = S ((S (jt_i_hlenumcopy)) * B)) /\\ exists fs_q_jt_hlenumcopysoundcode. A = fs_q_jt_hlenumcopysoundcode * S ((S (jt_i_hlenumcopy)) * B) + (jt_b_hlenumcopy))) /\\ (((exists fs_h_jt_hlenumcopysoundscale. fs_h_jt_hlenumcopysoundscale + S (jt_c_hlenumcopy) = S ((S (jt_i_hlenumcopy)) * D)) /\\ exists fs_q_jt_hlenumcopysoundscale. C = fs_q_jt_hlenumcopysoundscale * S ((S (jt_i_hlenumcopy)) * D) + (jt_c_hlenumcopy))))) /\\ (((forall jt_index_hlenumcopybound. (exists jt_gap_hlenumcopyboundindex. jt_gap_hlenumcopyboundindex+S (jt_index_hlenumcopybound)=(k)) -> exists jt_value_hlenumcopybound. ((((exists fs_h_jt_hlenumcopyboundat. fs_h_jt_hlenumcopyboundat + S (jt_value_hlenumcopybound) = S ((S (jt_index_hlenumcopybound)) * jt_c_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopyboundat. jt_b_hlenumcopy = fs_q_jt_hlenumcopyboundat * S ((S (jt_index_hlenumcopybound)) * jt_c_hlenumcopy) + (jt_value_hlenumcopybound))) /\\ (exists jt_gap_hlenumcopyboundvalue. jt_gap_hlenumcopyboundvalue+S (jt_value_hlenumcopybound)=(m)))) /\\ (forall jt_divisor_hlenumcopyprimitive. (exists jt_factor_hlenumcopyprimitivemodulus. (m)=(jt_divisor_hlenumcopyprimitive)*jt_factor_hlenumcopyprimitivemodulus) -> (forall jt_index_hlenumcopyprimitivecoordinates jt_value_hlenumcopyprimitivecoordinates. (exists jt_gap_hlenumcopyprimitivecoordinatesindex. jt_gap_hlenumcopyprimitivecoordinatesindex+S (jt_index_hlenumcopyprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hlenumcopyprimitivecoordinatesat. fs_h_jt_hlenumcopyprimitivecoordinatesat + S (jt_value_hlenumcopyprimitivecoordinates) = S ((S (jt_index_hlenumcopyprimitivecoordinates)) * jt_c_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopyprimitivecoordinatesat. jt_b_hlenumcopy = fs_q_jt_hlenumcopyprimitivecoordinatesat * S ((S (jt_index_hlenumcopyprimitivecoordinates)) * jt_c_hlenumcopy) + (jt_value_hlenumcopyprimitivecoordinates))) -> (exists jt_factor_hlenumcopyprimitivecoordinatesdivides. (jt_value_hlenumcopyprimitivecoordinates)=(jt_divisor_hlenumcopyprimitive)*jt_factor_hlenumcopyprimitivecoordinatesdivides)) -> jt_divisor_hlenumcopyprimitive=1))))) /\\ (((forall jt_b_hlenumcopy jt_c_hlenumcopy. (forall jt_index_hlenumcopyinputbound. (exists jt_gap_hlenumcopyinputboundindex. jt_gap_hlenumcopyinputboundindex+S (jt_index_hlenumcopyinputbound)=(k)) -> exists jt_value_hlenumcopyinputbound. ((((exists fs_h_jt_hlenumcopyinputboundat. fs_h_jt_hlenumcopyinputboundat + S (jt_value_hlenumcopyinputbound) = S ((S (jt_index_hlenumcopyinputbound)) * jt_c_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopyinputboundat. jt_b_hlenumcopy = fs_q_jt_hlenumcopyinputboundat * S ((S (jt_index_hlenumcopyinputbound)) * jt_c_hlenumcopy) + (jt_value_hlenumcopyinputbound))) /\\ (exists jt_gap_hlenumcopyinputboundvalue. jt_gap_hlenumcopyinputboundvalue+S (jt_value_hlenumcopyinputbound)=(m)))) -> (forall jt_divisor_hlenumcopyinputprimitive. (exists jt_factor_hlenumcopyinputprimitivemodulus. (m)=(jt_divisor_hlenumcopyinputprimitive)*jt_factor_hlenumcopyinputprimitivemodulus) -> (forall jt_index_hlenumcopyinputprimitivecoordinates jt_value_hlenumcopyinputprimitivecoordinates. (exists jt_gap_hlenumcopyinputprimitivecoordinatesindex. jt_gap_hlenumcopyinputprimitivecoordinatesindex+S (jt_index_hlenumcopyinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hlenumcopyinputprimitivecoordinatesat. fs_h_jt_hlenumcopyinputprimitivecoordinatesat + S (jt_value_hlenumcopyinputprimitivecoordinates) = S ((S (jt_index_hlenumcopyinputprimitivecoordinates)) * jt_c_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopyinputprimitivecoordinatesat. jt_b_hlenumcopy = fs_q_jt_hlenumcopyinputprimitivecoordinatesat * S ((S (jt_index_hlenumcopyinputprimitivecoordinates)) * jt_c_hlenumcopy) + (jt_value_hlenumcopyinputprimitivecoordinates))) -> (exists jt_factor_hlenumcopyinputprimitivecoordinatesdivides. (jt_value_hlenumcopyinputprimitivecoordinates)=(jt_divisor_hlenumcopyinputprimitive)*jt_factor_hlenumcopyinputprimitivecoordinatesdivides)) -> jt_divisor_hlenumcopyinputprimitive=1) -> exists jt_i_hlenumcopy jt_d_hlenumcopy jt_e_hlenumcopy. ((exists jt_gap_hlenumcopycompleteindex. jt_gap_hlenumcopycompleteindex+S (jt_i_hlenumcopy)=(u)) /\\ (((((((exists fs_h_jt_hlenumcopycompletecode. fs_h_jt_hlenumcopycompletecode + S (jt_d_hlenumcopy) = S ((S (jt_i_hlenumcopy)) * B)) /\\ exists fs_q_jt_hlenumcopycompletecode. A = fs_q_jt_hlenumcopycompletecode * S ((S (jt_i_hlenumcopy)) * B) + (jt_d_hlenumcopy))) /\\ (((exists fs_h_jt_hlenumcopycompletescale. fs_h_jt_hlenumcopycompletescale + S (jt_e_hlenumcopy) = S ((S (jt_i_hlenumcopy)) * D)) /\\ exists fs_q_jt_hlenumcopycompletescale. C = fs_q_jt_hlenumcopycompletescale * S ((S (jt_i_hlenumcopy)) * D) + (jt_e_hlenumcopy))))) /\\ (forall jt_index_hlenumcopyrepresented jt_left_hlenumcopyrepresented jt_right_hlenumcopyrepresented. (exists jt_gap_hlenumcopyrepresentedindex. jt_gap_hlenumcopyrepresentedindex+S (jt_index_hlenumcopyrepresented)=(k)) -> (((exists fs_h_jt_hlenumcopyrepresentedleft. fs_h_jt_hlenumcopyrepresentedleft + S (jt_left_hlenumcopyrepresented) = S ((S (jt_index_hlenumcopyrepresented)) * jt_c_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopyrepresentedleft. jt_b_hlenumcopy = fs_q_jt_hlenumcopyrepresentedleft * S ((S (jt_index_hlenumcopyrepresented)) * jt_c_hlenumcopy) + (jt_left_hlenumcopyrepresented))) -> (((exists fs_h_jt_hlenumcopyrepresentedright. fs_h_jt_hlenumcopyrepresentedright + S (jt_right_hlenumcopyrepresented) = S ((S (jt_index_hlenumcopyrepresented)) * jt_e_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopyrepresentedright. jt_d_hlenumcopy = fs_q_jt_hlenumcopyrepresentedright * S ((S (jt_index_hlenumcopyrepresented)) * jt_e_hlenumcopy) + (jt_right_hlenumcopyrepresented))) -> jt_left_hlenumcopyrepresented=jt_right_hlenumcopyrepresented))))) /\\ (forall jt_i_hlenumcopy jt_h_hlenumcopy jt_b_hlenumcopy jt_c_hlenumcopy jt_d_hlenumcopy jt_e_hlenumcopy. (exists jt_gap_hlenumcopyfirstindex. jt_gap_hlenumcopyfirstindex+S (jt_i_hlenumcopy)=(u)) -> (exists jt_gap_hlenumcopysecondindex. jt_gap_hlenumcopysecondindex+S (jt_h_hlenumcopy)=(u)) -> (((((exists fs_h_jt_hlenumcopyfirstcode. fs_h_jt_hlenumcopyfirstcode + S (jt_b_hlenumcopy) = S ((S (jt_i_hlenumcopy)) * B)) /\\ exists fs_q_jt_hlenumcopyfirstcode. A = fs_q_jt_hlenumcopyfirstcode * S ((S (jt_i_hlenumcopy)) * B) + (jt_b_hlenumcopy))) /\\ (((exists fs_h_jt_hlenumcopyfirstscale. fs_h_jt_hlenumcopyfirstscale + S (jt_c_hlenumcopy) = S ((S (jt_i_hlenumcopy)) * D)) /\\ exists fs_q_jt_hlenumcopyfirstscale. C = fs_q_jt_hlenumcopyfirstscale * S ((S (jt_i_hlenumcopy)) * D) + (jt_c_hlenumcopy))))) -> (((((exists fs_h_jt_hlenumcopysecondcode. fs_h_jt_hlenumcopysecondcode + S (jt_d_hlenumcopy) = S ((S (jt_h_hlenumcopy)) * B)) /\\ exists fs_q_jt_hlenumcopysecondcode. A = fs_q_jt_hlenumcopysecondcode * S ((S (jt_h_hlenumcopy)) * B) + (jt_d_hlenumcopy))) /\\ (((exists fs_h_jt_hlenumcopysecondscale. fs_h_jt_hlenumcopysecondscale + S (jt_e_hlenumcopy) = S ((S (jt_h_hlenumcopy)) * D)) /\\ exists fs_q_jt_hlenumcopysecondscale. C = fs_q_jt_hlenumcopysecondscale * S ((S (jt_h_hlenumcopy)) * D) + (jt_e_hlenumcopy))))) -> (forall jt_index_hlenumcopysame jt_left_hlenumcopysame jt_right_hlenumcopysame. (exists jt_gap_hlenumcopysameindex. jt_gap_hlenumcopysameindex+S (jt_index_hlenumcopysame)=(k)) -> (((exists fs_h_jt_hlenumcopysameleft. fs_h_jt_hlenumcopysameleft + S (jt_left_hlenumcopysame) = S ((S (jt_index_hlenumcopysame)) * jt_c_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopysameleft. jt_b_hlenumcopy = fs_q_jt_hlenumcopysameleft * S ((S (jt_index_hlenumcopysame)) * jt_c_hlenumcopy) + (jt_left_hlenumcopysame))) -> (((exists fs_h_jt_hlenumcopysameright. fs_h_jt_hlenumcopysameright + S (jt_right_hlenumcopysame) = S ((S (jt_index_hlenumcopysame)) * jt_e_hlenumcopy)) /\\ exists fs_q_jt_hlenumcopysameright. jt_d_hlenumcopy = fs_q_jt_hlenumcopysameright * S ((S (jt_index_hlenumcopysame)) * jt_e_hlenumcopy) + (jt_right_hlenumcopysame))) -> jt_left_hlenumcopysame=jt_right_hlenumcopysame) -> jt_i_hlenumcopy=jt_h_hlenumcopy))))", "exact hleft", "cases hcopy", "exact hcopy_left", "have hl : exists b c. ((((((exists fs_h_jt_hlentrycode. fs_h_jt_hlentrycode + S (b) = S ((S (x)) * B)) /\\ exists fs_q_jt_hlentrycode. A = fs_q_jt_hlentrycode * S ((S (x)) * B) + (b))) /\\ (((exists fs_h_jt_hlentryscale. fs_h_jt_hlentryscale + S (c) = S ((S (x)) * D)) /\\ exists fs_q_jt_hlentryscale. C = fs_q_jt_hlentryscale * S ((S (x)) * D) + (c))))) /\\ (((forall jt_index_hlbound. (exists jt_gap_hlboundindex. jt_gap_hlboundindex+S (jt_index_hlbound)=(k)) -> exists jt_value_hlbound. ((((exists fs_h_jt_hlboundat. fs_h_jt_hlboundat + S (jt_value_hlbound) = S ((S (jt_index_hlbound)) * c)) /\\ exists fs_q_jt_hlboundat. b = fs_q_jt_hlboundat * S ((S (jt_index_hlbound)) * c) + (jt_value_hlbound))) /\\ (exists jt_gap_hlboundvalue. jt_gap_hlboundvalue+S (jt_value_hlbound)=(m)))) /\\ (forall jt_divisor_hlprimitive. (exists jt_factor_hlprimitivemodulus. (m)=(jt_divisor_hlprimitive)*jt_factor_hlprimitivemodulus) -> (forall jt_index_hlprimitivecoordinates jt_value_hlprimitivecoordinates. (exists jt_gap_hlprimitivecoordinatesindex. jt_gap_hlprimitivecoordinatesindex+S (jt_index_hlprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hlprimitivecoordinatesat. fs_h_jt_hlprimitivecoordinatesat + S (jt_value_hlprimitivecoordinates) = S ((S (jt_index_hlprimitivecoordinates)) * c)) /\\ exists fs_q_jt_hlprimitivecoordinatesat. b = fs_q_jt_hlprimitivecoordinatesat * S ((S (jt_index_hlprimitivecoordinates)) * c) + (jt_value_hlprimitivecoordinates))) -> (exists jt_factor_hlprimitivecoordinatesdivides. (jt_value_hlprimitivecoordinates)=(jt_divisor_hlprimitive)*jt_factor_hlprimitivecoordinatesdivides)) -> jt_divisor_hlprimitive=1))))", "specialize hlsound (x)", "apply hlsound", "exact hrow", "cases hl", "cases hl_witness", "cases hl_witness_witness", "cases hl_witness_witness_right", "have hrsound : forall i. (exists jt_gap_hrsoundindex. jt_gap_hrsoundindex+S (i)=(v)) -> exists b c. ((((((exists fs_h_jt_hrsoundentrycode. fs_h_jt_hrsoundentrycode + S (b) = S ((S (i)) * F)) /\\ exists fs_q_jt_hrsoundentrycode. E = fs_q_jt_hrsoundentrycode * S ((S (i)) * F) + (b))) /\\ (((exists fs_h_jt_hrsoundentryscale. fs_h_jt_hrsoundentryscale + S (c) = S ((S (i)) * H)) /\\ exists fs_q_jt_hrsoundentryscale. G = fs_q_jt_hrsoundentryscale * S ((S (i)) * H) + (c))))) /\\ (((forall jt_index_hrsoundbound. (exists jt_gap_hrsoundboundindex. jt_gap_hrsoundboundindex+S (jt_index_hrsoundbound)=(k)) -> exists jt_value_hrsoundbound. ((((exists fs_h_jt_hrsoundboundat. fs_h_jt_hrsoundboundat + S (jt_value_hrsoundbound) = S ((S (jt_index_hrsoundbound)) * c)) /\\ exists fs_q_jt_hrsoundboundat. b = fs_q_jt_hrsoundboundat * S ((S (jt_index_hrsoundbound)) * c) + (jt_value_hrsoundbound))) /\\ (exists jt_gap_hrsoundboundvalue. jt_gap_hrsoundboundvalue+S (jt_value_hrsoundbound)=(n)))) /\\ (forall jt_divisor_hrsoundprimitive. (exists jt_factor_hrsoundprimitivemodulus. (n)=(jt_divisor_hrsoundprimitive)*jt_factor_hrsoundprimitivemodulus) -> (forall jt_index_hrsoundprimitivecoordinates jt_value_hrsoundprimitivecoordinates. (exists jt_gap_hrsoundprimitivecoordinatesindex. jt_gap_hrsoundprimitivecoordinatesindex+S (jt_index_hrsoundprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hrsoundprimitivecoordinatesat. fs_h_jt_hrsoundprimitivecoordinatesat + S (jt_value_hrsoundprimitivecoordinates) = S ((S (jt_index_hrsoundprimitivecoordinates)) * c)) /\\ exists fs_q_jt_hrsoundprimitivecoordinatesat. b = fs_q_jt_hrsoundprimitivecoordinatesat * S ((S (jt_index_hrsoundprimitivecoordinates)) * c) + (jt_value_hrsoundprimitivecoordinates))) -> (exists jt_factor_hrsoundprimitivecoordinatesdivides. (jt_value_hrsoundprimitivecoordinates)=(jt_divisor_hrsoundprimitive)*jt_factor_hrsoundprimitivecoordinatesdivides)) -> jt_divisor_hrsoundprimitive=1))))", "have hcopy : ((forall jt_i_hrenumcopy. (exists jt_gap_hrenumcopysoundindex. jt_gap_hrenumcopysoundindex+S (jt_i_hrenumcopy)=(v)) -> exists jt_b_hrenumcopy jt_c_hrenumcopy. ((((((exists fs_h_jt_hrenumcopysoundcode. fs_h_jt_hrenumcopysoundcode + S (jt_b_hrenumcopy) = S ((S (jt_i_hrenumcopy)) * F)) /\\ exists fs_q_jt_hrenumcopysoundcode. E = fs_q_jt_hrenumcopysoundcode * S ((S (jt_i_hrenumcopy)) * F) + (jt_b_hrenumcopy))) /\\ (((exists fs_h_jt_hrenumcopysoundscale. fs_h_jt_hrenumcopysoundscale + S (jt_c_hrenumcopy) = S ((S (jt_i_hrenumcopy)) * H)) /\\ exists fs_q_jt_hrenumcopysoundscale. G = fs_q_jt_hrenumcopysoundscale * S ((S (jt_i_hrenumcopy)) * H) + (jt_c_hrenumcopy))))) /\\ (((forall jt_index_hrenumcopybound. (exists jt_gap_hrenumcopyboundindex. jt_gap_hrenumcopyboundindex+S (jt_index_hrenumcopybound)=(k)) -> exists jt_value_hrenumcopybound. ((((exists fs_h_jt_hrenumcopyboundat. fs_h_jt_hrenumcopyboundat + S (jt_value_hrenumcopybound) = S ((S (jt_index_hrenumcopybound)) * jt_c_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopyboundat. jt_b_hrenumcopy = fs_q_jt_hrenumcopyboundat * S ((S (jt_index_hrenumcopybound)) * jt_c_hrenumcopy) + (jt_value_hrenumcopybound))) /\\ (exists jt_gap_hrenumcopyboundvalue. jt_gap_hrenumcopyboundvalue+S (jt_value_hrenumcopybound)=(n)))) /\\ (forall jt_divisor_hrenumcopyprimitive. (exists jt_factor_hrenumcopyprimitivemodulus. (n)=(jt_divisor_hrenumcopyprimitive)*jt_factor_hrenumcopyprimitivemodulus) -> (forall jt_index_hrenumcopyprimitivecoordinates jt_value_hrenumcopyprimitivecoordinates. (exists jt_gap_hrenumcopyprimitivecoordinatesindex. jt_gap_hrenumcopyprimitivecoordinatesindex+S (jt_index_hrenumcopyprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hrenumcopyprimitivecoordinatesat. fs_h_jt_hrenumcopyprimitivecoordinatesat + S (jt_value_hrenumcopyprimitivecoordinates) = S ((S (jt_index_hrenumcopyprimitivecoordinates)) * jt_c_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopyprimitivecoordinatesat. jt_b_hrenumcopy = fs_q_jt_hrenumcopyprimitivecoordinatesat * S ((S (jt_index_hrenumcopyprimitivecoordinates)) * jt_c_hrenumcopy) + (jt_value_hrenumcopyprimitivecoordinates))) -> (exists jt_factor_hrenumcopyprimitivecoordinatesdivides. (jt_value_hrenumcopyprimitivecoordinates)=(jt_divisor_hrenumcopyprimitive)*jt_factor_hrenumcopyprimitivecoordinatesdivides)) -> jt_divisor_hrenumcopyprimitive=1))))) /\\ (((forall jt_b_hrenumcopy jt_c_hrenumcopy. (forall jt_index_hrenumcopyinputbound. (exists jt_gap_hrenumcopyinputboundindex. jt_gap_hrenumcopyinputboundindex+S (jt_index_hrenumcopyinputbound)=(k)) -> exists jt_value_hrenumcopyinputbound. ((((exists fs_h_jt_hrenumcopyinputboundat. fs_h_jt_hrenumcopyinputboundat + S (jt_value_hrenumcopyinputbound) = S ((S (jt_index_hrenumcopyinputbound)) * jt_c_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopyinputboundat. jt_b_hrenumcopy = fs_q_jt_hrenumcopyinputboundat * S ((S (jt_index_hrenumcopyinputbound)) * jt_c_hrenumcopy) + (jt_value_hrenumcopyinputbound))) /\\ (exists jt_gap_hrenumcopyinputboundvalue. jt_gap_hrenumcopyinputboundvalue+S (jt_value_hrenumcopyinputbound)=(n)))) -> (forall jt_divisor_hrenumcopyinputprimitive. (exists jt_factor_hrenumcopyinputprimitivemodulus. (n)=(jt_divisor_hrenumcopyinputprimitive)*jt_factor_hrenumcopyinputprimitivemodulus) -> (forall jt_index_hrenumcopyinputprimitivecoordinates jt_value_hrenumcopyinputprimitivecoordinates. (exists jt_gap_hrenumcopyinputprimitivecoordinatesindex. jt_gap_hrenumcopyinputprimitivecoordinatesindex+S (jt_index_hrenumcopyinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hrenumcopyinputprimitivecoordinatesat. fs_h_jt_hrenumcopyinputprimitivecoordinatesat + S (jt_value_hrenumcopyinputprimitivecoordinates) = S ((S (jt_index_hrenumcopyinputprimitivecoordinates)) * jt_c_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopyinputprimitivecoordinatesat. jt_b_hrenumcopy = fs_q_jt_hrenumcopyinputprimitivecoordinatesat * S ((S (jt_index_hrenumcopyinputprimitivecoordinates)) * jt_c_hrenumcopy) + (jt_value_hrenumcopyinputprimitivecoordinates))) -> (exists jt_factor_hrenumcopyinputprimitivecoordinatesdivides. (jt_value_hrenumcopyinputprimitivecoordinates)=(jt_divisor_hrenumcopyinputprimitive)*jt_factor_hrenumcopyinputprimitivecoordinatesdivides)) -> jt_divisor_hrenumcopyinputprimitive=1) -> exists jt_i_hrenumcopy jt_d_hrenumcopy jt_e_hrenumcopy. ((exists jt_gap_hrenumcopycompleteindex. jt_gap_hrenumcopycompleteindex+S (jt_i_hrenumcopy)=(v)) /\\ (((((((exists fs_h_jt_hrenumcopycompletecode. fs_h_jt_hrenumcopycompletecode + S (jt_d_hrenumcopy) = S ((S (jt_i_hrenumcopy)) * F)) /\\ exists fs_q_jt_hrenumcopycompletecode. E = fs_q_jt_hrenumcopycompletecode * S ((S (jt_i_hrenumcopy)) * F) + (jt_d_hrenumcopy))) /\\ (((exists fs_h_jt_hrenumcopycompletescale. fs_h_jt_hrenumcopycompletescale + S (jt_e_hrenumcopy) = S ((S (jt_i_hrenumcopy)) * H)) /\\ exists fs_q_jt_hrenumcopycompletescale. G = fs_q_jt_hrenumcopycompletescale * S ((S (jt_i_hrenumcopy)) * H) + (jt_e_hrenumcopy))))) /\\ (forall jt_index_hrenumcopyrepresented jt_left_hrenumcopyrepresented jt_right_hrenumcopyrepresented. (exists jt_gap_hrenumcopyrepresentedindex. jt_gap_hrenumcopyrepresentedindex+S (jt_index_hrenumcopyrepresented)=(k)) -> (((exists fs_h_jt_hrenumcopyrepresentedleft. fs_h_jt_hrenumcopyrepresentedleft + S (jt_left_hrenumcopyrepresented) = S ((S (jt_index_hrenumcopyrepresented)) * jt_c_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopyrepresentedleft. jt_b_hrenumcopy = fs_q_jt_hrenumcopyrepresentedleft * S ((S (jt_index_hrenumcopyrepresented)) * jt_c_hrenumcopy) + (jt_left_hrenumcopyrepresented))) -> (((exists fs_h_jt_hrenumcopyrepresentedright. fs_h_jt_hrenumcopyrepresentedright + S (jt_right_hrenumcopyrepresented) = S ((S (jt_index_hrenumcopyrepresented)) * jt_e_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopyrepresentedright. jt_d_hrenumcopy = fs_q_jt_hrenumcopyrepresentedright * S ((S (jt_index_hrenumcopyrepresented)) * jt_e_hrenumcopy) + (jt_right_hrenumcopyrepresented))) -> jt_left_hrenumcopyrepresented=jt_right_hrenumcopyrepresented))))) /\\ (forall jt_i_hrenumcopy jt_h_hrenumcopy jt_b_hrenumcopy jt_c_hrenumcopy jt_d_hrenumcopy jt_e_hrenumcopy. (exists jt_gap_hrenumcopyfirstindex. jt_gap_hrenumcopyfirstindex+S (jt_i_hrenumcopy)=(v)) -> (exists jt_gap_hrenumcopysecondindex. jt_gap_hrenumcopysecondindex+S (jt_h_hrenumcopy)=(v)) -> (((((exists fs_h_jt_hrenumcopyfirstcode. fs_h_jt_hrenumcopyfirstcode + S (jt_b_hrenumcopy) = S ((S (jt_i_hrenumcopy)) * F)) /\\ exists fs_q_jt_hrenumcopyfirstcode. E = fs_q_jt_hrenumcopyfirstcode * S ((S (jt_i_hrenumcopy)) * F) + (jt_b_hrenumcopy))) /\\ (((exists fs_h_jt_hrenumcopyfirstscale. fs_h_jt_hrenumcopyfirstscale + S (jt_c_hrenumcopy) = S ((S (jt_i_hrenumcopy)) * H)) /\\ exists fs_q_jt_hrenumcopyfirstscale. G = fs_q_jt_hrenumcopyfirstscale * S ((S (jt_i_hrenumcopy)) * H) + (jt_c_hrenumcopy))))) -> (((((exists fs_h_jt_hrenumcopysecondcode. fs_h_jt_hrenumcopysecondcode + S (jt_d_hrenumcopy) = S ((S (jt_h_hrenumcopy)) * F)) /\\ exists fs_q_jt_hrenumcopysecondcode. E = fs_q_jt_hrenumcopysecondcode * S ((S (jt_h_hrenumcopy)) * F) + (jt_d_hrenumcopy))) /\\ (((exists fs_h_jt_hrenumcopysecondscale. fs_h_jt_hrenumcopysecondscale + S (jt_e_hrenumcopy) = S ((S (jt_h_hrenumcopy)) * H)) /\\ exists fs_q_jt_hrenumcopysecondscale. G = fs_q_jt_hrenumcopysecondscale * S ((S (jt_h_hrenumcopy)) * H) + (jt_e_hrenumcopy))))) -> (forall jt_index_hrenumcopysame jt_left_hrenumcopysame jt_right_hrenumcopysame. (exists jt_gap_hrenumcopysameindex. jt_gap_hrenumcopysameindex+S (jt_index_hrenumcopysame)=(k)) -> (((exists fs_h_jt_hrenumcopysameleft. fs_h_jt_hrenumcopysameleft + S (jt_left_hrenumcopysame) = S ((S (jt_index_hrenumcopysame)) * jt_c_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopysameleft. jt_b_hrenumcopy = fs_q_jt_hrenumcopysameleft * S ((S (jt_index_hrenumcopysame)) * jt_c_hrenumcopy) + (jt_left_hrenumcopysame))) -> (((exists fs_h_jt_hrenumcopysameright. fs_h_jt_hrenumcopysameright + S (jt_right_hrenumcopysame) = S ((S (jt_index_hrenumcopysame)) * jt_e_hrenumcopy)) /\\ exists fs_q_jt_hrenumcopysameright. jt_d_hrenumcopy = fs_q_jt_hrenumcopysameright * S ((S (jt_index_hrenumcopysame)) * jt_e_hrenumcopy) + (jt_right_hrenumcopysame))) -> jt_left_hrenumcopysame=jt_right_hrenumcopysame) -> jt_i_hrenumcopy=jt_h_hrenumcopy))))", "exact hright", "cases hcopy", "exact hcopy_left", "have hr : exists b c. ((((((exists fs_h_jt_hrentrycode. fs_h_jt_hrentrycode + S (b) = S ((S (x1)) * F)) /\\ exists fs_q_jt_hrentrycode. E = fs_q_jt_hrentrycode * S ((S (x1)) * F) + (b))) /\\ (((exists fs_h_jt_hrentryscale. fs_h_jt_hrentryscale + S (c) = S ((S (x1)) * H)) /\\ exists fs_q_jt_hrentryscale. G = fs_q_jt_hrentryscale * S ((S (x1)) * H) + (c))))) /\\ (((forall jt_index_hrbound. (exists jt_gap_hrboundindex. jt_gap_hrboundindex+S (jt_index_hrbound)=(k)) -> exists jt_value_hrbound. ((((exists fs_h_jt_hrboundat. fs_h_jt_hrboundat + S (jt_value_hrbound) = S ((S (jt_index_hrbound)) * c)) /\\ exists fs_q_jt_hrboundat. b = fs_q_jt_hrboundat * S ((S (jt_index_hrbound)) * c) + (jt_value_hrbound))) /\\ (exists jt_gap_hrboundvalue. jt_gap_hrboundvalue+S (jt_value_hrbound)=(n)))) /\\ (forall jt_divisor_hrprimitive. (exists jt_factor_hrprimitivemodulus. (n)=(jt_divisor_hrprimitive)*jt_factor_hrprimitivemodulus) -> (forall jt_index_hrprimitivecoordinates jt_value_hrprimitivecoordinates. (exists jt_gap_hrprimitivecoordinatesindex. jt_gap_hrprimitivecoordinatesindex+S (jt_index_hrprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hrprimitivecoordinatesat. fs_h_jt_hrprimitivecoordinatesat + S (jt_value_hrprimitivecoordinates) = S ((S (jt_index_hrprimitivecoordinates)) * c)) /\\ exists fs_q_jt_hrprimitivecoordinatesat. b = fs_q_jt_hrprimitivecoordinatesat * S ((S (jt_index_hrprimitivecoordinates)) * c) + (jt_value_hrprimitivecoordinates))) -> (exists jt_factor_hrprimitivecoordinatesdivides. (jt_value_hrprimitivecoordinates)=(jt_divisor_hrprimitive)*jt_factor_hrprimitivecoordinatesdivides)) -> jt_divisor_hrprimitive=1))))", "specialize hrsound (x1)", "apply hrsound", "exact hcoords_witness_witness_right", "cases hr", "cases hr_witness", "cases hr_witness_witness", "cases hr_witness_witness_right", "have hc : exists f g. ((((forall jt_index_rectchosencrtbound. (exists jt_gap_rectchosencrtboundindex. jt_gap_rectchosencrtboundindex+S (jt_index_rectchosencrtbound)=(k)) -> exists jt_value_rectchosencrtbound. ((((exists fs_h_jt_rectchosencrtboundat. fs_h_jt_rectchosencrtboundat + S (jt_value_rectchosencrtbound) = S ((S (jt_index_rectchosencrtbound)) * g)) /\\ exists fs_q_jt_rectchosencrtboundat. f = fs_q_jt_rectchosencrtboundat * S ((S (jt_index_rectchosencrtbound)) * g) + (jt_value_rectchosencrtbound))) /\\ (exists jt_gap_rectchosencrtboundvalue. jt_gap_rectchosencrtboundvalue+S (jt_value_rectchosencrtbound)=(m*n)))) /\\ (((forall jt_index_rectchosencrtleft jt_left_rectchosencrtleft jt_right_rectchosencrtleft. (exists jt_gap_rectchosencrtleftindex. jt_gap_rectchosencrtleftindex+S (jt_index_rectchosencrtleft)=(k)) -> (((exists fs_h_jt_rectchosencrtleftleft. fs_h_jt_rectchosencrtleftleft + S (jt_left_rectchosencrtleft) = S ((S (jt_index_rectchosencrtleft)) * g)) /\\ exists fs_q_jt_rectchosencrtleftleft. f = fs_q_jt_rectchosencrtleftleft * S ((S (jt_index_rectchosencrtleft)) * g) + (jt_left_rectchosencrtleft))) -> (((exists fs_h_jt_rectchosencrtleftright. fs_h_jt_rectchosencrtleftright + S (jt_right_rectchosencrtleft) = S ((S (jt_index_rectchosencrtleft)) * x3)) /\\ exists fs_q_jt_rectchosencrtleftright. x2 = fs_q_jt_rectchosencrtleftright * S ((S (jt_index_rectchosencrtleft)) * x3) + (jt_right_rectchosencrtleft))) -> (exists jt_left_rectchosencrtleftmod jt_right_rectchosencrtleftmod. (jt_left_rectchosencrtleft)+(m)*jt_left_rectchosencrtleftmod=(jt_right_rectchosencrtleft)+(m)*jt_right_rectchosencrtleftmod)) /\\ (forall jt_index_rectchosencrtright jt_left_rectchosencrtright jt_right_rectchosencrtright. (exists jt_gap_rectchosencrtrightindex. jt_gap_rectchosencrtrightindex+S (jt_index_rectchosencrtright)=(k)) -> (((exists fs_h_jt_rectchosencrtrightleft. fs_h_jt_rectchosencrtrightleft + S (jt_left_rectchosencrtright) = S ((S (jt_index_rectchosencrtright)) * g)) /\\ exists fs_q_jt_rectchosencrtrightleft. f = fs_q_jt_rectchosencrtrightleft * S ((S (jt_index_rectchosencrtright)) * g) + (jt_left_rectchosencrtright))) -> (((exists fs_h_jt_rectchosencrtrightright. fs_h_jt_rectchosencrtrightright + S (jt_right_rectchosencrtright) = S ((S (jt_index_rectchosencrtright)) * x5)) /\\ exists fs_q_jt_rectchosencrtrightright. x4 = fs_q_jt_rectchosencrtrightright * S ((S (jt_index_rectchosencrtright)) * x5) + (jt_right_rectchosencrtright))) -> (exists jt_left_rectchosencrtrightmod jt_right_rectchosencrtrightmod. (jt_left_rectchosencrtright)+(n)*jt_left_rectchosencrtrightmod=(jt_right_rectchosencrtright)+(n)*jt_right_rectchosencrtrightmod)))))) /\\ (forall jt_divisor_rectchosenprimitive. (exists jt_factor_rectchosenprimitivemodulus. (m*n)=(jt_divisor_rectchosenprimitive)*jt_factor_rectchosenprimitivemodulus) -> (forall jt_index_rectchosenprimitivecoordinates jt_value_rectchosenprimitivecoordinates. (exists jt_gap_rectchosenprimitivecoordinatesindex. jt_gap_rectchosenprimitivecoordinatesindex+S (jt_index_rectchosenprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectchosenprimitivecoordinatesat. fs_h_jt_rectchosenprimitivecoordinatesat + S (jt_value_rectchosenprimitivecoordinates) = S ((S (jt_index_rectchosenprimitivecoordinates)) * g)) /\\ exists fs_q_jt_rectchosenprimitivecoordinatesat. f = fs_q_jt_rectchosenprimitivecoordinatesat * S ((S (jt_index_rectchosenprimitivecoordinates)) * g) + (jt_value_rectchosenprimitivecoordinates))) -> (exists jt_factor_rectchosenprimitivecoordinatesdivides. (jt_value_rectchosenprimitivecoordinates)=(jt_divisor_rectchosenprimitive)*jt_factor_rectchosenprimitivecoordinatesdivides)) -> jt_divisor_rectchosenprimitive=1))", "specialize jordan_primitive_crt_tuple_exists (m)", "specialize jordan_primitive_crt_tuple_exists (n)", "specialize jordan_primitive_crt_tuple_exists (x2)", "specialize jordan_primitive_crt_tuple_exists (x3)", "specialize jordan_primitive_crt_tuple_exists (x4)", "specialize jordan_primitive_crt_tuple_exists (x5)", "specialize jordan_primitive_crt_tuple_exists (k)", "apply jordan_primitive_crt_tuple_exists", "exact hm", "exact hn", "exact hcop", "exact hl_witness_witness_right_right", "exact hr_witness_witness_right_right", "cases hc", "cases hc_witness", "cases hc_witness_witness", "have hext : exists U V W X. ((forall jt_index_rectpreservecodes jt_left_rectpreservecodes jt_right_rectpreservecodes. (exists jt_gap_rectpreservecodesindex. jt_gap_rectpreservecodesindex+S (jt_index_rectpreservecodes)=(q)) -> (((exists fs_h_jt_rectpreservecodesleft. fs_h_jt_rectpreservecodesleft + S (jt_left_rectpreservecodes) = S ((S (jt_index_rectpreservecodes)) * Q)) /\\ exists fs_q_jt_rectpreservecodesleft. P = fs_q_jt_rectpreservecodesleft * S ((S (jt_index_rectpreservecodes)) * Q) + (jt_left_rectpreservecodes))) -> (((exists fs_h_jt_rectpreservecodesright. fs_h_jt_rectpreservecodesright + S (jt_right_rectpreservecodes) = S ((S (jt_index_rectpreservecodes)) * V)) /\\ exists fs_q_jt_rectpreservecodesright. U = fs_q_jt_rectpreservecodesright * S ((S (jt_index_rectpreservecodes)) * V) + (jt_right_rectpreservecodes))) -> jt_left_rectpreservecodes=jt_right_rectpreservecodes) /\\ (((forall jt_index_rectpreservescales jt_left_rectpreservescales jt_right_rectpreservescales. (exists jt_gap_rectpreservescalesindex. jt_gap_rectpreservescalesindex+S (jt_index_rectpreservescales)=(q)) -> (((exists fs_h_jt_rectpreservescalesleft. fs_h_jt_rectpreservescalesleft + S (jt_left_rectpreservescales) = S ((S (jt_index_rectpreservescales)) * T)) /\\ exists fs_q_jt_rectpreservescalesleft. R = fs_q_jt_rectpreservescalesleft * S ((S (jt_index_rectpreservescales)) * T) + (jt_left_rectpreservescales))) -> (((exists fs_h_jt_rectpreservescalesright. fs_h_jt_rectpreservescalesright + S (jt_right_rectpreservescales) = S ((S (jt_index_rectpreservescales)) * X)) /\\ exists fs_q_jt_rectpreservescalesright. W = fs_q_jt_rectpreservescalesright * S ((S (jt_index_rectpreservescales)) * X) + (jt_right_rectpreservescales))) -> jt_left_rectpreservescales=jt_right_rectpreservescales) /\\ (((((exists fs_h_jt_rectlastcode. fs_h_jt_rectlastcode + S (x6) = S ((S (q)) * V)) /\\ exists fs_q_jt_rectlastcode. U = fs_q_jt_rectlastcode * S ((S (q)) * V) + (x6))) /\\ (((exists fs_h_jt_rectlastscale. fs_h_jt_rectlastscale + S (x7) = S ((S (q)) * X)) /\\ exists fs_q_jt_rectlastscale. W = fs_q_jt_rectlastscale * S ((S (q)) * X) + (x7))))))))", "specialize jordan_tuple_outer_append_exists (P)", "specialize jordan_tuple_outer_append_exists (Q)", "specialize jordan_tuple_outer_append_exists (R)", "specialize jordan_tuple_outer_append_exists (T)", "specialize jordan_tuple_outer_append_exists (q)", "specialize jordan_tuple_outer_append_exists (x6)", "specialize jordan_tuple_outer_append_exists (x7)", "apply jordan_tuple_outer_append_exists", "cases hext", "cases hext_witness", "cases hext_witness_witness", "cases hext_witness_witness_witness", "cases hext_witness_witness_witness_witness", "cases hext_witness_witness_witness_witness_right", "cases hext_witness_witness_witness_witness_right_right", "exists x8", "exists x9", "exists x10", "exists x11", "intro p", "intro hp", "have hpc : p=q \\/ (exists jt_gap_rectnewcase. jt_gap_rectnewcase+S (p)=(q))", "specialize finite_lt_succ_eq_or_lt (q)", "specialize finite_lt_succ_eq_or_lt (p)", "apply finite_lt_succ_eq_or_lt", "exact hp", "cases hpc", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "exists x6", "exists x7", "split", "exact hrow", "split", "exact hcoords_witness_witness_right", "split", "rewrite hpc_left", "exact hcoords_witness_witness_left", "split", "exact hl_witness_witness_left", "split", "exact hr_witness_witness_left", "split", "split", "rewrite hpc_left", "rewrite hpc_left", "exact hext_witness_witness_witness_witness_right_right_left", "rewrite hpc_left", "rewrite hpc_left", "exact hext_witness_witness_witness_witness_right_right_right", "split", "exact hc_witness_witness_left", "exact hc_witness_witness_right", "have hprev : exists i j b c d e f g. ((exists jt_gap_rectoldvaluerow. jt_gap_rectoldvaluerow+S (i)=(u)) /\\ (((exists jt_gap_rectoldvaluecolumn. jt_gap_rectoldvaluecolumn+S (j)=(v)) /\\ (((p=(v)*(i)+(j)) /\\ (((((((exists fs_h_jt_rectoldvalueleftcode. fs_h_jt_rectoldvalueleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_rectoldvalueleftcode. A = fs_q_jt_rectoldvalueleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_rectoldvalueleftscale. fs_h_jt_rectoldvalueleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_rectoldvalueleftscale. C = fs_q_jt_rectoldvalueleftscale * S ((S (i)) * D) + (c))))) /\\ (((((((exists fs_h_jt_rectoldvaluerightcode. fs_h_jt_rectoldvaluerightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_rectoldvaluerightcode. E = fs_q_jt_rectoldvaluerightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_rectoldvaluerightscale. fs_h_jt_rectoldvaluerightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_rectoldvaluerightscale. G = fs_q_jt_rectoldvaluerightscale * S ((S (j)) * H) + (e))))) /\\ (((((((exists fs_h_jt_rectoldvalueoutputcode. fs_h_jt_rectoldvalueoutputcode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_rectoldvalueoutputcode. P = fs_q_jt_rectoldvalueoutputcode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_rectoldvalueoutputscale. fs_h_jt_rectoldvalueoutputscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_rectoldvalueoutputscale. R = fs_q_jt_rectoldvalueoutputscale * S ((S (p)) * T) + (g))))) /\\ (((((forall jt_index_rectoldvaluecrtbound. (exists jt_gap_rectoldvaluecrtboundindex. jt_gap_rectoldvaluecrtboundindex+S (jt_index_rectoldvaluecrtbound)=(k)) -> exists jt_value_rectoldvaluecrtbound. ((((exists fs_h_jt_rectoldvaluecrtboundat. fs_h_jt_rectoldvaluecrtboundat + S (jt_value_rectoldvaluecrtbound) = S ((S (jt_index_rectoldvaluecrtbound)) * g)) /\\ exists fs_q_jt_rectoldvaluecrtboundat. f = fs_q_jt_rectoldvaluecrtboundat * S ((S (jt_index_rectoldvaluecrtbound)) * g) + (jt_value_rectoldvaluecrtbound))) /\\ (exists jt_gap_rectoldvaluecrtboundvalue. jt_gap_rectoldvaluecrtboundvalue+S (jt_value_rectoldvaluecrtbound)=(m*n)))) /\\ (((forall jt_index_rectoldvaluecrtleft jt_left_rectoldvaluecrtleft jt_right_rectoldvaluecrtleft. (exists jt_gap_rectoldvaluecrtleftindex. jt_gap_rectoldvaluecrtleftindex+S (jt_index_rectoldvaluecrtleft)=(k)) -> (((exists fs_h_jt_rectoldvaluecrtleftleft. fs_h_jt_rectoldvaluecrtleftleft + S (jt_left_rectoldvaluecrtleft) = S ((S (jt_index_rectoldvaluecrtleft)) * g)) /\\ exists fs_q_jt_rectoldvaluecrtleftleft. f = fs_q_jt_rectoldvaluecrtleftleft * S ((S (jt_index_rectoldvaluecrtleft)) * g) + (jt_left_rectoldvaluecrtleft))) -> (((exists fs_h_jt_rectoldvaluecrtleftright. fs_h_jt_rectoldvaluecrtleftright + S (jt_right_rectoldvaluecrtleft) = S ((S (jt_index_rectoldvaluecrtleft)) * c)) /\\ exists fs_q_jt_rectoldvaluecrtleftright. b = fs_q_jt_rectoldvaluecrtleftright * S ((S (jt_index_rectoldvaluecrtleft)) * c) + (jt_right_rectoldvaluecrtleft))) -> (exists jt_left_rectoldvaluecrtleftmod jt_right_rectoldvaluecrtleftmod. (jt_left_rectoldvaluecrtleft)+(m)*jt_left_rectoldvaluecrtleftmod=(jt_right_rectoldvaluecrtleft)+(m)*jt_right_rectoldvaluecrtleftmod)) /\\ (forall jt_index_rectoldvaluecrtright jt_left_rectoldvaluecrtright jt_right_rectoldvaluecrtright. (exists jt_gap_rectoldvaluecrtrightindex. jt_gap_rectoldvaluecrtrightindex+S (jt_index_rectoldvaluecrtright)=(k)) -> (((exists fs_h_jt_rectoldvaluecrtrightleft. fs_h_jt_rectoldvaluecrtrightleft + S (jt_left_rectoldvaluecrtright) = S ((S (jt_index_rectoldvaluecrtright)) * g)) /\\ exists fs_q_jt_rectoldvaluecrtrightleft. f = fs_q_jt_rectoldvaluecrtrightleft * S ((S (jt_index_rectoldvaluecrtright)) * g) + (jt_left_rectoldvaluecrtright))) -> (((exists fs_h_jt_rectoldvaluecrtrightright. fs_h_jt_rectoldvaluecrtrightright + S (jt_right_rectoldvaluecrtright) = S ((S (jt_index_rectoldvaluecrtright)) * e)) /\\ exists fs_q_jt_rectoldvaluecrtrightright. d = fs_q_jt_rectoldvaluecrtrightright * S ((S (jt_index_rectoldvaluecrtright)) * e) + (jt_right_rectoldvaluecrtright))) -> (exists jt_left_rectoldvaluecrtrightmod jt_right_rectoldvaluecrtrightmod. (jt_left_rectoldvaluecrtright)+(n)*jt_left_rectoldvaluecrtrightmod=(jt_right_rectoldvaluecrtright)+(n)*jt_right_rectoldvaluecrtrightmod)))))) /\\ (forall jt_divisor_rectoldvalueprimitive. (exists jt_factor_rectoldvalueprimitivemodulus. (m*n)=(jt_divisor_rectoldvalueprimitive)*jt_factor_rectoldvalueprimitivemodulus) -> (forall jt_index_rectoldvalueprimitivecoordinates jt_value_rectoldvalueprimitivecoordinates. (exists jt_gap_rectoldvalueprimitivecoordinatesindex. jt_gap_rectoldvalueprimitivecoordinatesindex+S (jt_index_rectoldvalueprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectoldvalueprimitivecoordinatesat. fs_h_jt_rectoldvalueprimitivecoordinatesat + S (jt_value_rectoldvalueprimitivecoordinates) = S ((S (jt_index_rectoldvalueprimitivecoordinates)) * g)) /\\ exists fs_q_jt_rectoldvalueprimitivecoordinatesat. f = fs_q_jt_rectoldvalueprimitivecoordinatesat * S ((S (jt_index_rectoldvalueprimitivecoordinates)) * g) + (jt_value_rectoldvalueprimitivecoordinates))) -> (exists jt_factor_rectoldvalueprimitivecoordinatesdivides. (jt_value_rectoldvalueprimitivecoordinates)=(jt_divisor_rectoldvalueprimitive)*jt_factor_rectoldvalueprimitivecoordinatesdivides)) -> jt_divisor_rectoldvalueprimitive=1))))))))))))))", "specialize hold (p)", "apply hold", "exact hpc_right", "cases hprev", "cases hprev_witness", "cases hprev_witness_witness", "cases hprev_witness_witness_witness", "cases hprev_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "cases hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "exists x12", "exists x13", "exists x14", "exists x15", "exists x16", "exists x17", "exists x18", "exists x19", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "split", "split", "specialize jordan_tuple_equal_entry (P)", "specialize jordan_tuple_equal_entry (Q)", "specialize jordan_tuple_equal_entry (x8)", "specialize jordan_tuple_equal_entry (x9)", "specialize jordan_tuple_equal_entry (q)", "specialize jordan_tuple_equal_entry (p)", "specialize jordan_tuple_equal_entry (x18)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_left", "exact hpc_right", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_left", "specialize jordan_tuple_equal_entry (R)", "specialize jordan_tuple_equal_entry (T)", "specialize jordan_tuple_equal_entry (x10)", "specialize jordan_tuple_equal_entry (x11)", "specialize jordan_tuple_equal_entry (q)", "specialize jordan_tuple_equal_entry (p)", "specialize jordan_tuple_equal_entry (x19)", "apply jordan_tuple_equal_entry", "exact hext_witness_witness_witness_witness_right_left", "exact hpc_right", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_right", "split", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "exact hprev_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" ], "script_sha256": "0026863a31acbaa16f0286cff83224d2b14a81f1ef238d676c927f517f671ce1", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "0026863a31acbaa16f0286cff83224d2b14a81f1ef238d676c927f517f671ce1", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "a41266d40b155ab1e76de188be6cd138158dc4478ba7a7cc8cbba1e85bd5215d" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v P Q R T q. ~(m=0) -> ~(n=0) -> (forall jt_divisor_rectappendcop. (exists jt_factor_rectappendcopa. (m)=(jt_divisor_rectappendcop)*jt_factor_rectappendcopa) -> (exists jt_factor_rectappendcopb. (n)=(jt_divisor_rectappendcop)*jt_factor_rectappendcopb) -> jt_divisor_rectappendcop=1) -> (((forall jt_i_rectappendleft. (exists jt_gap_rectappendleftsoundindex. jt_gap_rectappendleftsoundindex+S (jt_i_rectappendleft)=(u)) -> exists jt_b_rectappendleft jt_c_rectappendleft. ((((((exists fs_h_jt_rectappendleftsoundcode. fs_h_jt_rectappendleftsoundcode + S (jt_b_rectappendleft) = S ((S (jt_i_rectappendleft)) * B)) /\\ exists fs_q_jt_rectappendleftsoundcode. A = fs_q_jt_rectappendleftsoundcode * S ((S (jt_i_rectappendleft)) * B) + (jt_b_rectappendleft))) /\\ (((exists fs_h_jt_rectappendleftsoundscale. fs_h_jt_rectappendleftsoundscale + S (jt_c_rectappendleft) = S ((S (jt_i_rectappendleft)) * D)) /\\ exists fs_q_jt_rectappendleftsoundscale. C = fs_q_jt_rectappendleftsoundscale * S ((S (jt_i_rectappendleft)) * D) + (jt_c_rectappendleft))))) /\\ (((forall jt_index_rectappendleftbound. (exists jt_gap_rectappendleftboundindex. jt_gap_rectappendleftboundindex+S (jt_index_rectappendleftbound)=(k)) -> exists jt_value_rectappendleftbound. ((((exists fs_h_jt_rectappendleftboundat. fs_h_jt_rectappendleftboundat + S (jt_value_rectappendleftbound) = S ((S (jt_index_rectappendleftbound)) * jt_c_rectappendleft)) /\\ exists fs_q_jt_rectappendleftboundat. jt_b_rectappendleft = fs_q_jt_rectappendleftboundat * S ((S (jt_index_rectappendleftbound)) * jt_c_rectappendleft) + (jt_value_rectappendleftbound))) /\\ (exists jt_gap_rectappendleftboundvalue. jt_gap_rectappendleftboundvalue+S (jt_value_rectappendleftbound)=(m)))) /\\ (forall jt_divisor_rectappendleftprimitive. (exists jt_factor_rectappendleftprimitivemodulus. (m)=(jt_divisor_rectappendleftprimitive)*jt_factor_rectappendleftprimitivemodulus) -> (forall jt_index_rectappendleftprimitivecoordinates jt_value_rectappendleftprimitivecoordinates. (exists jt_gap_rectappendleftprimitivecoordinatesindex. jt_gap_rectappendleftprimitivecoordinatesindex+S (jt_index_rectappendleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectappendleftprimitivecoordinatesat. fs_h_jt_rectappendleftprimitivecoordinatesat + S (jt_value_rectappendleftprimitivecoordinates) = S ((S (jt_index_rectappendleftprimitivecoordinates)) * jt_c_rectappendleft)) /\\ exists fs_q_jt_rectappendleftprimitivecoordinatesat. jt_b_rectappendleft = fs_q_jt_rectappendleftprimitivecoordinatesat * S ((S (jt_index_rectappendleftprimitivecoordinates)) * jt_c_rectappendleft) + (jt_value_rectappendleftprimitivecoordinates))) -> (exists jt_factor_rectappendleftprimitivecoordinatesdivides. (jt_value_rectappendleftprimitivecoordinates)=(jt_divisor_rectappendleftprimitive)*jt_factor_rectappendleftprimitivecoordinatesdivides)) -> jt_divisor_rectappendleftprimitive=1))))) /\\ (((forall jt_b_rectappendleft jt_c_rectappendleft. (forall jt_index_rectappendleftinputbound. (exists jt_gap_rectappendleftinputboundindex. jt_gap_rectappendleftinputboundindex+S (jt_index_rectappendleftinputbound)=(k)) -> exists jt_value_rectappendleftinputbound. ((((exists fs_h_jt_rectappendleftinputboundat. fs_h_jt_rectappendleftinputboundat + S (jt_value_rectappendleftinputbound) = S ((S (jt_index_rectappendleftinputbound)) * jt_c_rectappendleft)) /\\ exists fs_q_jt_rectappendleftinputboundat. jt_b_rectappendleft = fs_q_jt_rectappendleftinputboundat * S ((S (jt_index_rectappendleftinputbound)) * jt_c_rectappendleft) + (jt_value_rectappendleftinputbound))) /\\ (exists jt_gap_rectappendleftinputboundvalue. jt_gap_rectappendleftinputboundvalue+S (jt_value_rectappendleftinputbound)=(m)))) -> (forall jt_divisor_rectappendleftinputprimitive. (exists jt_factor_rectappendleftinputprimitivemodulus. (m)=(jt_divisor_rectappendleftinputprimitive)*jt_factor_rectappendleftinputprimitivemodulus) -> (forall jt_index_rectappendleftinputprimitivecoordinates jt_value_rectappendleftinputprimitivecoordinates. (exists jt_gap_rectappendleftinputprimitivecoordinatesindex. jt_gap_rectappendleftinputprimitivecoordinatesindex+S (jt_index_rectappendleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectappendleftinputprimitivecoordinatesat. fs_h_jt_rectappendleftinputprimitivecoordinatesat + S (jt_value_rectappendleftinputprimitivecoordinates) = S ((S (jt_index_rectappendleftinputprimitivecoordinates)) * jt_c_rectappendleft)) /\\ exists fs_q_jt_rectappendleftinputprimitivecoordinatesat. jt_b_rectappendleft = fs_q_jt_rectappendleftinputprimitivecoordinatesat * S ((S (jt_index_rectappendleftinputprimitivecoordinates)) * jt_c_rectappendleft) + (jt_value_rectappendleftinputprimitivecoordinates))) -> (exists jt_factor_rectappendleftinputprimitivecoordinatesdivides. (jt_value_rectappendleftinputprimitivecoordinates)=(jt_divisor_rectappendleftinputprimitive)*jt_factor_rectappendleftinputprimitivecoordinatesdivides)) -> jt_divisor_rectappendleftinputprimitive=1) -> exists jt_i_rectappendleft jt_d_rectappendleft jt_e_rectappendleft. ((exists jt_gap_rectappendleftcompleteindex. jt_gap_rectappendleftcompleteindex+S (jt_i_rectappendleft)=(u)) /\\ (((((((exists fs_h_jt_rectappendleftcompletecode. fs_h_jt_rectappendleftcompletecode + S (jt_d_rectappendleft) = S ((S (jt_i_rectappendleft)) * B)) /\\ exists fs_q_jt_rectappendleftcompletecode. A = fs_q_jt_rectappendleftcompletecode * S ((S (jt_i_rectappendleft)) * B) + (jt_d_rectappendleft))) /\\ (((exists fs_h_jt_rectappendleftcompletescale. fs_h_jt_rectappendleftcompletescale + S (jt_e_rectappendleft) = S ((S (jt_i_rectappendleft)) * D)) /\\ exists fs_q_jt_rectappendleftcompletescale. C = fs_q_jt_rectappendleftcompletescale * S ((S (jt_i_rectappendleft)) * D) + (jt_e_rectappendleft))))) /\\ (forall jt_index_rectappendleftrepresented jt_left_rectappendleftrepresented jt_right_rectappendleftrepresented. (exists jt_gap_rectappendleftrepresentedindex. jt_gap_rectappendleftrepresentedindex+S (jt_index_rectappendleftrepresented)=(k)) -> (((exists fs_h_jt_rectappendleftrepresentedleft. fs_h_jt_rectappendleftrepresentedleft + S (jt_left_rectappendleftrepresented) = S ((S (jt_index_rectappendleftrepresented)) * jt_c_rectappendleft)) /\\ exists fs_q_jt_rectappendleftrepresentedleft. jt_b_rectappendleft = fs_q_jt_rectappendleftrepresentedleft * S ((S (jt_index_rectappendleftrepresented)) * jt_c_rectappendleft) + (jt_left_rectappendleftrepresented))) -> (((exists fs_h_jt_rectappendleftrepresentedright. fs_h_jt_rectappendleftrepresentedright + S (jt_right_rectappendleftrepresented) = S ((S (jt_index_rectappendleftrepresented)) * jt_e_rectappendleft)) /\\ exists fs_q_jt_rectappendleftrepresentedright. jt_d_rectappendleft = fs_q_jt_rectappendleftrepresentedright * S ((S (jt_index_rectappendleftrepresented)) * jt_e_rectappendleft) + (jt_right_rectappendleftrepresented))) -> jt_left_rectappendleftrepresented=jt_right_rectappendleftrepresented))))) /\\ (forall jt_i_rectappendleft jt_h_rectappendleft jt_b_rectappendleft jt_c_rectappendleft jt_d_rectappendleft jt_e_rectappendleft. (exists jt_gap_rectappendleftfirstindex. jt_gap_rectappendleftfirstindex+S (jt_i_rectappendleft)=(u)) -> (exists jt_gap_rectappendleftsecondindex. jt_gap_rectappendleftsecondindex+S (jt_h_rectappendleft)=(u)) -> (((((exists fs_h_jt_rectappendleftfirstcode. fs_h_jt_rectappendleftfirstcode + S (jt_b_rectappendleft) = S ((S (jt_i_rectappendleft)) * B)) /\\ exists fs_q_jt_rectappendleftfirstcode. A = fs_q_jt_rectappendleftfirstcode * S ((S (jt_i_rectappendleft)) * B) + (jt_b_rectappendleft))) /\\ (((exists fs_h_jt_rectappendleftfirstscale. fs_h_jt_rectappendleftfirstscale + S (jt_c_rectappendleft) = S ((S (jt_i_rectappendleft)) * D)) /\\ exists fs_q_jt_rectappendleftfirstscale. C = fs_q_jt_rectappendleftfirstscale * S ((S (jt_i_rectappendleft)) * D) + (jt_c_rectappendleft))))) -> (((((exists fs_h_jt_rectappendleftsecondcode. fs_h_jt_rectappendleftsecondcode + S (jt_d_rectappendleft) = S ((S (jt_h_rectappendleft)) * B)) /\\ exists fs_q_jt_rectappendleftsecondcode. A = fs_q_jt_rectappendleftsecondcode * S ((S (jt_h_rectappendleft)) * B) + (jt_d_rectappendleft))) /\\ (((exists fs_h_jt_rectappendleftsecondscale. fs_h_jt_rectappendleftsecondscale + S (jt_e_rectappendleft) = S ((S (jt_h_rectappendleft)) * D)) /\\ exists fs_q_jt_rectappendleftsecondscale. C = fs_q_jt_rectappendleftsecondscale * S ((S (jt_h_rectappendleft)) * D) + (jt_e_rectappendleft))))) -> (forall jt_index_rectappendleftsame jt_left_rectappendleftsame jt_right_rectappendleftsame. (exists jt_gap_rectappendleftsameindex. jt_gap_rectappendleftsameindex+S (jt_index_rectappendleftsame)=(k)) -> (((exists fs_h_jt_rectappendleftsameleft. fs_h_jt_rectappendleftsameleft + S (jt_left_rectappendleftsame) = S ((S (jt_index_rectappendleftsame)) * jt_c_rectappendleft)) /\\ exists fs_q_jt_rectappendleftsameleft. jt_b_rectappendleft = fs_q_jt_rectappendleftsameleft * S ((S (jt_index_rectappendleftsame)) * jt_c_rectappendleft) + (jt_left_rectappendleftsame))) -> (((exists fs_h_jt_rectappendleftsameright. fs_h_jt_rectappendleftsameright + S (jt_right_rectappendleftsame) = S ((S (jt_index_rectappendleftsame)) * jt_e_rectappendleft)) /\\ exists fs_q_jt_rectappendleftsameright. jt_d_rectappendleft = fs_q_jt_rectappendleftsameright * S ((S (jt_index_rectappendleftsame)) * jt_e_rectappendleft) + (jt_right_rectappendleftsame))) -> jt_left_rectappendleftsame=jt_right_rectappendleftsame) -> jt_i_rectappendleft=jt_h_rectappendleft))))) -> (((forall jt_i_rectappendright. (exists jt_gap_rectappendrightsoundindex. jt_gap_rectappendrightsoundindex+S (jt_i_rectappendright)=(v)) -> exists jt_b_rectappendright jt_c_rectappendright. ((((((exists fs_h_jt_rectappendrightsoundcode. fs_h_jt_rectappendrightsoundcode + S (jt_b_rectappendright) = S ((S (jt_i_rectappendright)) * F)) /\\ exists fs_q_jt_rectappendrightsoundcode. E = fs_q_jt_rectappendrightsoundcode * S ((S (jt_i_rectappendright)) * F) + (jt_b_rectappendright))) /\\ (((exists fs_h_jt_rectappendrightsoundscale. fs_h_jt_rectappendrightsoundscale + S (jt_c_rectappendright) = S ((S (jt_i_rectappendright)) * H)) /\\ exists fs_q_jt_rectappendrightsoundscale. G = fs_q_jt_rectappendrightsoundscale * S ((S (jt_i_rectappendright)) * H) + (jt_c_rectappendright))))) /\\ (((forall jt_index_rectappendrightbound. (exists jt_gap_rectappendrightboundindex. jt_gap_rectappendrightboundindex+S (jt_index_rectappendrightbound)=(k)) -> exists jt_value_rectappendrightbound. ((((exists fs_h_jt_rectappendrightboundat. fs_h_jt_rectappendrightboundat + S (jt_value_rectappendrightbound) = S ((S (jt_index_rectappendrightbound)) * jt_c_rectappendright)) /\\ exists fs_q_jt_rectappendrightboundat. jt_b_rectappendright = fs_q_jt_rectappendrightboundat * S ((S (jt_index_rectappendrightbound)) * jt_c_rectappendright) + (jt_value_rectappendrightbound))) /\\ (exists jt_gap_rectappendrightboundvalue. jt_gap_rectappendrightboundvalue+S (jt_value_rectappendrightbound)=(n)))) /\\ (forall jt_divisor_rectappendrightprimitive. (exists jt_factor_rectappendrightprimitivemodulus. (n)=(jt_divisor_rectappendrightprimitive)*jt_factor_rectappendrightprimitivemodulus) -> (forall jt_index_rectappendrightprimitivecoordinates jt_value_rectappendrightprimitivecoordinates. (exists jt_gap_rectappendrightprimitivecoordinatesindex. jt_gap_rectappendrightprimitivecoordinatesindex+S (jt_index_rectappendrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectappendrightprimitivecoordinatesat. fs_h_jt_rectappendrightprimitivecoordinatesat + S (jt_value_rectappendrightprimitivecoordinates) = S ((S (jt_index_rectappendrightprimitivecoordinates)) * jt_c_rectappendright)) /\\ exists fs_q_jt_rectappendrightprimitivecoordinatesat. jt_b_rectappendright = fs_q_jt_rectappendrightprimitivecoordinatesat * S ((S (jt_index_rectappendrightprimitivecoordinates)) * jt_c_rectappendright) + (jt_value_rectappendrightprimitivecoordinates))) -> (exists jt_factor_rectappendrightprimitivecoordinatesdivides. (jt_value_rectappendrightprimitivecoordinates)=(jt_divisor_rectappendrightprimitive)*jt_factor_rectappendrightprimitivecoordinatesdivides)) -> jt_divisor_rectappendrightprimitive=1))))) /\\ (((forall jt_b_rectappendright jt_c_rectappendright. (forall jt_index_rectappendrightinputbound. (exists jt_gap_rectappendrightinputboundindex. jt_gap_rectappendrightinputboundindex+S (jt_index_rectappendrightinputbound)=(k)) -> exists jt_value_rectappendrightinputbound. ((((exists fs_h_jt_rectappendrightinputboundat. fs_h_jt_rectappendrightinputboundat + S (jt_value_rectappendrightinputbound) = S ((S (jt_index_rectappendrightinputbound)) * jt_c_rectappendright)) /\\ exists fs_q_jt_rectappendrightinputboundat. jt_b_rectappendright = fs_q_jt_rectappendrightinputboundat * S ((S (jt_index_rectappendrightinputbound)) * jt_c_rectappendright) + (jt_value_rectappendrightinputbound))) /\\ (exists jt_gap_rectappendrightinputboundvalue. jt_gap_rectappendrightinputboundvalue+S (jt_value_rectappendrightinputbound)=(n)))) -> (forall jt_divisor_rectappendrightinputprimitive. (exists jt_factor_rectappendrightinputprimitivemodulus. (n)=(jt_divisor_rectappendrightinputprimitive)*jt_factor_rectappendrightinputprimitivemodulus) -> (forall jt_index_rectappendrightinputprimitivecoordinates jt_value_rectappendrightinputprimitivecoordinates. (exists jt_gap_rectappendrightinputprimitivecoordinatesindex. jt_gap_rectappendrightinputprimitivecoordinatesindex+S (jt_index_rectappendrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectappendrightinputprimitivecoordinatesat. fs_h_jt_rectappendrightinputprimitivecoordinatesat + S (jt_value_rectappendrightinputprimitivecoordinates) = S ((S (jt_index_rectappendrightinputprimitivecoordinates)) * jt_c_rectappendright)) /\\ exists fs_q_jt_rectappendrightinputprimitivecoordinatesat. jt_b_rectappendright = fs_q_jt_rectappendrightinputprimitivecoordinatesat * S ((S (jt_index_rectappendrightinputprimitivecoordinates)) * jt_c_rectappendright) + (jt_value_rectappendrightinputprimitivecoordinates))) -> (exists jt_factor_rectappendrightinputprimitivecoordinatesdivides. (jt_value_rectappendrightinputprimitivecoordinates)=(jt_divisor_rectappendrightinputprimitive)*jt_factor_rectappendrightinputprimitivecoordinatesdivides)) -> jt_divisor_rectappendrightinputprimitive=1) -> exists jt_i_rectappendright jt_d_rectappendright jt_e_rectappendright. ((exists jt_gap_rectappendrightcompleteindex. jt_gap_rectappendrightcompleteindex+S (jt_i_rectappendright)=(v)) /\\ (((((((exists fs_h_jt_rectappendrightcompletecode. fs_h_jt_rectappendrightcompletecode + S (jt_d_rectappendright) = S ((S (jt_i_rectappendright)) * F)) /\\ exists fs_q_jt_rectappendrightcompletecode. E = fs_q_jt_rectappendrightcompletecode * S ((S (jt_i_rectappendright)) * F) + (jt_d_rectappendright))) /\\ (((exists fs_h_jt_rectappendrightcompletescale. fs_h_jt_rectappendrightcompletescale + S (jt_e_rectappendright) = S ((S (jt_i_rectappendright)) * H)) /\\ exists fs_q_jt_rectappendrightcompletescale. G = fs_q_jt_rectappendrightcompletescale * S ((S (jt_i_rectappendright)) * H) + (jt_e_rectappendright))))) /\\ (forall jt_index_rectappendrightrepresented jt_left_rectappendrightrepresented jt_right_rectappendrightrepresented. (exists jt_gap_rectappendrightrepresentedindex. jt_gap_rectappendrightrepresentedindex+S (jt_index_rectappendrightrepresented)=(k)) -> (((exists fs_h_jt_rectappendrightrepresentedleft. fs_h_jt_rectappendrightrepresentedleft + S (jt_left_rectappendrightrepresented) = S ((S (jt_index_rectappendrightrepresented)) * jt_c_rectappendright)) /\\ exists fs_q_jt_rectappendrightrepresentedleft. jt_b_rectappendright = fs_q_jt_rectappendrightrepresentedleft * S ((S (jt_index_rectappendrightrepresented)) * jt_c_rectappendright) + (jt_left_rectappendrightrepresented))) -> (((exists fs_h_jt_rectappendrightrepresentedright. fs_h_jt_rectappendrightrepresentedright + S (jt_right_rectappendrightrepresented) = S ((S (jt_index_rectappendrightrepresented)) * jt_e_rectappendright)) /\\ exists fs_q_jt_rectappendrightrepresentedright. jt_d_rectappendright = fs_q_jt_rectappendrightrepresentedright * S ((S (jt_index_rectappendrightrepresented)) * jt_e_rectappendright) + (jt_right_rectappendrightrepresented))) -> jt_left_rectappendrightrepresented=jt_right_rectappendrightrepresented))))) /\\ (forall jt_i_rectappendright jt_h_rectappendright jt_b_rectappendright jt_c_rectappendright jt_d_rectappendright jt_e_rectappendright. (exists jt_gap_rectappendrightfirstindex. jt_gap_rectappendrightfirstindex+S (jt_i_rectappendright)=(v)) -> (exists jt_gap_rectappendrightsecondindex. jt_gap_rectappendrightsecondindex+S (jt_h_rectappendright)=(v)) -> (((((exists fs_h_jt_rectappendrightfirstcode. fs_h_jt_rectappendrightfirstcode + S (jt_b_rectappendright) = S ((S (jt_i_rectappendright)) * F)) /\\ exists fs_q_jt_rectappendrightfirstcode. E = fs_q_jt_rectappendrightfirstcode * S ((S (jt_i_rectappendright)) * F) + (jt_b_rectappendright))) /\\ (((exists fs_h_jt_rectappendrightfirstscale. fs_h_jt_rectappendrightfirstscale + S (jt_c_rectappendright) = S ((S (jt_i_rectappendright)) * H)) /\\ exists fs_q_jt_rectappendrightfirstscale. G = fs_q_jt_rectappendrightfirstscale * S ((S (jt_i_rectappendright)) * H) + (jt_c_rectappendright))))) -> (((((exists fs_h_jt_rectappendrightsecondcode. fs_h_jt_rectappendrightsecondcode + S (jt_d_rectappendright) = S ((S (jt_h_rectappendright)) * F)) /\\ exists fs_q_jt_rectappendrightsecondcode. E = fs_q_jt_rectappendrightsecondcode * S ((S (jt_h_rectappendright)) * F) + (jt_d_rectappendright))) /\\ (((exists fs_h_jt_rectappendrightsecondscale. fs_h_jt_rectappendrightsecondscale + S (jt_e_rectappendright) = S ((S (jt_h_rectappendright)) * H)) /\\ exists fs_q_jt_rectappendrightsecondscale. G = fs_q_jt_rectappendrightsecondscale * S ((S (jt_h_rectappendright)) * H) + (jt_e_rectappendright))))) -> (forall jt_index_rectappendrightsame jt_left_rectappendrightsame jt_right_rectappendrightsame. (exists jt_gap_rectappendrightsameindex. jt_gap_rectappendrightsameindex+S (jt_index_rectappendrightsame)=(k)) -> (((exists fs_h_jt_rectappendrightsameleft. fs_h_jt_rectappendrightsameleft + S (jt_left_rectappendrightsame) = S ((S (jt_index_rectappendrightsame)) * jt_c_rectappendright)) /\\ exists fs_q_jt_rectappendrightsameleft. jt_b_rectappendright = fs_q_jt_rectappendrightsameleft * S ((S (jt_index_rectappendrightsame)) * jt_c_rectappendright) + (jt_left_rectappendrightsame))) -> (((exists fs_h_jt_rectappendrightsameright. fs_h_jt_rectappendrightsameright + S (jt_right_rectappendrightsame) = S ((S (jt_index_rectappendrightsame)) * jt_e_rectappendright)) /\\ exists fs_q_jt_rectappendrightsameright. jt_d_rectappendright = fs_q_jt_rectappendrightsameright * S ((S (jt_index_rectappendrightsame)) * jt_e_rectappendright) + (jt_right_rectappendrightsame))) -> jt_left_rectappendrightsame=jt_right_rectappendrightsame) -> jt_i_rectappendright=jt_h_rectappendright))))) -> (forall jt_index_rectappendold. (exists jt_gap_rectappendoldindex. jt_gap_rectappendoldindex+S (jt_index_rectappendold)=(q)) -> exists jt_row_rectappendold jt_column_rectappendold jt_b_rectappendold jt_c_rectappendold jt_d_rectappendold jt_e_rectappendold jt_f_rectappendold jt_g_rectappendold. ((exists jt_gap_rectappendoldrow. jt_gap_rectappendoldrow+S (jt_row_rectappendold)=(u)) /\\ (((exists jt_gap_rectappendoldcolumn. jt_gap_rectappendoldcolumn+S (jt_column_rectappendold)=(v)) /\\ (((jt_index_rectappendold=(v)*jt_row_rectappendold+jt_column_rectappendold) /\\ (((((((exists fs_h_jt_rectappendoldleftcode. fs_h_jt_rectappendoldleftcode + S (jt_b_rectappendold) = S ((S (jt_row_rectappendold)) * B)) /\\ exists fs_q_jt_rectappendoldleftcode. A = fs_q_jt_rectappendoldleftcode * S ((S (jt_row_rectappendold)) * B) + (jt_b_rectappendold))) /\\ (((exists fs_h_jt_rectappendoldleftscale. fs_h_jt_rectappendoldleftscale + S (jt_c_rectappendold) = S ((S (jt_row_rectappendold)) * D)) /\\ exists fs_q_jt_rectappendoldleftscale. C = fs_q_jt_rectappendoldleftscale * S ((S (jt_row_rectappendold)) * D) + (jt_c_rectappendold))))) /\\ (((((((exists fs_h_jt_rectappendoldrightcode. fs_h_jt_rectappendoldrightcode + S (jt_d_rectappendold) = S ((S (jt_column_rectappendold)) * F)) /\\ exists fs_q_jt_rectappendoldrightcode. E = fs_q_jt_rectappendoldrightcode * S ((S (jt_column_rectappendold)) * F) + (jt_d_rectappendold))) /\\ (((exists fs_h_jt_rectappendoldrightscale. fs_h_jt_rectappendoldrightscale + S (jt_e_rectappendold) = S ((S (jt_column_rectappendold)) * H)) /\\ exists fs_q_jt_rectappendoldrightscale. G = fs_q_jt_rectappendoldrightscale * S ((S (jt_column_rectappendold)) * H) + (jt_e_rectappendold))))) /\\ (((((((exists fs_h_jt_rectappendoldoutputcode. fs_h_jt_rectappendoldoutputcode + S (jt_f_rectappendold) = S ((S (jt_index_rectappendold)) * Q)) /\\ exists fs_q_jt_rectappendoldoutputcode. P = fs_q_jt_rectappendoldoutputcode * S ((S (jt_index_rectappendold)) * Q) + (jt_f_rectappendold))) /\\ (((exists fs_h_jt_rectappendoldoutputscale. fs_h_jt_rectappendoldoutputscale + S (jt_g_rectappendold) = S ((S (jt_index_rectappendold)) * T)) /\\ exists fs_q_jt_rectappendoldoutputscale. R = fs_q_jt_rectappendoldoutputscale * S ((S (jt_index_rectappendold)) * T) + (jt_g_rectappendold))))) /\\ (((((forall jt_index_rectappendoldcrtbound. (exists jt_gap_rectappendoldcrtboundindex. jt_gap_rectappendoldcrtboundindex+S (jt_index_rectappendoldcrtbound)=(k)) -> exists jt_value_rectappendoldcrtbound. ((((exists fs_h_jt_rectappendoldcrtboundat. fs_h_jt_rectappendoldcrtboundat + S (jt_value_rectappendoldcrtbound) = S ((S (jt_index_rectappendoldcrtbound)) * jt_g_rectappendold)) /\\ exists fs_q_jt_rectappendoldcrtboundat. jt_f_rectappendold = fs_q_jt_rectappendoldcrtboundat * S ((S (jt_index_rectappendoldcrtbound)) * jt_g_rectappendold) + (jt_value_rectappendoldcrtbound))) /\\ (exists jt_gap_rectappendoldcrtboundvalue. jt_gap_rectappendoldcrtboundvalue+S (jt_value_rectappendoldcrtbound)=(m*n)))) /\\ (((forall jt_index_rectappendoldcrtleft jt_left_rectappendoldcrtleft jt_right_rectappendoldcrtleft. (exists jt_gap_rectappendoldcrtleftindex. jt_gap_rectappendoldcrtleftindex+S (jt_index_rectappendoldcrtleft)=(k)) -> (((exists fs_h_jt_rectappendoldcrtleftleft. fs_h_jt_rectappendoldcrtleftleft + S (jt_left_rectappendoldcrtleft) = S ((S (jt_index_rectappendoldcrtleft)) * jt_g_rectappendold)) /\\ exists fs_q_jt_rectappendoldcrtleftleft. jt_f_rectappendold = fs_q_jt_rectappendoldcrtleftleft * S ((S (jt_index_rectappendoldcrtleft)) * jt_g_rectappendold) + (jt_left_rectappendoldcrtleft))) -> (((exists fs_h_jt_rectappendoldcrtleftright. fs_h_jt_rectappendoldcrtleftright + S (jt_right_rectappendoldcrtleft) = S ((S (jt_index_rectappendoldcrtleft)) * jt_c_rectappendold)) /\\ exists fs_q_jt_rectappendoldcrtleftright. jt_b_rectappendold = fs_q_jt_rectappendoldcrtleftright * S ((S (jt_index_rectappendoldcrtleft)) * jt_c_rectappendold) + (jt_right_rectappendoldcrtleft))) -> (exists jt_left_rectappendoldcrtleftmod jt_right_rectappendoldcrtleftmod. (jt_left_rectappendoldcrtleft)+(m)*jt_left_rectappendoldcrtleftmod=(jt_right_rectappendoldcrtleft)+(m)*jt_right_rectappendoldcrtleftmod)) /\\ (forall jt_index_rectappendoldcrtright jt_left_rectappendoldcrtright jt_right_rectappendoldcrtright. (exists jt_gap_rectappendoldcrtrightindex. jt_gap_rectappendoldcrtrightindex+S (jt_index_rectappendoldcrtright)=(k)) -> (((exists fs_h_jt_rectappendoldcrtrightleft. fs_h_jt_rectappendoldcrtrightleft + S (jt_left_rectappendoldcrtright) = S ((S (jt_index_rectappendoldcrtright)) * jt_g_rectappendold)) /\\ exists fs_q_jt_rectappendoldcrtrightleft. jt_f_rectappendold = fs_q_jt_rectappendoldcrtrightleft * S ((S (jt_index_rectappendoldcrtright)) * jt_g_rectappendold) + (jt_left_rectappendoldcrtright))) -> (((exists fs_h_jt_rectappendoldcrtrightright. fs_h_jt_rectappendoldcrtrightright + S (jt_right_rectappendoldcrtright) = S ((S (jt_index_rectappendoldcrtright)) * jt_e_rectappendold)) /\\ exists fs_q_jt_rectappendoldcrtrightright. jt_d_rectappendold = fs_q_jt_rectappendoldcrtrightright * S ((S (jt_index_rectappendoldcrtright)) * jt_e_rectappendold) + (jt_right_rectappendoldcrtright))) -> (exists jt_left_rectappendoldcrtrightmod jt_right_rectappendoldcrtrightmod. (jt_left_rectappendoldcrtright)+(n)*jt_left_rectappendoldcrtrightmod=(jt_right_rectappendoldcrtright)+(n)*jt_right_rectappendoldcrtrightmod)))))) /\\ (forall jt_divisor_rectappendoldprimitive. (exists jt_factor_rectappendoldprimitivemodulus. (m*n)=(jt_divisor_rectappendoldprimitive)*jt_factor_rectappendoldprimitivemodulus) -> (forall jt_index_rectappendoldprimitivecoordinates jt_value_rectappendoldprimitivecoordinates. (exists jt_gap_rectappendoldprimitivecoordinatesindex. jt_gap_rectappendoldprimitivecoordinatesindex+S (jt_index_rectappendoldprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectappendoldprimitivecoordinatesat. fs_h_jt_rectappendoldprimitivecoordinatesat + S (jt_value_rectappendoldprimitivecoordinates) = S ((S (jt_index_rectappendoldprimitivecoordinates)) * jt_g_rectappendold)) /\\ exists fs_q_jt_rectappendoldprimitivecoordinatesat. jt_f_rectappendold = fs_q_jt_rectappendoldprimitivecoordinatesat * S ((S (jt_index_rectappendoldprimitivecoordinates)) * jt_g_rectappendold) + (jt_value_rectappendoldprimitivecoordinates))) -> (exists jt_factor_rectappendoldprimitivecoordinatesdivides. (jt_value_rectappendoldprimitivecoordinates)=(jt_divisor_rectappendoldprimitive)*jt_factor_rectappendoldprimitivecoordinatesdivides)) -> jt_divisor_rectappendoldprimitive=1))))))))))))))) -> (exists jt_gap_rectappendbound. jt_gap_rectappendbound+S (q)=(u*v)) -> exists U V W X. forall jt_index_rectappendnew. (exists jt_gap_rectappendnewindex. jt_gap_rectappendnewindex+S (jt_index_rectappendnew)=(S q)) -> exists jt_row_rectappendnew jt_column_rectappendnew jt_b_rectappendnew jt_c_rectappendnew jt_d_rectappendnew jt_e_rectappendnew jt_f_rectappendnew jt_g_rectappendnew. ((exists jt_gap_rectappendnewrow. jt_gap_rectappendnewrow+S (jt_row_rectappendnew)=(u)) /\\ (((exists jt_gap_rectappendnewcolumn. jt_gap_rectappendnewcolumn+S (jt_column_rectappendnew)=(v)) /\\ (((jt_index_rectappendnew=(v)*jt_row_rectappendnew+jt_column_rectappendnew) /\\ (((((((exists fs_h_jt_rectappendnewleftcode. fs_h_jt_rectappendnewleftcode + S (jt_b_rectappendnew) = S ((S (jt_row_rectappendnew)) * B)) /\\ exists fs_q_jt_rectappendnewleftcode. A = fs_q_jt_rectappendnewleftcode * S ((S (jt_row_rectappendnew)) * B) + (jt_b_rectappendnew))) /\\ (((exists fs_h_jt_rectappendnewleftscale. fs_h_jt_rectappendnewleftscale + S (jt_c_rectappendnew) = S ((S (jt_row_rectappendnew)) * D)) /\\ exists fs_q_jt_rectappendnewleftscale. C = fs_q_jt_rectappendnewleftscale * S ((S (jt_row_rectappendnew)) * D) + (jt_c_rectappendnew))))) /\\ (((((((exists fs_h_jt_rectappendnewrightcode. fs_h_jt_rectappendnewrightcode + S (jt_d_rectappendnew) = S ((S (jt_column_rectappendnew)) * F)) /\\ exists fs_q_jt_rectappendnewrightcode. E = fs_q_jt_rectappendnewrightcode * S ((S (jt_column_rectappendnew)) * F) + (jt_d_rectappendnew))) /\\ (((exists fs_h_jt_rectappendnewrightscale. fs_h_jt_rectappendnewrightscale + S (jt_e_rectappendnew) = S ((S (jt_column_rectappendnew)) * H)) /\\ exists fs_q_jt_rectappendnewrightscale. G = fs_q_jt_rectappendnewrightscale * S ((S (jt_column_rectappendnew)) * H) + (jt_e_rectappendnew))))) /\\ (((((((exists fs_h_jt_rectappendnewoutputcode. fs_h_jt_rectappendnewoutputcode + S (jt_f_rectappendnew) = S ((S (jt_index_rectappendnew)) * V)) /\\ exists fs_q_jt_rectappendnewoutputcode. U = fs_q_jt_rectappendnewoutputcode * S ((S (jt_index_rectappendnew)) * V) + (jt_f_rectappendnew))) /\\ (((exists fs_h_jt_rectappendnewoutputscale. fs_h_jt_rectappendnewoutputscale + S (jt_g_rectappendnew) = S ((S (jt_index_rectappendnew)) * X)) /\\ exists fs_q_jt_rectappendnewoutputscale. W = fs_q_jt_rectappendnewoutputscale * S ((S (jt_index_rectappendnew)) * X) + (jt_g_rectappendnew))))) /\\ (((((forall jt_index_rectappendnewcrtbound. (exists jt_gap_rectappendnewcrtboundindex. jt_gap_rectappendnewcrtboundindex+S (jt_index_rectappendnewcrtbound)=(k)) -> exists jt_value_rectappendnewcrtbound. ((((exists fs_h_jt_rectappendnewcrtboundat. fs_h_jt_rectappendnewcrtboundat + S (jt_value_rectappendnewcrtbound) = S ((S (jt_index_rectappendnewcrtbound)) * jt_g_rectappendnew)) /\\ exists fs_q_jt_rectappendnewcrtboundat. jt_f_rectappendnew = fs_q_jt_rectappendnewcrtboundat * S ((S (jt_index_rectappendnewcrtbound)) * jt_g_rectappendnew) + (jt_value_rectappendnewcrtbound))) /\\ (exists jt_gap_rectappendnewcrtboundvalue. jt_gap_rectappendnewcrtboundvalue+S (jt_value_rectappendnewcrtbound)=(m*n)))) /\\ (((forall jt_index_rectappendnewcrtleft jt_left_rectappendnewcrtleft jt_right_rectappendnewcrtleft. (exists jt_gap_rectappendnewcrtleftindex. jt_gap_rectappendnewcrtleftindex+S (jt_index_rectappendnewcrtleft)=(k)) -> (((exists fs_h_jt_rectappendnewcrtleftleft. fs_h_jt_rectappendnewcrtleftleft + S (jt_left_rectappendnewcrtleft) = S ((S (jt_index_rectappendnewcrtleft)) * jt_g_rectappendnew)) /\\ exists fs_q_jt_rectappendnewcrtleftleft. jt_f_rectappendnew = fs_q_jt_rectappendnewcrtleftleft * S ((S (jt_index_rectappendnewcrtleft)) * jt_g_rectappendnew) + (jt_left_rectappendnewcrtleft))) -> (((exists fs_h_jt_rectappendnewcrtleftright. fs_h_jt_rectappendnewcrtleftright + S (jt_right_rectappendnewcrtleft) = S ((S (jt_index_rectappendnewcrtleft)) * jt_c_rectappendnew)) /\\ exists fs_q_jt_rectappendnewcrtleftright. jt_b_rectappendnew = fs_q_jt_rectappendnewcrtleftright * S ((S (jt_index_rectappendnewcrtleft)) * jt_c_rectappendnew) + (jt_right_rectappendnewcrtleft))) -> (exists jt_left_rectappendnewcrtleftmod jt_right_rectappendnewcrtleftmod. (jt_left_rectappendnewcrtleft)+(m)*jt_left_rectappendnewcrtleftmod=(jt_right_rectappendnewcrtleft)+(m)*jt_right_rectappendnewcrtleftmod)) /\\ (forall jt_index_rectappendnewcrtright jt_left_rectappendnewcrtright jt_right_rectappendnewcrtright. (exists jt_gap_rectappendnewcrtrightindex. jt_gap_rectappendnewcrtrightindex+S (jt_index_rectappendnewcrtright)=(k)) -> (((exists fs_h_jt_rectappendnewcrtrightleft. fs_h_jt_rectappendnewcrtrightleft + S (jt_left_rectappendnewcrtright) = S ((S (jt_index_rectappendnewcrtright)) * jt_g_rectappendnew)) /\\ exists fs_q_jt_rectappendnewcrtrightleft. jt_f_rectappendnew = fs_q_jt_rectappendnewcrtrightleft * S ((S (jt_index_rectappendnewcrtright)) * jt_g_rectappendnew) + (jt_left_rectappendnewcrtright))) -> (((exists fs_h_jt_rectappendnewcrtrightright. fs_h_jt_rectappendnewcrtrightright + S (jt_right_rectappendnewcrtright) = S ((S (jt_index_rectappendnewcrtright)) * jt_e_rectappendnew)) /\\ exists fs_q_jt_rectappendnewcrtrightright. jt_d_rectappendnew = fs_q_jt_rectappendnewcrtrightright * S ((S (jt_index_rectappendnewcrtright)) * jt_e_rectappendnew) + (jt_right_rectappendnewcrtright))) -> (exists jt_left_rectappendnewcrtrightmod jt_right_rectappendnewcrtrightmod. (jt_left_rectappendnewcrtright)+(n)*jt_left_rectappendnewcrtrightmod=(jt_right_rectappendnewcrtright)+(n)*jt_right_rectappendnewcrtrightmod)))))) /\\ (forall jt_divisor_rectappendnewprimitive. (exists jt_factor_rectappendnewprimitivemodulus. (m*n)=(jt_divisor_rectappendnewprimitive)*jt_factor_rectappendnewprimitivemodulus) -> (forall jt_index_rectappendnewprimitivecoordinates jt_value_rectappendnewprimitivecoordinates. (exists jt_gap_rectappendnewprimitivecoordinatesindex. jt_gap_rectappendnewprimitivecoordinatesindex+S (jt_index_rectappendnewprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectappendnewprimitivecoordinatesat. fs_h_jt_rectappendnewprimitivecoordinatesat + S (jt_value_rectappendnewprimitivecoordinates) = S ((S (jt_index_rectappendnewprimitivecoordinates)) * jt_g_rectappendnew)) /\\ exists fs_q_jt_rectappendnewprimitivecoordinatesat. jt_f_rectappendnew = fs_q_jt_rectappendnewprimitivecoordinatesat * S ((S (jt_index_rectappendnewprimitivecoordinates)) * jt_g_rectappendnew) + (jt_value_rectappendnewprimitivecoordinates))) -> (exists jt_factor_rectappendnewprimitivecoordinatesdivides. (jt_value_rectappendnewprimitivecoordinates)=(jt_divisor_rectappendnewprimitive)*jt_factor_rectappendnewprimitivecoordinatesdivides)) -> jt_divisor_rectappendnewprimitive=1))))))))))))))", "statement_sha256": "a41266d40b155ab1e76de188be6cd138158dc4478ba7a7cc8cbba1e85bd5215d", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "At the next flat index, construct the actual primitive CRT tuple and append its code and scale to fresh beta lists." }, { "admission_dependencies": [ "jordan_rectangle_crt_append" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 70, "body_proof_nodes": 117, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro q", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "intro hold", "intro hq", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "specialize jordan_rectangle_crt_append (m)", "specialize jordan_rectangle_crt_append (n)", "specialize jordan_rectangle_crt_append (k)", "specialize jordan_rectangle_crt_append (A)", "specialize jordan_rectangle_crt_append (B)", "specialize jordan_rectangle_crt_append (C)", "specialize jordan_rectangle_crt_append (D)", "specialize jordan_rectangle_crt_append (u)", "specialize jordan_rectangle_crt_append (E)", "specialize jordan_rectangle_crt_append (F)", "specialize jordan_rectangle_crt_append (G)", "specialize jordan_rectangle_crt_append (H)", "specialize jordan_rectangle_crt_append (v)", "specialize jordan_rectangle_crt_append (x)", "specialize jordan_rectangle_crt_append (x1)", "specialize jordan_rectangle_crt_append (x2)", "specialize jordan_rectangle_crt_append (x3)", "specialize jordan_rectangle_crt_append (q)", "apply jordan_rectangle_crt_append", "exact hm", "exact hn", "exact hcop", "exact hleft", "exact hright", "exact hold_witness_witness_witness_witness", "exact hq" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ q. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → (∃ x. ∃ y. ∃ z. ∃ i. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,q)) → Lt(q,u · v) → ∃ x. ∃ y. ∃ z. ∃ i. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,S q)", "defined_statement_sha256": "6b9ff251be43daa4f217e3ab64e8c1cdd89c820942528ab815fdf6820105607a", "definition_uses": { "ND0374": 2, "ND0381": 2, "PD0002": 1, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "42ac7078d445af2cfd4afd2f146a21f9026ae6768a02920baa885d15ee467a6a", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hleft" } ], [ { "kind": "text", "text": "intro hright" } ], [ { "kind": "text", "text": "intro hold" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "cases hold" } ], [ { "kind": "text", "text": "cases hold_witness" } ], [ { "kind": "text", "text": "cases hold_witness_witness" } ], [ { "kind": "text", "text": "cases hold_witness_witness_witness" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (x)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (x1)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (x2)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (x3)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_append (q)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_append" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hleft" } ], [ { "kind": "text", "text": "exact hright" } ], [ { "kind": "text", "text": "exact hold_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "exact hq" } ] ], "statement_definition_uses": { "ND0374": 2, "ND0381": 2, "PD0002": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ q. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → (∃ x. ∃ y. ∃ z. ∃ i. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,q)" }, { "kind": "text", "text": ") → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(q,u · v)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ i. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,S q)" } ] }, "dependencies": [ "jordan_rectangle_crt_append" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0041", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_successor", "original_ha_bundle_verified": true, "proof_bundle_node_id": 326, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro q", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "intro hold", "intro hq", "cases hold", "cases hold_witness", "cases hold_witness_witness", "cases hold_witness_witness_witness", "specialize jordan_rectangle_crt_append (m)", "specialize jordan_rectangle_crt_append (n)", "specialize jordan_rectangle_crt_append (k)", "specialize jordan_rectangle_crt_append (A)", "specialize jordan_rectangle_crt_append (B)", "specialize jordan_rectangle_crt_append (C)", "specialize jordan_rectangle_crt_append (D)", "specialize jordan_rectangle_crt_append (u)", "specialize jordan_rectangle_crt_append (E)", "specialize jordan_rectangle_crt_append (F)", "specialize jordan_rectangle_crt_append (G)", "specialize jordan_rectangle_crt_append (H)", "specialize jordan_rectangle_crt_append (v)", "specialize jordan_rectangle_crt_append (x)", "specialize jordan_rectangle_crt_append (x1)", "specialize jordan_rectangle_crt_append (x2)", "specialize jordan_rectangle_crt_append (x3)", "specialize jordan_rectangle_crt_append (q)", "apply jordan_rectangle_crt_append", "exact hm", "exact hn", "exact hcop", "exact hleft", "exact hright", "exact hold_witness_witness_witness_witness", "exact hq" ], "script_sha256": "f0612f7874a430560260d5b7d23dd71312d92882fbcfbc0e52604be993f6c762", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "f0612f7874a430560260d5b7d23dd71312d92882fbcfbc0e52604be993f6c762", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "42ac7078d445af2cfd4afd2f146a21f9026ae6768a02920baa885d15ee467a6a" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v q. ~(m=0) -> ~(n=0) -> (forall jt_divisor_rectsuccessorcop. (exists jt_factor_rectsuccessorcopa. (m)=(jt_divisor_rectsuccessorcop)*jt_factor_rectsuccessorcopa) -> (exists jt_factor_rectsuccessorcopb. (n)=(jt_divisor_rectsuccessorcop)*jt_factor_rectsuccessorcopb) -> jt_divisor_rectsuccessorcop=1) -> (((forall jt_i_rectsuccessorleft. (exists jt_gap_rectsuccessorleftsoundindex. jt_gap_rectsuccessorleftsoundindex+S (jt_i_rectsuccessorleft)=(u)) -> exists jt_b_rectsuccessorleft jt_c_rectsuccessorleft. ((((((exists fs_h_jt_rectsuccessorleftsoundcode. fs_h_jt_rectsuccessorleftsoundcode + S (jt_b_rectsuccessorleft) = S ((S (jt_i_rectsuccessorleft)) * B)) /\\ exists fs_q_jt_rectsuccessorleftsoundcode. A = fs_q_jt_rectsuccessorleftsoundcode * S ((S (jt_i_rectsuccessorleft)) * B) + (jt_b_rectsuccessorleft))) /\\ (((exists fs_h_jt_rectsuccessorleftsoundscale. fs_h_jt_rectsuccessorleftsoundscale + S (jt_c_rectsuccessorleft) = S ((S (jt_i_rectsuccessorleft)) * D)) /\\ exists fs_q_jt_rectsuccessorleftsoundscale. C = fs_q_jt_rectsuccessorleftsoundscale * S ((S (jt_i_rectsuccessorleft)) * D) + (jt_c_rectsuccessorleft))))) /\\ (((forall jt_index_rectsuccessorleftbound. (exists jt_gap_rectsuccessorleftboundindex. jt_gap_rectsuccessorleftboundindex+S (jt_index_rectsuccessorleftbound)=(k)) -> exists jt_value_rectsuccessorleftbound. ((((exists fs_h_jt_rectsuccessorleftboundat. fs_h_jt_rectsuccessorleftboundat + S (jt_value_rectsuccessorleftbound) = S ((S (jt_index_rectsuccessorleftbound)) * jt_c_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftboundat. jt_b_rectsuccessorleft = fs_q_jt_rectsuccessorleftboundat * S ((S (jt_index_rectsuccessorleftbound)) * jt_c_rectsuccessorleft) + (jt_value_rectsuccessorleftbound))) /\\ (exists jt_gap_rectsuccessorleftboundvalue. jt_gap_rectsuccessorleftboundvalue+S (jt_value_rectsuccessorleftbound)=(m)))) /\\ (forall jt_divisor_rectsuccessorleftprimitive. (exists jt_factor_rectsuccessorleftprimitivemodulus. (m)=(jt_divisor_rectsuccessorleftprimitive)*jt_factor_rectsuccessorleftprimitivemodulus) -> (forall jt_index_rectsuccessorleftprimitivecoordinates jt_value_rectsuccessorleftprimitivecoordinates. (exists jt_gap_rectsuccessorleftprimitivecoordinatesindex. jt_gap_rectsuccessorleftprimitivecoordinatesindex+S (jt_index_rectsuccessorleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectsuccessorleftprimitivecoordinatesat. fs_h_jt_rectsuccessorleftprimitivecoordinatesat + S (jt_value_rectsuccessorleftprimitivecoordinates) = S ((S (jt_index_rectsuccessorleftprimitivecoordinates)) * jt_c_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftprimitivecoordinatesat. jt_b_rectsuccessorleft = fs_q_jt_rectsuccessorleftprimitivecoordinatesat * S ((S (jt_index_rectsuccessorleftprimitivecoordinates)) * jt_c_rectsuccessorleft) + (jt_value_rectsuccessorleftprimitivecoordinates))) -> (exists jt_factor_rectsuccessorleftprimitivecoordinatesdivides. (jt_value_rectsuccessorleftprimitivecoordinates)=(jt_divisor_rectsuccessorleftprimitive)*jt_factor_rectsuccessorleftprimitivecoordinatesdivides)) -> jt_divisor_rectsuccessorleftprimitive=1))))) /\\ (((forall jt_b_rectsuccessorleft jt_c_rectsuccessorleft. (forall jt_index_rectsuccessorleftinputbound. (exists jt_gap_rectsuccessorleftinputboundindex. jt_gap_rectsuccessorleftinputboundindex+S (jt_index_rectsuccessorleftinputbound)=(k)) -> exists jt_value_rectsuccessorleftinputbound. ((((exists fs_h_jt_rectsuccessorleftinputboundat. fs_h_jt_rectsuccessorleftinputboundat + S (jt_value_rectsuccessorleftinputbound) = S ((S (jt_index_rectsuccessorleftinputbound)) * jt_c_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftinputboundat. jt_b_rectsuccessorleft = fs_q_jt_rectsuccessorleftinputboundat * S ((S (jt_index_rectsuccessorleftinputbound)) * jt_c_rectsuccessorleft) + (jt_value_rectsuccessorleftinputbound))) /\\ (exists jt_gap_rectsuccessorleftinputboundvalue. jt_gap_rectsuccessorleftinputboundvalue+S (jt_value_rectsuccessorleftinputbound)=(m)))) -> (forall jt_divisor_rectsuccessorleftinputprimitive. (exists jt_factor_rectsuccessorleftinputprimitivemodulus. (m)=(jt_divisor_rectsuccessorleftinputprimitive)*jt_factor_rectsuccessorleftinputprimitivemodulus) -> (forall jt_index_rectsuccessorleftinputprimitivecoordinates jt_value_rectsuccessorleftinputprimitivecoordinates. (exists jt_gap_rectsuccessorleftinputprimitivecoordinatesindex. jt_gap_rectsuccessorleftinputprimitivecoordinatesindex+S (jt_index_rectsuccessorleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectsuccessorleftinputprimitivecoordinatesat. fs_h_jt_rectsuccessorleftinputprimitivecoordinatesat + S (jt_value_rectsuccessorleftinputprimitivecoordinates) = S ((S (jt_index_rectsuccessorleftinputprimitivecoordinates)) * jt_c_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftinputprimitivecoordinatesat. jt_b_rectsuccessorleft = fs_q_jt_rectsuccessorleftinputprimitivecoordinatesat * S ((S (jt_index_rectsuccessorleftinputprimitivecoordinates)) * jt_c_rectsuccessorleft) + (jt_value_rectsuccessorleftinputprimitivecoordinates))) -> (exists jt_factor_rectsuccessorleftinputprimitivecoordinatesdivides. (jt_value_rectsuccessorleftinputprimitivecoordinates)=(jt_divisor_rectsuccessorleftinputprimitive)*jt_factor_rectsuccessorleftinputprimitivecoordinatesdivides)) -> jt_divisor_rectsuccessorleftinputprimitive=1) -> exists jt_i_rectsuccessorleft jt_d_rectsuccessorleft jt_e_rectsuccessorleft. ((exists jt_gap_rectsuccessorleftcompleteindex. jt_gap_rectsuccessorleftcompleteindex+S (jt_i_rectsuccessorleft)=(u)) /\\ (((((((exists fs_h_jt_rectsuccessorleftcompletecode. fs_h_jt_rectsuccessorleftcompletecode + S (jt_d_rectsuccessorleft) = S ((S (jt_i_rectsuccessorleft)) * B)) /\\ exists fs_q_jt_rectsuccessorleftcompletecode. A = fs_q_jt_rectsuccessorleftcompletecode * S ((S (jt_i_rectsuccessorleft)) * B) + (jt_d_rectsuccessorleft))) /\\ (((exists fs_h_jt_rectsuccessorleftcompletescale. fs_h_jt_rectsuccessorleftcompletescale + S (jt_e_rectsuccessorleft) = S ((S (jt_i_rectsuccessorleft)) * D)) /\\ exists fs_q_jt_rectsuccessorleftcompletescale. C = fs_q_jt_rectsuccessorleftcompletescale * S ((S (jt_i_rectsuccessorleft)) * D) + (jt_e_rectsuccessorleft))))) /\\ (forall jt_index_rectsuccessorleftrepresented jt_left_rectsuccessorleftrepresented jt_right_rectsuccessorleftrepresented. (exists jt_gap_rectsuccessorleftrepresentedindex. jt_gap_rectsuccessorleftrepresentedindex+S (jt_index_rectsuccessorleftrepresented)=(k)) -> (((exists fs_h_jt_rectsuccessorleftrepresentedleft. fs_h_jt_rectsuccessorleftrepresentedleft + S (jt_left_rectsuccessorleftrepresented) = S ((S (jt_index_rectsuccessorleftrepresented)) * jt_c_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftrepresentedleft. jt_b_rectsuccessorleft = fs_q_jt_rectsuccessorleftrepresentedleft * S ((S (jt_index_rectsuccessorleftrepresented)) * jt_c_rectsuccessorleft) + (jt_left_rectsuccessorleftrepresented))) -> (((exists fs_h_jt_rectsuccessorleftrepresentedright. fs_h_jt_rectsuccessorleftrepresentedright + S (jt_right_rectsuccessorleftrepresented) = S ((S (jt_index_rectsuccessorleftrepresented)) * jt_e_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftrepresentedright. jt_d_rectsuccessorleft = fs_q_jt_rectsuccessorleftrepresentedright * S ((S (jt_index_rectsuccessorleftrepresented)) * jt_e_rectsuccessorleft) + (jt_right_rectsuccessorleftrepresented))) -> jt_left_rectsuccessorleftrepresented=jt_right_rectsuccessorleftrepresented))))) /\\ (forall jt_i_rectsuccessorleft jt_h_rectsuccessorleft jt_b_rectsuccessorleft jt_c_rectsuccessorleft jt_d_rectsuccessorleft jt_e_rectsuccessorleft. (exists jt_gap_rectsuccessorleftfirstindex. jt_gap_rectsuccessorleftfirstindex+S (jt_i_rectsuccessorleft)=(u)) -> (exists jt_gap_rectsuccessorleftsecondindex. jt_gap_rectsuccessorleftsecondindex+S (jt_h_rectsuccessorleft)=(u)) -> (((((exists fs_h_jt_rectsuccessorleftfirstcode. fs_h_jt_rectsuccessorleftfirstcode + S (jt_b_rectsuccessorleft) = S ((S (jt_i_rectsuccessorleft)) * B)) /\\ exists fs_q_jt_rectsuccessorleftfirstcode. A = fs_q_jt_rectsuccessorleftfirstcode * S ((S (jt_i_rectsuccessorleft)) * B) + (jt_b_rectsuccessorleft))) /\\ (((exists fs_h_jt_rectsuccessorleftfirstscale. fs_h_jt_rectsuccessorleftfirstscale + S (jt_c_rectsuccessorleft) = S ((S (jt_i_rectsuccessorleft)) * D)) /\\ exists fs_q_jt_rectsuccessorleftfirstscale. C = fs_q_jt_rectsuccessorleftfirstscale * S ((S (jt_i_rectsuccessorleft)) * D) + (jt_c_rectsuccessorleft))))) -> (((((exists fs_h_jt_rectsuccessorleftsecondcode. fs_h_jt_rectsuccessorleftsecondcode + S (jt_d_rectsuccessorleft) = S ((S (jt_h_rectsuccessorleft)) * B)) /\\ exists fs_q_jt_rectsuccessorleftsecondcode. A = fs_q_jt_rectsuccessorleftsecondcode * S ((S (jt_h_rectsuccessorleft)) * B) + (jt_d_rectsuccessorleft))) /\\ (((exists fs_h_jt_rectsuccessorleftsecondscale. fs_h_jt_rectsuccessorleftsecondscale + S (jt_e_rectsuccessorleft) = S ((S (jt_h_rectsuccessorleft)) * D)) /\\ exists fs_q_jt_rectsuccessorleftsecondscale. C = fs_q_jt_rectsuccessorleftsecondscale * S ((S (jt_h_rectsuccessorleft)) * D) + (jt_e_rectsuccessorleft))))) -> (forall jt_index_rectsuccessorleftsame jt_left_rectsuccessorleftsame jt_right_rectsuccessorleftsame. (exists jt_gap_rectsuccessorleftsameindex. jt_gap_rectsuccessorleftsameindex+S (jt_index_rectsuccessorleftsame)=(k)) -> (((exists fs_h_jt_rectsuccessorleftsameleft. fs_h_jt_rectsuccessorleftsameleft + S (jt_left_rectsuccessorleftsame) = S ((S (jt_index_rectsuccessorleftsame)) * jt_c_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftsameleft. jt_b_rectsuccessorleft = fs_q_jt_rectsuccessorleftsameleft * S ((S (jt_index_rectsuccessorleftsame)) * jt_c_rectsuccessorleft) + (jt_left_rectsuccessorleftsame))) -> (((exists fs_h_jt_rectsuccessorleftsameright. fs_h_jt_rectsuccessorleftsameright + S (jt_right_rectsuccessorleftsame) = S ((S (jt_index_rectsuccessorleftsame)) * jt_e_rectsuccessorleft)) /\\ exists fs_q_jt_rectsuccessorleftsameright. jt_d_rectsuccessorleft = fs_q_jt_rectsuccessorleftsameright * S ((S (jt_index_rectsuccessorleftsame)) * jt_e_rectsuccessorleft) + (jt_right_rectsuccessorleftsame))) -> jt_left_rectsuccessorleftsame=jt_right_rectsuccessorleftsame) -> jt_i_rectsuccessorleft=jt_h_rectsuccessorleft))))) -> (((forall jt_i_rectsuccessorright. (exists jt_gap_rectsuccessorrightsoundindex. jt_gap_rectsuccessorrightsoundindex+S (jt_i_rectsuccessorright)=(v)) -> exists jt_b_rectsuccessorright jt_c_rectsuccessorright. ((((((exists fs_h_jt_rectsuccessorrightsoundcode. fs_h_jt_rectsuccessorrightsoundcode + S (jt_b_rectsuccessorright) = S ((S (jt_i_rectsuccessorright)) * F)) /\\ exists fs_q_jt_rectsuccessorrightsoundcode. E = fs_q_jt_rectsuccessorrightsoundcode * S ((S (jt_i_rectsuccessorright)) * F) + (jt_b_rectsuccessorright))) /\\ (((exists fs_h_jt_rectsuccessorrightsoundscale. fs_h_jt_rectsuccessorrightsoundscale + S (jt_c_rectsuccessorright) = S ((S (jt_i_rectsuccessorright)) * H)) /\\ exists fs_q_jt_rectsuccessorrightsoundscale. G = fs_q_jt_rectsuccessorrightsoundscale * S ((S (jt_i_rectsuccessorright)) * H) + (jt_c_rectsuccessorright))))) /\\ (((forall jt_index_rectsuccessorrightbound. (exists jt_gap_rectsuccessorrightboundindex. jt_gap_rectsuccessorrightboundindex+S (jt_index_rectsuccessorrightbound)=(k)) -> exists jt_value_rectsuccessorrightbound. ((((exists fs_h_jt_rectsuccessorrightboundat. fs_h_jt_rectsuccessorrightboundat + S (jt_value_rectsuccessorrightbound) = S ((S (jt_index_rectsuccessorrightbound)) * jt_c_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightboundat. jt_b_rectsuccessorright = fs_q_jt_rectsuccessorrightboundat * S ((S (jt_index_rectsuccessorrightbound)) * jt_c_rectsuccessorright) + (jt_value_rectsuccessorrightbound))) /\\ (exists jt_gap_rectsuccessorrightboundvalue. jt_gap_rectsuccessorrightboundvalue+S (jt_value_rectsuccessorrightbound)=(n)))) /\\ (forall jt_divisor_rectsuccessorrightprimitive. (exists jt_factor_rectsuccessorrightprimitivemodulus. (n)=(jt_divisor_rectsuccessorrightprimitive)*jt_factor_rectsuccessorrightprimitivemodulus) -> (forall jt_index_rectsuccessorrightprimitivecoordinates jt_value_rectsuccessorrightprimitivecoordinates. (exists jt_gap_rectsuccessorrightprimitivecoordinatesindex. jt_gap_rectsuccessorrightprimitivecoordinatesindex+S (jt_index_rectsuccessorrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectsuccessorrightprimitivecoordinatesat. fs_h_jt_rectsuccessorrightprimitivecoordinatesat + S (jt_value_rectsuccessorrightprimitivecoordinates) = S ((S (jt_index_rectsuccessorrightprimitivecoordinates)) * jt_c_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightprimitivecoordinatesat. jt_b_rectsuccessorright = fs_q_jt_rectsuccessorrightprimitivecoordinatesat * S ((S (jt_index_rectsuccessorrightprimitivecoordinates)) * jt_c_rectsuccessorright) + (jt_value_rectsuccessorrightprimitivecoordinates))) -> (exists jt_factor_rectsuccessorrightprimitivecoordinatesdivides. (jt_value_rectsuccessorrightprimitivecoordinates)=(jt_divisor_rectsuccessorrightprimitive)*jt_factor_rectsuccessorrightprimitivecoordinatesdivides)) -> jt_divisor_rectsuccessorrightprimitive=1))))) /\\ (((forall jt_b_rectsuccessorright jt_c_rectsuccessorright. (forall jt_index_rectsuccessorrightinputbound. (exists jt_gap_rectsuccessorrightinputboundindex. jt_gap_rectsuccessorrightinputboundindex+S (jt_index_rectsuccessorrightinputbound)=(k)) -> exists jt_value_rectsuccessorrightinputbound. ((((exists fs_h_jt_rectsuccessorrightinputboundat. fs_h_jt_rectsuccessorrightinputboundat + S (jt_value_rectsuccessorrightinputbound) = S ((S (jt_index_rectsuccessorrightinputbound)) * jt_c_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightinputboundat. jt_b_rectsuccessorright = fs_q_jt_rectsuccessorrightinputboundat * S ((S (jt_index_rectsuccessorrightinputbound)) * jt_c_rectsuccessorright) + (jt_value_rectsuccessorrightinputbound))) /\\ (exists jt_gap_rectsuccessorrightinputboundvalue. jt_gap_rectsuccessorrightinputboundvalue+S (jt_value_rectsuccessorrightinputbound)=(n)))) -> (forall jt_divisor_rectsuccessorrightinputprimitive. (exists jt_factor_rectsuccessorrightinputprimitivemodulus. (n)=(jt_divisor_rectsuccessorrightinputprimitive)*jt_factor_rectsuccessorrightinputprimitivemodulus) -> (forall jt_index_rectsuccessorrightinputprimitivecoordinates jt_value_rectsuccessorrightinputprimitivecoordinates. (exists jt_gap_rectsuccessorrightinputprimitivecoordinatesindex. jt_gap_rectsuccessorrightinputprimitivecoordinatesindex+S (jt_index_rectsuccessorrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectsuccessorrightinputprimitivecoordinatesat. fs_h_jt_rectsuccessorrightinputprimitivecoordinatesat + S (jt_value_rectsuccessorrightinputprimitivecoordinates) = S ((S (jt_index_rectsuccessorrightinputprimitivecoordinates)) * jt_c_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightinputprimitivecoordinatesat. jt_b_rectsuccessorright = fs_q_jt_rectsuccessorrightinputprimitivecoordinatesat * S ((S (jt_index_rectsuccessorrightinputprimitivecoordinates)) * jt_c_rectsuccessorright) + (jt_value_rectsuccessorrightinputprimitivecoordinates))) -> (exists jt_factor_rectsuccessorrightinputprimitivecoordinatesdivides. (jt_value_rectsuccessorrightinputprimitivecoordinates)=(jt_divisor_rectsuccessorrightinputprimitive)*jt_factor_rectsuccessorrightinputprimitivecoordinatesdivides)) -> jt_divisor_rectsuccessorrightinputprimitive=1) -> exists jt_i_rectsuccessorright jt_d_rectsuccessorright jt_e_rectsuccessorright. ((exists jt_gap_rectsuccessorrightcompleteindex. jt_gap_rectsuccessorrightcompleteindex+S (jt_i_rectsuccessorright)=(v)) /\\ (((((((exists fs_h_jt_rectsuccessorrightcompletecode. fs_h_jt_rectsuccessorrightcompletecode + S (jt_d_rectsuccessorright) = S ((S (jt_i_rectsuccessorright)) * F)) /\\ exists fs_q_jt_rectsuccessorrightcompletecode. E = fs_q_jt_rectsuccessorrightcompletecode * S ((S (jt_i_rectsuccessorright)) * F) + (jt_d_rectsuccessorright))) /\\ (((exists fs_h_jt_rectsuccessorrightcompletescale. fs_h_jt_rectsuccessorrightcompletescale + S (jt_e_rectsuccessorright) = S ((S (jt_i_rectsuccessorright)) * H)) /\\ exists fs_q_jt_rectsuccessorrightcompletescale. G = fs_q_jt_rectsuccessorrightcompletescale * S ((S (jt_i_rectsuccessorright)) * H) + (jt_e_rectsuccessorright))))) /\\ (forall jt_index_rectsuccessorrightrepresented jt_left_rectsuccessorrightrepresented jt_right_rectsuccessorrightrepresented. (exists jt_gap_rectsuccessorrightrepresentedindex. jt_gap_rectsuccessorrightrepresentedindex+S (jt_index_rectsuccessorrightrepresented)=(k)) -> (((exists fs_h_jt_rectsuccessorrightrepresentedleft. fs_h_jt_rectsuccessorrightrepresentedleft + S (jt_left_rectsuccessorrightrepresented) = S ((S (jt_index_rectsuccessorrightrepresented)) * jt_c_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightrepresentedleft. jt_b_rectsuccessorright = fs_q_jt_rectsuccessorrightrepresentedleft * S ((S (jt_index_rectsuccessorrightrepresented)) * jt_c_rectsuccessorright) + (jt_left_rectsuccessorrightrepresented))) -> (((exists fs_h_jt_rectsuccessorrightrepresentedright. fs_h_jt_rectsuccessorrightrepresentedright + S (jt_right_rectsuccessorrightrepresented) = S ((S (jt_index_rectsuccessorrightrepresented)) * jt_e_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightrepresentedright. jt_d_rectsuccessorright = fs_q_jt_rectsuccessorrightrepresentedright * S ((S (jt_index_rectsuccessorrightrepresented)) * jt_e_rectsuccessorright) + (jt_right_rectsuccessorrightrepresented))) -> jt_left_rectsuccessorrightrepresented=jt_right_rectsuccessorrightrepresented))))) /\\ (forall jt_i_rectsuccessorright jt_h_rectsuccessorright jt_b_rectsuccessorright jt_c_rectsuccessorright jt_d_rectsuccessorright jt_e_rectsuccessorright. (exists jt_gap_rectsuccessorrightfirstindex. jt_gap_rectsuccessorrightfirstindex+S (jt_i_rectsuccessorright)=(v)) -> (exists jt_gap_rectsuccessorrightsecondindex. jt_gap_rectsuccessorrightsecondindex+S (jt_h_rectsuccessorright)=(v)) -> (((((exists fs_h_jt_rectsuccessorrightfirstcode. fs_h_jt_rectsuccessorrightfirstcode + S (jt_b_rectsuccessorright) = S ((S (jt_i_rectsuccessorright)) * F)) /\\ exists fs_q_jt_rectsuccessorrightfirstcode. E = fs_q_jt_rectsuccessorrightfirstcode * S ((S (jt_i_rectsuccessorright)) * F) + (jt_b_rectsuccessorright))) /\\ (((exists fs_h_jt_rectsuccessorrightfirstscale. fs_h_jt_rectsuccessorrightfirstscale + S (jt_c_rectsuccessorright) = S ((S (jt_i_rectsuccessorright)) * H)) /\\ exists fs_q_jt_rectsuccessorrightfirstscale. G = fs_q_jt_rectsuccessorrightfirstscale * S ((S (jt_i_rectsuccessorright)) * H) + (jt_c_rectsuccessorright))))) -> (((((exists fs_h_jt_rectsuccessorrightsecondcode. fs_h_jt_rectsuccessorrightsecondcode + S (jt_d_rectsuccessorright) = S ((S (jt_h_rectsuccessorright)) * F)) /\\ exists fs_q_jt_rectsuccessorrightsecondcode. E = fs_q_jt_rectsuccessorrightsecondcode * S ((S (jt_h_rectsuccessorright)) * F) + (jt_d_rectsuccessorright))) /\\ (((exists fs_h_jt_rectsuccessorrightsecondscale. fs_h_jt_rectsuccessorrightsecondscale + S (jt_e_rectsuccessorright) = S ((S (jt_h_rectsuccessorright)) * H)) /\\ exists fs_q_jt_rectsuccessorrightsecondscale. G = fs_q_jt_rectsuccessorrightsecondscale * S ((S (jt_h_rectsuccessorright)) * H) + (jt_e_rectsuccessorright))))) -> (forall jt_index_rectsuccessorrightsame jt_left_rectsuccessorrightsame jt_right_rectsuccessorrightsame. (exists jt_gap_rectsuccessorrightsameindex. jt_gap_rectsuccessorrightsameindex+S (jt_index_rectsuccessorrightsame)=(k)) -> (((exists fs_h_jt_rectsuccessorrightsameleft. fs_h_jt_rectsuccessorrightsameleft + S (jt_left_rectsuccessorrightsame) = S ((S (jt_index_rectsuccessorrightsame)) * jt_c_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightsameleft. jt_b_rectsuccessorright = fs_q_jt_rectsuccessorrightsameleft * S ((S (jt_index_rectsuccessorrightsame)) * jt_c_rectsuccessorright) + (jt_left_rectsuccessorrightsame))) -> (((exists fs_h_jt_rectsuccessorrightsameright. fs_h_jt_rectsuccessorrightsameright + S (jt_right_rectsuccessorrightsame) = S ((S (jt_index_rectsuccessorrightsame)) * jt_e_rectsuccessorright)) /\\ exists fs_q_jt_rectsuccessorrightsameright. jt_d_rectsuccessorright = fs_q_jt_rectsuccessorrightsameright * S ((S (jt_index_rectsuccessorrightsame)) * jt_e_rectsuccessorright) + (jt_right_rectsuccessorrightsame))) -> jt_left_rectsuccessorrightsame=jt_right_rectsuccessorrightsame) -> jt_i_rectsuccessorright=jt_h_rectsuccessorright))))) -> (exists P Q R T. forall jt_index_rectsuccessorold. (exists jt_gap_rectsuccessoroldindex. jt_gap_rectsuccessoroldindex+S (jt_index_rectsuccessorold)=(q)) -> exists jt_row_rectsuccessorold jt_column_rectsuccessorold jt_b_rectsuccessorold jt_c_rectsuccessorold jt_d_rectsuccessorold jt_e_rectsuccessorold jt_f_rectsuccessorold jt_g_rectsuccessorold. ((exists jt_gap_rectsuccessoroldrow. jt_gap_rectsuccessoroldrow+S (jt_row_rectsuccessorold)=(u)) /\\ (((exists jt_gap_rectsuccessoroldcolumn. jt_gap_rectsuccessoroldcolumn+S (jt_column_rectsuccessorold)=(v)) /\\ (((jt_index_rectsuccessorold=(v)*jt_row_rectsuccessorold+jt_column_rectsuccessorold) /\\ (((((((exists fs_h_jt_rectsuccessoroldleftcode. fs_h_jt_rectsuccessoroldleftcode + S (jt_b_rectsuccessorold) = S ((S (jt_row_rectsuccessorold)) * B)) /\\ exists fs_q_jt_rectsuccessoroldleftcode. A = fs_q_jt_rectsuccessoroldleftcode * S ((S (jt_row_rectsuccessorold)) * B) + (jt_b_rectsuccessorold))) /\\ (((exists fs_h_jt_rectsuccessoroldleftscale. fs_h_jt_rectsuccessoroldleftscale + S (jt_c_rectsuccessorold) = S ((S (jt_row_rectsuccessorold)) * D)) /\\ exists fs_q_jt_rectsuccessoroldleftscale. C = fs_q_jt_rectsuccessoroldleftscale * S ((S (jt_row_rectsuccessorold)) * D) + (jt_c_rectsuccessorold))))) /\\ (((((((exists fs_h_jt_rectsuccessoroldrightcode. fs_h_jt_rectsuccessoroldrightcode + S (jt_d_rectsuccessorold) = S ((S (jt_column_rectsuccessorold)) * F)) /\\ exists fs_q_jt_rectsuccessoroldrightcode. E = fs_q_jt_rectsuccessoroldrightcode * S ((S (jt_column_rectsuccessorold)) * F) + (jt_d_rectsuccessorold))) /\\ (((exists fs_h_jt_rectsuccessoroldrightscale. fs_h_jt_rectsuccessoroldrightscale + S (jt_e_rectsuccessorold) = S ((S (jt_column_rectsuccessorold)) * H)) /\\ exists fs_q_jt_rectsuccessoroldrightscale. G = fs_q_jt_rectsuccessoroldrightscale * S ((S (jt_column_rectsuccessorold)) * H) + (jt_e_rectsuccessorold))))) /\\ (((((((exists fs_h_jt_rectsuccessoroldoutputcode. fs_h_jt_rectsuccessoroldoutputcode + S (jt_f_rectsuccessorold) = S ((S (jt_index_rectsuccessorold)) * Q)) /\\ exists fs_q_jt_rectsuccessoroldoutputcode. P = fs_q_jt_rectsuccessoroldoutputcode * S ((S (jt_index_rectsuccessorold)) * Q) + (jt_f_rectsuccessorold))) /\\ (((exists fs_h_jt_rectsuccessoroldoutputscale. fs_h_jt_rectsuccessoroldoutputscale + S (jt_g_rectsuccessorold) = S ((S (jt_index_rectsuccessorold)) * T)) /\\ exists fs_q_jt_rectsuccessoroldoutputscale. R = fs_q_jt_rectsuccessoroldoutputscale * S ((S (jt_index_rectsuccessorold)) * T) + (jt_g_rectsuccessorold))))) /\\ (((((forall jt_index_rectsuccessoroldcrtbound. (exists jt_gap_rectsuccessoroldcrtboundindex. jt_gap_rectsuccessoroldcrtboundindex+S (jt_index_rectsuccessoroldcrtbound)=(k)) -> exists jt_value_rectsuccessoroldcrtbound. ((((exists fs_h_jt_rectsuccessoroldcrtboundat. fs_h_jt_rectsuccessoroldcrtboundat + S (jt_value_rectsuccessoroldcrtbound) = S ((S (jt_index_rectsuccessoroldcrtbound)) * jt_g_rectsuccessorold)) /\\ exists fs_q_jt_rectsuccessoroldcrtboundat. jt_f_rectsuccessorold = fs_q_jt_rectsuccessoroldcrtboundat * S ((S (jt_index_rectsuccessoroldcrtbound)) * jt_g_rectsuccessorold) + (jt_value_rectsuccessoroldcrtbound))) /\\ (exists jt_gap_rectsuccessoroldcrtboundvalue. jt_gap_rectsuccessoroldcrtboundvalue+S (jt_value_rectsuccessoroldcrtbound)=(m*n)))) /\\ (((forall jt_index_rectsuccessoroldcrtleft jt_left_rectsuccessoroldcrtleft jt_right_rectsuccessoroldcrtleft. (exists jt_gap_rectsuccessoroldcrtleftindex. jt_gap_rectsuccessoroldcrtleftindex+S (jt_index_rectsuccessoroldcrtleft)=(k)) -> (((exists fs_h_jt_rectsuccessoroldcrtleftleft. fs_h_jt_rectsuccessoroldcrtleftleft + S (jt_left_rectsuccessoroldcrtleft) = S ((S (jt_index_rectsuccessoroldcrtleft)) * jt_g_rectsuccessorold)) /\\ exists fs_q_jt_rectsuccessoroldcrtleftleft. jt_f_rectsuccessorold = fs_q_jt_rectsuccessoroldcrtleftleft * S ((S (jt_index_rectsuccessoroldcrtleft)) * jt_g_rectsuccessorold) + (jt_left_rectsuccessoroldcrtleft))) -> (((exists fs_h_jt_rectsuccessoroldcrtleftright. fs_h_jt_rectsuccessoroldcrtleftright + S (jt_right_rectsuccessoroldcrtleft) = S ((S (jt_index_rectsuccessoroldcrtleft)) * jt_c_rectsuccessorold)) /\\ exists fs_q_jt_rectsuccessoroldcrtleftright. jt_b_rectsuccessorold = fs_q_jt_rectsuccessoroldcrtleftright * S ((S (jt_index_rectsuccessoroldcrtleft)) * jt_c_rectsuccessorold) + (jt_right_rectsuccessoroldcrtleft))) -> (exists jt_left_rectsuccessoroldcrtleftmod jt_right_rectsuccessoroldcrtleftmod. (jt_left_rectsuccessoroldcrtleft)+(m)*jt_left_rectsuccessoroldcrtleftmod=(jt_right_rectsuccessoroldcrtleft)+(m)*jt_right_rectsuccessoroldcrtleftmod)) /\\ (forall jt_index_rectsuccessoroldcrtright jt_left_rectsuccessoroldcrtright jt_right_rectsuccessoroldcrtright. (exists jt_gap_rectsuccessoroldcrtrightindex. jt_gap_rectsuccessoroldcrtrightindex+S (jt_index_rectsuccessoroldcrtright)=(k)) -> (((exists fs_h_jt_rectsuccessoroldcrtrightleft. fs_h_jt_rectsuccessoroldcrtrightleft + S (jt_left_rectsuccessoroldcrtright) = S ((S (jt_index_rectsuccessoroldcrtright)) * jt_g_rectsuccessorold)) /\\ exists fs_q_jt_rectsuccessoroldcrtrightleft. jt_f_rectsuccessorold = fs_q_jt_rectsuccessoroldcrtrightleft * S ((S (jt_index_rectsuccessoroldcrtright)) * jt_g_rectsuccessorold) + (jt_left_rectsuccessoroldcrtright))) -> (((exists fs_h_jt_rectsuccessoroldcrtrightright. fs_h_jt_rectsuccessoroldcrtrightright + S (jt_right_rectsuccessoroldcrtright) = S ((S (jt_index_rectsuccessoroldcrtright)) * jt_e_rectsuccessorold)) /\\ exists fs_q_jt_rectsuccessoroldcrtrightright. jt_d_rectsuccessorold = fs_q_jt_rectsuccessoroldcrtrightright * S ((S (jt_index_rectsuccessoroldcrtright)) * jt_e_rectsuccessorold) + (jt_right_rectsuccessoroldcrtright))) -> (exists jt_left_rectsuccessoroldcrtrightmod jt_right_rectsuccessoroldcrtrightmod. (jt_left_rectsuccessoroldcrtright)+(n)*jt_left_rectsuccessoroldcrtrightmod=(jt_right_rectsuccessoroldcrtright)+(n)*jt_right_rectsuccessoroldcrtrightmod)))))) /\\ (forall jt_divisor_rectsuccessoroldprimitive. (exists jt_factor_rectsuccessoroldprimitivemodulus. (m*n)=(jt_divisor_rectsuccessoroldprimitive)*jt_factor_rectsuccessoroldprimitivemodulus) -> (forall jt_index_rectsuccessoroldprimitivecoordinates jt_value_rectsuccessoroldprimitivecoordinates. (exists jt_gap_rectsuccessoroldprimitivecoordinatesindex. jt_gap_rectsuccessoroldprimitivecoordinatesindex+S (jt_index_rectsuccessoroldprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectsuccessoroldprimitivecoordinatesat. fs_h_jt_rectsuccessoroldprimitivecoordinatesat + S (jt_value_rectsuccessoroldprimitivecoordinates) = S ((S (jt_index_rectsuccessoroldprimitivecoordinates)) * jt_g_rectsuccessorold)) /\\ exists fs_q_jt_rectsuccessoroldprimitivecoordinatesat. jt_f_rectsuccessorold = fs_q_jt_rectsuccessoroldprimitivecoordinatesat * S ((S (jt_index_rectsuccessoroldprimitivecoordinates)) * jt_g_rectsuccessorold) + (jt_value_rectsuccessoroldprimitivecoordinates))) -> (exists jt_factor_rectsuccessoroldprimitivecoordinatesdivides. (jt_value_rectsuccessoroldprimitivecoordinates)=(jt_divisor_rectsuccessoroldprimitive)*jt_factor_rectsuccessoroldprimitivecoordinatesdivides)) -> jt_divisor_rectsuccessoroldprimitive=1))))))))))))))) -> (exists jt_gap_rectsuccessorbound. jt_gap_rectsuccessorbound+S (q)=(u*v)) -> exists P Q R T. forall jt_index_rectsuccessornew. (exists jt_gap_rectsuccessornewindex. jt_gap_rectsuccessornewindex+S (jt_index_rectsuccessornew)=(S q)) -> exists jt_row_rectsuccessornew jt_column_rectsuccessornew jt_b_rectsuccessornew jt_c_rectsuccessornew jt_d_rectsuccessornew jt_e_rectsuccessornew jt_f_rectsuccessornew jt_g_rectsuccessornew. ((exists jt_gap_rectsuccessornewrow. jt_gap_rectsuccessornewrow+S (jt_row_rectsuccessornew)=(u)) /\\ (((exists jt_gap_rectsuccessornewcolumn. jt_gap_rectsuccessornewcolumn+S (jt_column_rectsuccessornew)=(v)) /\\ (((jt_index_rectsuccessornew=(v)*jt_row_rectsuccessornew+jt_column_rectsuccessornew) /\\ (((((((exists fs_h_jt_rectsuccessornewleftcode. fs_h_jt_rectsuccessornewleftcode + S (jt_b_rectsuccessornew) = S ((S (jt_row_rectsuccessornew)) * B)) /\\ exists fs_q_jt_rectsuccessornewleftcode. A = fs_q_jt_rectsuccessornewleftcode * S ((S (jt_row_rectsuccessornew)) * B) + (jt_b_rectsuccessornew))) /\\ (((exists fs_h_jt_rectsuccessornewleftscale. fs_h_jt_rectsuccessornewleftscale + S (jt_c_rectsuccessornew) = S ((S (jt_row_rectsuccessornew)) * D)) /\\ exists fs_q_jt_rectsuccessornewleftscale. C = fs_q_jt_rectsuccessornewleftscale * S ((S (jt_row_rectsuccessornew)) * D) + (jt_c_rectsuccessornew))))) /\\ (((((((exists fs_h_jt_rectsuccessornewrightcode. fs_h_jt_rectsuccessornewrightcode + S (jt_d_rectsuccessornew) = S ((S (jt_column_rectsuccessornew)) * F)) /\\ exists fs_q_jt_rectsuccessornewrightcode. E = fs_q_jt_rectsuccessornewrightcode * S ((S (jt_column_rectsuccessornew)) * F) + (jt_d_rectsuccessornew))) /\\ (((exists fs_h_jt_rectsuccessornewrightscale. fs_h_jt_rectsuccessornewrightscale + S (jt_e_rectsuccessornew) = S ((S (jt_column_rectsuccessornew)) * H)) /\\ exists fs_q_jt_rectsuccessornewrightscale. G = fs_q_jt_rectsuccessornewrightscale * S ((S (jt_column_rectsuccessornew)) * H) + (jt_e_rectsuccessornew))))) /\\ (((((((exists fs_h_jt_rectsuccessornewoutputcode. fs_h_jt_rectsuccessornewoutputcode + S (jt_f_rectsuccessornew) = S ((S (jt_index_rectsuccessornew)) * Q)) /\\ exists fs_q_jt_rectsuccessornewoutputcode. P = fs_q_jt_rectsuccessornewoutputcode * S ((S (jt_index_rectsuccessornew)) * Q) + (jt_f_rectsuccessornew))) /\\ (((exists fs_h_jt_rectsuccessornewoutputscale. fs_h_jt_rectsuccessornewoutputscale + S (jt_g_rectsuccessornew) = S ((S (jt_index_rectsuccessornew)) * T)) /\\ exists fs_q_jt_rectsuccessornewoutputscale. R = fs_q_jt_rectsuccessornewoutputscale * S ((S (jt_index_rectsuccessornew)) * T) + (jt_g_rectsuccessornew))))) /\\ (((((forall jt_index_rectsuccessornewcrtbound. (exists jt_gap_rectsuccessornewcrtboundindex. jt_gap_rectsuccessornewcrtboundindex+S (jt_index_rectsuccessornewcrtbound)=(k)) -> exists jt_value_rectsuccessornewcrtbound. ((((exists fs_h_jt_rectsuccessornewcrtboundat. fs_h_jt_rectsuccessornewcrtboundat + S (jt_value_rectsuccessornewcrtbound) = S ((S (jt_index_rectsuccessornewcrtbound)) * jt_g_rectsuccessornew)) /\\ exists fs_q_jt_rectsuccessornewcrtboundat. jt_f_rectsuccessornew = fs_q_jt_rectsuccessornewcrtboundat * S ((S (jt_index_rectsuccessornewcrtbound)) * jt_g_rectsuccessornew) + (jt_value_rectsuccessornewcrtbound))) /\\ (exists jt_gap_rectsuccessornewcrtboundvalue. jt_gap_rectsuccessornewcrtboundvalue+S (jt_value_rectsuccessornewcrtbound)=(m*n)))) /\\ (((forall jt_index_rectsuccessornewcrtleft jt_left_rectsuccessornewcrtleft jt_right_rectsuccessornewcrtleft. (exists jt_gap_rectsuccessornewcrtleftindex. jt_gap_rectsuccessornewcrtleftindex+S (jt_index_rectsuccessornewcrtleft)=(k)) -> (((exists fs_h_jt_rectsuccessornewcrtleftleft. fs_h_jt_rectsuccessornewcrtleftleft + S (jt_left_rectsuccessornewcrtleft) = S ((S (jt_index_rectsuccessornewcrtleft)) * jt_g_rectsuccessornew)) /\\ exists fs_q_jt_rectsuccessornewcrtleftleft. jt_f_rectsuccessornew = fs_q_jt_rectsuccessornewcrtleftleft * S ((S (jt_index_rectsuccessornewcrtleft)) * jt_g_rectsuccessornew) + (jt_left_rectsuccessornewcrtleft))) -> (((exists fs_h_jt_rectsuccessornewcrtleftright. fs_h_jt_rectsuccessornewcrtleftright + S (jt_right_rectsuccessornewcrtleft) = S ((S (jt_index_rectsuccessornewcrtleft)) * jt_c_rectsuccessornew)) /\\ exists fs_q_jt_rectsuccessornewcrtleftright. jt_b_rectsuccessornew = fs_q_jt_rectsuccessornewcrtleftright * S ((S (jt_index_rectsuccessornewcrtleft)) * jt_c_rectsuccessornew) + (jt_right_rectsuccessornewcrtleft))) -> (exists jt_left_rectsuccessornewcrtleftmod jt_right_rectsuccessornewcrtleftmod. (jt_left_rectsuccessornewcrtleft)+(m)*jt_left_rectsuccessornewcrtleftmod=(jt_right_rectsuccessornewcrtleft)+(m)*jt_right_rectsuccessornewcrtleftmod)) /\\ (forall jt_index_rectsuccessornewcrtright jt_left_rectsuccessornewcrtright jt_right_rectsuccessornewcrtright. (exists jt_gap_rectsuccessornewcrtrightindex. jt_gap_rectsuccessornewcrtrightindex+S (jt_index_rectsuccessornewcrtright)=(k)) -> (((exists fs_h_jt_rectsuccessornewcrtrightleft. fs_h_jt_rectsuccessornewcrtrightleft + S (jt_left_rectsuccessornewcrtright) = S ((S (jt_index_rectsuccessornewcrtright)) * jt_g_rectsuccessornew)) /\\ exists fs_q_jt_rectsuccessornewcrtrightleft. jt_f_rectsuccessornew = fs_q_jt_rectsuccessornewcrtrightleft * S ((S (jt_index_rectsuccessornewcrtright)) * jt_g_rectsuccessornew) + (jt_left_rectsuccessornewcrtright))) -> (((exists fs_h_jt_rectsuccessornewcrtrightright. fs_h_jt_rectsuccessornewcrtrightright + S (jt_right_rectsuccessornewcrtright) = S ((S (jt_index_rectsuccessornewcrtright)) * jt_e_rectsuccessornew)) /\\ exists fs_q_jt_rectsuccessornewcrtrightright. jt_d_rectsuccessornew = fs_q_jt_rectsuccessornewcrtrightright * S ((S (jt_index_rectsuccessornewcrtright)) * jt_e_rectsuccessornew) + (jt_right_rectsuccessornewcrtright))) -> (exists jt_left_rectsuccessornewcrtrightmod jt_right_rectsuccessornewcrtrightmod. (jt_left_rectsuccessornewcrtright)+(n)*jt_left_rectsuccessornewcrtrightmod=(jt_right_rectsuccessornewcrtright)+(n)*jt_right_rectsuccessornewcrtrightmod)))))) /\\ (forall jt_divisor_rectsuccessornewprimitive. (exists jt_factor_rectsuccessornewprimitivemodulus. (m*n)=(jt_divisor_rectsuccessornewprimitive)*jt_factor_rectsuccessornewprimitivemodulus) -> (forall jt_index_rectsuccessornewprimitivecoordinates jt_value_rectsuccessornewprimitivecoordinates. (exists jt_gap_rectsuccessornewprimitivecoordinatesindex. jt_gap_rectsuccessornewprimitivecoordinatesindex+S (jt_index_rectsuccessornewprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectsuccessornewprimitivecoordinatesat. fs_h_jt_rectsuccessornewprimitivecoordinatesat + S (jt_value_rectsuccessornewprimitivecoordinates) = S ((S (jt_index_rectsuccessornewprimitivecoordinates)) * jt_g_rectsuccessornew)) /\\ exists fs_q_jt_rectsuccessornewprimitivecoordinatesat. jt_f_rectsuccessornew = fs_q_jt_rectsuccessornewprimitivecoordinatesat * S ((S (jt_index_rectsuccessornewprimitivecoordinates)) * jt_g_rectsuccessornew) + (jt_value_rectsuccessornewprimitivecoordinates))) -> (exists jt_factor_rectsuccessornewprimitivecoordinatesdivides. (jt_value_rectsuccessornewprimitivecoordinates)=(jt_divisor_rectsuccessornewprimitive)*jt_factor_rectsuccessornewprimitivecoordinatesdivides)) -> jt_divisor_rectsuccessornewprimitive=1))))))))))))))", "statement_sha256": "42ac7078d445af2cfd4afd2f146a21f9026ae6768a02920baa885d15ee467a6a", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Eliminate the four actual prefix witnesses and append one CRT output in a separate constructive proof scope." }, { "admission_dependencies": [ "lt_not_le", "zero_le", "le_trans", "le_succ_self", "jordan_rectangle_crt_successor" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 49, "body_proof_nodes": 82, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "induction q", "intro hq", "exists 0", "exists 0", "exists 0", "exists 0", "intro p", "intro hp", "exfalso", "specialize lt_not_le (p)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hp", "specialize zero_le (p)", "apply zero_le", "intro hq", "have hold : ∃ P. ∃ Q. ∃ R. ∃ T. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q)", "apply IH", "specialize le_trans (q)", "specialize le_trans (S q)", "specialize le_trans (u*v)", "apply le_trans", "specialize le_succ_self (q)", "apply le_succ_self", "exact hq", "specialize jordan_rectangle_crt_successor (m)", "specialize jordan_rectangle_crt_successor (n)", "specialize jordan_rectangle_crt_successor (k)", "specialize jordan_rectangle_crt_successor (A)", "specialize jordan_rectangle_crt_successor (B)", "specialize jordan_rectangle_crt_successor (C)", "specialize jordan_rectangle_crt_successor (D)", "specialize jordan_rectangle_crt_successor (u)", "specialize jordan_rectangle_crt_successor (E)", "specialize jordan_rectangle_crt_successor (F)", "specialize jordan_rectangle_crt_successor (G)", "specialize jordan_rectangle_crt_successor (H)", "specialize jordan_rectangle_crt_successor (v)", "specialize jordan_rectangle_crt_successor (q)", "apply jordan_rectangle_crt_successor", "exact hm", "exact hn", "exact hcop", "exact hleft", "exact hright", "exact hold", "exact hq" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → ∀ x. Le(x,u · v) → ∃ y. ∃ z. ∃ i. ∃ j. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,y,z,i,j,x)", "defined_statement_sha256": "45bd8c60555819532fbfd7944d917c8bc03c2edc7e6874fd7a3a645e20dbe559", "definition_uses": { "ND0374": 2, "ND0381": 2, "PD0001": 1, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "1f53595b967e27a64fc239e07328651a8b697fa048e9a5075f31613b7e8241c6", "free_names": [], "script_definition_uses": { "ND0381": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hleft" } ], [ { "kind": "text", "text": "intro hright" } ], [ { "kind": "text", "text": "induction q" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (p)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "specialize zero_le (p)" } ], [ { "kind": "text", "text": "apply zero_le" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "have hold : " }, { "kind": "text", "text": "∃ P. ∃ Q. ∃ R. ∃ T. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q)" } ], [ { "kind": "text", "text": "apply IH" } ], [ { "kind": "text", "text": "specialize le_trans (q)" } ], [ { "kind": "text", "text": "specialize le_trans (S q)" } ], [ { "kind": "text", "text": "specialize le_trans (u*v)" } ], [ { "kind": "text", "text": "apply le_trans" } ], [ { "kind": "text", "text": "specialize le_succ_self (q)" } ], [ { "kind": "text", "text": "apply le_succ_self" } ], [ { "kind": "text", "text": "exact hq" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_successor (q)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_successor" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hleft" } ], [ { "kind": "text", "text": "exact hright" } ], [ { "kind": "text", "text": "exact hold" } ], [ { "kind": "text", "text": "exact hq" } ] ], "statement_definition_uses": { "ND0374": 2, "ND0381": 1, "PD0001": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → ∀ x. " }, { "definition": "PD0001", "kind": "definition", "text": "Le(x,u · v)" }, { "kind": "text", "text": " → ∃ y. ∃ z. ∃ i. ∃ j. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,y,z,i,j,x)" } ] }, "dependencies": [ "lt_not_le", "zero_le", "le_trans", "le_succ_self", "jordan_rectangle_crt_successor" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0042", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 327, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hm", "intro hn", "intro hcop", "intro hleft", "intro hright", "induction q", "intro hq", "exists 0", "exists 0", "exists 0", "exists 0", "intro p", "intro hp", "exfalso", "specialize lt_not_le (p)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hp", "specialize zero_le (p)", "apply zero_le", "intro hq", "have hold : exists P Q R T. forall jt_index_rectinductionold. (exists jt_gap_rectinductionoldindex. jt_gap_rectinductionoldindex+S (jt_index_rectinductionold)=(q)) -> exists jt_row_rectinductionold jt_column_rectinductionold jt_b_rectinductionold jt_c_rectinductionold jt_d_rectinductionold jt_e_rectinductionold jt_f_rectinductionold jt_g_rectinductionold. ((exists jt_gap_rectinductionoldrow. jt_gap_rectinductionoldrow+S (jt_row_rectinductionold)=(u)) /\\ (((exists jt_gap_rectinductionoldcolumn. jt_gap_rectinductionoldcolumn+S (jt_column_rectinductionold)=(v)) /\\ (((jt_index_rectinductionold=(v)*jt_row_rectinductionold+jt_column_rectinductionold) /\\ (((((((exists fs_h_jt_rectinductionoldleftcode. fs_h_jt_rectinductionoldleftcode + S (jt_b_rectinductionold) = S ((S (jt_row_rectinductionold)) * B)) /\\ exists fs_q_jt_rectinductionoldleftcode. A = fs_q_jt_rectinductionoldleftcode * S ((S (jt_row_rectinductionold)) * B) + (jt_b_rectinductionold))) /\\ (((exists fs_h_jt_rectinductionoldleftscale. fs_h_jt_rectinductionoldleftscale + S (jt_c_rectinductionold) = S ((S (jt_row_rectinductionold)) * D)) /\\ exists fs_q_jt_rectinductionoldleftscale. C = fs_q_jt_rectinductionoldleftscale * S ((S (jt_row_rectinductionold)) * D) + (jt_c_rectinductionold))))) /\\ (((((((exists fs_h_jt_rectinductionoldrightcode. fs_h_jt_rectinductionoldrightcode + S (jt_d_rectinductionold) = S ((S (jt_column_rectinductionold)) * F)) /\\ exists fs_q_jt_rectinductionoldrightcode. E = fs_q_jt_rectinductionoldrightcode * S ((S (jt_column_rectinductionold)) * F) + (jt_d_rectinductionold))) /\\ (((exists fs_h_jt_rectinductionoldrightscale. fs_h_jt_rectinductionoldrightscale + S (jt_e_rectinductionold) = S ((S (jt_column_rectinductionold)) * H)) /\\ exists fs_q_jt_rectinductionoldrightscale. G = fs_q_jt_rectinductionoldrightscale * S ((S (jt_column_rectinductionold)) * H) + (jt_e_rectinductionold))))) /\\ (((((((exists fs_h_jt_rectinductionoldoutputcode. fs_h_jt_rectinductionoldoutputcode + S (jt_f_rectinductionold) = S ((S (jt_index_rectinductionold)) * Q)) /\\ exists fs_q_jt_rectinductionoldoutputcode. P = fs_q_jt_rectinductionoldoutputcode * S ((S (jt_index_rectinductionold)) * Q) + (jt_f_rectinductionold))) /\\ (((exists fs_h_jt_rectinductionoldoutputscale. fs_h_jt_rectinductionoldoutputscale + S (jt_g_rectinductionold) = S ((S (jt_index_rectinductionold)) * T)) /\\ exists fs_q_jt_rectinductionoldoutputscale. R = fs_q_jt_rectinductionoldoutputscale * S ((S (jt_index_rectinductionold)) * T) + (jt_g_rectinductionold))))) /\\ (((((forall jt_index_rectinductionoldcrtbound. (exists jt_gap_rectinductionoldcrtboundindex. jt_gap_rectinductionoldcrtboundindex+S (jt_index_rectinductionoldcrtbound)=(k)) -> exists jt_value_rectinductionoldcrtbound. ((((exists fs_h_jt_rectinductionoldcrtboundat. fs_h_jt_rectinductionoldcrtboundat + S (jt_value_rectinductionoldcrtbound) = S ((S (jt_index_rectinductionoldcrtbound)) * jt_g_rectinductionold)) /\\ exists fs_q_jt_rectinductionoldcrtboundat. jt_f_rectinductionold = fs_q_jt_rectinductionoldcrtboundat * S ((S (jt_index_rectinductionoldcrtbound)) * jt_g_rectinductionold) + (jt_value_rectinductionoldcrtbound))) /\\ (exists jt_gap_rectinductionoldcrtboundvalue. jt_gap_rectinductionoldcrtboundvalue+S (jt_value_rectinductionoldcrtbound)=(m*n)))) /\\ (((forall jt_index_rectinductionoldcrtleft jt_left_rectinductionoldcrtleft jt_right_rectinductionoldcrtleft. (exists jt_gap_rectinductionoldcrtleftindex. jt_gap_rectinductionoldcrtleftindex+S (jt_index_rectinductionoldcrtleft)=(k)) -> (((exists fs_h_jt_rectinductionoldcrtleftleft. fs_h_jt_rectinductionoldcrtleftleft + S (jt_left_rectinductionoldcrtleft) = S ((S (jt_index_rectinductionoldcrtleft)) * jt_g_rectinductionold)) /\\ exists fs_q_jt_rectinductionoldcrtleftleft. jt_f_rectinductionold = fs_q_jt_rectinductionoldcrtleftleft * S ((S (jt_index_rectinductionoldcrtleft)) * jt_g_rectinductionold) + (jt_left_rectinductionoldcrtleft))) -> (((exists fs_h_jt_rectinductionoldcrtleftright. fs_h_jt_rectinductionoldcrtleftright + S (jt_right_rectinductionoldcrtleft) = S ((S (jt_index_rectinductionoldcrtleft)) * jt_c_rectinductionold)) /\\ exists fs_q_jt_rectinductionoldcrtleftright. jt_b_rectinductionold = fs_q_jt_rectinductionoldcrtleftright * S ((S (jt_index_rectinductionoldcrtleft)) * jt_c_rectinductionold) + (jt_right_rectinductionoldcrtleft))) -> (exists jt_left_rectinductionoldcrtleftmod jt_right_rectinductionoldcrtleftmod. (jt_left_rectinductionoldcrtleft)+(m)*jt_left_rectinductionoldcrtleftmod=(jt_right_rectinductionoldcrtleft)+(m)*jt_right_rectinductionoldcrtleftmod)) /\\ (forall jt_index_rectinductionoldcrtright jt_left_rectinductionoldcrtright jt_right_rectinductionoldcrtright. (exists jt_gap_rectinductionoldcrtrightindex. jt_gap_rectinductionoldcrtrightindex+S (jt_index_rectinductionoldcrtright)=(k)) -> (((exists fs_h_jt_rectinductionoldcrtrightleft. fs_h_jt_rectinductionoldcrtrightleft + S (jt_left_rectinductionoldcrtright) = S ((S (jt_index_rectinductionoldcrtright)) * jt_g_rectinductionold)) /\\ exists fs_q_jt_rectinductionoldcrtrightleft. jt_f_rectinductionold = fs_q_jt_rectinductionoldcrtrightleft * S ((S (jt_index_rectinductionoldcrtright)) * jt_g_rectinductionold) + (jt_left_rectinductionoldcrtright))) -> (((exists fs_h_jt_rectinductionoldcrtrightright. fs_h_jt_rectinductionoldcrtrightright + S (jt_right_rectinductionoldcrtright) = S ((S (jt_index_rectinductionoldcrtright)) * jt_e_rectinductionold)) /\\ exists fs_q_jt_rectinductionoldcrtrightright. jt_d_rectinductionold = fs_q_jt_rectinductionoldcrtrightright * S ((S (jt_index_rectinductionoldcrtright)) * jt_e_rectinductionold) + (jt_right_rectinductionoldcrtright))) -> (exists jt_left_rectinductionoldcrtrightmod jt_right_rectinductionoldcrtrightmod. (jt_left_rectinductionoldcrtright)+(n)*jt_left_rectinductionoldcrtrightmod=(jt_right_rectinductionoldcrtright)+(n)*jt_right_rectinductionoldcrtrightmod)))))) /\\ (forall jt_divisor_rectinductionoldprimitive. (exists jt_factor_rectinductionoldprimitivemodulus. (m*n)=(jt_divisor_rectinductionoldprimitive)*jt_factor_rectinductionoldprimitivemodulus) -> (forall jt_index_rectinductionoldprimitivecoordinates jt_value_rectinductionoldprimitivecoordinates. (exists jt_gap_rectinductionoldprimitivecoordinatesindex. jt_gap_rectinductionoldprimitivecoordinatesindex+S (jt_index_rectinductionoldprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectinductionoldprimitivecoordinatesat. fs_h_jt_rectinductionoldprimitivecoordinatesat + S (jt_value_rectinductionoldprimitivecoordinates) = S ((S (jt_index_rectinductionoldprimitivecoordinates)) * jt_g_rectinductionold)) /\\ exists fs_q_jt_rectinductionoldprimitivecoordinatesat. jt_f_rectinductionold = fs_q_jt_rectinductionoldprimitivecoordinatesat * S ((S (jt_index_rectinductionoldprimitivecoordinates)) * jt_g_rectinductionold) + (jt_value_rectinductionoldprimitivecoordinates))) -> (exists jt_factor_rectinductionoldprimitivecoordinatesdivides. (jt_value_rectinductionoldprimitivecoordinates)=(jt_divisor_rectinductionoldprimitive)*jt_factor_rectinductionoldprimitivecoordinatesdivides)) -> jt_divisor_rectinductionoldprimitive=1))))))))))))))", "apply IH", "specialize le_trans (q)", "specialize le_trans (S q)", "specialize le_trans (u*v)", "apply le_trans", "specialize le_succ_self (q)", "apply le_succ_self", "exact hq", "specialize jordan_rectangle_crt_successor (m)", "specialize jordan_rectangle_crt_successor (n)", "specialize jordan_rectangle_crt_successor (k)", "specialize jordan_rectangle_crt_successor (A)", "specialize jordan_rectangle_crt_successor (B)", "specialize jordan_rectangle_crt_successor (C)", "specialize jordan_rectangle_crt_successor (D)", "specialize jordan_rectangle_crt_successor (u)", "specialize jordan_rectangle_crt_successor (E)", "specialize jordan_rectangle_crt_successor (F)", "specialize jordan_rectangle_crt_successor (G)", "specialize jordan_rectangle_crt_successor (H)", "specialize jordan_rectangle_crt_successor (v)", "specialize jordan_rectangle_crt_successor (q)", "apply jordan_rectangle_crt_successor", "exact hm", "exact hn", "exact hcop", "exact hleft", "exact hright", "exact hold", "exact hq" ], "script_sha256": "35a0c2030ffbaa7822908d875665ef1d5896aff61bd2be84ac45cbbd247be53c", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "35a0c2030ffbaa7822908d875665ef1d5896aff61bd2be84ac45cbbd247be53c", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "1f53595b967e27a64fc239e07328651a8b697fa048e9a5075f31613b7e8241c6" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v. ~(m=0) -> ~(n=0) -> (forall jt_divisor_rectexistscop. (exists jt_factor_rectexistscopa. (m)=(jt_divisor_rectexistscop)*jt_factor_rectexistscopa) -> (exists jt_factor_rectexistscopb. (n)=(jt_divisor_rectexistscop)*jt_factor_rectexistscopb) -> jt_divisor_rectexistscop=1) -> (((forall jt_i_rectexistsleft. (exists jt_gap_rectexistsleftsoundindex. jt_gap_rectexistsleftsoundindex+S (jt_i_rectexistsleft)=(u)) -> exists jt_b_rectexistsleft jt_c_rectexistsleft. ((((((exists fs_h_jt_rectexistsleftsoundcode. fs_h_jt_rectexistsleftsoundcode + S (jt_b_rectexistsleft) = S ((S (jt_i_rectexistsleft)) * B)) /\\ exists fs_q_jt_rectexistsleftsoundcode. A = fs_q_jt_rectexistsleftsoundcode * S ((S (jt_i_rectexistsleft)) * B) + (jt_b_rectexistsleft))) /\\ (((exists fs_h_jt_rectexistsleftsoundscale. fs_h_jt_rectexistsleftsoundscale + S (jt_c_rectexistsleft) = S ((S (jt_i_rectexistsleft)) * D)) /\\ exists fs_q_jt_rectexistsleftsoundscale. C = fs_q_jt_rectexistsleftsoundscale * S ((S (jt_i_rectexistsleft)) * D) + (jt_c_rectexistsleft))))) /\\ (((forall jt_index_rectexistsleftbound. (exists jt_gap_rectexistsleftboundindex. jt_gap_rectexistsleftboundindex+S (jt_index_rectexistsleftbound)=(k)) -> exists jt_value_rectexistsleftbound. ((((exists fs_h_jt_rectexistsleftboundat. fs_h_jt_rectexistsleftboundat + S (jt_value_rectexistsleftbound) = S ((S (jt_index_rectexistsleftbound)) * jt_c_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftboundat. jt_b_rectexistsleft = fs_q_jt_rectexistsleftboundat * S ((S (jt_index_rectexistsleftbound)) * jt_c_rectexistsleft) + (jt_value_rectexistsleftbound))) /\\ (exists jt_gap_rectexistsleftboundvalue. jt_gap_rectexistsleftboundvalue+S (jt_value_rectexistsleftbound)=(m)))) /\\ (forall jt_divisor_rectexistsleftprimitive. (exists jt_factor_rectexistsleftprimitivemodulus. (m)=(jt_divisor_rectexistsleftprimitive)*jt_factor_rectexistsleftprimitivemodulus) -> (forall jt_index_rectexistsleftprimitivecoordinates jt_value_rectexistsleftprimitivecoordinates. (exists jt_gap_rectexistsleftprimitivecoordinatesindex. jt_gap_rectexistsleftprimitivecoordinatesindex+S (jt_index_rectexistsleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectexistsleftprimitivecoordinatesat. fs_h_jt_rectexistsleftprimitivecoordinatesat + S (jt_value_rectexistsleftprimitivecoordinates) = S ((S (jt_index_rectexistsleftprimitivecoordinates)) * jt_c_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftprimitivecoordinatesat. jt_b_rectexistsleft = fs_q_jt_rectexistsleftprimitivecoordinatesat * S ((S (jt_index_rectexistsleftprimitivecoordinates)) * jt_c_rectexistsleft) + (jt_value_rectexistsleftprimitivecoordinates))) -> (exists jt_factor_rectexistsleftprimitivecoordinatesdivides. (jt_value_rectexistsleftprimitivecoordinates)=(jt_divisor_rectexistsleftprimitive)*jt_factor_rectexistsleftprimitivecoordinatesdivides)) -> jt_divisor_rectexistsleftprimitive=1))))) /\\ (((forall jt_b_rectexistsleft jt_c_rectexistsleft. (forall jt_index_rectexistsleftinputbound. (exists jt_gap_rectexistsleftinputboundindex. jt_gap_rectexistsleftinputboundindex+S (jt_index_rectexistsleftinputbound)=(k)) -> exists jt_value_rectexistsleftinputbound. ((((exists fs_h_jt_rectexistsleftinputboundat. fs_h_jt_rectexistsleftinputboundat + S (jt_value_rectexistsleftinputbound) = S ((S (jt_index_rectexistsleftinputbound)) * jt_c_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftinputboundat. jt_b_rectexistsleft = fs_q_jt_rectexistsleftinputboundat * S ((S (jt_index_rectexistsleftinputbound)) * jt_c_rectexistsleft) + (jt_value_rectexistsleftinputbound))) /\\ (exists jt_gap_rectexistsleftinputboundvalue. jt_gap_rectexistsleftinputboundvalue+S (jt_value_rectexistsleftinputbound)=(m)))) -> (forall jt_divisor_rectexistsleftinputprimitive. (exists jt_factor_rectexistsleftinputprimitivemodulus. (m)=(jt_divisor_rectexistsleftinputprimitive)*jt_factor_rectexistsleftinputprimitivemodulus) -> (forall jt_index_rectexistsleftinputprimitivecoordinates jt_value_rectexistsleftinputprimitivecoordinates. (exists jt_gap_rectexistsleftinputprimitivecoordinatesindex. jt_gap_rectexistsleftinputprimitivecoordinatesindex+S (jt_index_rectexistsleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectexistsleftinputprimitivecoordinatesat. fs_h_jt_rectexistsleftinputprimitivecoordinatesat + S (jt_value_rectexistsleftinputprimitivecoordinates) = S ((S (jt_index_rectexistsleftinputprimitivecoordinates)) * jt_c_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftinputprimitivecoordinatesat. jt_b_rectexistsleft = fs_q_jt_rectexistsleftinputprimitivecoordinatesat * S ((S (jt_index_rectexistsleftinputprimitivecoordinates)) * jt_c_rectexistsleft) + (jt_value_rectexistsleftinputprimitivecoordinates))) -> (exists jt_factor_rectexistsleftinputprimitivecoordinatesdivides. (jt_value_rectexistsleftinputprimitivecoordinates)=(jt_divisor_rectexistsleftinputprimitive)*jt_factor_rectexistsleftinputprimitivecoordinatesdivides)) -> jt_divisor_rectexistsleftinputprimitive=1) -> exists jt_i_rectexistsleft jt_d_rectexistsleft jt_e_rectexistsleft. ((exists jt_gap_rectexistsleftcompleteindex. jt_gap_rectexistsleftcompleteindex+S (jt_i_rectexistsleft)=(u)) /\\ (((((((exists fs_h_jt_rectexistsleftcompletecode. fs_h_jt_rectexistsleftcompletecode + S (jt_d_rectexistsleft) = S ((S (jt_i_rectexistsleft)) * B)) /\\ exists fs_q_jt_rectexistsleftcompletecode. A = fs_q_jt_rectexistsleftcompletecode * S ((S (jt_i_rectexistsleft)) * B) + (jt_d_rectexistsleft))) /\\ (((exists fs_h_jt_rectexistsleftcompletescale. fs_h_jt_rectexistsleftcompletescale + S (jt_e_rectexistsleft) = S ((S (jt_i_rectexistsleft)) * D)) /\\ exists fs_q_jt_rectexistsleftcompletescale. C = fs_q_jt_rectexistsleftcompletescale * S ((S (jt_i_rectexistsleft)) * D) + (jt_e_rectexistsleft))))) /\\ (forall jt_index_rectexistsleftrepresented jt_left_rectexistsleftrepresented jt_right_rectexistsleftrepresented. (exists jt_gap_rectexistsleftrepresentedindex. jt_gap_rectexistsleftrepresentedindex+S (jt_index_rectexistsleftrepresented)=(k)) -> (((exists fs_h_jt_rectexistsleftrepresentedleft. fs_h_jt_rectexistsleftrepresentedleft + S (jt_left_rectexistsleftrepresented) = S ((S (jt_index_rectexistsleftrepresented)) * jt_c_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftrepresentedleft. jt_b_rectexistsleft = fs_q_jt_rectexistsleftrepresentedleft * S ((S (jt_index_rectexistsleftrepresented)) * jt_c_rectexistsleft) + (jt_left_rectexistsleftrepresented))) -> (((exists fs_h_jt_rectexistsleftrepresentedright. fs_h_jt_rectexistsleftrepresentedright + S (jt_right_rectexistsleftrepresented) = S ((S (jt_index_rectexistsleftrepresented)) * jt_e_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftrepresentedright. jt_d_rectexistsleft = fs_q_jt_rectexistsleftrepresentedright * S ((S (jt_index_rectexistsleftrepresented)) * jt_e_rectexistsleft) + (jt_right_rectexistsleftrepresented))) -> jt_left_rectexistsleftrepresented=jt_right_rectexistsleftrepresented))))) /\\ (forall jt_i_rectexistsleft jt_h_rectexistsleft jt_b_rectexistsleft jt_c_rectexistsleft jt_d_rectexistsleft jt_e_rectexistsleft. (exists jt_gap_rectexistsleftfirstindex. jt_gap_rectexistsleftfirstindex+S (jt_i_rectexistsleft)=(u)) -> (exists jt_gap_rectexistsleftsecondindex. jt_gap_rectexistsleftsecondindex+S (jt_h_rectexistsleft)=(u)) -> (((((exists fs_h_jt_rectexistsleftfirstcode. fs_h_jt_rectexistsleftfirstcode + S (jt_b_rectexistsleft) = S ((S (jt_i_rectexistsleft)) * B)) /\\ exists fs_q_jt_rectexistsleftfirstcode. A = fs_q_jt_rectexistsleftfirstcode * S ((S (jt_i_rectexistsleft)) * B) + (jt_b_rectexistsleft))) /\\ (((exists fs_h_jt_rectexistsleftfirstscale. fs_h_jt_rectexistsleftfirstscale + S (jt_c_rectexistsleft) = S ((S (jt_i_rectexistsleft)) * D)) /\\ exists fs_q_jt_rectexistsleftfirstscale. C = fs_q_jt_rectexistsleftfirstscale * S ((S (jt_i_rectexistsleft)) * D) + (jt_c_rectexistsleft))))) -> (((((exists fs_h_jt_rectexistsleftsecondcode. fs_h_jt_rectexistsleftsecondcode + S (jt_d_rectexistsleft) = S ((S (jt_h_rectexistsleft)) * B)) /\\ exists fs_q_jt_rectexistsleftsecondcode. A = fs_q_jt_rectexistsleftsecondcode * S ((S (jt_h_rectexistsleft)) * B) + (jt_d_rectexistsleft))) /\\ (((exists fs_h_jt_rectexistsleftsecondscale. fs_h_jt_rectexistsleftsecondscale + S (jt_e_rectexistsleft) = S ((S (jt_h_rectexistsleft)) * D)) /\\ exists fs_q_jt_rectexistsleftsecondscale. C = fs_q_jt_rectexistsleftsecondscale * S ((S (jt_h_rectexistsleft)) * D) + (jt_e_rectexistsleft))))) -> (forall jt_index_rectexistsleftsame jt_left_rectexistsleftsame jt_right_rectexistsleftsame. (exists jt_gap_rectexistsleftsameindex. jt_gap_rectexistsleftsameindex+S (jt_index_rectexistsleftsame)=(k)) -> (((exists fs_h_jt_rectexistsleftsameleft. fs_h_jt_rectexistsleftsameleft + S (jt_left_rectexistsleftsame) = S ((S (jt_index_rectexistsleftsame)) * jt_c_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftsameleft. jt_b_rectexistsleft = fs_q_jt_rectexistsleftsameleft * S ((S (jt_index_rectexistsleftsame)) * jt_c_rectexistsleft) + (jt_left_rectexistsleftsame))) -> (((exists fs_h_jt_rectexistsleftsameright. fs_h_jt_rectexistsleftsameright + S (jt_right_rectexistsleftsame) = S ((S (jt_index_rectexistsleftsame)) * jt_e_rectexistsleft)) /\\ exists fs_q_jt_rectexistsleftsameright. jt_d_rectexistsleft = fs_q_jt_rectexistsleftsameright * S ((S (jt_index_rectexistsleftsame)) * jt_e_rectexistsleft) + (jt_right_rectexistsleftsame))) -> jt_left_rectexistsleftsame=jt_right_rectexistsleftsame) -> jt_i_rectexistsleft=jt_h_rectexistsleft))))) -> (((forall jt_i_rectexistsright. (exists jt_gap_rectexistsrightsoundindex. jt_gap_rectexistsrightsoundindex+S (jt_i_rectexistsright)=(v)) -> exists jt_b_rectexistsright jt_c_rectexistsright. ((((((exists fs_h_jt_rectexistsrightsoundcode. fs_h_jt_rectexistsrightsoundcode + S (jt_b_rectexistsright) = S ((S (jt_i_rectexistsright)) * F)) /\\ exists fs_q_jt_rectexistsrightsoundcode. E = fs_q_jt_rectexistsrightsoundcode * S ((S (jt_i_rectexistsright)) * F) + (jt_b_rectexistsright))) /\\ (((exists fs_h_jt_rectexistsrightsoundscale. fs_h_jt_rectexistsrightsoundscale + S (jt_c_rectexistsright) = S ((S (jt_i_rectexistsright)) * H)) /\\ exists fs_q_jt_rectexistsrightsoundscale. G = fs_q_jt_rectexistsrightsoundscale * S ((S (jt_i_rectexistsright)) * H) + (jt_c_rectexistsright))))) /\\ (((forall jt_index_rectexistsrightbound. (exists jt_gap_rectexistsrightboundindex. jt_gap_rectexistsrightboundindex+S (jt_index_rectexistsrightbound)=(k)) -> exists jt_value_rectexistsrightbound. ((((exists fs_h_jt_rectexistsrightboundat. fs_h_jt_rectexistsrightboundat + S (jt_value_rectexistsrightbound) = S ((S (jt_index_rectexistsrightbound)) * jt_c_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightboundat. jt_b_rectexistsright = fs_q_jt_rectexistsrightboundat * S ((S (jt_index_rectexistsrightbound)) * jt_c_rectexistsright) + (jt_value_rectexistsrightbound))) /\\ (exists jt_gap_rectexistsrightboundvalue. jt_gap_rectexistsrightboundvalue+S (jt_value_rectexistsrightbound)=(n)))) /\\ (forall jt_divisor_rectexistsrightprimitive. (exists jt_factor_rectexistsrightprimitivemodulus. (n)=(jt_divisor_rectexistsrightprimitive)*jt_factor_rectexistsrightprimitivemodulus) -> (forall jt_index_rectexistsrightprimitivecoordinates jt_value_rectexistsrightprimitivecoordinates. (exists jt_gap_rectexistsrightprimitivecoordinatesindex. jt_gap_rectexistsrightprimitivecoordinatesindex+S (jt_index_rectexistsrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectexistsrightprimitivecoordinatesat. fs_h_jt_rectexistsrightprimitivecoordinatesat + S (jt_value_rectexistsrightprimitivecoordinates) = S ((S (jt_index_rectexistsrightprimitivecoordinates)) * jt_c_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightprimitivecoordinatesat. jt_b_rectexistsright = fs_q_jt_rectexistsrightprimitivecoordinatesat * S ((S (jt_index_rectexistsrightprimitivecoordinates)) * jt_c_rectexistsright) + (jt_value_rectexistsrightprimitivecoordinates))) -> (exists jt_factor_rectexistsrightprimitivecoordinatesdivides. (jt_value_rectexistsrightprimitivecoordinates)=(jt_divisor_rectexistsrightprimitive)*jt_factor_rectexistsrightprimitivecoordinatesdivides)) -> jt_divisor_rectexistsrightprimitive=1))))) /\\ (((forall jt_b_rectexistsright jt_c_rectexistsright. (forall jt_index_rectexistsrightinputbound. (exists jt_gap_rectexistsrightinputboundindex. jt_gap_rectexistsrightinputboundindex+S (jt_index_rectexistsrightinputbound)=(k)) -> exists jt_value_rectexistsrightinputbound. ((((exists fs_h_jt_rectexistsrightinputboundat. fs_h_jt_rectexistsrightinputboundat + S (jt_value_rectexistsrightinputbound) = S ((S (jt_index_rectexistsrightinputbound)) * jt_c_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightinputboundat. jt_b_rectexistsright = fs_q_jt_rectexistsrightinputboundat * S ((S (jt_index_rectexistsrightinputbound)) * jt_c_rectexistsright) + (jt_value_rectexistsrightinputbound))) /\\ (exists jt_gap_rectexistsrightinputboundvalue. jt_gap_rectexistsrightinputboundvalue+S (jt_value_rectexistsrightinputbound)=(n)))) -> (forall jt_divisor_rectexistsrightinputprimitive. (exists jt_factor_rectexistsrightinputprimitivemodulus. (n)=(jt_divisor_rectexistsrightinputprimitive)*jt_factor_rectexistsrightinputprimitivemodulus) -> (forall jt_index_rectexistsrightinputprimitivecoordinates jt_value_rectexistsrightinputprimitivecoordinates. (exists jt_gap_rectexistsrightinputprimitivecoordinatesindex. jt_gap_rectexistsrightinputprimitivecoordinatesindex+S (jt_index_rectexistsrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectexistsrightinputprimitivecoordinatesat. fs_h_jt_rectexistsrightinputprimitivecoordinatesat + S (jt_value_rectexistsrightinputprimitivecoordinates) = S ((S (jt_index_rectexistsrightinputprimitivecoordinates)) * jt_c_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightinputprimitivecoordinatesat. jt_b_rectexistsright = fs_q_jt_rectexistsrightinputprimitivecoordinatesat * S ((S (jt_index_rectexistsrightinputprimitivecoordinates)) * jt_c_rectexistsright) + (jt_value_rectexistsrightinputprimitivecoordinates))) -> (exists jt_factor_rectexistsrightinputprimitivecoordinatesdivides. (jt_value_rectexistsrightinputprimitivecoordinates)=(jt_divisor_rectexistsrightinputprimitive)*jt_factor_rectexistsrightinputprimitivecoordinatesdivides)) -> jt_divisor_rectexistsrightinputprimitive=1) -> exists jt_i_rectexistsright jt_d_rectexistsright jt_e_rectexistsright. ((exists jt_gap_rectexistsrightcompleteindex. jt_gap_rectexistsrightcompleteindex+S (jt_i_rectexistsright)=(v)) /\\ (((((((exists fs_h_jt_rectexistsrightcompletecode. fs_h_jt_rectexistsrightcompletecode + S (jt_d_rectexistsright) = S ((S (jt_i_rectexistsright)) * F)) /\\ exists fs_q_jt_rectexistsrightcompletecode. E = fs_q_jt_rectexistsrightcompletecode * S ((S (jt_i_rectexistsright)) * F) + (jt_d_rectexistsright))) /\\ (((exists fs_h_jt_rectexistsrightcompletescale. fs_h_jt_rectexistsrightcompletescale + S (jt_e_rectexistsright) = S ((S (jt_i_rectexistsright)) * H)) /\\ exists fs_q_jt_rectexistsrightcompletescale. G = fs_q_jt_rectexistsrightcompletescale * S ((S (jt_i_rectexistsright)) * H) + (jt_e_rectexistsright))))) /\\ (forall jt_index_rectexistsrightrepresented jt_left_rectexistsrightrepresented jt_right_rectexistsrightrepresented. (exists jt_gap_rectexistsrightrepresentedindex. jt_gap_rectexistsrightrepresentedindex+S (jt_index_rectexistsrightrepresented)=(k)) -> (((exists fs_h_jt_rectexistsrightrepresentedleft. fs_h_jt_rectexistsrightrepresentedleft + S (jt_left_rectexistsrightrepresented) = S ((S (jt_index_rectexistsrightrepresented)) * jt_c_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightrepresentedleft. jt_b_rectexistsright = fs_q_jt_rectexistsrightrepresentedleft * S ((S (jt_index_rectexistsrightrepresented)) * jt_c_rectexistsright) + (jt_left_rectexistsrightrepresented))) -> (((exists fs_h_jt_rectexistsrightrepresentedright. fs_h_jt_rectexistsrightrepresentedright + S (jt_right_rectexistsrightrepresented) = S ((S (jt_index_rectexistsrightrepresented)) * jt_e_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightrepresentedright. jt_d_rectexistsright = fs_q_jt_rectexistsrightrepresentedright * S ((S (jt_index_rectexistsrightrepresented)) * jt_e_rectexistsright) + (jt_right_rectexistsrightrepresented))) -> jt_left_rectexistsrightrepresented=jt_right_rectexistsrightrepresented))))) /\\ (forall jt_i_rectexistsright jt_h_rectexistsright jt_b_rectexistsright jt_c_rectexistsright jt_d_rectexistsright jt_e_rectexistsright. (exists jt_gap_rectexistsrightfirstindex. jt_gap_rectexistsrightfirstindex+S (jt_i_rectexistsright)=(v)) -> (exists jt_gap_rectexistsrightsecondindex. jt_gap_rectexistsrightsecondindex+S (jt_h_rectexistsright)=(v)) -> (((((exists fs_h_jt_rectexistsrightfirstcode. fs_h_jt_rectexistsrightfirstcode + S (jt_b_rectexistsright) = S ((S (jt_i_rectexistsright)) * F)) /\\ exists fs_q_jt_rectexistsrightfirstcode. E = fs_q_jt_rectexistsrightfirstcode * S ((S (jt_i_rectexistsright)) * F) + (jt_b_rectexistsright))) /\\ (((exists fs_h_jt_rectexistsrightfirstscale. fs_h_jt_rectexistsrightfirstscale + S (jt_c_rectexistsright) = S ((S (jt_i_rectexistsright)) * H)) /\\ exists fs_q_jt_rectexistsrightfirstscale. G = fs_q_jt_rectexistsrightfirstscale * S ((S (jt_i_rectexistsright)) * H) + (jt_c_rectexistsright))))) -> (((((exists fs_h_jt_rectexistsrightsecondcode. fs_h_jt_rectexistsrightsecondcode + S (jt_d_rectexistsright) = S ((S (jt_h_rectexistsright)) * F)) /\\ exists fs_q_jt_rectexistsrightsecondcode. E = fs_q_jt_rectexistsrightsecondcode * S ((S (jt_h_rectexistsright)) * F) + (jt_d_rectexistsright))) /\\ (((exists fs_h_jt_rectexistsrightsecondscale. fs_h_jt_rectexistsrightsecondscale + S (jt_e_rectexistsright) = S ((S (jt_h_rectexistsright)) * H)) /\\ exists fs_q_jt_rectexistsrightsecondscale. G = fs_q_jt_rectexistsrightsecondscale * S ((S (jt_h_rectexistsright)) * H) + (jt_e_rectexistsright))))) -> (forall jt_index_rectexistsrightsame jt_left_rectexistsrightsame jt_right_rectexistsrightsame. (exists jt_gap_rectexistsrightsameindex. jt_gap_rectexistsrightsameindex+S (jt_index_rectexistsrightsame)=(k)) -> (((exists fs_h_jt_rectexistsrightsameleft. fs_h_jt_rectexistsrightsameleft + S (jt_left_rectexistsrightsame) = S ((S (jt_index_rectexistsrightsame)) * jt_c_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightsameleft. jt_b_rectexistsright = fs_q_jt_rectexistsrightsameleft * S ((S (jt_index_rectexistsrightsame)) * jt_c_rectexistsright) + (jt_left_rectexistsrightsame))) -> (((exists fs_h_jt_rectexistsrightsameright. fs_h_jt_rectexistsrightsameright + S (jt_right_rectexistsrightsame) = S ((S (jt_index_rectexistsrightsame)) * jt_e_rectexistsright)) /\\ exists fs_q_jt_rectexistsrightsameright. jt_d_rectexistsright = fs_q_jt_rectexistsrightsameright * S ((S (jt_index_rectexistsrightsame)) * jt_e_rectexistsright) + (jt_right_rectexistsrightsame))) -> jt_left_rectexistsrightsame=jt_right_rectexistsrightsame) -> jt_i_rectexistsright=jt_h_rectexistsright))))) -> forall q. (exists jt_gap_rectexistsbound. jt_gap_rectexistsbound+(q)=(u*v)) -> exists P Q R T. forall jt_index_rectexiststarget. (exists jt_gap_rectexiststargetindex. jt_gap_rectexiststargetindex+S (jt_index_rectexiststarget)=(q)) -> exists jt_row_rectexiststarget jt_column_rectexiststarget jt_b_rectexiststarget jt_c_rectexiststarget jt_d_rectexiststarget jt_e_rectexiststarget jt_f_rectexiststarget jt_g_rectexiststarget. ((exists jt_gap_rectexiststargetrow. jt_gap_rectexiststargetrow+S (jt_row_rectexiststarget)=(u)) /\\ (((exists jt_gap_rectexiststargetcolumn. jt_gap_rectexiststargetcolumn+S (jt_column_rectexiststarget)=(v)) /\\ (((jt_index_rectexiststarget=(v)*jt_row_rectexiststarget+jt_column_rectexiststarget) /\\ (((((((exists fs_h_jt_rectexiststargetleftcode. fs_h_jt_rectexiststargetleftcode + S (jt_b_rectexiststarget) = S ((S (jt_row_rectexiststarget)) * B)) /\\ exists fs_q_jt_rectexiststargetleftcode. A = fs_q_jt_rectexiststargetleftcode * S ((S (jt_row_rectexiststarget)) * B) + (jt_b_rectexiststarget))) /\\ (((exists fs_h_jt_rectexiststargetleftscale. fs_h_jt_rectexiststargetleftscale + S (jt_c_rectexiststarget) = S ((S (jt_row_rectexiststarget)) * D)) /\\ exists fs_q_jt_rectexiststargetleftscale. C = fs_q_jt_rectexiststargetleftscale * S ((S (jt_row_rectexiststarget)) * D) + (jt_c_rectexiststarget))))) /\\ (((((((exists fs_h_jt_rectexiststargetrightcode. fs_h_jt_rectexiststargetrightcode + S (jt_d_rectexiststarget) = S ((S (jt_column_rectexiststarget)) * F)) /\\ exists fs_q_jt_rectexiststargetrightcode. E = fs_q_jt_rectexiststargetrightcode * S ((S (jt_column_rectexiststarget)) * F) + (jt_d_rectexiststarget))) /\\ (((exists fs_h_jt_rectexiststargetrightscale. fs_h_jt_rectexiststargetrightscale + S (jt_e_rectexiststarget) = S ((S (jt_column_rectexiststarget)) * H)) /\\ exists fs_q_jt_rectexiststargetrightscale. G = fs_q_jt_rectexiststargetrightscale * S ((S (jt_column_rectexiststarget)) * H) + (jt_e_rectexiststarget))))) /\\ (((((((exists fs_h_jt_rectexiststargetoutputcode. fs_h_jt_rectexiststargetoutputcode + S (jt_f_rectexiststarget) = S ((S (jt_index_rectexiststarget)) * Q)) /\\ exists fs_q_jt_rectexiststargetoutputcode. P = fs_q_jt_rectexiststargetoutputcode * S ((S (jt_index_rectexiststarget)) * Q) + (jt_f_rectexiststarget))) /\\ (((exists fs_h_jt_rectexiststargetoutputscale. fs_h_jt_rectexiststargetoutputscale + S (jt_g_rectexiststarget) = S ((S (jt_index_rectexiststarget)) * T)) /\\ exists fs_q_jt_rectexiststargetoutputscale. R = fs_q_jt_rectexiststargetoutputscale * S ((S (jt_index_rectexiststarget)) * T) + (jt_g_rectexiststarget))))) /\\ (((((forall jt_index_rectexiststargetcrtbound. (exists jt_gap_rectexiststargetcrtboundindex. jt_gap_rectexiststargetcrtboundindex+S (jt_index_rectexiststargetcrtbound)=(k)) -> exists jt_value_rectexiststargetcrtbound. ((((exists fs_h_jt_rectexiststargetcrtboundat. fs_h_jt_rectexiststargetcrtboundat + S (jt_value_rectexiststargetcrtbound) = S ((S (jt_index_rectexiststargetcrtbound)) * jt_g_rectexiststarget)) /\\ exists fs_q_jt_rectexiststargetcrtboundat. jt_f_rectexiststarget = fs_q_jt_rectexiststargetcrtboundat * S ((S (jt_index_rectexiststargetcrtbound)) * jt_g_rectexiststarget) + (jt_value_rectexiststargetcrtbound))) /\\ (exists jt_gap_rectexiststargetcrtboundvalue. jt_gap_rectexiststargetcrtboundvalue+S (jt_value_rectexiststargetcrtbound)=(m*n)))) /\\ (((forall jt_index_rectexiststargetcrtleft jt_left_rectexiststargetcrtleft jt_right_rectexiststargetcrtleft. (exists jt_gap_rectexiststargetcrtleftindex. jt_gap_rectexiststargetcrtleftindex+S (jt_index_rectexiststargetcrtleft)=(k)) -> (((exists fs_h_jt_rectexiststargetcrtleftleft. fs_h_jt_rectexiststargetcrtleftleft + S (jt_left_rectexiststargetcrtleft) = S ((S (jt_index_rectexiststargetcrtleft)) * jt_g_rectexiststarget)) /\\ exists fs_q_jt_rectexiststargetcrtleftleft. jt_f_rectexiststarget = fs_q_jt_rectexiststargetcrtleftleft * S ((S (jt_index_rectexiststargetcrtleft)) * jt_g_rectexiststarget) + (jt_left_rectexiststargetcrtleft))) -> (((exists fs_h_jt_rectexiststargetcrtleftright. fs_h_jt_rectexiststargetcrtleftright + S (jt_right_rectexiststargetcrtleft) = S ((S (jt_index_rectexiststargetcrtleft)) * jt_c_rectexiststarget)) /\\ exists fs_q_jt_rectexiststargetcrtleftright. jt_b_rectexiststarget = fs_q_jt_rectexiststargetcrtleftright * S ((S (jt_index_rectexiststargetcrtleft)) * jt_c_rectexiststarget) + (jt_right_rectexiststargetcrtleft))) -> (exists jt_left_rectexiststargetcrtleftmod jt_right_rectexiststargetcrtleftmod. (jt_left_rectexiststargetcrtleft)+(m)*jt_left_rectexiststargetcrtleftmod=(jt_right_rectexiststargetcrtleft)+(m)*jt_right_rectexiststargetcrtleftmod)) /\\ (forall jt_index_rectexiststargetcrtright jt_left_rectexiststargetcrtright jt_right_rectexiststargetcrtright. (exists jt_gap_rectexiststargetcrtrightindex. jt_gap_rectexiststargetcrtrightindex+S (jt_index_rectexiststargetcrtright)=(k)) -> (((exists fs_h_jt_rectexiststargetcrtrightleft. fs_h_jt_rectexiststargetcrtrightleft + S (jt_left_rectexiststargetcrtright) = S ((S (jt_index_rectexiststargetcrtright)) * jt_g_rectexiststarget)) /\\ exists fs_q_jt_rectexiststargetcrtrightleft. jt_f_rectexiststarget = fs_q_jt_rectexiststargetcrtrightleft * S ((S (jt_index_rectexiststargetcrtright)) * jt_g_rectexiststarget) + (jt_left_rectexiststargetcrtright))) -> (((exists fs_h_jt_rectexiststargetcrtrightright. fs_h_jt_rectexiststargetcrtrightright + S (jt_right_rectexiststargetcrtright) = S ((S (jt_index_rectexiststargetcrtright)) * jt_e_rectexiststarget)) /\\ exists fs_q_jt_rectexiststargetcrtrightright. jt_d_rectexiststarget = fs_q_jt_rectexiststargetcrtrightright * S ((S (jt_index_rectexiststargetcrtright)) * jt_e_rectexiststarget) + (jt_right_rectexiststargetcrtright))) -> (exists jt_left_rectexiststargetcrtrightmod jt_right_rectexiststargetcrtrightmod. (jt_left_rectexiststargetcrtright)+(n)*jt_left_rectexiststargetcrtrightmod=(jt_right_rectexiststargetcrtright)+(n)*jt_right_rectexiststargetcrtrightmod)))))) /\\ (forall jt_divisor_rectexiststargetprimitive. (exists jt_factor_rectexiststargetprimitivemodulus. (m*n)=(jt_divisor_rectexiststargetprimitive)*jt_factor_rectexiststargetprimitivemodulus) -> (forall jt_index_rectexiststargetprimitivecoordinates jt_value_rectexiststargetprimitivecoordinates. (exists jt_gap_rectexiststargetprimitivecoordinatesindex. jt_gap_rectexiststargetprimitivecoordinatesindex+S (jt_index_rectexiststargetprimitivecoordinates)=(k)) -> (((exists fs_h_jt_rectexiststargetprimitivecoordinatesat. fs_h_jt_rectexiststargetprimitivecoordinatesat + S (jt_value_rectexiststargetprimitivecoordinates) = S ((S (jt_index_rectexiststargetprimitivecoordinates)) * jt_g_rectexiststarget)) /\\ exists fs_q_jt_rectexiststargetprimitivecoordinatesat. jt_f_rectexiststarget = fs_q_jt_rectexiststargetprimitivecoordinatesat * S ((S (jt_index_rectexiststargetprimitivecoordinates)) * jt_g_rectexiststarget) + (jt_value_rectexiststargetprimitivecoordinates))) -> (exists jt_factor_rectexiststargetprimitivecoordinatesdivides. (jt_value_rectexiststargetprimitivecoordinates)=(jt_divisor_rectexiststargetprimitive)*jt_factor_rectexiststargetprimitivecoordinatesdivides)) -> jt_divisor_rectexiststargetprimitive=1))))))))))))))", "statement_sha256": "1f53595b967e27a64fc239e07328651a8b697fa048e9a5075f31613b7e8241c6", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "HA induction constructs the genuine rectangular CRT output table at every prefix through u*v, including zero-width rectangles." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 65, "body_proof_nodes": 326, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro q", "intro p", "intro f", "intro g", "intro hr", "intro hp", "intro he", "have hv : ∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. ∃ f. ∃ g. Lt(i,u) ∧ (Lt(j,v) ∧ (p = v · i + j ∧ (BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) ∧ (BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) ∧ (JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k)))))))", "specialize hr (p)", "apply hr", "exact hp", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "cases he", "have hf : x6=f", "specialize beta_at_unique (P)", "specialize beta_at_unique (Q)", "specialize beta_at_unique (p)", "specialize beta_at_unique (x6)", "specialize beta_at_unique (f)", "apply beta_at_unique", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_left", "exact he_left", "have hg : x7=g", "specialize beta_at_unique (R)", "specialize beta_at_unique (T)", "specialize beta_at_unique (p)", "specialize beta_at_unique (x7)", "specialize beta_at_unique (g)", "apply beta_at_unique", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_right", "exact he_right", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "split", "split", "exact he_left", "exact he_right", "split", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ q. ∀ p. ∀ f. ∀ g. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q) → Lt(p,q) → BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) → ∃ x. ∃ y. ∃ z. ∃ i. ∃ j. ∃ w. Lt(x,u) ∧ (Lt(y,v) ∧ (p = v · x + y ∧ (BetaAt(A,B,x,z) ∧ BetaAt(C,D,x,i) ∧ (BetaAt(E,F,y,j) ∧ BetaAt(G,H,y,w) ∧ (BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) ∧ (JordanCanonicalTupleCRT(m,n,z,i,j,w,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k)))))))", "defined_statement_sha256": "633133f49aec2b3a94f76064b52a37a1420614a0d3751c2730037ba6f9bfa086", "definition_uses": { "ND0372": 2, "ND0380": 2, "ND0381": 1, "PD0002": 5, "PD0013": 14 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "b547d1e9b633e9df584f30bdcff96c4bc96cc4e52a4c33fbdd909b5ef4b3565d", "free_names": [], "script_definition_uses": { "ND0372": 1, "ND0380": 1, "PD0002": 2, "PD0013": 6 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro P" } ], [ { "kind": "text", "text": "intro Q" } ], [ { "kind": "text", "text": "intro R" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. ∃ f. ∃ g. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (p = v · i + j ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" }, { "kind": "text", "text": "))))))" } ], [ { "kind": "text", "text": "specialize hr (p)" } ], [ { "kind": "text", "text": "apply hr" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases he" } ], [ { "kind": "text", "text": "have hf : x6=f" } ], [ { "kind": "text", "text": "specialize beta_at_unique (P)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (Q)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (p)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x6)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (f)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_left" } ], [ { "kind": "text", "text": "exact he_left" } ], [ { "kind": "text", "text": "have hg : x7=g" } ], [ { "kind": "text", "text": "specialize beta_at_unique (R)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (T)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (p)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x7)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (g)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_right" } ], [ { "kind": "text", "text": "exact he_right" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "exists x5" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact he_left" } ], [ { "kind": "text", "text": "exact he_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ] ], "statement_definition_uses": { "ND0372": 1, "ND0380": 1, "ND0381": 1, "PD0002": 3, "PD0013": 8 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ q. ∀ p. ∀ f. ∀ g. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,q)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(p,q)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ i. ∃ j. ∃ w. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,v)" }, { "kind": "text", "text": " ∧ (p = v · x + y ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,i)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,y,j)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,y,w)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,z,i,j,w,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" }, { "kind": "text", "text": "))))))" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0043", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_actual_entry", "original_ha_bundle_verified": true, "proof_bundle_node_id": 328, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro q", "intro p", "intro f", "intro g", "intro hr", "intro hp", "intro he", "have hv : exists i j b c d e f g. ((exists jt_gap_locatedrow. jt_gap_locatedrow+S (i)=(u)) /\\ (((exists jt_gap_locatedcolumn. jt_gap_locatedcolumn+S (j)=(v)) /\\ (((p=(v)*(i)+(j)) /\\ (((((((exists fs_h_jt_locatedleftcode. fs_h_jt_locatedleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_locatedleftcode. A = fs_q_jt_locatedleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_locatedleftscale. fs_h_jt_locatedleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_locatedleftscale. C = fs_q_jt_locatedleftscale * S ((S (i)) * D) + (c))))) /\\ (((((((exists fs_h_jt_locatedrightcode. fs_h_jt_locatedrightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_locatedrightcode. E = fs_q_jt_locatedrightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_locatedrightscale. fs_h_jt_locatedrightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_locatedrightscale. G = fs_q_jt_locatedrightscale * S ((S (j)) * H) + (e))))) /\\ (((((((exists fs_h_jt_locatedoutputcode. fs_h_jt_locatedoutputcode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_locatedoutputcode. P = fs_q_jt_locatedoutputcode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_locatedoutputscale. fs_h_jt_locatedoutputscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_locatedoutputscale. R = fs_q_jt_locatedoutputscale * S ((S (p)) * T) + (g))))) /\\ (((((forall jt_index_locatedcrtbound. (exists jt_gap_locatedcrtboundindex. jt_gap_locatedcrtboundindex+S (jt_index_locatedcrtbound)=(k)) -> exists jt_value_locatedcrtbound. ((((exists fs_h_jt_locatedcrtboundat. fs_h_jt_locatedcrtboundat + S (jt_value_locatedcrtbound) = S ((S (jt_index_locatedcrtbound)) * g)) /\\ exists fs_q_jt_locatedcrtboundat. f = fs_q_jt_locatedcrtboundat * S ((S (jt_index_locatedcrtbound)) * g) + (jt_value_locatedcrtbound))) /\\ (exists jt_gap_locatedcrtboundvalue. jt_gap_locatedcrtboundvalue+S (jt_value_locatedcrtbound)=(m*n)))) /\\ (((forall jt_index_locatedcrtleft jt_left_locatedcrtleft jt_right_locatedcrtleft. (exists jt_gap_locatedcrtleftindex. jt_gap_locatedcrtleftindex+S (jt_index_locatedcrtleft)=(k)) -> (((exists fs_h_jt_locatedcrtleftleft. fs_h_jt_locatedcrtleftleft + S (jt_left_locatedcrtleft) = S ((S (jt_index_locatedcrtleft)) * g)) /\\ exists fs_q_jt_locatedcrtleftleft. f = fs_q_jt_locatedcrtleftleft * S ((S (jt_index_locatedcrtleft)) * g) + (jt_left_locatedcrtleft))) -> (((exists fs_h_jt_locatedcrtleftright. fs_h_jt_locatedcrtleftright + S (jt_right_locatedcrtleft) = S ((S (jt_index_locatedcrtleft)) * c)) /\\ exists fs_q_jt_locatedcrtleftright. b = fs_q_jt_locatedcrtleftright * S ((S (jt_index_locatedcrtleft)) * c) + (jt_right_locatedcrtleft))) -> (exists jt_left_locatedcrtleftmod jt_right_locatedcrtleftmod. (jt_left_locatedcrtleft)+(m)*jt_left_locatedcrtleftmod=(jt_right_locatedcrtleft)+(m)*jt_right_locatedcrtleftmod)) /\\ (forall jt_index_locatedcrtright jt_left_locatedcrtright jt_right_locatedcrtright. (exists jt_gap_locatedcrtrightindex. jt_gap_locatedcrtrightindex+S (jt_index_locatedcrtright)=(k)) -> (((exists fs_h_jt_locatedcrtrightleft. fs_h_jt_locatedcrtrightleft + S (jt_left_locatedcrtright) = S ((S (jt_index_locatedcrtright)) * g)) /\\ exists fs_q_jt_locatedcrtrightleft. f = fs_q_jt_locatedcrtrightleft * S ((S (jt_index_locatedcrtright)) * g) + (jt_left_locatedcrtright))) -> (((exists fs_h_jt_locatedcrtrightright. fs_h_jt_locatedcrtrightright + S (jt_right_locatedcrtright) = S ((S (jt_index_locatedcrtright)) * e)) /\\ exists fs_q_jt_locatedcrtrightright. d = fs_q_jt_locatedcrtrightright * S ((S (jt_index_locatedcrtright)) * e) + (jt_right_locatedcrtright))) -> (exists jt_left_locatedcrtrightmod jt_right_locatedcrtrightmod. (jt_left_locatedcrtright)+(n)*jt_left_locatedcrtrightmod=(jt_right_locatedcrtright)+(n)*jt_right_locatedcrtrightmod)))))) /\\ (forall jt_divisor_locatedprimitive. (exists jt_factor_locatedprimitivemodulus. (m*n)=(jt_divisor_locatedprimitive)*jt_factor_locatedprimitivemodulus) -> (forall jt_index_locatedprimitivecoordinates jt_value_locatedprimitivecoordinates. (exists jt_gap_locatedprimitivecoordinatesindex. jt_gap_locatedprimitivecoordinatesindex+S (jt_index_locatedprimitivecoordinates)=(k)) -> (((exists fs_h_jt_locatedprimitivecoordinatesat. fs_h_jt_locatedprimitivecoordinatesat + S (jt_value_locatedprimitivecoordinates) = S ((S (jt_index_locatedprimitivecoordinates)) * g)) /\\ exists fs_q_jt_locatedprimitivecoordinatesat. f = fs_q_jt_locatedprimitivecoordinatesat * S ((S (jt_index_locatedprimitivecoordinates)) * g) + (jt_value_locatedprimitivecoordinates))) -> (exists jt_factor_locatedprimitivecoordinatesdivides. (jt_value_locatedprimitivecoordinates)=(jt_divisor_locatedprimitive)*jt_factor_locatedprimitivecoordinatesdivides)) -> jt_divisor_locatedprimitive=1))))))))))))))", "specialize hr (p)", "apply hr", "exact hp", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "cases he", "have hf : x6=f", "specialize beta_at_unique (P)", "specialize beta_at_unique (Q)", "specialize beta_at_unique (p)", "specialize beta_at_unique (x6)", "specialize beta_at_unique (f)", "apply beta_at_unique", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_left", "exact he_left", "have hg : x7=g", "specialize beta_at_unique (R)", "specialize beta_at_unique (T)", "specialize beta_at_unique (p)", "specialize beta_at_unique (x7)", "specialize beta_at_unique (g)", "apply beta_at_unique", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left_right", "exact he_right", "exists x", "exists x1", "exists x2", "exists x3", "exists x4", "exists x5", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "split", "split", "exact he_left", "exact he_right", "split", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "rewrite hf at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "rewrite hg at hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" ], "script_sha256": "588359471df924a22bc42debf030019aafccaebd0a477940cf75f453be859781", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "588359471df924a22bc42debf030019aafccaebd0a477940cf75f453be859781", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "b547d1e9b633e9df584f30bdcff96c4bc96cc4e52a4c33fbdd909b5ef4b3565d" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v P Q R T q p f g. (forall jt_index_locrect. (exists jt_gap_locrectindex. jt_gap_locrectindex+S (jt_index_locrect)=(q)) -> exists jt_row_locrect jt_column_locrect jt_b_locrect jt_c_locrect jt_d_locrect jt_e_locrect jt_f_locrect jt_g_locrect. ((exists jt_gap_locrectrow. jt_gap_locrectrow+S (jt_row_locrect)=(u)) /\\ (((exists jt_gap_locrectcolumn. jt_gap_locrectcolumn+S (jt_column_locrect)=(v)) /\\ (((jt_index_locrect=(v)*jt_row_locrect+jt_column_locrect) /\\ (((((((exists fs_h_jt_locrectleftcode. fs_h_jt_locrectleftcode + S (jt_b_locrect) = S ((S (jt_row_locrect)) * B)) /\\ exists fs_q_jt_locrectleftcode. A = fs_q_jt_locrectleftcode * S ((S (jt_row_locrect)) * B) + (jt_b_locrect))) /\\ (((exists fs_h_jt_locrectleftscale. fs_h_jt_locrectleftscale + S (jt_c_locrect) = S ((S (jt_row_locrect)) * D)) /\\ exists fs_q_jt_locrectleftscale. C = fs_q_jt_locrectleftscale * S ((S (jt_row_locrect)) * D) + (jt_c_locrect))))) /\\ (((((((exists fs_h_jt_locrectrightcode. fs_h_jt_locrectrightcode + S (jt_d_locrect) = S ((S (jt_column_locrect)) * F)) /\\ exists fs_q_jt_locrectrightcode. E = fs_q_jt_locrectrightcode * S ((S (jt_column_locrect)) * F) + (jt_d_locrect))) /\\ (((exists fs_h_jt_locrectrightscale. fs_h_jt_locrectrightscale + S (jt_e_locrect) = S ((S (jt_column_locrect)) * H)) /\\ exists fs_q_jt_locrectrightscale. G = fs_q_jt_locrectrightscale * S ((S (jt_column_locrect)) * H) + (jt_e_locrect))))) /\\ (((((((exists fs_h_jt_locrectoutputcode. fs_h_jt_locrectoutputcode + S (jt_f_locrect) = S ((S (jt_index_locrect)) * Q)) /\\ exists fs_q_jt_locrectoutputcode. P = fs_q_jt_locrectoutputcode * S ((S (jt_index_locrect)) * Q) + (jt_f_locrect))) /\\ (((exists fs_h_jt_locrectoutputscale. fs_h_jt_locrectoutputscale + S (jt_g_locrect) = S ((S (jt_index_locrect)) * T)) /\\ exists fs_q_jt_locrectoutputscale. R = fs_q_jt_locrectoutputscale * S ((S (jt_index_locrect)) * T) + (jt_g_locrect))))) /\\ (((((forall jt_index_locrectcrtbound. (exists jt_gap_locrectcrtboundindex. jt_gap_locrectcrtboundindex+S (jt_index_locrectcrtbound)=(k)) -> exists jt_value_locrectcrtbound. ((((exists fs_h_jt_locrectcrtboundat. fs_h_jt_locrectcrtboundat + S (jt_value_locrectcrtbound) = S ((S (jt_index_locrectcrtbound)) * jt_g_locrect)) /\\ exists fs_q_jt_locrectcrtboundat. jt_f_locrect = fs_q_jt_locrectcrtboundat * S ((S (jt_index_locrectcrtbound)) * jt_g_locrect) + (jt_value_locrectcrtbound))) /\\ (exists jt_gap_locrectcrtboundvalue. jt_gap_locrectcrtboundvalue+S (jt_value_locrectcrtbound)=(m*n)))) /\\ (((forall jt_index_locrectcrtleft jt_left_locrectcrtleft jt_right_locrectcrtleft. (exists jt_gap_locrectcrtleftindex. jt_gap_locrectcrtleftindex+S (jt_index_locrectcrtleft)=(k)) -> (((exists fs_h_jt_locrectcrtleftleft. fs_h_jt_locrectcrtleftleft + S (jt_left_locrectcrtleft) = S ((S (jt_index_locrectcrtleft)) * jt_g_locrect)) /\\ exists fs_q_jt_locrectcrtleftleft. jt_f_locrect = fs_q_jt_locrectcrtleftleft * S ((S (jt_index_locrectcrtleft)) * jt_g_locrect) + (jt_left_locrectcrtleft))) -> (((exists fs_h_jt_locrectcrtleftright. fs_h_jt_locrectcrtleftright + S (jt_right_locrectcrtleft) = S ((S (jt_index_locrectcrtleft)) * jt_c_locrect)) /\\ exists fs_q_jt_locrectcrtleftright. jt_b_locrect = fs_q_jt_locrectcrtleftright * S ((S (jt_index_locrectcrtleft)) * jt_c_locrect) + (jt_right_locrectcrtleft))) -> (exists jt_left_locrectcrtleftmod jt_right_locrectcrtleftmod. (jt_left_locrectcrtleft)+(m)*jt_left_locrectcrtleftmod=(jt_right_locrectcrtleft)+(m)*jt_right_locrectcrtleftmod)) /\\ (forall jt_index_locrectcrtright jt_left_locrectcrtright jt_right_locrectcrtright. (exists jt_gap_locrectcrtrightindex. jt_gap_locrectcrtrightindex+S (jt_index_locrectcrtright)=(k)) -> (((exists fs_h_jt_locrectcrtrightleft. fs_h_jt_locrectcrtrightleft + S (jt_left_locrectcrtright) = S ((S (jt_index_locrectcrtright)) * jt_g_locrect)) /\\ exists fs_q_jt_locrectcrtrightleft. jt_f_locrect = fs_q_jt_locrectcrtrightleft * S ((S (jt_index_locrectcrtright)) * jt_g_locrect) + (jt_left_locrectcrtright))) -> (((exists fs_h_jt_locrectcrtrightright. fs_h_jt_locrectcrtrightright + S (jt_right_locrectcrtright) = S ((S (jt_index_locrectcrtright)) * jt_e_locrect)) /\\ exists fs_q_jt_locrectcrtrightright. jt_d_locrect = fs_q_jt_locrectcrtrightright * S ((S (jt_index_locrectcrtright)) * jt_e_locrect) + (jt_right_locrectcrtright))) -> (exists jt_left_locrectcrtrightmod jt_right_locrectcrtrightmod. (jt_left_locrectcrtright)+(n)*jt_left_locrectcrtrightmod=(jt_right_locrectcrtright)+(n)*jt_right_locrectcrtrightmod)))))) /\\ (forall jt_divisor_locrectprimitive. (exists jt_factor_locrectprimitivemodulus. (m*n)=(jt_divisor_locrectprimitive)*jt_factor_locrectprimitivemodulus) -> (forall jt_index_locrectprimitivecoordinates jt_value_locrectprimitivecoordinates. (exists jt_gap_locrectprimitivecoordinatesindex. jt_gap_locrectprimitivecoordinatesindex+S (jt_index_locrectprimitivecoordinates)=(k)) -> (((exists fs_h_jt_locrectprimitivecoordinatesat. fs_h_jt_locrectprimitivecoordinatesat + S (jt_value_locrectprimitivecoordinates) = S ((S (jt_index_locrectprimitivecoordinates)) * jt_g_locrect)) /\\ exists fs_q_jt_locrectprimitivecoordinatesat. jt_f_locrect = fs_q_jt_locrectprimitivecoordinatesat * S ((S (jt_index_locrectprimitivecoordinates)) * jt_g_locrect) + (jt_value_locrectprimitivecoordinates))) -> (exists jt_factor_locrectprimitivecoordinatesdivides. (jt_value_locrectprimitivecoordinates)=(jt_divisor_locrectprimitive)*jt_factor_locrectprimitivecoordinatesdivides)) -> jt_divisor_locrectprimitive=1))))))))))))))) -> (exists jt_gap_locbound. jt_gap_locbound+S (p)=(q)) -> (((((exists fs_h_jt_locentrycode. fs_h_jt_locentrycode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_locentrycode. P = fs_q_jt_locentrycode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_locentryscale. fs_h_jt_locentryscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_locentryscale. R = fs_q_jt_locentryscale * S ((S (p)) * T) + (g))))) -> exists i j b c d e. ((exists jt_gap_locresultrow. jt_gap_locresultrow+S (i)=(u)) /\\ (((exists jt_gap_locresultcolumn. jt_gap_locresultcolumn+S (j)=(v)) /\\ (((p=(v)*(i)+(j)) /\\ (((((((exists fs_h_jt_locresultleftcode. fs_h_jt_locresultleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_locresultleftcode. A = fs_q_jt_locresultleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_locresultleftscale. fs_h_jt_locresultleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_locresultleftscale. C = fs_q_jt_locresultleftscale * S ((S (i)) * D) + (c))))) /\\ (((((((exists fs_h_jt_locresultrightcode. fs_h_jt_locresultrightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_locresultrightcode. E = fs_q_jt_locresultrightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_locresultrightscale. fs_h_jt_locresultrightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_locresultrightscale. G = fs_q_jt_locresultrightscale * S ((S (j)) * H) + (e))))) /\\ (((((((exists fs_h_jt_locresultoutputcode. fs_h_jt_locresultoutputcode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_locresultoutputcode. P = fs_q_jt_locresultoutputcode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_locresultoutputscale. fs_h_jt_locresultoutputscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_locresultoutputscale. R = fs_q_jt_locresultoutputscale * S ((S (p)) * T) + (g))))) /\\ (((((forall jt_index_locresultcrtbound. (exists jt_gap_locresultcrtboundindex. jt_gap_locresultcrtboundindex+S (jt_index_locresultcrtbound)=(k)) -> exists jt_value_locresultcrtbound. ((((exists fs_h_jt_locresultcrtboundat. fs_h_jt_locresultcrtboundat + S (jt_value_locresultcrtbound) = S ((S (jt_index_locresultcrtbound)) * g)) /\\ exists fs_q_jt_locresultcrtboundat. f = fs_q_jt_locresultcrtboundat * S ((S (jt_index_locresultcrtbound)) * g) + (jt_value_locresultcrtbound))) /\\ (exists jt_gap_locresultcrtboundvalue. jt_gap_locresultcrtboundvalue+S (jt_value_locresultcrtbound)=(m*n)))) /\\ (((forall jt_index_locresultcrtleft jt_left_locresultcrtleft jt_right_locresultcrtleft. (exists jt_gap_locresultcrtleftindex. jt_gap_locresultcrtleftindex+S (jt_index_locresultcrtleft)=(k)) -> (((exists fs_h_jt_locresultcrtleftleft. fs_h_jt_locresultcrtleftleft + S (jt_left_locresultcrtleft) = S ((S (jt_index_locresultcrtleft)) * g)) /\\ exists fs_q_jt_locresultcrtleftleft. f = fs_q_jt_locresultcrtleftleft * S ((S (jt_index_locresultcrtleft)) * g) + (jt_left_locresultcrtleft))) -> (((exists fs_h_jt_locresultcrtleftright. fs_h_jt_locresultcrtleftright + S (jt_right_locresultcrtleft) = S ((S (jt_index_locresultcrtleft)) * c)) /\\ exists fs_q_jt_locresultcrtleftright. b = fs_q_jt_locresultcrtleftright * S ((S (jt_index_locresultcrtleft)) * c) + (jt_right_locresultcrtleft))) -> (exists jt_left_locresultcrtleftmod jt_right_locresultcrtleftmod. (jt_left_locresultcrtleft)+(m)*jt_left_locresultcrtleftmod=(jt_right_locresultcrtleft)+(m)*jt_right_locresultcrtleftmod)) /\\ (forall jt_index_locresultcrtright jt_left_locresultcrtright jt_right_locresultcrtright. (exists jt_gap_locresultcrtrightindex. jt_gap_locresultcrtrightindex+S (jt_index_locresultcrtright)=(k)) -> (((exists fs_h_jt_locresultcrtrightleft. fs_h_jt_locresultcrtrightleft + S (jt_left_locresultcrtright) = S ((S (jt_index_locresultcrtright)) * g)) /\\ exists fs_q_jt_locresultcrtrightleft. f = fs_q_jt_locresultcrtrightleft * S ((S (jt_index_locresultcrtright)) * g) + (jt_left_locresultcrtright))) -> (((exists fs_h_jt_locresultcrtrightright. fs_h_jt_locresultcrtrightright + S (jt_right_locresultcrtright) = S ((S (jt_index_locresultcrtright)) * e)) /\\ exists fs_q_jt_locresultcrtrightright. d = fs_q_jt_locresultcrtrightright * S ((S (jt_index_locresultcrtright)) * e) + (jt_right_locresultcrtright))) -> (exists jt_left_locresultcrtrightmod jt_right_locresultcrtrightmod. (jt_left_locresultcrtright)+(n)*jt_left_locresultcrtrightmod=(jt_right_locresultcrtright)+(n)*jt_right_locresultcrtrightmod)))))) /\\ (forall jt_divisor_locresultprimitive. (exists jt_factor_locresultprimitivemodulus. (m*n)=(jt_divisor_locresultprimitive)*jt_factor_locresultprimitivemodulus) -> (forall jt_index_locresultprimitivecoordinates jt_value_locresultprimitivecoordinates. (exists jt_gap_locresultprimitivecoordinatesindex. jt_gap_locresultprimitivecoordinatesindex+S (jt_index_locresultprimitivecoordinates)=(k)) -> (((exists fs_h_jt_locresultprimitivecoordinatesat. fs_h_jt_locresultprimitivecoordinatesat + S (jt_value_locresultprimitivecoordinates) = S ((S (jt_index_locresultprimitivecoordinates)) * g)) /\\ exists fs_q_jt_locresultprimitivecoordinatesat. f = fs_q_jt_locresultprimitivecoordinatesat * S ((S (jt_index_locresultprimitivecoordinates)) * g) + (jt_value_locresultprimitivecoordinates))) -> (exists jt_factor_locresultprimitivecoordinatesdivides. (jt_value_locresultprimitivecoordinates)=(jt_divisor_locresultprimitive)*jt_factor_locresultprimitivecoordinatesdivides)) -> jt_divisor_locresultprimitive=1))))))))))))))", "statement_sha256": "b547d1e9b633e9df584f30bdcff96c4bc96cc4e52a4c33fbdd909b5ef4b3565d", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Decode an arbitrary actual output entry without identifying distinct beta representations." }, { "admission_dependencies": [ "jordan_rectangle_flat_bound", "jordan_rectangle_pair_unique", "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 64, "body_proof_nodes": 424, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro i", "intro j", "intro b", "intro c", "intro d", "intro e", "intro hr", "intro hi", "intro hj", "intro hl", "intro hh", "have hp : Lt(v · i + j,u · v)", "specialize jordan_rectangle_flat_bound (u)", "specialize jordan_rectangle_flat_bound (v)", "specialize jordan_rectangle_flat_bound (i)", "specialize jordan_rectangle_flat_bound (j)", "apply jordan_rectangle_flat_bound", "exact hi", "exact hj", "have hv : ∃ ri. ∃ rj. ∃ rb. ∃ rc. ∃ rd. ∃ re. ∃ rf. ∃ rg. Lt(ri,u) ∧ (Lt(rj,v) ∧ (v · i + j = v · ri + rj ∧ (BetaAt(A,B,ri,rb) ∧ BetaAt(C,D,ri,rc) ∧ (BetaAt(E,F,rj,rd) ∧ BetaAt(G,H,rj,re) ∧ (MatrixAt(P,Q,i,v,j,rf) ∧ MatrixAt(R,T,i,v,j,rg) ∧ (JordanCanonicalTupleCRT(m,n,rb,rc,rd,re,rf,rg,k) ∧ JordanPrimitiveTuple(m · n,rf,rg,k)))))))", "specialize hr (v*i+j)", "apply hr", "exact hp", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "have hij : i=x /\\ j=x1", "specialize jordan_rectangle_pair_unique (v)", "specialize jordan_rectangle_pair_unique (i)", "specialize jordan_rectangle_pair_unique (j)", "specialize jordan_rectangle_pair_unique (x)", "specialize jordan_rectangle_pair_unique (x1)", "apply jordan_rectangle_pair_unique", "exact hj", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "cases hij", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "cases hl", "cases hh", "have hb : b=x2", "specialize beta_at_unique (A)", "specialize beta_at_unique (B)", "specialize beta_at_unique (x)", "specialize beta_at_unique (b)", "specialize beta_at_unique (x2)", "apply beta_at_unique", "rewrite hij_left at hl_left", "rewrite hij_left at hl_left", "exact hl_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left_left", "have hc : c=x3", "specialize beta_at_unique (C)", "specialize beta_at_unique (D)", "specialize beta_at_unique (x)", "specialize beta_at_unique (c)", "specialize beta_at_unique (x3)", "apply beta_at_unique", "rewrite hij_left at hl_right", "rewrite hij_left at hl_right", "exact hl_right", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left_right", "have hd : d=x4", "specialize beta_at_unique (E)", "specialize beta_at_unique (F)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (d)", "specialize beta_at_unique (x4)", "apply beta_at_unique", "rewrite hij_right at hh_left", "rewrite hij_right at hh_left", "exact hh_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left_left", "have he : e=x5", "specialize beta_at_unique (G)", "specialize beta_at_unique (H)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (e)", "specialize beta_at_unique (x5)", "apply beta_at_unique", "rewrite hij_right at hh_right", "rewrite hij_right at hh_right", "exact hh_right", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left_right", "exists x6", "exists x7", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "split", "rewrite hb", "rewrite hc", "rewrite hc", "rewrite hd", "rewrite he", "rewrite he", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ i. ∀ j. ∀ b. ∀ c. ∀ d. ∀ e. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v) → Lt(i,u) → Lt(j,v) → BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) → BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) → ∃ x. ∃ y. MatrixAt(P,Q,i,v,j,x) ∧ MatrixAt(R,T,i,v,j,y) ∧ (JordanCanonicalTupleCRT(m,n,b,c,d,e,x,y,k) ∧ JordanPrimitiveTuple(m · n,x,y,k))", "defined_statement_sha256": "aaf141a7a44c121c4a6559205e2ba22c1256cc277746fd95854aac9b9996b15d", "definition_uses": { "ND0003": 4, "ND0372": 2, "ND0380": 2, "ND0381": 1, "PD0002": 5, "PD0013": 8 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "45e148fcc8b2ba0052a57a1589c89f05db6503059cb467686840757102a15696", "free_names": [], "script_definition_uses": { "ND0003": 2, "ND0372": 1, "ND0380": 1, "PD0002": 3, "PD0013": 4 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro P" } ], [ { "kind": "text", "text": "intro Q" } ], [ { "kind": "text", "text": "intro R" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "have hp : " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(v · i + j,u · v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (i)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (j)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_flat_bound" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ ri. ∃ rj. ∃ rb. ∃ rc. ∃ rd. ∃ re. ∃ rf. ∃ rg. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(ri,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(rj,v)" }, { "kind": "text", "text": " ∧ (v · i + j = v · ri + rj ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,ri,rb)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,ri,rc)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,rj,rd)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,rj,re)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0003", "kind": "definition", "text": "MatrixAt(P,Q,i,v,j,rf)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0003", "kind": "definition", "text": "MatrixAt(R,T,i,v,j,rg)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,rb,rc,rd,re,rf,rg,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,rf,rg,k)" }, { "kind": "text", "text": "))))))" } ], [ { "kind": "text", "text": "specialize hr (v*i+j)" } ], [ { "kind": "text", "text": "apply hr" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "have hij : i=x /\\ j=x1" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_pair_unique (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_pair_unique (i)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_pair_unique (j)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_pair_unique (x)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_pair_unique (x1)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_pair_unique" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "cases hij" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hh" } ], [ { "kind": "text", "text": "have hb : b=x2" } ], [ { "kind": "text", "text": "specialize beta_at_unique (A)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (B)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x2)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hij_left at hl_left" } ], [ { "kind": "text", "text": "rewrite hij_left at hl_left" } ], [ { "kind": "text", "text": "exact hl_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left_left" } ], [ { "kind": "text", "text": "have hc : c=x3" } ], [ { "kind": "text", "text": "specialize beta_at_unique (C)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (D)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x3)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hij_left at hl_right" } ], [ { "kind": "text", "text": "rewrite hij_left at hl_right" } ], [ { "kind": "text", "text": "exact hl_right" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left_right" } ], [ { "kind": "text", "text": "have hd : d=x4" } ], [ { "kind": "text", "text": "specialize beta_at_unique (E)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (F)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (d)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x4)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hij_right at hh_left" } ], [ { "kind": "text", "text": "rewrite hij_right at hh_left" } ], [ { "kind": "text", "text": "exact hh_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left_left" } ], [ { "kind": "text", "text": "have he : e=x5" } ], [ { "kind": "text", "text": "specialize beta_at_unique (G)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (H)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x1)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (e)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x5)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "rewrite hij_right at hh_right" } ], [ { "kind": "text", "text": "rewrite hij_right at hh_right" } ], [ { "kind": "text", "text": "exact hh_right" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left_right" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "exists x7" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hb" } ], [ { "kind": "text", "text": "rewrite hc" } ], [ { "kind": "text", "text": "rewrite hc" } ], [ { "kind": "text", "text": "rewrite hd" } ], [ { "kind": "text", "text": "rewrite he" } ], [ { "kind": "text", "text": "rewrite he" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ] ], "statement_definition_uses": { "ND0003": 2, "ND0372": 1, "ND0380": 1, "ND0381": 1, "PD0002": 2, "PD0013": 4 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ i. ∀ j. ∀ b. ∀ c. ∀ d. ∀ e. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " → ∃ x. ∃ y. " }, { "definition": "ND0003", "kind": "definition", "text": "MatrixAt(P,Q,i,v,j,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0003", "kind": "definition", "text": "MatrixAt(R,T,i,v,j,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,x,y,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,x,y,k)" }, { "kind": "text", "text": ")" } ] }, "dependencies": [ "jordan_rectangle_flat_bound", "jordan_rectangle_pair_unique", "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0044", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_pair_value", "original_ha_bundle_verified": true, "proof_bundle_node_id": 329, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro i", "intro j", "intro b", "intro c", "intro d", "intro e", "intro hr", "intro hi", "intro hj", "intro hl", "intro hh", "have hp : exists jt_gap_pairbound. jt_gap_pairbound+S (v*i+j)=(u*v)", "specialize jordan_rectangle_flat_bound (u)", "specialize jordan_rectangle_flat_bound (v)", "specialize jordan_rectangle_flat_bound (i)", "specialize jordan_rectangle_flat_bound (j)", "apply jordan_rectangle_flat_bound", "exact hi", "exact hj", "have hv : exists ri rj rb rc rd re rf rg. ((exists jt_gap_pairvaluerow. jt_gap_pairvaluerow+S (ri)=(u)) /\\ (((exists jt_gap_pairvaluecolumn. jt_gap_pairvaluecolumn+S (rj)=(v)) /\\ (((v*i+j=(v)*(ri)+(rj)) /\\ (((((((exists fs_h_jt_pairvalueleftcode. fs_h_jt_pairvalueleftcode + S (rb) = S ((S (ri)) * B)) /\\ exists fs_q_jt_pairvalueleftcode. A = fs_q_jt_pairvalueleftcode * S ((S (ri)) * B) + (rb))) /\\ (((exists fs_h_jt_pairvalueleftscale. fs_h_jt_pairvalueleftscale + S (rc) = S ((S (ri)) * D)) /\\ exists fs_q_jt_pairvalueleftscale. C = fs_q_jt_pairvalueleftscale * S ((S (ri)) * D) + (rc))))) /\\ (((((((exists fs_h_jt_pairvaluerightcode. fs_h_jt_pairvaluerightcode + S (rd) = S ((S (rj)) * F)) /\\ exists fs_q_jt_pairvaluerightcode. E = fs_q_jt_pairvaluerightcode * S ((S (rj)) * F) + (rd))) /\\ (((exists fs_h_jt_pairvaluerightscale. fs_h_jt_pairvaluerightscale + S (re) = S ((S (rj)) * H)) /\\ exists fs_q_jt_pairvaluerightscale. G = fs_q_jt_pairvaluerightscale * S ((S (rj)) * H) + (re))))) /\\ (((((((exists fs_h_jt_pairvalueoutputcode. fs_h_jt_pairvalueoutputcode + S (rf) = S ((S (v*i+j)) * Q)) /\\ exists fs_q_jt_pairvalueoutputcode. P = fs_q_jt_pairvalueoutputcode * S ((S (v*i+j)) * Q) + (rf))) /\\ (((exists fs_h_jt_pairvalueoutputscale. fs_h_jt_pairvalueoutputscale + S (rg) = S ((S (v*i+j)) * T)) /\\ exists fs_q_jt_pairvalueoutputscale. R = fs_q_jt_pairvalueoutputscale * S ((S (v*i+j)) * T) + (rg))))) /\\ (((((forall jt_index_pairvaluecrtbound. (exists jt_gap_pairvaluecrtboundindex. jt_gap_pairvaluecrtboundindex+S (jt_index_pairvaluecrtbound)=(k)) -> exists jt_value_pairvaluecrtbound. ((((exists fs_h_jt_pairvaluecrtboundat. fs_h_jt_pairvaluecrtboundat + S (jt_value_pairvaluecrtbound) = S ((S (jt_index_pairvaluecrtbound)) * rg)) /\\ exists fs_q_jt_pairvaluecrtboundat. rf = fs_q_jt_pairvaluecrtboundat * S ((S (jt_index_pairvaluecrtbound)) * rg) + (jt_value_pairvaluecrtbound))) /\\ (exists jt_gap_pairvaluecrtboundvalue. jt_gap_pairvaluecrtboundvalue+S (jt_value_pairvaluecrtbound)=(m*n)))) /\\ (((forall jt_index_pairvaluecrtleft jt_left_pairvaluecrtleft jt_right_pairvaluecrtleft. (exists jt_gap_pairvaluecrtleftindex. jt_gap_pairvaluecrtleftindex+S (jt_index_pairvaluecrtleft)=(k)) -> (((exists fs_h_jt_pairvaluecrtleftleft. fs_h_jt_pairvaluecrtleftleft + S (jt_left_pairvaluecrtleft) = S ((S (jt_index_pairvaluecrtleft)) * rg)) /\\ exists fs_q_jt_pairvaluecrtleftleft. rf = fs_q_jt_pairvaluecrtleftleft * S ((S (jt_index_pairvaluecrtleft)) * rg) + (jt_left_pairvaluecrtleft))) -> (((exists fs_h_jt_pairvaluecrtleftright. fs_h_jt_pairvaluecrtleftright + S (jt_right_pairvaluecrtleft) = S ((S (jt_index_pairvaluecrtleft)) * rc)) /\\ exists fs_q_jt_pairvaluecrtleftright. rb = fs_q_jt_pairvaluecrtleftright * S ((S (jt_index_pairvaluecrtleft)) * rc) + (jt_right_pairvaluecrtleft))) -> (exists jt_left_pairvaluecrtleftmod jt_right_pairvaluecrtleftmod. (jt_left_pairvaluecrtleft)+(m)*jt_left_pairvaluecrtleftmod=(jt_right_pairvaluecrtleft)+(m)*jt_right_pairvaluecrtleftmod)) /\\ (forall jt_index_pairvaluecrtright jt_left_pairvaluecrtright jt_right_pairvaluecrtright. (exists jt_gap_pairvaluecrtrightindex. jt_gap_pairvaluecrtrightindex+S (jt_index_pairvaluecrtright)=(k)) -> (((exists fs_h_jt_pairvaluecrtrightleft. fs_h_jt_pairvaluecrtrightleft + S (jt_left_pairvaluecrtright) = S ((S (jt_index_pairvaluecrtright)) * rg)) /\\ exists fs_q_jt_pairvaluecrtrightleft. rf = fs_q_jt_pairvaluecrtrightleft * S ((S (jt_index_pairvaluecrtright)) * rg) + (jt_left_pairvaluecrtright))) -> (((exists fs_h_jt_pairvaluecrtrightright. fs_h_jt_pairvaluecrtrightright + S (jt_right_pairvaluecrtright) = S ((S (jt_index_pairvaluecrtright)) * re)) /\\ exists fs_q_jt_pairvaluecrtrightright. rd = fs_q_jt_pairvaluecrtrightright * S ((S (jt_index_pairvaluecrtright)) * re) + (jt_right_pairvaluecrtright))) -> (exists jt_left_pairvaluecrtrightmod jt_right_pairvaluecrtrightmod. (jt_left_pairvaluecrtright)+(n)*jt_left_pairvaluecrtrightmod=(jt_right_pairvaluecrtright)+(n)*jt_right_pairvaluecrtrightmod)))))) /\\ (forall jt_divisor_pairvalueprimitive. (exists jt_factor_pairvalueprimitivemodulus. (m*n)=(jt_divisor_pairvalueprimitive)*jt_factor_pairvalueprimitivemodulus) -> (forall jt_index_pairvalueprimitivecoordinates jt_value_pairvalueprimitivecoordinates. (exists jt_gap_pairvalueprimitivecoordinatesindex. jt_gap_pairvalueprimitivecoordinatesindex+S (jt_index_pairvalueprimitivecoordinates)=(k)) -> (((exists fs_h_jt_pairvalueprimitivecoordinatesat. fs_h_jt_pairvalueprimitivecoordinatesat + S (jt_value_pairvalueprimitivecoordinates) = S ((S (jt_index_pairvalueprimitivecoordinates)) * rg)) /\\ exists fs_q_jt_pairvalueprimitivecoordinatesat. rf = fs_q_jt_pairvalueprimitivecoordinatesat * S ((S (jt_index_pairvalueprimitivecoordinates)) * rg) + (jt_value_pairvalueprimitivecoordinates))) -> (exists jt_factor_pairvalueprimitivecoordinatesdivides. (jt_value_pairvalueprimitivecoordinates)=(jt_divisor_pairvalueprimitive)*jt_factor_pairvalueprimitivecoordinatesdivides)) -> jt_divisor_pairvalueprimitive=1))))))))))))))", "specialize hr (v*i+j)", "apply hr", "exact hp", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "have hij : i=x /\\ j=x1", "specialize jordan_rectangle_pair_unique (v)", "specialize jordan_rectangle_pair_unique (i)", "specialize jordan_rectangle_pair_unique (j)", "specialize jordan_rectangle_pair_unique (x)", "specialize jordan_rectangle_pair_unique (x1)", "apply jordan_rectangle_pair_unique", "exact hj", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_left", "cases hij", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "cases hl", "cases hh", "have hb : b=x2", "specialize beta_at_unique (A)", "specialize beta_at_unique (B)", "specialize beta_at_unique (x)", "specialize beta_at_unique (b)", "specialize beta_at_unique (x2)", "apply beta_at_unique", "rewrite hij_left at hl_left", "rewrite hij_left at hl_left", "exact hl_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left_left", "have hc : c=x3", "specialize beta_at_unique (C)", "specialize beta_at_unique (D)", "specialize beta_at_unique (x)", "specialize beta_at_unique (c)", "specialize beta_at_unique (x3)", "apply beta_at_unique", "rewrite hij_left at hl_right", "rewrite hij_left at hl_right", "exact hl_right", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_left_right", "have hd : d=x4", "specialize beta_at_unique (E)", "specialize beta_at_unique (F)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (d)", "specialize beta_at_unique (x4)", "apply beta_at_unique", "rewrite hij_right at hh_left", "rewrite hij_right at hh_left", "exact hh_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left_left", "have he : e=x5", "specialize beta_at_unique (G)", "specialize beta_at_unique (H)", "specialize beta_at_unique (x1)", "specialize beta_at_unique (e)", "specialize beta_at_unique (x5)", "apply beta_at_unique", "rewrite hij_right at hh_right", "rewrite hij_right at hh_right", "exact hh_right", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_left_right", "exists x6", "exists x7", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "split", "rewrite hb", "rewrite hc", "rewrite hc", "rewrite hd", "rewrite he", "rewrite he", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" ], "script_sha256": "ffe7c39bccee9596bfd8a32ec5ad03802fadbea5bac300d8f6e5be6aa5f2685d", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "ffe7c39bccee9596bfd8a32ec5ad03802fadbea5bac300d8f6e5be6aa5f2685d", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "45e148fcc8b2ba0052a57a1589c89f05db6503059cb467686840757102a15696" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v P Q R T i j b c d e. (forall jt_index_pairrect. (exists jt_gap_pairrectindex. jt_gap_pairrectindex+S (jt_index_pairrect)=(u*v)) -> exists jt_row_pairrect jt_column_pairrect jt_b_pairrect jt_c_pairrect jt_d_pairrect jt_e_pairrect jt_f_pairrect jt_g_pairrect. ((exists jt_gap_pairrectrow. jt_gap_pairrectrow+S (jt_row_pairrect)=(u)) /\\ (((exists jt_gap_pairrectcolumn. jt_gap_pairrectcolumn+S (jt_column_pairrect)=(v)) /\\ (((jt_index_pairrect=(v)*jt_row_pairrect+jt_column_pairrect) /\\ (((((((exists fs_h_jt_pairrectleftcode. fs_h_jt_pairrectleftcode + S (jt_b_pairrect) = S ((S (jt_row_pairrect)) * B)) /\\ exists fs_q_jt_pairrectleftcode. A = fs_q_jt_pairrectleftcode * S ((S (jt_row_pairrect)) * B) + (jt_b_pairrect))) /\\ (((exists fs_h_jt_pairrectleftscale. fs_h_jt_pairrectleftscale + S (jt_c_pairrect) = S ((S (jt_row_pairrect)) * D)) /\\ exists fs_q_jt_pairrectleftscale. C = fs_q_jt_pairrectleftscale * S ((S (jt_row_pairrect)) * D) + (jt_c_pairrect))))) /\\ (((((((exists fs_h_jt_pairrectrightcode. fs_h_jt_pairrectrightcode + S (jt_d_pairrect) = S ((S (jt_column_pairrect)) * F)) /\\ exists fs_q_jt_pairrectrightcode. E = fs_q_jt_pairrectrightcode * S ((S (jt_column_pairrect)) * F) + (jt_d_pairrect))) /\\ (((exists fs_h_jt_pairrectrightscale. fs_h_jt_pairrectrightscale + S (jt_e_pairrect) = S ((S (jt_column_pairrect)) * H)) /\\ exists fs_q_jt_pairrectrightscale. G = fs_q_jt_pairrectrightscale * S ((S (jt_column_pairrect)) * H) + (jt_e_pairrect))))) /\\ (((((((exists fs_h_jt_pairrectoutputcode. fs_h_jt_pairrectoutputcode + S (jt_f_pairrect) = S ((S (jt_index_pairrect)) * Q)) /\\ exists fs_q_jt_pairrectoutputcode. P = fs_q_jt_pairrectoutputcode * S ((S (jt_index_pairrect)) * Q) + (jt_f_pairrect))) /\\ (((exists fs_h_jt_pairrectoutputscale. fs_h_jt_pairrectoutputscale + S (jt_g_pairrect) = S ((S (jt_index_pairrect)) * T)) /\\ exists fs_q_jt_pairrectoutputscale. R = fs_q_jt_pairrectoutputscale * S ((S (jt_index_pairrect)) * T) + (jt_g_pairrect))))) /\\ (((((forall jt_index_pairrectcrtbound. (exists jt_gap_pairrectcrtboundindex. jt_gap_pairrectcrtboundindex+S (jt_index_pairrectcrtbound)=(k)) -> exists jt_value_pairrectcrtbound. ((((exists fs_h_jt_pairrectcrtboundat. fs_h_jt_pairrectcrtboundat + S (jt_value_pairrectcrtbound) = S ((S (jt_index_pairrectcrtbound)) * jt_g_pairrect)) /\\ exists fs_q_jt_pairrectcrtboundat. jt_f_pairrect = fs_q_jt_pairrectcrtboundat * S ((S (jt_index_pairrectcrtbound)) * jt_g_pairrect) + (jt_value_pairrectcrtbound))) /\\ (exists jt_gap_pairrectcrtboundvalue. jt_gap_pairrectcrtboundvalue+S (jt_value_pairrectcrtbound)=(m*n)))) /\\ (((forall jt_index_pairrectcrtleft jt_left_pairrectcrtleft jt_right_pairrectcrtleft. (exists jt_gap_pairrectcrtleftindex. jt_gap_pairrectcrtleftindex+S (jt_index_pairrectcrtleft)=(k)) -> (((exists fs_h_jt_pairrectcrtleftleft. fs_h_jt_pairrectcrtleftleft + S (jt_left_pairrectcrtleft) = S ((S (jt_index_pairrectcrtleft)) * jt_g_pairrect)) /\\ exists fs_q_jt_pairrectcrtleftleft. jt_f_pairrect = fs_q_jt_pairrectcrtleftleft * S ((S (jt_index_pairrectcrtleft)) * jt_g_pairrect) + (jt_left_pairrectcrtleft))) -> (((exists fs_h_jt_pairrectcrtleftright. fs_h_jt_pairrectcrtleftright + S (jt_right_pairrectcrtleft) = S ((S (jt_index_pairrectcrtleft)) * jt_c_pairrect)) /\\ exists fs_q_jt_pairrectcrtleftright. jt_b_pairrect = fs_q_jt_pairrectcrtleftright * S ((S (jt_index_pairrectcrtleft)) * jt_c_pairrect) + (jt_right_pairrectcrtleft))) -> (exists jt_left_pairrectcrtleftmod jt_right_pairrectcrtleftmod. (jt_left_pairrectcrtleft)+(m)*jt_left_pairrectcrtleftmod=(jt_right_pairrectcrtleft)+(m)*jt_right_pairrectcrtleftmod)) /\\ (forall jt_index_pairrectcrtright jt_left_pairrectcrtright jt_right_pairrectcrtright. (exists jt_gap_pairrectcrtrightindex. jt_gap_pairrectcrtrightindex+S (jt_index_pairrectcrtright)=(k)) -> (((exists fs_h_jt_pairrectcrtrightleft. fs_h_jt_pairrectcrtrightleft + S (jt_left_pairrectcrtright) = S ((S (jt_index_pairrectcrtright)) * jt_g_pairrect)) /\\ exists fs_q_jt_pairrectcrtrightleft. jt_f_pairrect = fs_q_jt_pairrectcrtrightleft * S ((S (jt_index_pairrectcrtright)) * jt_g_pairrect) + (jt_left_pairrectcrtright))) -> (((exists fs_h_jt_pairrectcrtrightright. fs_h_jt_pairrectcrtrightright + S (jt_right_pairrectcrtright) = S ((S (jt_index_pairrectcrtright)) * jt_e_pairrect)) /\\ exists fs_q_jt_pairrectcrtrightright. jt_d_pairrect = fs_q_jt_pairrectcrtrightright * S ((S (jt_index_pairrectcrtright)) * jt_e_pairrect) + (jt_right_pairrectcrtright))) -> (exists jt_left_pairrectcrtrightmod jt_right_pairrectcrtrightmod. (jt_left_pairrectcrtright)+(n)*jt_left_pairrectcrtrightmod=(jt_right_pairrectcrtright)+(n)*jt_right_pairrectcrtrightmod)))))) /\\ (forall jt_divisor_pairrectprimitive. (exists jt_factor_pairrectprimitivemodulus. (m*n)=(jt_divisor_pairrectprimitive)*jt_factor_pairrectprimitivemodulus) -> (forall jt_index_pairrectprimitivecoordinates jt_value_pairrectprimitivecoordinates. (exists jt_gap_pairrectprimitivecoordinatesindex. jt_gap_pairrectprimitivecoordinatesindex+S (jt_index_pairrectprimitivecoordinates)=(k)) -> (((exists fs_h_jt_pairrectprimitivecoordinatesat. fs_h_jt_pairrectprimitivecoordinatesat + S (jt_value_pairrectprimitivecoordinates) = S ((S (jt_index_pairrectprimitivecoordinates)) * jt_g_pairrect)) /\\ exists fs_q_jt_pairrectprimitivecoordinatesat. jt_f_pairrect = fs_q_jt_pairrectprimitivecoordinatesat * S ((S (jt_index_pairrectprimitivecoordinates)) * jt_g_pairrect) + (jt_value_pairrectprimitivecoordinates))) -> (exists jt_factor_pairrectprimitivecoordinatesdivides. (jt_value_pairrectprimitivecoordinates)=(jt_divisor_pairrectprimitive)*jt_factor_pairrectprimitivecoordinatesdivides)) -> jt_divisor_pairrectprimitive=1))))))))))))))) -> (exists jt_gap_pairrow. jt_gap_pairrow+S (i)=(u)) -> (exists jt_gap_paircol. jt_gap_paircol+S (j)=(v)) -> (((((exists fs_h_jt_pairleftcode. fs_h_jt_pairleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_pairleftcode. A = fs_q_jt_pairleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_pairleftscale. fs_h_jt_pairleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_pairleftscale. C = fs_q_jt_pairleftscale * S ((S (i)) * D) + (c))))) -> (((((exists fs_h_jt_pairrightcode. fs_h_jt_pairrightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_pairrightcode. E = fs_q_jt_pairrightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_pairrightscale. fs_h_jt_pairrightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_pairrightscale. G = fs_q_jt_pairrightscale * S ((S (j)) * H) + (e))))) -> exists f g. ((((((exists fs_h_jt_pairoutputcode. fs_h_jt_pairoutputcode + S (f) = S ((S (v*i+j)) * Q)) /\\ exists fs_q_jt_pairoutputcode. P = fs_q_jt_pairoutputcode * S ((S (v*i+j)) * Q) + (f))) /\\ (((exists fs_h_jt_pairoutputscale. fs_h_jt_pairoutputscale + S (g) = S ((S (v*i+j)) * T)) /\\ exists fs_q_jt_pairoutputscale. R = fs_q_jt_pairoutputscale * S ((S (v*i+j)) * T) + (g))))) /\\ (((((forall jt_index_paircrtbound. (exists jt_gap_paircrtboundindex. jt_gap_paircrtboundindex+S (jt_index_paircrtbound)=(k)) -> exists jt_value_paircrtbound. ((((exists fs_h_jt_paircrtboundat. fs_h_jt_paircrtboundat + S (jt_value_paircrtbound) = S ((S (jt_index_paircrtbound)) * g)) /\\ exists fs_q_jt_paircrtboundat. f = fs_q_jt_paircrtboundat * S ((S (jt_index_paircrtbound)) * g) + (jt_value_paircrtbound))) /\\ (exists jt_gap_paircrtboundvalue. jt_gap_paircrtboundvalue+S (jt_value_paircrtbound)=(m*n)))) /\\ (((forall jt_index_paircrtleft jt_left_paircrtleft jt_right_paircrtleft. (exists jt_gap_paircrtleftindex. jt_gap_paircrtleftindex+S (jt_index_paircrtleft)=(k)) -> (((exists fs_h_jt_paircrtleftleft. fs_h_jt_paircrtleftleft + S (jt_left_paircrtleft) = S ((S (jt_index_paircrtleft)) * g)) /\\ exists fs_q_jt_paircrtleftleft. f = fs_q_jt_paircrtleftleft * S ((S (jt_index_paircrtleft)) * g) + (jt_left_paircrtleft))) -> (((exists fs_h_jt_paircrtleftright. fs_h_jt_paircrtleftright + S (jt_right_paircrtleft) = S ((S (jt_index_paircrtleft)) * c)) /\\ exists fs_q_jt_paircrtleftright. b = fs_q_jt_paircrtleftright * S ((S (jt_index_paircrtleft)) * c) + (jt_right_paircrtleft))) -> (exists jt_left_paircrtleftmod jt_right_paircrtleftmod. (jt_left_paircrtleft)+(m)*jt_left_paircrtleftmod=(jt_right_paircrtleft)+(m)*jt_right_paircrtleftmod)) /\\ (forall jt_index_paircrtright jt_left_paircrtright jt_right_paircrtright. (exists jt_gap_paircrtrightindex. jt_gap_paircrtrightindex+S (jt_index_paircrtright)=(k)) -> (((exists fs_h_jt_paircrtrightleft. fs_h_jt_paircrtrightleft + S (jt_left_paircrtright) = S ((S (jt_index_paircrtright)) * g)) /\\ exists fs_q_jt_paircrtrightleft. f = fs_q_jt_paircrtrightleft * S ((S (jt_index_paircrtright)) * g) + (jt_left_paircrtright))) -> (((exists fs_h_jt_paircrtrightright. fs_h_jt_paircrtrightright + S (jt_right_paircrtright) = S ((S (jt_index_paircrtright)) * e)) /\\ exists fs_q_jt_paircrtrightright. d = fs_q_jt_paircrtrightright * S ((S (jt_index_paircrtright)) * e) + (jt_right_paircrtright))) -> (exists jt_left_paircrtrightmod jt_right_paircrtrightmod. (jt_left_paircrtright)+(n)*jt_left_paircrtrightmod=(jt_right_paircrtright)+(n)*jt_right_paircrtrightmod)))))) /\\ (forall jt_divisor_pairprimitive. (exists jt_factor_pairprimitivemodulus. (m*n)=(jt_divisor_pairprimitive)*jt_factor_pairprimitivemodulus) -> (forall jt_index_pairprimitivecoordinates jt_value_pairprimitivecoordinates. (exists jt_gap_pairprimitivecoordinatesindex. jt_gap_pairprimitivecoordinatesindex+S (jt_index_pairprimitivecoordinates)=(k)) -> (((exists fs_h_jt_pairprimitivecoordinatesat. fs_h_jt_pairprimitivecoordinatesat + S (jt_value_pairprimitivecoordinates) = S ((S (jt_index_pairprimitivecoordinates)) * g)) /\\ exists fs_q_jt_pairprimitivecoordinatesat. f = fs_q_jt_pairprimitivecoordinatesat * S ((S (jt_index_pairprimitivecoordinates)) * g) + (jt_value_pairprimitivecoordinates))) -> (exists jt_factor_pairprimitivecoordinatesdivides. (jt_value_pairprimitivecoordinates)=(jt_divisor_pairprimitive)*jt_factor_pairprimitivecoordinatesdivides)) -> jt_divisor_pairprimitive=1))))", "statement_sha256": "45e148fcc8b2ba0052a57a1589c89f05db6503059cb467686840757102a15696", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The actual rectangular table realizes every prescribed pair of source entries at its unique flat index." }, { "admission_dependencies": [ "jordan_tuple_normalize_exists", "jordan_primitive_tuple_congruence_transport", "jordan_enumeration_complete", "jordan_tuple_congruence_trans", "jordan_tuple_equal_congruence" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 38, "body_proof_nodes": 94, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro j", "intro b", "intro c", "intro hn", "intro he", "intro hp", "have hnrm : ∃ d. ∃ e. BetaPrefixInto(d,e,k,n) ∧ JordanTupleCongruence(n,b,c,d,e,k)", "specialize jordan_tuple_normalize_exists (n)", "specialize jordan_tuple_normalize_exists (b)", "specialize jordan_tuple_normalize_exists (c)", "specialize jordan_tuple_normalize_exists (k)", "apply jordan_tuple_normalize_exists", "exact hn", "cases hnrm", "cases hnrm_witness", "cases hnrm_witness_witness", "have hprim : JordanPrimitiveTuple(n,x,x1,k)", "specialize jordan_primitive_tuple_congruence_transport (n)", "specialize jordan_primitive_tuple_congruence_transport (b)", "specialize jordan_primitive_tuple_congruence_transport (c)", "specialize jordan_primitive_tuple_congruence_transport (x)", "specialize jordan_primitive_tuple_congruence_transport (x1)", "specialize jordan_primitive_tuple_congruence_transport (k)", "apply jordan_primitive_tuple_congruence_transport", "exact hnrm_witness_witness_right", "exact hp", "have hl : JordanTupleListed(x,x1,k,A,B,C,D,j)", "specialize jordan_enumeration_complete (k)", "specialize jordan_enumeration_complete (n)", "specialize jordan_enumeration_complete (A)", "specialize jordan_enumeration_complete (B)", "specialize jordan_enumeration_complete (C)", "specialize jordan_enumeration_complete (D)", "specialize jordan_enumeration_complete (j)", "specialize jordan_enumeration_complete (x)", "specialize jordan_enumeration_complete (x1)", "apply jordan_enumeration_complete", "exact he", "exact hnrm_witness_witness_left", "exact hprim", "cases hl", "cases hl_witness", "cases hl_witness_witness", "cases hl_witness_witness_witness", "cases hl_witness_witness_witness_right", "exists x2", "exists x3", "exists x4", "split", "exact hl_witness_witness_witness_left", "split", "exact hl_witness_witness_witness_right_left", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (x)", "specialize jordan_tuple_congruence_trans (x1)", "specialize jordan_tuple_congruence_trans (x3)", "specialize jordan_tuple_congruence_trans (x4)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "exact hnrm_witness_witness_right", "specialize jordan_tuple_equal_congruence (n)", "specialize jordan_tuple_equal_congruence (x)", "specialize jordan_tuple_equal_congruence (x1)", "specialize jordan_tuple_equal_congruence (x3)", "specialize jordan_tuple_equal_congruence (x4)", "specialize jordan_tuple_equal_congruence (k)", "apply jordan_tuple_equal_congruence", "exact hl_witness_witness_witness_right_right" ], "defined_statement": "∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ b. ∀ c. ¬n = 0 → JordanTupleEnumeration(k,n,A,B,C,D,j) → JordanPrimitiveTuple(n,b,c,k) → ∃ x. ∃ y. ∃ z. Lt(x,j) ∧ (BetaAt(A,B,x,y) ∧ BetaAt(C,D,x,z) ∧ JordanTupleCongruence(n,b,c,y,z,k))", "defined_statement_sha256": "1191d93e3a252292d72201b20d862ffff3c3f144a45d9cefda7c9642b2a61891", "definition_uses": { "ND0262": 1, "ND0372": 2, "ND0373": 2, "ND0374": 1, "ND0376": 1, "PD0002": 1, "PD0013": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "d0ab5dc30ecbdb8c8c59525e1f13e7d9b4008da10704f38f736c0267b32d7447", "free_names": [], "script_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0373": 1, "ND0376": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "have hnrm : " }, { "kind": "text", "text": "∃ d. ∃ e. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(d,e,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_normalize_exists (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_normalize_exists" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "cases hnrm" } ], [ { "kind": "text", "text": "cases hnrm_witness" } ], [ { "kind": "text", "text": "cases hnrm_witness_witness" } ], [ { "kind": "text", "text": "have hprim : " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,x,x1,k)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (x)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (x1)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_congruence_transport (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_congruence_transport" } ], [ { "kind": "text", "text": "exact hnrm_witness_witness_right" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "have hl : " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(x,x1,k,A,B,C,D,j)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (j)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (x1)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_complete" } ], [ { "kind": "text", "text": "exact he" } ], [ { "kind": "text", "text": "exact hnrm_witness_witness_left" } ], [ { "kind": "text", "text": "exact hprim" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hl_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hl_witness_witness_witness_right" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "exists x4" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hl_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hl_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_congruence_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_congruence_trans" } ], [ { "kind": "text", "text": "exact hnrm_witness_witness_right" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (n)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_congruence (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_congruence" } ], [ { "kind": "text", "text": "exact hl_witness_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0372": 1, "ND0373": 1, "ND0374": 1, "PD0002": 1, "PD0013": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ j. ∀ b. ∀ c. ¬n = 0 → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,y,z,k)" }, { "kind": "text", "text": ")" } ] }, "dependencies": [ "jordan_tuple_normalize_exists", "jordan_primitive_tuple_congruence_transport", "jordan_enumeration_complete", "jordan_tuple_congruence_trans", "jordan_tuple_equal_congruence" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0045", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_reduce_primitive", "original_ha_bundle_verified": true, "proof_bundle_node_id": 330, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro j", "intro b", "intro c", "intro hn", "intro he", "intro hp", "have hnrm : exists d e. ((forall jt_index_reducebound. (exists jt_gap_reduceboundindex. jt_gap_reduceboundindex+S (jt_index_reducebound)=(k)) -> exists jt_value_reducebound. ((((exists fs_h_jt_reduceboundat. fs_h_jt_reduceboundat + S (jt_value_reducebound) = S ((S (jt_index_reducebound)) * e)) /\\ exists fs_q_jt_reduceboundat. d = fs_q_jt_reduceboundat * S ((S (jt_index_reducebound)) * e) + (jt_value_reducebound))) /\\ (exists jt_gap_reduceboundvalue. jt_gap_reduceboundvalue+S (jt_value_reducebound)=(n)))) /\\ (forall jt_index_reducemod jt_left_reducemod jt_right_reducemod. (exists jt_gap_reducemodindex. jt_gap_reducemodindex+S (jt_index_reducemod)=(k)) -> (((exists fs_h_jt_reducemodleft. fs_h_jt_reducemodleft + S (jt_left_reducemod) = S ((S (jt_index_reducemod)) * c)) /\\ exists fs_q_jt_reducemodleft. b = fs_q_jt_reducemodleft * S ((S (jt_index_reducemod)) * c) + (jt_left_reducemod))) -> (((exists fs_h_jt_reducemodright. fs_h_jt_reducemodright + S (jt_right_reducemod) = S ((S (jt_index_reducemod)) * e)) /\\ exists fs_q_jt_reducemodright. d = fs_q_jt_reducemodright * S ((S (jt_index_reducemod)) * e) + (jt_right_reducemod))) -> (exists jt_left_reducemodmod jt_right_reducemodmod. (jt_left_reducemod)+(n)*jt_left_reducemodmod=(jt_right_reducemod)+(n)*jt_right_reducemodmod)))", "specialize jordan_tuple_normalize_exists (n)", "specialize jordan_tuple_normalize_exists (b)", "specialize jordan_tuple_normalize_exists (c)", "specialize jordan_tuple_normalize_exists (k)", "apply jordan_tuple_normalize_exists", "exact hn", "cases hnrm", "cases hnrm_witness", "cases hnrm_witness_witness", "have hprim : forall jt_divisor_reduceprim. (exists jt_factor_reduceprimmodulus. (n)=(jt_divisor_reduceprim)*jt_factor_reduceprimmodulus) -> (forall jt_index_reduceprimcoordinates jt_value_reduceprimcoordinates. (exists jt_gap_reduceprimcoordinatesindex. jt_gap_reduceprimcoordinatesindex+S (jt_index_reduceprimcoordinates)=(k)) -> (((exists fs_h_jt_reduceprimcoordinatesat. fs_h_jt_reduceprimcoordinatesat + S (jt_value_reduceprimcoordinates) = S ((S (jt_index_reduceprimcoordinates)) * x1)) /\\ exists fs_q_jt_reduceprimcoordinatesat. x = fs_q_jt_reduceprimcoordinatesat * S ((S (jt_index_reduceprimcoordinates)) * x1) + (jt_value_reduceprimcoordinates))) -> (exists jt_factor_reduceprimcoordinatesdivides. (jt_value_reduceprimcoordinates)=(jt_divisor_reduceprim)*jt_factor_reduceprimcoordinatesdivides)) -> jt_divisor_reduceprim=1", "specialize jordan_primitive_tuple_congruence_transport (n)", "specialize jordan_primitive_tuple_congruence_transport (b)", "specialize jordan_primitive_tuple_congruence_transport (c)", "specialize jordan_primitive_tuple_congruence_transport (x)", "specialize jordan_primitive_tuple_congruence_transport (x1)", "specialize jordan_primitive_tuple_congruence_transport (k)", "apply jordan_primitive_tuple_congruence_transport", "exact hnrm_witness_witness_right", "exact hp", "have hl : exists jt_index_reducelisted jt_code_reducelisted jt_scale_reducelisted. ((exists jt_gap_reducelistedindex. jt_gap_reducelistedindex+S (jt_index_reducelisted)=(j)) /\\ (((((((exists fs_h_jt_reducelistedcode. fs_h_jt_reducelistedcode + S (jt_code_reducelisted) = S ((S (jt_index_reducelisted)) * B)) /\\ exists fs_q_jt_reducelistedcode. A = fs_q_jt_reducelistedcode * S ((S (jt_index_reducelisted)) * B) + (jt_code_reducelisted))) /\\ (((exists fs_h_jt_reducelistedscale. fs_h_jt_reducelistedscale + S (jt_scale_reducelisted) = S ((S (jt_index_reducelisted)) * D)) /\\ exists fs_q_jt_reducelistedscale. C = fs_q_jt_reducelistedscale * S ((S (jt_index_reducelisted)) * D) + (jt_scale_reducelisted))))) /\\ (forall jt_index_reducelistedequal jt_left_reducelistedequal jt_right_reducelistedequal. (exists jt_gap_reducelistedequalindex. jt_gap_reducelistedequalindex+S (jt_index_reducelistedequal)=(k)) -> (((exists fs_h_jt_reducelistedequalleft. fs_h_jt_reducelistedequalleft + S (jt_left_reducelistedequal) = S ((S (jt_index_reducelistedequal)) * x1)) /\\ exists fs_q_jt_reducelistedequalleft. x = fs_q_jt_reducelistedequalleft * S ((S (jt_index_reducelistedequal)) * x1) + (jt_left_reducelistedequal))) -> (((exists fs_h_jt_reducelistedequalright. fs_h_jt_reducelistedequalright + S (jt_right_reducelistedequal) = S ((S (jt_index_reducelistedequal)) * jt_scale_reducelisted)) /\\ exists fs_q_jt_reducelistedequalright. jt_code_reducelisted = fs_q_jt_reducelistedequalright * S ((S (jt_index_reducelistedequal)) * jt_scale_reducelisted) + (jt_right_reducelistedequal))) -> jt_left_reducelistedequal=jt_right_reducelistedequal))))", "specialize jordan_enumeration_complete (k)", "specialize jordan_enumeration_complete (n)", "specialize jordan_enumeration_complete (A)", "specialize jordan_enumeration_complete (B)", "specialize jordan_enumeration_complete (C)", "specialize jordan_enumeration_complete (D)", "specialize jordan_enumeration_complete (j)", "specialize jordan_enumeration_complete (x)", "specialize jordan_enumeration_complete (x1)", "apply jordan_enumeration_complete", "exact he", "exact hnrm_witness_witness_left", "exact hprim", "cases hl", "cases hl_witness", "cases hl_witness_witness", "cases hl_witness_witness_witness", "cases hl_witness_witness_witness_right", "exists x2", "exists x3", "exists x4", "split", "exact hl_witness_witness_witness_left", "split", "exact hl_witness_witness_witness_right_left", "specialize jordan_tuple_congruence_trans (n)", "specialize jordan_tuple_congruence_trans (b)", "specialize jordan_tuple_congruence_trans (c)", "specialize jordan_tuple_congruence_trans (x)", "specialize jordan_tuple_congruence_trans (x1)", "specialize jordan_tuple_congruence_trans (x3)", "specialize jordan_tuple_congruence_trans (x4)", "specialize jordan_tuple_congruence_trans (k)", "apply jordan_tuple_congruence_trans", "exact hnrm_witness_witness_right", "specialize jordan_tuple_equal_congruence (n)", "specialize jordan_tuple_equal_congruence (x)", "specialize jordan_tuple_equal_congruence (x1)", "specialize jordan_tuple_equal_congruence (x3)", "specialize jordan_tuple_equal_congruence (x4)", "specialize jordan_tuple_equal_congruence (k)", "apply jordan_tuple_equal_congruence", "exact hl_witness_witness_witness_right_right" ], "script_sha256": "b3f9cd3b935e17e5f31124bfe338be5a3d6c51e1674ba33121ba347b663dda96", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "b3f9cd3b935e17e5f31124bfe338be5a3d6c51e1674ba33121ba347b663dda96", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "d0ab5dc30ecbdb8c8c59525e1f13e7d9b4008da10704f38f736c0267b32d7447" } ], "stable_member": false, "statement": "forall n k A B C D j b c. ~(n=0) -> (((forall jt_i_reduceenum. (exists jt_gap_reduceenumsoundindex. jt_gap_reduceenumsoundindex+S (jt_i_reduceenum)=(j)) -> exists jt_b_reduceenum jt_c_reduceenum. ((((((exists fs_h_jt_reduceenumsoundcode. fs_h_jt_reduceenumsoundcode + S (jt_b_reduceenum) = S ((S (jt_i_reduceenum)) * B)) /\\ exists fs_q_jt_reduceenumsoundcode. A = fs_q_jt_reduceenumsoundcode * S ((S (jt_i_reduceenum)) * B) + (jt_b_reduceenum))) /\\ (((exists fs_h_jt_reduceenumsoundscale. fs_h_jt_reduceenumsoundscale + S (jt_c_reduceenum) = S ((S (jt_i_reduceenum)) * D)) /\\ exists fs_q_jt_reduceenumsoundscale. C = fs_q_jt_reduceenumsoundscale * S ((S (jt_i_reduceenum)) * D) + (jt_c_reduceenum))))) /\\ (((forall jt_index_reduceenumbound. (exists jt_gap_reduceenumboundindex. jt_gap_reduceenumboundindex+S (jt_index_reduceenumbound)=(k)) -> exists jt_value_reduceenumbound. ((((exists fs_h_jt_reduceenumboundat. fs_h_jt_reduceenumboundat + S (jt_value_reduceenumbound) = S ((S (jt_index_reduceenumbound)) * jt_c_reduceenum)) /\\ exists fs_q_jt_reduceenumboundat. jt_b_reduceenum = fs_q_jt_reduceenumboundat * S ((S (jt_index_reduceenumbound)) * jt_c_reduceenum) + (jt_value_reduceenumbound))) /\\ (exists jt_gap_reduceenumboundvalue. jt_gap_reduceenumboundvalue+S (jt_value_reduceenumbound)=(n)))) /\\ (forall jt_divisor_reduceenumprimitive. (exists jt_factor_reduceenumprimitivemodulus. (n)=(jt_divisor_reduceenumprimitive)*jt_factor_reduceenumprimitivemodulus) -> (forall jt_index_reduceenumprimitivecoordinates jt_value_reduceenumprimitivecoordinates. (exists jt_gap_reduceenumprimitivecoordinatesindex. jt_gap_reduceenumprimitivecoordinatesindex+S (jt_index_reduceenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_reduceenumprimitivecoordinatesat. fs_h_jt_reduceenumprimitivecoordinatesat + S (jt_value_reduceenumprimitivecoordinates) = S ((S (jt_index_reduceenumprimitivecoordinates)) * jt_c_reduceenum)) /\\ exists fs_q_jt_reduceenumprimitivecoordinatesat. jt_b_reduceenum = fs_q_jt_reduceenumprimitivecoordinatesat * S ((S (jt_index_reduceenumprimitivecoordinates)) * jt_c_reduceenum) + (jt_value_reduceenumprimitivecoordinates))) -> (exists jt_factor_reduceenumprimitivecoordinatesdivides. (jt_value_reduceenumprimitivecoordinates)=(jt_divisor_reduceenumprimitive)*jt_factor_reduceenumprimitivecoordinatesdivides)) -> jt_divisor_reduceenumprimitive=1))))) /\\ (((forall jt_b_reduceenum jt_c_reduceenum. (forall jt_index_reduceenuminputbound. (exists jt_gap_reduceenuminputboundindex. jt_gap_reduceenuminputboundindex+S (jt_index_reduceenuminputbound)=(k)) -> exists jt_value_reduceenuminputbound. ((((exists fs_h_jt_reduceenuminputboundat. fs_h_jt_reduceenuminputboundat + S (jt_value_reduceenuminputbound) = S ((S (jt_index_reduceenuminputbound)) * jt_c_reduceenum)) /\\ exists fs_q_jt_reduceenuminputboundat. jt_b_reduceenum = fs_q_jt_reduceenuminputboundat * S ((S (jt_index_reduceenuminputbound)) * jt_c_reduceenum) + (jt_value_reduceenuminputbound))) /\\ (exists jt_gap_reduceenuminputboundvalue. jt_gap_reduceenuminputboundvalue+S (jt_value_reduceenuminputbound)=(n)))) -> (forall jt_divisor_reduceenuminputprimitive. (exists jt_factor_reduceenuminputprimitivemodulus. (n)=(jt_divisor_reduceenuminputprimitive)*jt_factor_reduceenuminputprimitivemodulus) -> (forall jt_index_reduceenuminputprimitivecoordinates jt_value_reduceenuminputprimitivecoordinates. (exists jt_gap_reduceenuminputprimitivecoordinatesindex. jt_gap_reduceenuminputprimitivecoordinatesindex+S (jt_index_reduceenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_reduceenuminputprimitivecoordinatesat. fs_h_jt_reduceenuminputprimitivecoordinatesat + S (jt_value_reduceenuminputprimitivecoordinates) = S ((S (jt_index_reduceenuminputprimitivecoordinates)) * jt_c_reduceenum)) /\\ exists fs_q_jt_reduceenuminputprimitivecoordinatesat. jt_b_reduceenum = fs_q_jt_reduceenuminputprimitivecoordinatesat * S ((S (jt_index_reduceenuminputprimitivecoordinates)) * jt_c_reduceenum) + (jt_value_reduceenuminputprimitivecoordinates))) -> (exists jt_factor_reduceenuminputprimitivecoordinatesdivides. (jt_value_reduceenuminputprimitivecoordinates)=(jt_divisor_reduceenuminputprimitive)*jt_factor_reduceenuminputprimitivecoordinatesdivides)) -> jt_divisor_reduceenuminputprimitive=1) -> exists jt_i_reduceenum jt_d_reduceenum jt_e_reduceenum. ((exists jt_gap_reduceenumcompleteindex. jt_gap_reduceenumcompleteindex+S (jt_i_reduceenum)=(j)) /\\ (((((((exists fs_h_jt_reduceenumcompletecode. fs_h_jt_reduceenumcompletecode + S (jt_d_reduceenum) = S ((S (jt_i_reduceenum)) * B)) /\\ exists fs_q_jt_reduceenumcompletecode. A = fs_q_jt_reduceenumcompletecode * S ((S (jt_i_reduceenum)) * B) + (jt_d_reduceenum))) /\\ (((exists fs_h_jt_reduceenumcompletescale. fs_h_jt_reduceenumcompletescale + S (jt_e_reduceenum) = S ((S (jt_i_reduceenum)) * D)) /\\ exists fs_q_jt_reduceenumcompletescale. C = fs_q_jt_reduceenumcompletescale * S ((S (jt_i_reduceenum)) * D) + (jt_e_reduceenum))))) /\\ (forall jt_index_reduceenumrepresented jt_left_reduceenumrepresented jt_right_reduceenumrepresented. (exists jt_gap_reduceenumrepresentedindex. jt_gap_reduceenumrepresentedindex+S (jt_index_reduceenumrepresented)=(k)) -> (((exists fs_h_jt_reduceenumrepresentedleft. fs_h_jt_reduceenumrepresentedleft + S (jt_left_reduceenumrepresented) = S ((S (jt_index_reduceenumrepresented)) * jt_c_reduceenum)) /\\ exists fs_q_jt_reduceenumrepresentedleft. jt_b_reduceenum = fs_q_jt_reduceenumrepresentedleft * S ((S (jt_index_reduceenumrepresented)) * jt_c_reduceenum) + (jt_left_reduceenumrepresented))) -> (((exists fs_h_jt_reduceenumrepresentedright. fs_h_jt_reduceenumrepresentedright + S (jt_right_reduceenumrepresented) = S ((S (jt_index_reduceenumrepresented)) * jt_e_reduceenum)) /\\ exists fs_q_jt_reduceenumrepresentedright. jt_d_reduceenum = fs_q_jt_reduceenumrepresentedright * S ((S (jt_index_reduceenumrepresented)) * jt_e_reduceenum) + (jt_right_reduceenumrepresented))) -> jt_left_reduceenumrepresented=jt_right_reduceenumrepresented))))) /\\ (forall jt_i_reduceenum jt_h_reduceenum jt_b_reduceenum jt_c_reduceenum jt_d_reduceenum jt_e_reduceenum. (exists jt_gap_reduceenumfirstindex. jt_gap_reduceenumfirstindex+S (jt_i_reduceenum)=(j)) -> (exists jt_gap_reduceenumsecondindex. jt_gap_reduceenumsecondindex+S (jt_h_reduceenum)=(j)) -> (((((exists fs_h_jt_reduceenumfirstcode. fs_h_jt_reduceenumfirstcode + S (jt_b_reduceenum) = S ((S (jt_i_reduceenum)) * B)) /\\ exists fs_q_jt_reduceenumfirstcode. A = fs_q_jt_reduceenumfirstcode * S ((S (jt_i_reduceenum)) * B) + (jt_b_reduceenum))) /\\ (((exists fs_h_jt_reduceenumfirstscale. fs_h_jt_reduceenumfirstscale + S (jt_c_reduceenum) = S ((S (jt_i_reduceenum)) * D)) /\\ exists fs_q_jt_reduceenumfirstscale. C = fs_q_jt_reduceenumfirstscale * S ((S (jt_i_reduceenum)) * D) + (jt_c_reduceenum))))) -> (((((exists fs_h_jt_reduceenumsecondcode. fs_h_jt_reduceenumsecondcode + S (jt_d_reduceenum) = S ((S (jt_h_reduceenum)) * B)) /\\ exists fs_q_jt_reduceenumsecondcode. A = fs_q_jt_reduceenumsecondcode * S ((S (jt_h_reduceenum)) * B) + (jt_d_reduceenum))) /\\ (((exists fs_h_jt_reduceenumsecondscale. fs_h_jt_reduceenumsecondscale + S (jt_e_reduceenum) = S ((S (jt_h_reduceenum)) * D)) /\\ exists fs_q_jt_reduceenumsecondscale. C = fs_q_jt_reduceenumsecondscale * S ((S (jt_h_reduceenum)) * D) + (jt_e_reduceenum))))) -> (forall jt_index_reduceenumsame jt_left_reduceenumsame jt_right_reduceenumsame. (exists jt_gap_reduceenumsameindex. jt_gap_reduceenumsameindex+S (jt_index_reduceenumsame)=(k)) -> (((exists fs_h_jt_reduceenumsameleft. fs_h_jt_reduceenumsameleft + S (jt_left_reduceenumsame) = S ((S (jt_index_reduceenumsame)) * jt_c_reduceenum)) /\\ exists fs_q_jt_reduceenumsameleft. jt_b_reduceenum = fs_q_jt_reduceenumsameleft * S ((S (jt_index_reduceenumsame)) * jt_c_reduceenum) + (jt_left_reduceenumsame))) -> (((exists fs_h_jt_reduceenumsameright. fs_h_jt_reduceenumsameright + S (jt_right_reduceenumsame) = S ((S (jt_index_reduceenumsame)) * jt_e_reduceenum)) /\\ exists fs_q_jt_reduceenumsameright. jt_d_reduceenum = fs_q_jt_reduceenumsameright * S ((S (jt_index_reduceenumsame)) * jt_e_reduceenum) + (jt_right_reduceenumsame))) -> jt_left_reduceenumsame=jt_right_reduceenumsame) -> jt_i_reduceenum=jt_h_reduceenum))))) -> (forall jt_divisor_reduceinput. (exists jt_factor_reduceinputmodulus. (n)=(jt_divisor_reduceinput)*jt_factor_reduceinputmodulus) -> (forall jt_index_reduceinputcoordinates jt_value_reduceinputcoordinates. (exists jt_gap_reduceinputcoordinatesindex. jt_gap_reduceinputcoordinatesindex+S (jt_index_reduceinputcoordinates)=(k)) -> (((exists fs_h_jt_reduceinputcoordinatesat. fs_h_jt_reduceinputcoordinatesat + S (jt_value_reduceinputcoordinates) = S ((S (jt_index_reduceinputcoordinates)) * c)) /\\ exists fs_q_jt_reduceinputcoordinatesat. b = fs_q_jt_reduceinputcoordinatesat * S ((S (jt_index_reduceinputcoordinates)) * c) + (jt_value_reduceinputcoordinates))) -> (exists jt_factor_reduceinputcoordinatesdivides. (jt_value_reduceinputcoordinates)=(jt_divisor_reduceinput)*jt_factor_reduceinputcoordinatesdivides)) -> jt_divisor_reduceinput=1) -> exists i d e. ((exists jt_gap_reduceindex. jt_gap_reduceindex+S (i)=(j)) /\\ (((((((exists fs_h_jt_reduceentrycode. fs_h_jt_reduceentrycode + S (d) = S ((S (i)) * B)) /\\ exists fs_q_jt_reduceentrycode. A = fs_q_jt_reduceentrycode * S ((S (i)) * B) + (d))) /\\ (((exists fs_h_jt_reduceentryscale. fs_h_jt_reduceentryscale + S (e) = S ((S (i)) * D)) /\\ exists fs_q_jt_reduceentryscale. C = fs_q_jt_reduceentryscale * S ((S (i)) * D) + (e))))) /\\ (forall jt_index_reduceresult jt_left_reduceresult jt_right_reduceresult. (exists jt_gap_reduceresultindex. jt_gap_reduceresultindex+S (jt_index_reduceresult)=(k)) -> (((exists fs_h_jt_reduceresultleft. fs_h_jt_reduceresultleft + S (jt_left_reduceresult) = S ((S (jt_index_reduceresult)) * c)) /\\ exists fs_q_jt_reduceresultleft. b = fs_q_jt_reduceresultleft * S ((S (jt_index_reduceresult)) * c) + (jt_left_reduceresult))) -> (((exists fs_h_jt_reduceresultright. fs_h_jt_reduceresultright + S (jt_right_reduceresult) = S ((S (jt_index_reduceresult)) * e)) /\\ exists fs_q_jt_reduceresultright. d = fs_q_jt_reduceresultright * S ((S (jt_index_reduceresult)) * e) + (jt_right_reduceresult))) -> (exists jt_left_reduceresultmod jt_right_reduceresultmod. (jt_left_reduceresult)+(n)*jt_left_reduceresultmod=(jt_right_reduceresult)+(n)*jt_right_reduceresultmod)))))", "statement_sha256": "d0ab5dc30ecbdb8c8c59525e1f13e7d9b4008da10704f38f736c0267b32d7447", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Reduce any primitive tuple to an actual canonical enumeration entry, retaining coordinate congruences and the genuine index." }, { "admission_dependencies": [ "jordan_rectangle_crt_actual_entry", "jordan_enumeration_actual_value", "jordan_crt_component_recovery", "jordan_enumeration_distinct" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 71, "body_proof_nodes": 369, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro p", "intro z", "intro f", "intro g", "intro h", "intro s", "intro hl", "intro hh", "intro hr", "intro hp", "intro hz", "intro he", "intro hf", "intro hsame", "have ha : ∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. Lt(i,u) ∧ (Lt(j,v) ∧ (p = v · i + j ∧ (BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) ∧ (BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) ∧ (JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k)))))))", "specialize jordan_rectangle_crt_actual_entry (m)", "specialize jordan_rectangle_crt_actual_entry (n)", "specialize jordan_rectangle_crt_actual_entry (k)", "specialize jordan_rectangle_crt_actual_entry (A)", "specialize jordan_rectangle_crt_actual_entry (B)", "specialize jordan_rectangle_crt_actual_entry (C)", "specialize jordan_rectangle_crt_actual_entry (D)", "specialize jordan_rectangle_crt_actual_entry (u)", "specialize jordan_rectangle_crt_actual_entry (E)", "specialize jordan_rectangle_crt_actual_entry (F)", "specialize jordan_rectangle_crt_actual_entry (G)", "specialize jordan_rectangle_crt_actual_entry (H)", "specialize jordan_rectangle_crt_actual_entry (v)", "specialize jordan_rectangle_crt_actual_entry (P)", "specialize jordan_rectangle_crt_actual_entry (Q)", "specialize jordan_rectangle_crt_actual_entry (R)", "specialize jordan_rectangle_crt_actual_entry (T)", "specialize jordan_rectangle_crt_actual_entry (u*v)", "specialize jordan_rectangle_crt_actual_entry (p)", "specialize jordan_rectangle_crt_actual_entry (f)", "specialize jordan_rectangle_crt_actual_entry (g)", "apply jordan_rectangle_crt_actual_entry", "exact hr", "exact hp", "exact he", "cases ha", "cases ha_witness", "cases ha_witness_witness", "cases ha_witness_witness_witness", "cases ha_witness_witness_witness_witness", "cases ha_witness_witness_witness_witness_witness", "cases ha_witness_witness_witness_witness_witness_witness", "cases ha_witness_witness_witness_witness_witness_witness_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "have hacrt : JordanCanonicalTupleCRT(m,n,x2,x3,x4,x5,f,g,k)", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "cases hacrt", "cases hacrt_right", "have hb : ∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. Lt(i,u) ∧ (Lt(j,v) ∧ (z = v · i + j ∧ (BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) ∧ (BetaAt(P,Q,z,h) ∧ BetaAt(R,T,z,s) ∧ (JordanCanonicalTupleCRT(m,n,b,c,d,e,h,s,k) ∧ JordanPrimitiveTuple(m · n,h,s,k)))))))", "specialize jordan_rectangle_crt_actual_entry (m)", "specialize jordan_rectangle_crt_actual_entry (n)", "specialize jordan_rectangle_crt_actual_entry (k)", "specialize jordan_rectangle_crt_actual_entry (A)", "specialize jordan_rectangle_crt_actual_entry (B)", "specialize jordan_rectangle_crt_actual_entry (C)", "specialize jordan_rectangle_crt_actual_entry (D)", "specialize jordan_rectangle_crt_actual_entry (u)", "specialize jordan_rectangle_crt_actual_entry (E)", "specialize jordan_rectangle_crt_actual_entry (F)", "specialize jordan_rectangle_crt_actual_entry (G)", "specialize jordan_rectangle_crt_actual_entry (H)", "specialize jordan_rectangle_crt_actual_entry (v)", "specialize jordan_rectangle_crt_actual_entry (P)", "specialize jordan_rectangle_crt_actual_entry (Q)", "specialize jordan_rectangle_crt_actual_entry (R)", "specialize jordan_rectangle_crt_actual_entry (T)", "specialize jordan_rectangle_crt_actual_entry (u*v)", "specialize jordan_rectangle_crt_actual_entry (z)", "specialize jordan_rectangle_crt_actual_entry (h)", "specialize jordan_rectangle_crt_actual_entry (s)", "apply jordan_rectangle_crt_actual_entry", "exact hr", "exact hz", "exact hf", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_witness", "cases hb_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness_witness_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "have hbcrt : JordanCanonicalTupleCRT(m,n,x8,x9,x10,x11,h,s,k)", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "cases hbcrt", "cases hbcrt_right", "have left0 : BetaPrefixInto(x2,x3,k,m) ∧ JordanPrimitiveTuple(m,x2,x3,k)", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (m)", "specialize jordan_enumeration_actual_value (A)", "specialize jordan_enumeration_actual_value (B)", "specialize jordan_enumeration_actual_value (C)", "specialize jordan_enumeration_actual_value (D)", "specialize jordan_enumeration_actual_value (u)", "specialize jordan_enumeration_actual_value (x)", "specialize jordan_enumeration_actual_value (x2)", "specialize jordan_enumeration_actual_value (x3)", "apply jordan_enumeration_actual_value", "exact hl", "exact ha_witness_witness_witness_witness_witness_witness_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_left", "cases left0", "have left1 : BetaPrefixInto(x8,x9,k,m) ∧ JordanPrimitiveTuple(m,x8,x9,k)", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (m)", "specialize jordan_enumeration_actual_value (A)", "specialize jordan_enumeration_actual_value (B)", "specialize jordan_enumeration_actual_value (C)", "specialize jordan_enumeration_actual_value (D)", "specialize jordan_enumeration_actual_value (u)", "specialize jordan_enumeration_actual_value (x6)", "specialize jordan_enumeration_actual_value (x8)", "specialize jordan_enumeration_actual_value (x9)", "apply jordan_enumeration_actual_value", "exact hl", "exact hb_witness_witness_witness_witness_witness_witness_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_left", "cases left1", "have leftsame : IntegerVectorZero(x2,x3,x8,x9,k)", "specialize jordan_crt_component_recovery (m)", "specialize jordan_crt_component_recovery (x2)", "specialize jordan_crt_component_recovery (x3)", "specialize jordan_crt_component_recovery (x8)", "specialize jordan_crt_component_recovery (x9)", "specialize jordan_crt_component_recovery (f)", "specialize jordan_crt_component_recovery (g)", "specialize jordan_crt_component_recovery (h)", "specialize jordan_crt_component_recovery (s)", "specialize jordan_crt_component_recovery (k)", "apply jordan_crt_component_recovery", "exact left0_left", "exact left1_left", "exact hacrt_right_left", "exact hbcrt_right_left", "exact hsame", "have leftindex : x=x6", "specialize jordan_enumeration_distinct (k)", "specialize jordan_enumeration_distinct (m)", "specialize jordan_enumeration_distinct (A)", "specialize jordan_enumeration_distinct (B)", "specialize jordan_enumeration_distinct (C)", "specialize jordan_enumeration_distinct (D)", "specialize jordan_enumeration_distinct (u)", "specialize jordan_enumeration_distinct (x)", "specialize jordan_enumeration_distinct (x6)", "specialize jordan_enumeration_distinct (x2)", "specialize jordan_enumeration_distinct (x3)", "specialize jordan_enumeration_distinct (x8)", "specialize jordan_enumeration_distinct (x9)", "apply jordan_enumeration_distinct", "exact hl", "exact ha_witness_witness_witness_witness_witness_witness_left", "exact hb_witness_witness_witness_witness_witness_witness_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact leftsame", "have right0 : BetaPrefixInto(x4,x5,k,n) ∧ JordanPrimitiveTuple(n,x4,x5,k)", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (n)", "specialize jordan_enumeration_actual_value (E)", "specialize jordan_enumeration_actual_value (F)", "specialize jordan_enumeration_actual_value (G)", "specialize jordan_enumeration_actual_value (H)", "specialize jordan_enumeration_actual_value (v)", "specialize jordan_enumeration_actual_value (x1)", "specialize jordan_enumeration_actual_value (x4)", "specialize jordan_enumeration_actual_value (x5)", "apply jordan_enumeration_actual_value", "exact hh", "exact ha_witness_witness_witness_witness_witness_witness_right_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "cases right0", "have right1 : BetaPrefixInto(x10,x11,k,n) ∧ JordanPrimitiveTuple(n,x10,x11,k)", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (n)", "specialize jordan_enumeration_actual_value (E)", "specialize jordan_enumeration_actual_value (F)", "specialize jordan_enumeration_actual_value (G)", "specialize jordan_enumeration_actual_value (H)", "specialize jordan_enumeration_actual_value (v)", "specialize jordan_enumeration_actual_value (x7)", "specialize jordan_enumeration_actual_value (x10)", "specialize jordan_enumeration_actual_value (x11)", "apply jordan_enumeration_actual_value", "exact hh", "exact hb_witness_witness_witness_witness_witness_witness_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "cases right1", "have rightsame : IntegerVectorZero(x4,x5,x10,x11,k)", "specialize jordan_crt_component_recovery (n)", "specialize jordan_crt_component_recovery (x4)", "specialize jordan_crt_component_recovery (x5)", "specialize jordan_crt_component_recovery (x10)", "specialize jordan_crt_component_recovery (x11)", "specialize jordan_crt_component_recovery (f)", "specialize jordan_crt_component_recovery (g)", "specialize jordan_crt_component_recovery (h)", "specialize jordan_crt_component_recovery (s)", "specialize jordan_crt_component_recovery (k)", "apply jordan_crt_component_recovery", "exact right0_left", "exact right1_left", "exact hacrt_right_right", "exact hbcrt_right_right", "exact hsame", "have rightindex : x1=x7", "specialize jordan_enumeration_distinct (k)", "specialize jordan_enumeration_distinct (n)", "specialize jordan_enumeration_distinct (E)", "specialize jordan_enumeration_distinct (F)", "specialize jordan_enumeration_distinct (G)", "specialize jordan_enumeration_distinct (H)", "specialize jordan_enumeration_distinct (v)", "specialize jordan_enumeration_distinct (x1)", "specialize jordan_enumeration_distinct (x7)", "specialize jordan_enumeration_distinct (x4)", "specialize jordan_enumeration_distinct (x5)", "specialize jordan_enumeration_distinct (x10)", "specialize jordan_enumeration_distinct (x11)", "apply jordan_enumeration_distinct", "exact hh", "exact ha_witness_witness_witness_witness_witness_witness_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact rightsame", "have hpos : p=v*x+x1", "exact ha_witness_witness_witness_witness_witness_witness_right_right_left", "rewrite leftindex at hpos", "rewrite rightindex at hpos", "trans v*x6+x7", "exact hpos", "symm", "exact hb_witness_witness_witness_witness_witness_witness_right_right_left" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ p. ∀ z. ∀ f. ∀ g. ∀ h. ∀ s. JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v) → Lt(p,u · v) → Lt(z,u · v) → BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) → BetaAt(P,Q,z,h) ∧ BetaAt(R,T,z,s) → IntegerVectorZero(f,g,h,s,k) → p = z", "defined_statement_sha256": "356feda7d83d5be5c2a499364dc65ce6cd71e74e9e4793600cacce5be1b4fb8c", "definition_uses": { "ND0121": 3, "ND0262": 4, "ND0372": 6, "ND0374": 2, "ND0380": 4, "ND0381": 1, "PD0002": 6, "PD0013": 16 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "cab9c7262834a02f7624af6c4d5ceebdb6eb91a1e71ff58f0d86fbf218c4d239", "free_names": [], "script_definition_uses": { "ND0121": 2, "ND0262": 4, "ND0372": 6, "ND0380": 4, "PD0002": 4, "PD0013": 12 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro P" } ], [ { "kind": "text", "text": "intro Q" } ], [ { "kind": "text", "text": "intro R" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro z" } ], [ { "kind": "text", "text": "intro f" } ], [ { "kind": "text", "text": "intro g" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro s" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hf" } ], [ { "kind": "text", "text": "intro hsame" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (p = v · i + j ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" }, { "kind": "text", "text": "))))))" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (P)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (Q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (R)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (T)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (u*v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (p)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (f)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (g)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_actual_entry" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "exact he" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "cases ha_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "have hacrt : " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,x2,x3,x4,x5,f,g,k)" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases hacrt" } ], [ { "kind": "text", "text": "cases hacrt_right" } ], [ { "kind": "text", "text": "have hb : " }, { "kind": "text", "text": "∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (z = v · i + j ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,z,h)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,z,s)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,h,s,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,h,s,k)" }, { "kind": "text", "text": "))))))" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (P)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (Q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (R)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (T)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (u*v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (z)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (h)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_actual_entry (s)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_actual_entry" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "exact hf" } ], [ { "kind": "text", "text": "cases hb" } ], [ { "kind": "text", "text": "cases hb_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "have hbcrt : " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,x8,x9,x10,x11,h,s,k)" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases hbcrt" } ], [ { "kind": "text", "text": "cases hbcrt_right" } ], [ { "kind": "text", "text": "have left0 : " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x2,x3,k,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,x2,x3,k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (m)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x2)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x3)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_actual_value" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "cases left0" } ], [ { "kind": "text", "text": "have left1 : " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x8,x9,k,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,x8,x9,k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (m)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x6)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x8)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x9)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_actual_value" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "cases left1" } ], [ { "kind": "text", "text": "have leftsame : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x2,x3,x8,x9,k)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (m)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x2)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x3)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x8)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x9)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (f)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (g)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (h)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (s)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (k)" } ], [ { "kind": "text", "text": "apply jordan_crt_component_recovery" } ], [ { "kind": "text", "text": "exact left0_left" } ], [ { "kind": "text", "text": "exact left1_left" } ], [ { "kind": "text", "text": "exact hacrt_right_left" } ], [ { "kind": "text", "text": "exact hbcrt_right_left" } ], [ { "kind": "text", "text": "exact hsame" } ], [ { "kind": "text", "text": "have leftindex : x=x6" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (m)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x6)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x2)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x3)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x8)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x9)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_distinct" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_left" } ], [ { "kind": "text", "text": "exact leftsame" } ], [ { "kind": "text", "text": "have right0 : " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x4,x5,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,x4,x5,k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x1)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x4)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x5)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_actual_value" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases right0" } ], [ { "kind": "text", "text": "have right1 : " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(x10,x11,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,x10,x11,k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x7)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x10)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_actual_value (x11)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_actual_value" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases right1" } ], [ { "kind": "text", "text": "have rightsame : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x4,x5,x10,x11,k)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (n)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x4)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x5)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x10)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (x11)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (f)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (g)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (h)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (s)" } ], [ { "kind": "text", "text": "specialize jordan_crt_component_recovery (k)" } ], [ { "kind": "text", "text": "apply jordan_crt_component_recovery" } ], [ { "kind": "text", "text": "exact right0_left" } ], [ { "kind": "text", "text": "exact right1_left" } ], [ { "kind": "text", "text": "exact hacrt_right_right" } ], [ { "kind": "text", "text": "exact hbcrt_right_right" } ], [ { "kind": "text", "text": "exact hsame" } ], [ { "kind": "text", "text": "have rightindex : x1=x7" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x1)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x7)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x4)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x5)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x10)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x11)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_distinct" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_left" } ], [ { "kind": "text", "text": "exact rightsame" } ], [ { "kind": "text", "text": "have hpos : p=v*x+x1" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_witness_witness_witness_right_right_left" } ], [ { "kind": "text", "text": "rewrite leftindex at hpos" } ], [ { "kind": "text", "text": "rewrite rightindex at hpos" } ], [ { "kind": "text", "text": "trans v*x6+x7" } ], [ { "kind": "text", "text": "exact hpos" } ], [ { "kind": "text", "text": "symm" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_witness_witness_witness_right_right_left" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0374": 2, "ND0381": 1, "PD0002": 2, "PD0013": 4 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ p. ∀ z. ∀ f. ∀ g. ∀ h. ∀ s. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(p,u · v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,u · v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,z,h)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,z,s)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(f,g,h,s,k)" }, { "kind": "text", "text": " → p = z" } ] }, "dependencies": [ "jordan_rectangle_crt_actual_entry", "jordan_enumeration_actual_value", "jordan_crt_component_recovery", "jordan_enumeration_distinct" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0046", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_distinct", "original_ha_bundle_verified": true, "proof_bundle_node_id": 331, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro p", "intro z", "intro f", "intro g", "intro h", "intro s", "intro hl", "intro hh", "intro hr", "intro hp", "intro hz", "intro he", "intro hf", "intro hsame", "have ha : exists i j b c d e. ((exists jt_gap_havaluerow. jt_gap_havaluerow+S (i)=(u)) /\\ (((exists jt_gap_havaluecolumn. jt_gap_havaluecolumn+S (j)=(v)) /\\ (((p=(v)*(i)+(j)) /\\ (((((((exists fs_h_jt_havalueleftcode. fs_h_jt_havalueleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_havalueleftcode. A = fs_q_jt_havalueleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_havalueleftscale. fs_h_jt_havalueleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_havalueleftscale. C = fs_q_jt_havalueleftscale * S ((S (i)) * D) + (c))))) /\\ (((((((exists fs_h_jt_havaluerightcode. fs_h_jt_havaluerightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_havaluerightcode. E = fs_q_jt_havaluerightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_havaluerightscale. fs_h_jt_havaluerightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_havaluerightscale. G = fs_q_jt_havaluerightscale * S ((S (j)) * H) + (e))))) /\\ (((((((exists fs_h_jt_havalueoutputcode. fs_h_jt_havalueoutputcode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_havalueoutputcode. P = fs_q_jt_havalueoutputcode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_havalueoutputscale. fs_h_jt_havalueoutputscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_havalueoutputscale. R = fs_q_jt_havalueoutputscale * S ((S (p)) * T) + (g))))) /\\ (((((forall jt_index_havaluecrtbound. (exists jt_gap_havaluecrtboundindex. jt_gap_havaluecrtboundindex+S (jt_index_havaluecrtbound)=(k)) -> exists jt_value_havaluecrtbound. ((((exists fs_h_jt_havaluecrtboundat. fs_h_jt_havaluecrtboundat + S (jt_value_havaluecrtbound) = S ((S (jt_index_havaluecrtbound)) * g)) /\\ exists fs_q_jt_havaluecrtboundat. f = fs_q_jt_havaluecrtboundat * S ((S (jt_index_havaluecrtbound)) * g) + (jt_value_havaluecrtbound))) /\\ (exists jt_gap_havaluecrtboundvalue. jt_gap_havaluecrtboundvalue+S (jt_value_havaluecrtbound)=(m*n)))) /\\ (((forall jt_index_havaluecrtleft jt_left_havaluecrtleft jt_right_havaluecrtleft. (exists jt_gap_havaluecrtleftindex. jt_gap_havaluecrtleftindex+S (jt_index_havaluecrtleft)=(k)) -> (((exists fs_h_jt_havaluecrtleftleft. fs_h_jt_havaluecrtleftleft + S (jt_left_havaluecrtleft) = S ((S (jt_index_havaluecrtleft)) * g)) /\\ exists fs_q_jt_havaluecrtleftleft. f = fs_q_jt_havaluecrtleftleft * S ((S (jt_index_havaluecrtleft)) * g) + (jt_left_havaluecrtleft))) -> (((exists fs_h_jt_havaluecrtleftright. fs_h_jt_havaluecrtleftright + S (jt_right_havaluecrtleft) = S ((S (jt_index_havaluecrtleft)) * c)) /\\ exists fs_q_jt_havaluecrtleftright. b = fs_q_jt_havaluecrtleftright * S ((S (jt_index_havaluecrtleft)) * c) + (jt_right_havaluecrtleft))) -> (exists jt_left_havaluecrtleftmod jt_right_havaluecrtleftmod. (jt_left_havaluecrtleft)+(m)*jt_left_havaluecrtleftmod=(jt_right_havaluecrtleft)+(m)*jt_right_havaluecrtleftmod)) /\\ (forall jt_index_havaluecrtright jt_left_havaluecrtright jt_right_havaluecrtright. (exists jt_gap_havaluecrtrightindex. jt_gap_havaluecrtrightindex+S (jt_index_havaluecrtright)=(k)) -> (((exists fs_h_jt_havaluecrtrightleft. fs_h_jt_havaluecrtrightleft + S (jt_left_havaluecrtright) = S ((S (jt_index_havaluecrtright)) * g)) /\\ exists fs_q_jt_havaluecrtrightleft. f = fs_q_jt_havaluecrtrightleft * S ((S (jt_index_havaluecrtright)) * g) + (jt_left_havaluecrtright))) -> (((exists fs_h_jt_havaluecrtrightright. fs_h_jt_havaluecrtrightright + S (jt_right_havaluecrtright) = S ((S (jt_index_havaluecrtright)) * e)) /\\ exists fs_q_jt_havaluecrtrightright. d = fs_q_jt_havaluecrtrightright * S ((S (jt_index_havaluecrtright)) * e) + (jt_right_havaluecrtright))) -> (exists jt_left_havaluecrtrightmod jt_right_havaluecrtrightmod. (jt_left_havaluecrtright)+(n)*jt_left_havaluecrtrightmod=(jt_right_havaluecrtright)+(n)*jt_right_havaluecrtrightmod)))))) /\\ (forall jt_divisor_havalueprimitive. (exists jt_factor_havalueprimitivemodulus. (m*n)=(jt_divisor_havalueprimitive)*jt_factor_havalueprimitivemodulus) -> (forall jt_index_havalueprimitivecoordinates jt_value_havalueprimitivecoordinates. (exists jt_gap_havalueprimitivecoordinatesindex. jt_gap_havalueprimitivecoordinatesindex+S (jt_index_havalueprimitivecoordinates)=(k)) -> (((exists fs_h_jt_havalueprimitivecoordinatesat. fs_h_jt_havalueprimitivecoordinatesat + S (jt_value_havalueprimitivecoordinates) = S ((S (jt_index_havalueprimitivecoordinates)) * g)) /\\ exists fs_q_jt_havalueprimitivecoordinatesat. f = fs_q_jt_havalueprimitivecoordinatesat * S ((S (jt_index_havalueprimitivecoordinates)) * g) + (jt_value_havalueprimitivecoordinates))) -> (exists jt_factor_havalueprimitivecoordinatesdivides. (jt_value_havalueprimitivecoordinates)=(jt_divisor_havalueprimitive)*jt_factor_havalueprimitivecoordinatesdivides)) -> jt_divisor_havalueprimitive=1))))))))))))))", "specialize jordan_rectangle_crt_actual_entry (m)", "specialize jordan_rectangle_crt_actual_entry (n)", "specialize jordan_rectangle_crt_actual_entry (k)", "specialize jordan_rectangle_crt_actual_entry (A)", "specialize jordan_rectangle_crt_actual_entry (B)", "specialize jordan_rectangle_crt_actual_entry (C)", "specialize jordan_rectangle_crt_actual_entry (D)", "specialize jordan_rectangle_crt_actual_entry (u)", "specialize jordan_rectangle_crt_actual_entry (E)", "specialize jordan_rectangle_crt_actual_entry (F)", "specialize jordan_rectangle_crt_actual_entry (G)", "specialize jordan_rectangle_crt_actual_entry (H)", "specialize jordan_rectangle_crt_actual_entry (v)", "specialize jordan_rectangle_crt_actual_entry (P)", "specialize jordan_rectangle_crt_actual_entry (Q)", "specialize jordan_rectangle_crt_actual_entry (R)", "specialize jordan_rectangle_crt_actual_entry (T)", "specialize jordan_rectangle_crt_actual_entry (u*v)", "specialize jordan_rectangle_crt_actual_entry (p)", "specialize jordan_rectangle_crt_actual_entry (f)", "specialize jordan_rectangle_crt_actual_entry (g)", "apply jordan_rectangle_crt_actual_entry", "exact hr", "exact hp", "exact he", "cases ha", "cases ha_witness", "cases ha_witness_witness", "cases ha_witness_witness_witness", "cases ha_witness_witness_witness_witness", "cases ha_witness_witness_witness_witness_witness", "cases ha_witness_witness_witness_witness_witness_witness", "cases ha_witness_witness_witness_witness_witness_witness_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "have hacrt : ((forall jt_index_hacrtbound. (exists jt_gap_hacrtboundindex. jt_gap_hacrtboundindex+S (jt_index_hacrtbound)=(k)) -> exists jt_value_hacrtbound. ((((exists fs_h_jt_hacrtboundat. fs_h_jt_hacrtboundat + S (jt_value_hacrtbound) = S ((S (jt_index_hacrtbound)) * g)) /\\ exists fs_q_jt_hacrtboundat. f = fs_q_jt_hacrtboundat * S ((S (jt_index_hacrtbound)) * g) + (jt_value_hacrtbound))) /\\ (exists jt_gap_hacrtboundvalue. jt_gap_hacrtboundvalue+S (jt_value_hacrtbound)=(m*n)))) /\\ (((forall jt_index_hacrtleft jt_left_hacrtleft jt_right_hacrtleft. (exists jt_gap_hacrtleftindex. jt_gap_hacrtleftindex+S (jt_index_hacrtleft)=(k)) -> (((exists fs_h_jt_hacrtleftleft. fs_h_jt_hacrtleftleft + S (jt_left_hacrtleft) = S ((S (jt_index_hacrtleft)) * g)) /\\ exists fs_q_jt_hacrtleftleft. f = fs_q_jt_hacrtleftleft * S ((S (jt_index_hacrtleft)) * g) + (jt_left_hacrtleft))) -> (((exists fs_h_jt_hacrtleftright. fs_h_jt_hacrtleftright + S (jt_right_hacrtleft) = S ((S (jt_index_hacrtleft)) * x3)) /\\ exists fs_q_jt_hacrtleftright. x2 = fs_q_jt_hacrtleftright * S ((S (jt_index_hacrtleft)) * x3) + (jt_right_hacrtleft))) -> (exists jt_left_hacrtleftmod jt_right_hacrtleftmod. (jt_left_hacrtleft)+(m)*jt_left_hacrtleftmod=(jt_right_hacrtleft)+(m)*jt_right_hacrtleftmod)) /\\ (forall jt_index_hacrtright jt_left_hacrtright jt_right_hacrtright. (exists jt_gap_hacrtrightindex. jt_gap_hacrtrightindex+S (jt_index_hacrtright)=(k)) -> (((exists fs_h_jt_hacrtrightleft. fs_h_jt_hacrtrightleft + S (jt_left_hacrtright) = S ((S (jt_index_hacrtright)) * g)) /\\ exists fs_q_jt_hacrtrightleft. f = fs_q_jt_hacrtrightleft * S ((S (jt_index_hacrtright)) * g) + (jt_left_hacrtright))) -> (((exists fs_h_jt_hacrtrightright. fs_h_jt_hacrtrightright + S (jt_right_hacrtright) = S ((S (jt_index_hacrtright)) * x5)) /\\ exists fs_q_jt_hacrtrightright. x4 = fs_q_jt_hacrtrightright * S ((S (jt_index_hacrtright)) * x5) + (jt_right_hacrtright))) -> (exists jt_left_hacrtrightmod jt_right_hacrtrightmod. (jt_left_hacrtright)+(n)*jt_left_hacrtrightmod=(jt_right_hacrtright)+(n)*jt_right_hacrtrightmod)))))", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "cases hacrt", "cases hacrt_right", "have hb : exists i j b c d e. ((exists jt_gap_hbvaluerow. jt_gap_hbvaluerow+S (i)=(u)) /\\ (((exists jt_gap_hbvaluecolumn. jt_gap_hbvaluecolumn+S (j)=(v)) /\\ (((z=(v)*(i)+(j)) /\\ (((((((exists fs_h_jt_hbvalueleftcode. fs_h_jt_hbvalueleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_hbvalueleftcode. A = fs_q_jt_hbvalueleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_hbvalueleftscale. fs_h_jt_hbvalueleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_hbvalueleftscale. C = fs_q_jt_hbvalueleftscale * S ((S (i)) * D) + (c))))) /\\ (((((((exists fs_h_jt_hbvaluerightcode. fs_h_jt_hbvaluerightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_hbvaluerightcode. E = fs_q_jt_hbvaluerightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_hbvaluerightscale. fs_h_jt_hbvaluerightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_hbvaluerightscale. G = fs_q_jt_hbvaluerightscale * S ((S (j)) * H) + (e))))) /\\ (((((((exists fs_h_jt_hbvalueoutputcode. fs_h_jt_hbvalueoutputcode + S (h) = S ((S (z)) * Q)) /\\ exists fs_q_jt_hbvalueoutputcode. P = fs_q_jt_hbvalueoutputcode * S ((S (z)) * Q) + (h))) /\\ (((exists fs_h_jt_hbvalueoutputscale. fs_h_jt_hbvalueoutputscale + S (s) = S ((S (z)) * T)) /\\ exists fs_q_jt_hbvalueoutputscale. R = fs_q_jt_hbvalueoutputscale * S ((S (z)) * T) + (s))))) /\\ (((((forall jt_index_hbvaluecrtbound. (exists jt_gap_hbvaluecrtboundindex. jt_gap_hbvaluecrtboundindex+S (jt_index_hbvaluecrtbound)=(k)) -> exists jt_value_hbvaluecrtbound. ((((exists fs_h_jt_hbvaluecrtboundat. fs_h_jt_hbvaluecrtboundat + S (jt_value_hbvaluecrtbound) = S ((S (jt_index_hbvaluecrtbound)) * s)) /\\ exists fs_q_jt_hbvaluecrtboundat. h = fs_q_jt_hbvaluecrtboundat * S ((S (jt_index_hbvaluecrtbound)) * s) + (jt_value_hbvaluecrtbound))) /\\ (exists jt_gap_hbvaluecrtboundvalue. jt_gap_hbvaluecrtboundvalue+S (jt_value_hbvaluecrtbound)=(m*n)))) /\\ (((forall jt_index_hbvaluecrtleft jt_left_hbvaluecrtleft jt_right_hbvaluecrtleft. (exists jt_gap_hbvaluecrtleftindex. jt_gap_hbvaluecrtleftindex+S (jt_index_hbvaluecrtleft)=(k)) -> (((exists fs_h_jt_hbvaluecrtleftleft. fs_h_jt_hbvaluecrtleftleft + S (jt_left_hbvaluecrtleft) = S ((S (jt_index_hbvaluecrtleft)) * s)) /\\ exists fs_q_jt_hbvaluecrtleftleft. h = fs_q_jt_hbvaluecrtleftleft * S ((S (jt_index_hbvaluecrtleft)) * s) + (jt_left_hbvaluecrtleft))) -> (((exists fs_h_jt_hbvaluecrtleftright. fs_h_jt_hbvaluecrtleftright + S (jt_right_hbvaluecrtleft) = S ((S (jt_index_hbvaluecrtleft)) * c)) /\\ exists fs_q_jt_hbvaluecrtleftright. b = fs_q_jt_hbvaluecrtleftright * S ((S (jt_index_hbvaluecrtleft)) * c) + (jt_right_hbvaluecrtleft))) -> (exists jt_left_hbvaluecrtleftmod jt_right_hbvaluecrtleftmod. (jt_left_hbvaluecrtleft)+(m)*jt_left_hbvaluecrtleftmod=(jt_right_hbvaluecrtleft)+(m)*jt_right_hbvaluecrtleftmod)) /\\ (forall jt_index_hbvaluecrtright jt_left_hbvaluecrtright jt_right_hbvaluecrtright. (exists jt_gap_hbvaluecrtrightindex. jt_gap_hbvaluecrtrightindex+S (jt_index_hbvaluecrtright)=(k)) -> (((exists fs_h_jt_hbvaluecrtrightleft. fs_h_jt_hbvaluecrtrightleft + S (jt_left_hbvaluecrtright) = S ((S (jt_index_hbvaluecrtright)) * s)) /\\ exists fs_q_jt_hbvaluecrtrightleft. h = fs_q_jt_hbvaluecrtrightleft * S ((S (jt_index_hbvaluecrtright)) * s) + (jt_left_hbvaluecrtright))) -> (((exists fs_h_jt_hbvaluecrtrightright. fs_h_jt_hbvaluecrtrightright + S (jt_right_hbvaluecrtright) = S ((S (jt_index_hbvaluecrtright)) * e)) /\\ exists fs_q_jt_hbvaluecrtrightright. d = fs_q_jt_hbvaluecrtrightright * S ((S (jt_index_hbvaluecrtright)) * e) + (jt_right_hbvaluecrtright))) -> (exists jt_left_hbvaluecrtrightmod jt_right_hbvaluecrtrightmod. (jt_left_hbvaluecrtright)+(n)*jt_left_hbvaluecrtrightmod=(jt_right_hbvaluecrtright)+(n)*jt_right_hbvaluecrtrightmod)))))) /\\ (forall jt_divisor_hbvalueprimitive. (exists jt_factor_hbvalueprimitivemodulus. (m*n)=(jt_divisor_hbvalueprimitive)*jt_factor_hbvalueprimitivemodulus) -> (forall jt_index_hbvalueprimitivecoordinates jt_value_hbvalueprimitivecoordinates. (exists jt_gap_hbvalueprimitivecoordinatesindex. jt_gap_hbvalueprimitivecoordinatesindex+S (jt_index_hbvalueprimitivecoordinates)=(k)) -> (((exists fs_h_jt_hbvalueprimitivecoordinatesat. fs_h_jt_hbvalueprimitivecoordinatesat + S (jt_value_hbvalueprimitivecoordinates) = S ((S (jt_index_hbvalueprimitivecoordinates)) * s)) /\\ exists fs_q_jt_hbvalueprimitivecoordinatesat. h = fs_q_jt_hbvalueprimitivecoordinatesat * S ((S (jt_index_hbvalueprimitivecoordinates)) * s) + (jt_value_hbvalueprimitivecoordinates))) -> (exists jt_factor_hbvalueprimitivecoordinatesdivides. (jt_value_hbvalueprimitivecoordinates)=(jt_divisor_hbvalueprimitive)*jt_factor_hbvalueprimitivecoordinatesdivides)) -> jt_divisor_hbvalueprimitive=1))))))))))))))", "specialize jordan_rectangle_crt_actual_entry (m)", "specialize jordan_rectangle_crt_actual_entry (n)", "specialize jordan_rectangle_crt_actual_entry (k)", "specialize jordan_rectangle_crt_actual_entry (A)", "specialize jordan_rectangle_crt_actual_entry (B)", "specialize jordan_rectangle_crt_actual_entry (C)", "specialize jordan_rectangle_crt_actual_entry (D)", "specialize jordan_rectangle_crt_actual_entry (u)", "specialize jordan_rectangle_crt_actual_entry (E)", "specialize jordan_rectangle_crt_actual_entry (F)", "specialize jordan_rectangle_crt_actual_entry (G)", "specialize jordan_rectangle_crt_actual_entry (H)", "specialize jordan_rectangle_crt_actual_entry (v)", "specialize jordan_rectangle_crt_actual_entry (P)", "specialize jordan_rectangle_crt_actual_entry (Q)", "specialize jordan_rectangle_crt_actual_entry (R)", "specialize jordan_rectangle_crt_actual_entry (T)", "specialize jordan_rectangle_crt_actual_entry (u*v)", "specialize jordan_rectangle_crt_actual_entry (z)", "specialize jordan_rectangle_crt_actual_entry (h)", "specialize jordan_rectangle_crt_actual_entry (s)", "apply jordan_rectangle_crt_actual_entry", "exact hr", "exact hz", "exact hf", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_witness", "cases hb_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness_witness", "cases hb_witness_witness_witness_witness_witness_witness_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "have hbcrt : ((forall jt_index_hbcrtbound. (exists jt_gap_hbcrtboundindex. jt_gap_hbcrtboundindex+S (jt_index_hbcrtbound)=(k)) -> exists jt_value_hbcrtbound. ((((exists fs_h_jt_hbcrtboundat. fs_h_jt_hbcrtboundat + S (jt_value_hbcrtbound) = S ((S (jt_index_hbcrtbound)) * s)) /\\ exists fs_q_jt_hbcrtboundat. h = fs_q_jt_hbcrtboundat * S ((S (jt_index_hbcrtbound)) * s) + (jt_value_hbcrtbound))) /\\ (exists jt_gap_hbcrtboundvalue. jt_gap_hbcrtboundvalue+S (jt_value_hbcrtbound)=(m*n)))) /\\ (((forall jt_index_hbcrtleft jt_left_hbcrtleft jt_right_hbcrtleft. (exists jt_gap_hbcrtleftindex. jt_gap_hbcrtleftindex+S (jt_index_hbcrtleft)=(k)) -> (((exists fs_h_jt_hbcrtleftleft. fs_h_jt_hbcrtleftleft + S (jt_left_hbcrtleft) = S ((S (jt_index_hbcrtleft)) * s)) /\\ exists fs_q_jt_hbcrtleftleft. h = fs_q_jt_hbcrtleftleft * S ((S (jt_index_hbcrtleft)) * s) + (jt_left_hbcrtleft))) -> (((exists fs_h_jt_hbcrtleftright. fs_h_jt_hbcrtleftright + S (jt_right_hbcrtleft) = S ((S (jt_index_hbcrtleft)) * x9)) /\\ exists fs_q_jt_hbcrtleftright. x8 = fs_q_jt_hbcrtleftright * S ((S (jt_index_hbcrtleft)) * x9) + (jt_right_hbcrtleft))) -> (exists jt_left_hbcrtleftmod jt_right_hbcrtleftmod. (jt_left_hbcrtleft)+(m)*jt_left_hbcrtleftmod=(jt_right_hbcrtleft)+(m)*jt_right_hbcrtleftmod)) /\\ (forall jt_index_hbcrtright jt_left_hbcrtright jt_right_hbcrtright. (exists jt_gap_hbcrtrightindex. jt_gap_hbcrtrightindex+S (jt_index_hbcrtright)=(k)) -> (((exists fs_h_jt_hbcrtrightleft. fs_h_jt_hbcrtrightleft + S (jt_left_hbcrtright) = S ((S (jt_index_hbcrtright)) * s)) /\\ exists fs_q_jt_hbcrtrightleft. h = fs_q_jt_hbcrtrightleft * S ((S (jt_index_hbcrtright)) * s) + (jt_left_hbcrtright))) -> (((exists fs_h_jt_hbcrtrightright. fs_h_jt_hbcrtrightright + S (jt_right_hbcrtright) = S ((S (jt_index_hbcrtright)) * x11)) /\\ exists fs_q_jt_hbcrtrightright. x10 = fs_q_jt_hbcrtrightright * S ((S (jt_index_hbcrtright)) * x11) + (jt_right_hbcrtright))) -> (exists jt_left_hbcrtrightmod jt_right_hbcrtrightmod. (jt_left_hbcrtright)+(n)*jt_left_hbcrtrightmod=(jt_right_hbcrtright)+(n)*jt_right_hbcrtrightmod)))))", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "cases hbcrt", "cases hbcrt_right", "have left0 : ((forall jt_index_left0bound. (exists jt_gap_left0boundindex. jt_gap_left0boundindex+S (jt_index_left0bound)=(k)) -> exists jt_value_left0bound. ((((exists fs_h_jt_left0boundat. fs_h_jt_left0boundat + S (jt_value_left0bound) = S ((S (jt_index_left0bound)) * x3)) /\\ exists fs_q_jt_left0boundat. x2 = fs_q_jt_left0boundat * S ((S (jt_index_left0bound)) * x3) + (jt_value_left0bound))) /\\ (exists jt_gap_left0boundvalue. jt_gap_left0boundvalue+S (jt_value_left0bound)=(m)))) /\\ (forall jt_divisor_left0primitive. (exists jt_factor_left0primitivemodulus. (m)=(jt_divisor_left0primitive)*jt_factor_left0primitivemodulus) -> (forall jt_index_left0primitivecoordinates jt_value_left0primitivecoordinates. (exists jt_gap_left0primitivecoordinatesindex. jt_gap_left0primitivecoordinatesindex+S (jt_index_left0primitivecoordinates)=(k)) -> (((exists fs_h_jt_left0primitivecoordinatesat. fs_h_jt_left0primitivecoordinatesat + S (jt_value_left0primitivecoordinates) = S ((S (jt_index_left0primitivecoordinates)) * x3)) /\\ exists fs_q_jt_left0primitivecoordinatesat. x2 = fs_q_jt_left0primitivecoordinatesat * S ((S (jt_index_left0primitivecoordinates)) * x3) + (jt_value_left0primitivecoordinates))) -> (exists jt_factor_left0primitivecoordinatesdivides. (jt_value_left0primitivecoordinates)=(jt_divisor_left0primitive)*jt_factor_left0primitivecoordinatesdivides)) -> jt_divisor_left0primitive=1))", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (m)", "specialize jordan_enumeration_actual_value (A)", "specialize jordan_enumeration_actual_value (B)", "specialize jordan_enumeration_actual_value (C)", "specialize jordan_enumeration_actual_value (D)", "specialize jordan_enumeration_actual_value (u)", "specialize jordan_enumeration_actual_value (x)", "specialize jordan_enumeration_actual_value (x2)", "specialize jordan_enumeration_actual_value (x3)", "apply jordan_enumeration_actual_value", "exact hl", "exact ha_witness_witness_witness_witness_witness_witness_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_left", "cases left0", "have left1 : ((forall jt_index_left1bound. (exists jt_gap_left1boundindex. jt_gap_left1boundindex+S (jt_index_left1bound)=(k)) -> exists jt_value_left1bound. ((((exists fs_h_jt_left1boundat. fs_h_jt_left1boundat + S (jt_value_left1bound) = S ((S (jt_index_left1bound)) * x9)) /\\ exists fs_q_jt_left1boundat. x8 = fs_q_jt_left1boundat * S ((S (jt_index_left1bound)) * x9) + (jt_value_left1bound))) /\\ (exists jt_gap_left1boundvalue. jt_gap_left1boundvalue+S (jt_value_left1bound)=(m)))) /\\ (forall jt_divisor_left1primitive. (exists jt_factor_left1primitivemodulus. (m)=(jt_divisor_left1primitive)*jt_factor_left1primitivemodulus) -> (forall jt_index_left1primitivecoordinates jt_value_left1primitivecoordinates. (exists jt_gap_left1primitivecoordinatesindex. jt_gap_left1primitivecoordinatesindex+S (jt_index_left1primitivecoordinates)=(k)) -> (((exists fs_h_jt_left1primitivecoordinatesat. fs_h_jt_left1primitivecoordinatesat + S (jt_value_left1primitivecoordinates) = S ((S (jt_index_left1primitivecoordinates)) * x9)) /\\ exists fs_q_jt_left1primitivecoordinatesat. x8 = fs_q_jt_left1primitivecoordinatesat * S ((S (jt_index_left1primitivecoordinates)) * x9) + (jt_value_left1primitivecoordinates))) -> (exists jt_factor_left1primitivecoordinatesdivides. (jt_value_left1primitivecoordinates)=(jt_divisor_left1primitive)*jt_factor_left1primitivecoordinatesdivides)) -> jt_divisor_left1primitive=1))", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (m)", "specialize jordan_enumeration_actual_value (A)", "specialize jordan_enumeration_actual_value (B)", "specialize jordan_enumeration_actual_value (C)", "specialize jordan_enumeration_actual_value (D)", "specialize jordan_enumeration_actual_value (u)", "specialize jordan_enumeration_actual_value (x6)", "specialize jordan_enumeration_actual_value (x8)", "specialize jordan_enumeration_actual_value (x9)", "apply jordan_enumeration_actual_value", "exact hl", "exact hb_witness_witness_witness_witness_witness_witness_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_left", "cases left1", "have leftsame : forall jt_index_leftsame jt_left_leftsame jt_right_leftsame. (exists jt_gap_leftsameindex. jt_gap_leftsameindex+S (jt_index_leftsame)=(k)) -> (((exists fs_h_jt_leftsameleft. fs_h_jt_leftsameleft + S (jt_left_leftsame) = S ((S (jt_index_leftsame)) * x3)) /\\ exists fs_q_jt_leftsameleft. x2 = fs_q_jt_leftsameleft * S ((S (jt_index_leftsame)) * x3) + (jt_left_leftsame))) -> (((exists fs_h_jt_leftsameright. fs_h_jt_leftsameright + S (jt_right_leftsame) = S ((S (jt_index_leftsame)) * x9)) /\\ exists fs_q_jt_leftsameright. x8 = fs_q_jt_leftsameright * S ((S (jt_index_leftsame)) * x9) + (jt_right_leftsame))) -> jt_left_leftsame=jt_right_leftsame", "specialize jordan_crt_component_recovery (m)", "specialize jordan_crt_component_recovery (x2)", "specialize jordan_crt_component_recovery (x3)", "specialize jordan_crt_component_recovery (x8)", "specialize jordan_crt_component_recovery (x9)", "specialize jordan_crt_component_recovery (f)", "specialize jordan_crt_component_recovery (g)", "specialize jordan_crt_component_recovery (h)", "specialize jordan_crt_component_recovery (s)", "specialize jordan_crt_component_recovery (k)", "apply jordan_crt_component_recovery", "exact left0_left", "exact left1_left", "exact hacrt_right_left", "exact hbcrt_right_left", "exact hsame", "have leftindex : x=x6", "specialize jordan_enumeration_distinct (k)", "specialize jordan_enumeration_distinct (m)", "specialize jordan_enumeration_distinct (A)", "specialize jordan_enumeration_distinct (B)", "specialize jordan_enumeration_distinct (C)", "specialize jordan_enumeration_distinct (D)", "specialize jordan_enumeration_distinct (u)", "specialize jordan_enumeration_distinct (x)", "specialize jordan_enumeration_distinct (x6)", "specialize jordan_enumeration_distinct (x2)", "specialize jordan_enumeration_distinct (x3)", "specialize jordan_enumeration_distinct (x8)", "specialize jordan_enumeration_distinct (x9)", "apply jordan_enumeration_distinct", "exact hl", "exact ha_witness_witness_witness_witness_witness_witness_left", "exact hb_witness_witness_witness_witness_witness_witness_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_left", "exact leftsame", "have right0 : ((forall jt_index_right0bound. (exists jt_gap_right0boundindex. jt_gap_right0boundindex+S (jt_index_right0bound)=(k)) -> exists jt_value_right0bound. ((((exists fs_h_jt_right0boundat. fs_h_jt_right0boundat + S (jt_value_right0bound) = S ((S (jt_index_right0bound)) * x5)) /\\ exists fs_q_jt_right0boundat. x4 = fs_q_jt_right0boundat * S ((S (jt_index_right0bound)) * x5) + (jt_value_right0bound))) /\\ (exists jt_gap_right0boundvalue. jt_gap_right0boundvalue+S (jt_value_right0bound)=(n)))) /\\ (forall jt_divisor_right0primitive. (exists jt_factor_right0primitivemodulus. (n)=(jt_divisor_right0primitive)*jt_factor_right0primitivemodulus) -> (forall jt_index_right0primitivecoordinates jt_value_right0primitivecoordinates. (exists jt_gap_right0primitivecoordinatesindex. jt_gap_right0primitivecoordinatesindex+S (jt_index_right0primitivecoordinates)=(k)) -> (((exists fs_h_jt_right0primitivecoordinatesat. fs_h_jt_right0primitivecoordinatesat + S (jt_value_right0primitivecoordinates) = S ((S (jt_index_right0primitivecoordinates)) * x5)) /\\ exists fs_q_jt_right0primitivecoordinatesat. x4 = fs_q_jt_right0primitivecoordinatesat * S ((S (jt_index_right0primitivecoordinates)) * x5) + (jt_value_right0primitivecoordinates))) -> (exists jt_factor_right0primitivecoordinatesdivides. (jt_value_right0primitivecoordinates)=(jt_divisor_right0primitive)*jt_factor_right0primitivecoordinatesdivides)) -> jt_divisor_right0primitive=1))", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (n)", "specialize jordan_enumeration_actual_value (E)", "specialize jordan_enumeration_actual_value (F)", "specialize jordan_enumeration_actual_value (G)", "specialize jordan_enumeration_actual_value (H)", "specialize jordan_enumeration_actual_value (v)", "specialize jordan_enumeration_actual_value (x1)", "specialize jordan_enumeration_actual_value (x4)", "specialize jordan_enumeration_actual_value (x5)", "apply jordan_enumeration_actual_value", "exact hh", "exact ha_witness_witness_witness_witness_witness_witness_right_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "cases right0", "have right1 : ((forall jt_index_right1bound. (exists jt_gap_right1boundindex. jt_gap_right1boundindex+S (jt_index_right1bound)=(k)) -> exists jt_value_right1bound. ((((exists fs_h_jt_right1boundat. fs_h_jt_right1boundat + S (jt_value_right1bound) = S ((S (jt_index_right1bound)) * x11)) /\\ exists fs_q_jt_right1boundat. x10 = fs_q_jt_right1boundat * S ((S (jt_index_right1bound)) * x11) + (jt_value_right1bound))) /\\ (exists jt_gap_right1boundvalue. jt_gap_right1boundvalue+S (jt_value_right1bound)=(n)))) /\\ (forall jt_divisor_right1primitive. (exists jt_factor_right1primitivemodulus. (n)=(jt_divisor_right1primitive)*jt_factor_right1primitivemodulus) -> (forall jt_index_right1primitivecoordinates jt_value_right1primitivecoordinates. (exists jt_gap_right1primitivecoordinatesindex. jt_gap_right1primitivecoordinatesindex+S (jt_index_right1primitivecoordinates)=(k)) -> (((exists fs_h_jt_right1primitivecoordinatesat. fs_h_jt_right1primitivecoordinatesat + S (jt_value_right1primitivecoordinates) = S ((S (jt_index_right1primitivecoordinates)) * x11)) /\\ exists fs_q_jt_right1primitivecoordinatesat. x10 = fs_q_jt_right1primitivecoordinatesat * S ((S (jt_index_right1primitivecoordinates)) * x11) + (jt_value_right1primitivecoordinates))) -> (exists jt_factor_right1primitivecoordinatesdivides. (jt_value_right1primitivecoordinates)=(jt_divisor_right1primitive)*jt_factor_right1primitivecoordinatesdivides)) -> jt_divisor_right1primitive=1))", "specialize jordan_enumeration_actual_value (k)", "specialize jordan_enumeration_actual_value (n)", "specialize jordan_enumeration_actual_value (E)", "specialize jordan_enumeration_actual_value (F)", "specialize jordan_enumeration_actual_value (G)", "specialize jordan_enumeration_actual_value (H)", "specialize jordan_enumeration_actual_value (v)", "specialize jordan_enumeration_actual_value (x7)", "specialize jordan_enumeration_actual_value (x10)", "specialize jordan_enumeration_actual_value (x11)", "apply jordan_enumeration_actual_value", "exact hh", "exact hb_witness_witness_witness_witness_witness_witness_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "cases right1", "have rightsame : forall jt_index_rightsame jt_left_rightsame jt_right_rightsame. (exists jt_gap_rightsameindex. jt_gap_rightsameindex+S (jt_index_rightsame)=(k)) -> (((exists fs_h_jt_rightsameleft. fs_h_jt_rightsameleft + S (jt_left_rightsame) = S ((S (jt_index_rightsame)) * x5)) /\\ exists fs_q_jt_rightsameleft. x4 = fs_q_jt_rightsameleft * S ((S (jt_index_rightsame)) * x5) + (jt_left_rightsame))) -> (((exists fs_h_jt_rightsameright. fs_h_jt_rightsameright + S (jt_right_rightsame) = S ((S (jt_index_rightsame)) * x11)) /\\ exists fs_q_jt_rightsameright. x10 = fs_q_jt_rightsameright * S ((S (jt_index_rightsame)) * x11) + (jt_right_rightsame))) -> jt_left_rightsame=jt_right_rightsame", "specialize jordan_crt_component_recovery (n)", "specialize jordan_crt_component_recovery (x4)", "specialize jordan_crt_component_recovery (x5)", "specialize jordan_crt_component_recovery (x10)", "specialize jordan_crt_component_recovery (x11)", "specialize jordan_crt_component_recovery (f)", "specialize jordan_crt_component_recovery (g)", "specialize jordan_crt_component_recovery (h)", "specialize jordan_crt_component_recovery (s)", "specialize jordan_crt_component_recovery (k)", "apply jordan_crt_component_recovery", "exact right0_left", "exact right1_left", "exact hacrt_right_right", "exact hbcrt_right_right", "exact hsame", "have rightindex : x1=x7", "specialize jordan_enumeration_distinct (k)", "specialize jordan_enumeration_distinct (n)", "specialize jordan_enumeration_distinct (E)", "specialize jordan_enumeration_distinct (F)", "specialize jordan_enumeration_distinct (G)", "specialize jordan_enumeration_distinct (H)", "specialize jordan_enumeration_distinct (v)", "specialize jordan_enumeration_distinct (x1)", "specialize jordan_enumeration_distinct (x7)", "specialize jordan_enumeration_distinct (x4)", "specialize jordan_enumeration_distinct (x5)", "specialize jordan_enumeration_distinct (x10)", "specialize jordan_enumeration_distinct (x11)", "apply jordan_enumeration_distinct", "exact hh", "exact ha_witness_witness_witness_witness_witness_witness_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_left", "exact ha_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact hb_witness_witness_witness_witness_witness_witness_right_right_right_right_left", "exact rightsame", "have hpos : p=v*x+x1", "exact ha_witness_witness_witness_witness_witness_witness_right_right_left", "rewrite leftindex at hpos", "rewrite rightindex at hpos", "trans v*x6+x7", "exact hpos", "symm", "exact hb_witness_witness_witness_witness_witness_witness_right_right_left" ], "script_sha256": "e6f49493eb94d80b5bb3196f7a311eddb47b22d035d90b95c12c8ffd83132bfd", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "e6f49493eb94d80b5bb3196f7a311eddb47b22d035d90b95c12c8ffd83132bfd", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "cab9c7262834a02f7624af6c4d5ceebdb6eb91a1e71ff58f0d86fbf218c4d239" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v P Q R T p z f g h s. (((forall jt_i_enumleft. (exists jt_gap_enumleftsoundindex. jt_gap_enumleftsoundindex+S (jt_i_enumleft)=(u)) -> exists jt_b_enumleft jt_c_enumleft. ((((((exists fs_h_jt_enumleftsoundcode. fs_h_jt_enumleftsoundcode + S (jt_b_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftsoundcode. A = fs_q_jt_enumleftsoundcode * S ((S (jt_i_enumleft)) * B) + (jt_b_enumleft))) /\\ (((exists fs_h_jt_enumleftsoundscale. fs_h_jt_enumleftsoundscale + S (jt_c_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftsoundscale. C = fs_q_jt_enumleftsoundscale * S ((S (jt_i_enumleft)) * D) + (jt_c_enumleft))))) /\\ (((forall jt_index_enumleftbound. (exists jt_gap_enumleftboundindex. jt_gap_enumleftboundindex+S (jt_index_enumleftbound)=(k)) -> exists jt_value_enumleftbound. ((((exists fs_h_jt_enumleftboundat. fs_h_jt_enumleftboundat + S (jt_value_enumleftbound) = S ((S (jt_index_enumleftbound)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftboundat. jt_b_enumleft = fs_q_jt_enumleftboundat * S ((S (jt_index_enumleftbound)) * jt_c_enumleft) + (jt_value_enumleftbound))) /\\ (exists jt_gap_enumleftboundvalue. jt_gap_enumleftboundvalue+S (jt_value_enumleftbound)=(m)))) /\\ (forall jt_divisor_enumleftprimitive. (exists jt_factor_enumleftprimitivemodulus. (m)=(jt_divisor_enumleftprimitive)*jt_factor_enumleftprimitivemodulus) -> (forall jt_index_enumleftprimitivecoordinates jt_value_enumleftprimitivecoordinates. (exists jt_gap_enumleftprimitivecoordinatesindex. jt_gap_enumleftprimitivecoordinatesindex+S (jt_index_enumleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumleftprimitivecoordinatesat. fs_h_jt_enumleftprimitivecoordinatesat + S (jt_value_enumleftprimitivecoordinates) = S ((S (jt_index_enumleftprimitivecoordinates)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftprimitivecoordinatesat. jt_b_enumleft = fs_q_jt_enumleftprimitivecoordinatesat * S ((S (jt_index_enumleftprimitivecoordinates)) * jt_c_enumleft) + (jt_value_enumleftprimitivecoordinates))) -> (exists jt_factor_enumleftprimitivecoordinatesdivides. (jt_value_enumleftprimitivecoordinates)=(jt_divisor_enumleftprimitive)*jt_factor_enumleftprimitivecoordinatesdivides)) -> jt_divisor_enumleftprimitive=1))))) /\\ (((forall jt_b_enumleft jt_c_enumleft. (forall jt_index_enumleftinputbound. (exists jt_gap_enumleftinputboundindex. jt_gap_enumleftinputboundindex+S (jt_index_enumleftinputbound)=(k)) -> exists jt_value_enumleftinputbound. ((((exists fs_h_jt_enumleftinputboundat. fs_h_jt_enumleftinputboundat + S (jt_value_enumleftinputbound) = S ((S (jt_index_enumleftinputbound)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftinputboundat. jt_b_enumleft = fs_q_jt_enumleftinputboundat * S ((S (jt_index_enumleftinputbound)) * jt_c_enumleft) + (jt_value_enumleftinputbound))) /\\ (exists jt_gap_enumleftinputboundvalue. jt_gap_enumleftinputboundvalue+S (jt_value_enumleftinputbound)=(m)))) -> (forall jt_divisor_enumleftinputprimitive. (exists jt_factor_enumleftinputprimitivemodulus. (m)=(jt_divisor_enumleftinputprimitive)*jt_factor_enumleftinputprimitivemodulus) -> (forall jt_index_enumleftinputprimitivecoordinates jt_value_enumleftinputprimitivecoordinates. (exists jt_gap_enumleftinputprimitivecoordinatesindex. jt_gap_enumleftinputprimitivecoordinatesindex+S (jt_index_enumleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumleftinputprimitivecoordinatesat. fs_h_jt_enumleftinputprimitivecoordinatesat + S (jt_value_enumleftinputprimitivecoordinates) = S ((S (jt_index_enumleftinputprimitivecoordinates)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftinputprimitivecoordinatesat. jt_b_enumleft = fs_q_jt_enumleftinputprimitivecoordinatesat * S ((S (jt_index_enumleftinputprimitivecoordinates)) * jt_c_enumleft) + (jt_value_enumleftinputprimitivecoordinates))) -> (exists jt_factor_enumleftinputprimitivecoordinatesdivides. (jt_value_enumleftinputprimitivecoordinates)=(jt_divisor_enumleftinputprimitive)*jt_factor_enumleftinputprimitivecoordinatesdivides)) -> jt_divisor_enumleftinputprimitive=1) -> exists jt_i_enumleft jt_d_enumleft jt_e_enumleft. ((exists jt_gap_enumleftcompleteindex. jt_gap_enumleftcompleteindex+S (jt_i_enumleft)=(u)) /\\ (((((((exists fs_h_jt_enumleftcompletecode. fs_h_jt_enumleftcompletecode + S (jt_d_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftcompletecode. A = fs_q_jt_enumleftcompletecode * S ((S (jt_i_enumleft)) * B) + (jt_d_enumleft))) /\\ (((exists fs_h_jt_enumleftcompletescale. fs_h_jt_enumleftcompletescale + S (jt_e_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftcompletescale. C = fs_q_jt_enumleftcompletescale * S ((S (jt_i_enumleft)) * D) + (jt_e_enumleft))))) /\\ (forall jt_index_enumleftrepresented jt_left_enumleftrepresented jt_right_enumleftrepresented. (exists jt_gap_enumleftrepresentedindex. jt_gap_enumleftrepresentedindex+S (jt_index_enumleftrepresented)=(k)) -> (((exists fs_h_jt_enumleftrepresentedleft. fs_h_jt_enumleftrepresentedleft + S (jt_left_enumleftrepresented) = S ((S (jt_index_enumleftrepresented)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftrepresentedleft. jt_b_enumleft = fs_q_jt_enumleftrepresentedleft * S ((S (jt_index_enumleftrepresented)) * jt_c_enumleft) + (jt_left_enumleftrepresented))) -> (((exists fs_h_jt_enumleftrepresentedright. fs_h_jt_enumleftrepresentedright + S (jt_right_enumleftrepresented) = S ((S (jt_index_enumleftrepresented)) * jt_e_enumleft)) /\\ exists fs_q_jt_enumleftrepresentedright. jt_d_enumleft = fs_q_jt_enumleftrepresentedright * S ((S (jt_index_enumleftrepresented)) * jt_e_enumleft) + (jt_right_enumleftrepresented))) -> jt_left_enumleftrepresented=jt_right_enumleftrepresented))))) /\\ (forall jt_i_enumleft jt_h_enumleft jt_b_enumleft jt_c_enumleft jt_d_enumleft jt_e_enumleft. (exists jt_gap_enumleftfirstindex. jt_gap_enumleftfirstindex+S (jt_i_enumleft)=(u)) -> (exists jt_gap_enumleftsecondindex. jt_gap_enumleftsecondindex+S (jt_h_enumleft)=(u)) -> (((((exists fs_h_jt_enumleftfirstcode. fs_h_jt_enumleftfirstcode + S (jt_b_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftfirstcode. A = fs_q_jt_enumleftfirstcode * S ((S (jt_i_enumleft)) * B) + (jt_b_enumleft))) /\\ (((exists fs_h_jt_enumleftfirstscale. fs_h_jt_enumleftfirstscale + S (jt_c_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftfirstscale. C = fs_q_jt_enumleftfirstscale * S ((S (jt_i_enumleft)) * D) + (jt_c_enumleft))))) -> (((((exists fs_h_jt_enumleftsecondcode. fs_h_jt_enumleftsecondcode + S (jt_d_enumleft) = S ((S (jt_h_enumleft)) * B)) /\\ exists fs_q_jt_enumleftsecondcode. A = fs_q_jt_enumleftsecondcode * S ((S (jt_h_enumleft)) * B) + (jt_d_enumleft))) /\\ (((exists fs_h_jt_enumleftsecondscale. fs_h_jt_enumleftsecondscale + S (jt_e_enumleft) = S ((S (jt_h_enumleft)) * D)) /\\ exists fs_q_jt_enumleftsecondscale. C = fs_q_jt_enumleftsecondscale * S ((S (jt_h_enumleft)) * D) + (jt_e_enumleft))))) -> (forall jt_index_enumleftsame jt_left_enumleftsame jt_right_enumleftsame. (exists jt_gap_enumleftsameindex. jt_gap_enumleftsameindex+S (jt_index_enumleftsame)=(k)) -> (((exists fs_h_jt_enumleftsameleft. fs_h_jt_enumleftsameleft + S (jt_left_enumleftsame) = S ((S (jt_index_enumleftsame)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftsameleft. jt_b_enumleft = fs_q_jt_enumleftsameleft * S ((S (jt_index_enumleftsame)) * jt_c_enumleft) + (jt_left_enumleftsame))) -> (((exists fs_h_jt_enumleftsameright. fs_h_jt_enumleftsameright + S (jt_right_enumleftsame) = S ((S (jt_index_enumleftsame)) * jt_e_enumleft)) /\\ exists fs_q_jt_enumleftsameright. jt_d_enumleft = fs_q_jt_enumleftsameright * S ((S (jt_index_enumleftsame)) * jt_e_enumleft) + (jt_right_enumleftsame))) -> jt_left_enumleftsame=jt_right_enumleftsame) -> jt_i_enumleft=jt_h_enumleft))))) -> (((forall jt_i_enumright. (exists jt_gap_enumrightsoundindex. jt_gap_enumrightsoundindex+S (jt_i_enumright)=(v)) -> exists jt_b_enumright jt_c_enumright. ((((((exists fs_h_jt_enumrightsoundcode. fs_h_jt_enumrightsoundcode + S (jt_b_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightsoundcode. E = fs_q_jt_enumrightsoundcode * S ((S (jt_i_enumright)) * F) + (jt_b_enumright))) /\\ (((exists fs_h_jt_enumrightsoundscale. fs_h_jt_enumrightsoundscale + S (jt_c_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightsoundscale. G = fs_q_jt_enumrightsoundscale * S ((S (jt_i_enumright)) * H) + (jt_c_enumright))))) /\\ (((forall jt_index_enumrightbound. (exists jt_gap_enumrightboundindex. jt_gap_enumrightboundindex+S (jt_index_enumrightbound)=(k)) -> exists jt_value_enumrightbound. ((((exists fs_h_jt_enumrightboundat. fs_h_jt_enumrightboundat + S (jt_value_enumrightbound) = S ((S (jt_index_enumrightbound)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightboundat. jt_b_enumright = fs_q_jt_enumrightboundat * S ((S (jt_index_enumrightbound)) * jt_c_enumright) + (jt_value_enumrightbound))) /\\ (exists jt_gap_enumrightboundvalue. jt_gap_enumrightboundvalue+S (jt_value_enumrightbound)=(n)))) /\\ (forall jt_divisor_enumrightprimitive. (exists jt_factor_enumrightprimitivemodulus. (n)=(jt_divisor_enumrightprimitive)*jt_factor_enumrightprimitivemodulus) -> (forall jt_index_enumrightprimitivecoordinates jt_value_enumrightprimitivecoordinates. (exists jt_gap_enumrightprimitivecoordinatesindex. jt_gap_enumrightprimitivecoordinatesindex+S (jt_index_enumrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrightprimitivecoordinatesat. fs_h_jt_enumrightprimitivecoordinatesat + S (jt_value_enumrightprimitivecoordinates) = S ((S (jt_index_enumrightprimitivecoordinates)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightprimitivecoordinatesat. jt_b_enumright = fs_q_jt_enumrightprimitivecoordinatesat * S ((S (jt_index_enumrightprimitivecoordinates)) * jt_c_enumright) + (jt_value_enumrightprimitivecoordinates))) -> (exists jt_factor_enumrightprimitivecoordinatesdivides. (jt_value_enumrightprimitivecoordinates)=(jt_divisor_enumrightprimitive)*jt_factor_enumrightprimitivecoordinatesdivides)) -> jt_divisor_enumrightprimitive=1))))) /\\ (((forall jt_b_enumright jt_c_enumright. (forall jt_index_enumrightinputbound. (exists jt_gap_enumrightinputboundindex. jt_gap_enumrightinputboundindex+S (jt_index_enumrightinputbound)=(k)) -> exists jt_value_enumrightinputbound. ((((exists fs_h_jt_enumrightinputboundat. fs_h_jt_enumrightinputboundat + S (jt_value_enumrightinputbound) = S ((S (jt_index_enumrightinputbound)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightinputboundat. jt_b_enumright = fs_q_jt_enumrightinputboundat * S ((S (jt_index_enumrightinputbound)) * jt_c_enumright) + (jt_value_enumrightinputbound))) /\\ (exists jt_gap_enumrightinputboundvalue. jt_gap_enumrightinputboundvalue+S (jt_value_enumrightinputbound)=(n)))) -> (forall jt_divisor_enumrightinputprimitive. (exists jt_factor_enumrightinputprimitivemodulus. (n)=(jt_divisor_enumrightinputprimitive)*jt_factor_enumrightinputprimitivemodulus) -> (forall jt_index_enumrightinputprimitivecoordinates jt_value_enumrightinputprimitivecoordinates. (exists jt_gap_enumrightinputprimitivecoordinatesindex. jt_gap_enumrightinputprimitivecoordinatesindex+S (jt_index_enumrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrightinputprimitivecoordinatesat. fs_h_jt_enumrightinputprimitivecoordinatesat + S (jt_value_enumrightinputprimitivecoordinates) = S ((S (jt_index_enumrightinputprimitivecoordinates)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightinputprimitivecoordinatesat. jt_b_enumright = fs_q_jt_enumrightinputprimitivecoordinatesat * S ((S (jt_index_enumrightinputprimitivecoordinates)) * jt_c_enumright) + (jt_value_enumrightinputprimitivecoordinates))) -> (exists jt_factor_enumrightinputprimitivecoordinatesdivides. (jt_value_enumrightinputprimitivecoordinates)=(jt_divisor_enumrightinputprimitive)*jt_factor_enumrightinputprimitivecoordinatesdivides)) -> jt_divisor_enumrightinputprimitive=1) -> exists jt_i_enumright jt_d_enumright jt_e_enumright. ((exists jt_gap_enumrightcompleteindex. jt_gap_enumrightcompleteindex+S (jt_i_enumright)=(v)) /\\ (((((((exists fs_h_jt_enumrightcompletecode. fs_h_jt_enumrightcompletecode + S (jt_d_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightcompletecode. E = fs_q_jt_enumrightcompletecode * S ((S (jt_i_enumright)) * F) + (jt_d_enumright))) /\\ (((exists fs_h_jt_enumrightcompletescale. fs_h_jt_enumrightcompletescale + S (jt_e_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightcompletescale. G = fs_q_jt_enumrightcompletescale * S ((S (jt_i_enumright)) * H) + (jt_e_enumright))))) /\\ (forall jt_index_enumrightrepresented jt_left_enumrightrepresented jt_right_enumrightrepresented. (exists jt_gap_enumrightrepresentedindex. jt_gap_enumrightrepresentedindex+S (jt_index_enumrightrepresented)=(k)) -> (((exists fs_h_jt_enumrightrepresentedleft. fs_h_jt_enumrightrepresentedleft + S (jt_left_enumrightrepresented) = S ((S (jt_index_enumrightrepresented)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightrepresentedleft. jt_b_enumright = fs_q_jt_enumrightrepresentedleft * S ((S (jt_index_enumrightrepresented)) * jt_c_enumright) + (jt_left_enumrightrepresented))) -> (((exists fs_h_jt_enumrightrepresentedright. fs_h_jt_enumrightrepresentedright + S (jt_right_enumrightrepresented) = S ((S (jt_index_enumrightrepresented)) * jt_e_enumright)) /\\ exists fs_q_jt_enumrightrepresentedright. jt_d_enumright = fs_q_jt_enumrightrepresentedright * S ((S (jt_index_enumrightrepresented)) * jt_e_enumright) + (jt_right_enumrightrepresented))) -> jt_left_enumrightrepresented=jt_right_enumrightrepresented))))) /\\ (forall jt_i_enumright jt_h_enumright jt_b_enumright jt_c_enumright jt_d_enumright jt_e_enumright. (exists jt_gap_enumrightfirstindex. jt_gap_enumrightfirstindex+S (jt_i_enumright)=(v)) -> (exists jt_gap_enumrightsecondindex. jt_gap_enumrightsecondindex+S (jt_h_enumright)=(v)) -> (((((exists fs_h_jt_enumrightfirstcode. fs_h_jt_enumrightfirstcode + S (jt_b_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightfirstcode. E = fs_q_jt_enumrightfirstcode * S ((S (jt_i_enumright)) * F) + (jt_b_enumright))) /\\ (((exists fs_h_jt_enumrightfirstscale. fs_h_jt_enumrightfirstscale + S (jt_c_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightfirstscale. G = fs_q_jt_enumrightfirstscale * S ((S (jt_i_enumright)) * H) + (jt_c_enumright))))) -> (((((exists fs_h_jt_enumrightsecondcode. fs_h_jt_enumrightsecondcode + S (jt_d_enumright) = S ((S (jt_h_enumright)) * F)) /\\ exists fs_q_jt_enumrightsecondcode. E = fs_q_jt_enumrightsecondcode * S ((S (jt_h_enumright)) * F) + (jt_d_enumright))) /\\ (((exists fs_h_jt_enumrightsecondscale. fs_h_jt_enumrightsecondscale + S (jt_e_enumright) = S ((S (jt_h_enumright)) * H)) /\\ exists fs_q_jt_enumrightsecondscale. G = fs_q_jt_enumrightsecondscale * S ((S (jt_h_enumright)) * H) + (jt_e_enumright))))) -> (forall jt_index_enumrightsame jt_left_enumrightsame jt_right_enumrightsame. (exists jt_gap_enumrightsameindex. jt_gap_enumrightsameindex+S (jt_index_enumrightsame)=(k)) -> (((exists fs_h_jt_enumrightsameleft. fs_h_jt_enumrightsameleft + S (jt_left_enumrightsame) = S ((S (jt_index_enumrightsame)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightsameleft. jt_b_enumright = fs_q_jt_enumrightsameleft * S ((S (jt_index_enumrightsame)) * jt_c_enumright) + (jt_left_enumrightsame))) -> (((exists fs_h_jt_enumrightsameright. fs_h_jt_enumrightsameright + S (jt_right_enumrightsame) = S ((S (jt_index_enumrightsame)) * jt_e_enumright)) /\\ exists fs_q_jt_enumrightsameright. jt_d_enumright = fs_q_jt_enumrightsameright * S ((S (jt_index_enumrightsame)) * jt_e_enumright) + (jt_right_enumrightsame))) -> jt_left_enumrightsame=jt_right_enumrightsame) -> jt_i_enumright=jt_h_enumright))))) -> (forall jt_index_enumrect. (exists jt_gap_enumrectindex. jt_gap_enumrectindex+S (jt_index_enumrect)=(u*v)) -> exists jt_row_enumrect jt_column_enumrect jt_b_enumrect jt_c_enumrect jt_d_enumrect jt_e_enumrect jt_f_enumrect jt_g_enumrect. ((exists jt_gap_enumrectrow. jt_gap_enumrectrow+S (jt_row_enumrect)=(u)) /\\ (((exists jt_gap_enumrectcolumn. jt_gap_enumrectcolumn+S (jt_column_enumrect)=(v)) /\\ (((jt_index_enumrect=(v)*jt_row_enumrect+jt_column_enumrect) /\\ (((((((exists fs_h_jt_enumrectleftcode. fs_h_jt_enumrectleftcode + S (jt_b_enumrect) = S ((S (jt_row_enumrect)) * B)) /\\ exists fs_q_jt_enumrectleftcode. A = fs_q_jt_enumrectleftcode * S ((S (jt_row_enumrect)) * B) + (jt_b_enumrect))) /\\ (((exists fs_h_jt_enumrectleftscale. fs_h_jt_enumrectleftscale + S (jt_c_enumrect) = S ((S (jt_row_enumrect)) * D)) /\\ exists fs_q_jt_enumrectleftscale. C = fs_q_jt_enumrectleftscale * S ((S (jt_row_enumrect)) * D) + (jt_c_enumrect))))) /\\ (((((((exists fs_h_jt_enumrectrightcode. fs_h_jt_enumrectrightcode + S (jt_d_enumrect) = S ((S (jt_column_enumrect)) * F)) /\\ exists fs_q_jt_enumrectrightcode. E = fs_q_jt_enumrectrightcode * S ((S (jt_column_enumrect)) * F) + (jt_d_enumrect))) /\\ (((exists fs_h_jt_enumrectrightscale. fs_h_jt_enumrectrightscale + S (jt_e_enumrect) = S ((S (jt_column_enumrect)) * H)) /\\ exists fs_q_jt_enumrectrightscale. G = fs_q_jt_enumrectrightscale * S ((S (jt_column_enumrect)) * H) + (jt_e_enumrect))))) /\\ (((((((exists fs_h_jt_enumrectoutputcode. fs_h_jt_enumrectoutputcode + S (jt_f_enumrect) = S ((S (jt_index_enumrect)) * Q)) /\\ exists fs_q_jt_enumrectoutputcode. P = fs_q_jt_enumrectoutputcode * S ((S (jt_index_enumrect)) * Q) + (jt_f_enumrect))) /\\ (((exists fs_h_jt_enumrectoutputscale. fs_h_jt_enumrectoutputscale + S (jt_g_enumrect) = S ((S (jt_index_enumrect)) * T)) /\\ exists fs_q_jt_enumrectoutputscale. R = fs_q_jt_enumrectoutputscale * S ((S (jt_index_enumrect)) * T) + (jt_g_enumrect))))) /\\ (((((forall jt_index_enumrectcrtbound. (exists jt_gap_enumrectcrtboundindex. jt_gap_enumrectcrtboundindex+S (jt_index_enumrectcrtbound)=(k)) -> exists jt_value_enumrectcrtbound. ((((exists fs_h_jt_enumrectcrtboundat. fs_h_jt_enumrectcrtboundat + S (jt_value_enumrectcrtbound) = S ((S (jt_index_enumrectcrtbound)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtboundat. jt_f_enumrect = fs_q_jt_enumrectcrtboundat * S ((S (jt_index_enumrectcrtbound)) * jt_g_enumrect) + (jt_value_enumrectcrtbound))) /\\ (exists jt_gap_enumrectcrtboundvalue. jt_gap_enumrectcrtboundvalue+S (jt_value_enumrectcrtbound)=(m*n)))) /\\ (((forall jt_index_enumrectcrtleft jt_left_enumrectcrtleft jt_right_enumrectcrtleft. (exists jt_gap_enumrectcrtleftindex. jt_gap_enumrectcrtleftindex+S (jt_index_enumrectcrtleft)=(k)) -> (((exists fs_h_jt_enumrectcrtleftleft. fs_h_jt_enumrectcrtleftleft + S (jt_left_enumrectcrtleft) = S ((S (jt_index_enumrectcrtleft)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtleftleft. jt_f_enumrect = fs_q_jt_enumrectcrtleftleft * S ((S (jt_index_enumrectcrtleft)) * jt_g_enumrect) + (jt_left_enumrectcrtleft))) -> (((exists fs_h_jt_enumrectcrtleftright. fs_h_jt_enumrectcrtleftright + S (jt_right_enumrectcrtleft) = S ((S (jt_index_enumrectcrtleft)) * jt_c_enumrect)) /\\ exists fs_q_jt_enumrectcrtleftright. jt_b_enumrect = fs_q_jt_enumrectcrtleftright * S ((S (jt_index_enumrectcrtleft)) * jt_c_enumrect) + (jt_right_enumrectcrtleft))) -> (exists jt_left_enumrectcrtleftmod jt_right_enumrectcrtleftmod. (jt_left_enumrectcrtleft)+(m)*jt_left_enumrectcrtleftmod=(jt_right_enumrectcrtleft)+(m)*jt_right_enumrectcrtleftmod)) /\\ (forall jt_index_enumrectcrtright jt_left_enumrectcrtright jt_right_enumrectcrtright. (exists jt_gap_enumrectcrtrightindex. jt_gap_enumrectcrtrightindex+S (jt_index_enumrectcrtright)=(k)) -> (((exists fs_h_jt_enumrectcrtrightleft. fs_h_jt_enumrectcrtrightleft + S (jt_left_enumrectcrtright) = S ((S (jt_index_enumrectcrtright)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtrightleft. jt_f_enumrect = fs_q_jt_enumrectcrtrightleft * S ((S (jt_index_enumrectcrtright)) * jt_g_enumrect) + (jt_left_enumrectcrtright))) -> (((exists fs_h_jt_enumrectcrtrightright. fs_h_jt_enumrectcrtrightright + S (jt_right_enumrectcrtright) = S ((S (jt_index_enumrectcrtright)) * jt_e_enumrect)) /\\ exists fs_q_jt_enumrectcrtrightright. jt_d_enumrect = fs_q_jt_enumrectcrtrightright * S ((S (jt_index_enumrectcrtright)) * jt_e_enumrect) + (jt_right_enumrectcrtright))) -> (exists jt_left_enumrectcrtrightmod jt_right_enumrectcrtrightmod. (jt_left_enumrectcrtright)+(n)*jt_left_enumrectcrtrightmod=(jt_right_enumrectcrtright)+(n)*jt_right_enumrectcrtrightmod)))))) /\\ (forall jt_divisor_enumrectprimitive. (exists jt_factor_enumrectprimitivemodulus. (m*n)=(jt_divisor_enumrectprimitive)*jt_factor_enumrectprimitivemodulus) -> (forall jt_index_enumrectprimitivecoordinates jt_value_enumrectprimitivecoordinates. (exists jt_gap_enumrectprimitivecoordinatesindex. jt_gap_enumrectprimitivecoordinatesindex+S (jt_index_enumrectprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrectprimitivecoordinatesat. fs_h_jt_enumrectprimitivecoordinatesat + S (jt_value_enumrectprimitivecoordinates) = S ((S (jt_index_enumrectprimitivecoordinates)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectprimitivecoordinatesat. jt_f_enumrect = fs_q_jt_enumrectprimitivecoordinatesat * S ((S (jt_index_enumrectprimitivecoordinates)) * jt_g_enumrect) + (jt_value_enumrectprimitivecoordinates))) -> (exists jt_factor_enumrectprimitivecoordinatesdivides. (jt_value_enumrectprimitivecoordinates)=(jt_divisor_enumrectprimitive)*jt_factor_enumrectprimitivecoordinatesdivides)) -> jt_divisor_enumrectprimitive=1))))))))))))))) -> (exists jt_gap_distinctp. jt_gap_distinctp+S (p)=(u*v)) -> (exists jt_gap_distinctz. jt_gap_distinctz+S (z)=(u*v)) -> (((((exists fs_h_jt_distinctentrypcode. fs_h_jt_distinctentrypcode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_distinctentrypcode. P = fs_q_jt_distinctentrypcode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_distinctentrypscale. fs_h_jt_distinctentrypscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_distinctentrypscale. R = fs_q_jt_distinctentrypscale * S ((S (p)) * T) + (g))))) -> (((((exists fs_h_jt_distinctentryzcode. fs_h_jt_distinctentryzcode + S (h) = S ((S (z)) * Q)) /\\ exists fs_q_jt_distinctentryzcode. P = fs_q_jt_distinctentryzcode * S ((S (z)) * Q) + (h))) /\\ (((exists fs_h_jt_distinctentryzscale. fs_h_jt_distinctentryzscale + S (s) = S ((S (z)) * T)) /\\ exists fs_q_jt_distinctentryzscale. R = fs_q_jt_distinctentryzscale * S ((S (z)) * T) + (s))))) -> (forall jt_index_distinctoutputs jt_left_distinctoutputs jt_right_distinctoutputs. (exists jt_gap_distinctoutputsindex. jt_gap_distinctoutputsindex+S (jt_index_distinctoutputs)=(k)) -> (((exists fs_h_jt_distinctoutputsleft. fs_h_jt_distinctoutputsleft + S (jt_left_distinctoutputs) = S ((S (jt_index_distinctoutputs)) * g)) /\\ exists fs_q_jt_distinctoutputsleft. f = fs_q_jt_distinctoutputsleft * S ((S (jt_index_distinctoutputs)) * g) + (jt_left_distinctoutputs))) -> (((exists fs_h_jt_distinctoutputsright. fs_h_jt_distinctoutputsright + S (jt_right_distinctoutputs) = S ((S (jt_index_distinctoutputs)) * s)) /\\ exists fs_q_jt_distinctoutputsright. h = fs_q_jt_distinctoutputsright * S ((S (jt_index_distinctoutputs)) * s) + (jt_right_distinctoutputs))) -> jt_left_distinctoutputs=jt_right_distinctoutputs) -> p=z", "statement_sha256": "cab9c7262834a02f7624af6c4d5ceebdb6eb91a1e71ff58f0d86fbf218c4d239", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Equal decoded CRT output tuples recover equal source positions and hence the same flat index." }, { "admission_dependencies": [ "jordan_primitive_tuple_product_components", "jordan_enumeration_reduce_primitive", "jordan_rectangle_crt_pair_value", "jordan_rectangle_flat_bound", "jordan_canonical_crt_tuple_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 68, "body_proof_nodes": 179, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro b", "intro c", "intro hm", "intro hn", "intro hcop", "intro hl", "intro hh", "intro hr", "intro hbound", "intro hprim", "have hcomponents : JordanPrimitiveTuple(m,b,c,k) ∧ JordanPrimitiveTuple(n,b,c,k)", "specialize jordan_primitive_tuple_product_components (m)", "specialize jordan_primitive_tuple_product_components (n)", "specialize jordan_primitive_tuple_product_components (b)", "specialize jordan_primitive_tuple_product_components (c)", "specialize jordan_primitive_tuple_product_components (k)", "apply jordan_primitive_tuple_product_components", "exact hprim", "cases hcomponents", "have ha : ∃ i. ∃ d. ∃ e. Lt(i,u) ∧ (BetaAt(A,B,i,d) ∧ BetaAt(C,D,i,e) ∧ JordanTupleCongruence(m,b,c,d,e,k))", "specialize jordan_enumeration_reduce_primitive (m)", "specialize jordan_enumeration_reduce_primitive (k)", "specialize jordan_enumeration_reduce_primitive (A)", "specialize jordan_enumeration_reduce_primitive (B)", "specialize jordan_enumeration_reduce_primitive (C)", "specialize jordan_enumeration_reduce_primitive (D)", "specialize jordan_enumeration_reduce_primitive (u)", "specialize jordan_enumeration_reduce_primitive (b)", "specialize jordan_enumeration_reduce_primitive (c)", "apply jordan_enumeration_reduce_primitive", "exact hm", "exact hl", "exact hcomponents_left", "cases ha", "cases ha_witness", "cases ha_witness_witness", "cases ha_witness_witness_witness", "cases ha_witness_witness_witness_right", "have hb : ∃ i. ∃ d. ∃ e. Lt(i,v) ∧ (BetaAt(E,F,i,d) ∧ BetaAt(G,H,i,e) ∧ JordanTupleCongruence(n,b,c,d,e,k))", "specialize jordan_enumeration_reduce_primitive (n)", "specialize jordan_enumeration_reduce_primitive (k)", "specialize jordan_enumeration_reduce_primitive (E)", "specialize jordan_enumeration_reduce_primitive (F)", "specialize jordan_enumeration_reduce_primitive (G)", "specialize jordan_enumeration_reduce_primitive (H)", "specialize jordan_enumeration_reduce_primitive (v)", "specialize jordan_enumeration_reduce_primitive (b)", "specialize jordan_enumeration_reduce_primitive (c)", "apply jordan_enumeration_reduce_primitive", "exact hn", "exact hh", "exact hcomponents_right", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_witness", "cases hb_witness_witness_witness_right", "have hout : ∃ f. ∃ g. MatrixAt(P,Q,x,v,x3,f) ∧ MatrixAt(R,T,x,v,x3,g) ∧ (JordanCanonicalTupleCRT(m,n,x1,x2,x4,x5,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k))", "specialize jordan_rectangle_crt_pair_value (m)", "specialize jordan_rectangle_crt_pair_value (n)", "specialize jordan_rectangle_crt_pair_value (k)", "specialize jordan_rectangle_crt_pair_value (A)", "specialize jordan_rectangle_crt_pair_value (B)", "specialize jordan_rectangle_crt_pair_value (C)", "specialize jordan_rectangle_crt_pair_value (D)", "specialize jordan_rectangle_crt_pair_value (u)", "specialize jordan_rectangle_crt_pair_value (E)", "specialize jordan_rectangle_crt_pair_value (F)", "specialize jordan_rectangle_crt_pair_value (G)", "specialize jordan_rectangle_crt_pair_value (H)", "specialize jordan_rectangle_crt_pair_value (v)", "specialize jordan_rectangle_crt_pair_value (P)", "specialize jordan_rectangle_crt_pair_value (Q)", "specialize jordan_rectangle_crt_pair_value (R)", "specialize jordan_rectangle_crt_pair_value (T)", "specialize jordan_rectangle_crt_pair_value (x)", "specialize jordan_rectangle_crt_pair_value (x3)", "specialize jordan_rectangle_crt_pair_value (x1)", "specialize jordan_rectangle_crt_pair_value (x2)", "specialize jordan_rectangle_crt_pair_value (x4)", "specialize jordan_rectangle_crt_pair_value (x5)", "apply jordan_rectangle_crt_pair_value", "exact hr", "exact ha_witness_witness_witness_left", "exact hb_witness_witness_witness_left", "exact ha_witness_witness_witness_right_left", "exact hb_witness_witness_witness_right_left", "cases hout", "cases hout_witness", "cases hout_witness_witness", "cases hout_witness_witness_right", "exists v*x+x3", "exists x6", "exists x7", "split", "specialize jordan_rectangle_flat_bound (u)", "specialize jordan_rectangle_flat_bound (v)", "specialize jordan_rectangle_flat_bound (x)", "specialize jordan_rectangle_flat_bound (x3)", "apply jordan_rectangle_flat_bound", "exact ha_witness_witness_witness_left", "exact hb_witness_witness_witness_left", "split", "exact hout_witness_witness_left", "specialize jordan_canonical_crt_tuple_unique (m)", "specialize jordan_canonical_crt_tuple_unique (n)", "specialize jordan_canonical_crt_tuple_unique (x1)", "specialize jordan_canonical_crt_tuple_unique (x2)", "specialize jordan_canonical_crt_tuple_unique (x4)", "specialize jordan_canonical_crt_tuple_unique (x5)", "specialize jordan_canonical_crt_tuple_unique (b)", "specialize jordan_canonical_crt_tuple_unique (c)", "specialize jordan_canonical_crt_tuple_unique (x6)", "specialize jordan_canonical_crt_tuple_unique (x7)", "specialize jordan_canonical_crt_tuple_unique (k)", "apply jordan_canonical_crt_tuple_unique", "exact hcop", "split", "exact hbound", "split", "exact ha_witness_witness_witness_right_right", "exact hb_witness_witness_witness_right_right", "exact hout_witness_witness_right_left" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ b. ∀ c. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v) → BetaPrefixInto(b,c,k,m · n) → JordanPrimitiveTuple(m · n,b,c,k) → JordanTupleListed(b,c,k,P,Q,R,T,u · v)", "defined_statement_sha256": "af2e5d190145b8dc77b033102217b06fd0f220be0146b017177d9c0eb4e08d28", "definition_uses": { "ND0003": 2, "ND0262": 1, "ND0372": 4, "ND0373": 2, "ND0374": 2, "ND0376": 1, "ND0380": 1, "ND0381": 1, "PD0002": 2, "PD0005": 1, "PD0013": 4 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "4cbf8e3e92b8d86002bdc9c90b51f74fe27a49f231d60bdb5b7858f2f56cd1f4", "free_names": [], "script_definition_uses": { "ND0003": 2, "ND0372": 3, "ND0373": 2, "ND0380": 1, "PD0002": 2, "PD0013": 4 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro P" } ], [ { "kind": "text", "text": "intro Q" } ], [ { "kind": "text", "text": "intro R" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hbound" } ], [ { "kind": "text", "text": "intro hprim" } ], [ { "kind": "text", "text": "have hcomponents : " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,b,c,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_product_components (m)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_product_components (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_product_components (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_product_components (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_product_components (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_product_components" } ], [ { "kind": "text", "text": "exact hprim" } ], [ { "kind": "text", "text": "cases hcomponents" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ i. ∃ d. ∃ e. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,e)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(m,b,c,d,e,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (m)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (b)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (c)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_reduce_primitive" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hcomponents_left" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "cases ha_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_witness_right" } ], [ { "kind": "text", "text": "have hb : " }, { "kind": "text", "text": "∃ i. ∃ d. ∃ e. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,v)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,i,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,i,e)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0373", "kind": "definition", "text": "JordanTupleCongruence(n,b,c,d,e,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (b)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_reduce_primitive (c)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_reduce_primitive" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact hcomponents_right" } ], [ { "kind": "text", "text": "cases hb" } ], [ { "kind": "text", "text": "cases hb_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_witness_right" } ], [ { "kind": "text", "text": "have hout : " }, { "kind": "text", "text": "∃ f. ∃ g. " }, { "definition": "ND0003", "kind": "definition", "text": "MatrixAt(P,Q,x,v,x3,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0003", "kind": "definition", "text": "MatrixAt(R,T,x,v,x3,g)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,x1,x2,x4,x5,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (P)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (Q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (R)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (T)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (x)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (x3)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (x1)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (x2)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (x4)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_pair_value (x5)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_pair_value" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "cases hout" } ], [ { "kind": "text", "text": "cases hout_witness" } ], [ { "kind": "text", "text": "cases hout_witness_witness" } ], [ { "kind": "text", "text": "cases hout_witness_witness_right" } ], [ { "kind": "text", "text": "exists v*x+x3" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "exists x7" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (x)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_flat_bound (x3)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_flat_bound" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hout_witness_witness_left" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (m)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (n)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (x1)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (x2)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (x4)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (x5)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (b)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (c)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (x6)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (x7)" } ], [ { "kind": "text", "text": "specialize jordan_canonical_crt_tuple_unique (k)" } ], [ { "kind": "text", "text": "apply jordan_canonical_crt_tuple_unique" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hbound" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact ha_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "exact hb_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "exact hout_witness_witness_right_left" } ] ], "statement_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0374": 2, "ND0376": 1, "ND0381": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ∀ b. ∀ c. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,m · n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(b,c,k,P,Q,R,T,u · v)" } ] }, "dependencies": [ "jordan_primitive_tuple_product_components", "jordan_enumeration_reduce_primitive", "jordan_rectangle_crt_pair_value", "jordan_rectangle_flat_bound", "jordan_canonical_crt_tuple_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0047", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_covers", "original_ha_bundle_verified": true, "proof_bundle_node_id": 332, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro b", "intro c", "intro hm", "intro hn", "intro hcop", "intro hl", "intro hh", "intro hr", "intro hbound", "intro hprim", "have hcomponents : ((forall jt_divisor_coverm. (exists jt_factor_covermmodulus. (m)=(jt_divisor_coverm)*jt_factor_covermmodulus) -> (forall jt_index_covermcoordinates jt_value_covermcoordinates. (exists jt_gap_covermcoordinatesindex. jt_gap_covermcoordinatesindex+S (jt_index_covermcoordinates)=(k)) -> (((exists fs_h_jt_covermcoordinatesat. fs_h_jt_covermcoordinatesat + S (jt_value_covermcoordinates) = S ((S (jt_index_covermcoordinates)) * c)) /\\ exists fs_q_jt_covermcoordinatesat. b = fs_q_jt_covermcoordinatesat * S ((S (jt_index_covermcoordinates)) * c) + (jt_value_covermcoordinates))) -> (exists jt_factor_covermcoordinatesdivides. (jt_value_covermcoordinates)=(jt_divisor_coverm)*jt_factor_covermcoordinatesdivides)) -> jt_divisor_coverm=1) /\\ (forall jt_divisor_covern. (exists jt_factor_covernmodulus. (n)=(jt_divisor_covern)*jt_factor_covernmodulus) -> (forall jt_index_coverncoordinates jt_value_coverncoordinates. (exists jt_gap_coverncoordinatesindex. jt_gap_coverncoordinatesindex+S (jt_index_coverncoordinates)=(k)) -> (((exists fs_h_jt_coverncoordinatesat. fs_h_jt_coverncoordinatesat + S (jt_value_coverncoordinates) = S ((S (jt_index_coverncoordinates)) * c)) /\\ exists fs_q_jt_coverncoordinatesat. b = fs_q_jt_coverncoordinatesat * S ((S (jt_index_coverncoordinates)) * c) + (jt_value_coverncoordinates))) -> (exists jt_factor_coverncoordinatesdivides. (jt_value_coverncoordinates)=(jt_divisor_covern)*jt_factor_coverncoordinatesdivides)) -> jt_divisor_covern=1))", "specialize jordan_primitive_tuple_product_components (m)", "specialize jordan_primitive_tuple_product_components (n)", "specialize jordan_primitive_tuple_product_components (b)", "specialize jordan_primitive_tuple_product_components (c)", "specialize jordan_primitive_tuple_product_components (k)", "apply jordan_primitive_tuple_product_components", "exact hprim", "cases hcomponents", "have ha : exists i d e. ((exists jt_gap_haindex. jt_gap_haindex+S (i)=(u)) /\\ (((((((exists fs_h_jt_haentrycode. fs_h_jt_haentrycode + S (d) = S ((S (i)) * B)) /\\ exists fs_q_jt_haentrycode. A = fs_q_jt_haentrycode * S ((S (i)) * B) + (d))) /\\ (((exists fs_h_jt_haentryscale. fs_h_jt_haentryscale + S (e) = S ((S (i)) * D)) /\\ exists fs_q_jt_haentryscale. C = fs_q_jt_haentryscale * S ((S (i)) * D) + (e))))) /\\ (forall jt_index_hamod jt_left_hamod jt_right_hamod. (exists jt_gap_hamodindex. jt_gap_hamodindex+S (jt_index_hamod)=(k)) -> (((exists fs_h_jt_hamodleft. fs_h_jt_hamodleft + S (jt_left_hamod) = S ((S (jt_index_hamod)) * c)) /\\ exists fs_q_jt_hamodleft. b = fs_q_jt_hamodleft * S ((S (jt_index_hamod)) * c) + (jt_left_hamod))) -> (((exists fs_h_jt_hamodright. fs_h_jt_hamodright + S (jt_right_hamod) = S ((S (jt_index_hamod)) * e)) /\\ exists fs_q_jt_hamodright. d = fs_q_jt_hamodright * S ((S (jt_index_hamod)) * e) + (jt_right_hamod))) -> (exists jt_left_hamodmod jt_right_hamodmod. (jt_left_hamod)+(m)*jt_left_hamodmod=(jt_right_hamod)+(m)*jt_right_hamodmod)))))", "specialize jordan_enumeration_reduce_primitive (m)", "specialize jordan_enumeration_reduce_primitive (k)", "specialize jordan_enumeration_reduce_primitive (A)", "specialize jordan_enumeration_reduce_primitive (B)", "specialize jordan_enumeration_reduce_primitive (C)", "specialize jordan_enumeration_reduce_primitive (D)", "specialize jordan_enumeration_reduce_primitive (u)", "specialize jordan_enumeration_reduce_primitive (b)", "specialize jordan_enumeration_reduce_primitive (c)", "apply jordan_enumeration_reduce_primitive", "exact hm", "exact hl", "exact hcomponents_left", "cases ha", "cases ha_witness", "cases ha_witness_witness", "cases ha_witness_witness_witness", "cases ha_witness_witness_witness_right", "have hb : exists i d e. ((exists jt_gap_hbindex. jt_gap_hbindex+S (i)=(v)) /\\ (((((((exists fs_h_jt_hbentrycode. fs_h_jt_hbentrycode + S (d) = S ((S (i)) * F)) /\\ exists fs_q_jt_hbentrycode. E = fs_q_jt_hbentrycode * S ((S (i)) * F) + (d))) /\\ (((exists fs_h_jt_hbentryscale. fs_h_jt_hbentryscale + S (e) = S ((S (i)) * H)) /\\ exists fs_q_jt_hbentryscale. G = fs_q_jt_hbentryscale * S ((S (i)) * H) + (e))))) /\\ (forall jt_index_hbmod jt_left_hbmod jt_right_hbmod. (exists jt_gap_hbmodindex. jt_gap_hbmodindex+S (jt_index_hbmod)=(k)) -> (((exists fs_h_jt_hbmodleft. fs_h_jt_hbmodleft + S (jt_left_hbmod) = S ((S (jt_index_hbmod)) * c)) /\\ exists fs_q_jt_hbmodleft. b = fs_q_jt_hbmodleft * S ((S (jt_index_hbmod)) * c) + (jt_left_hbmod))) -> (((exists fs_h_jt_hbmodright. fs_h_jt_hbmodright + S (jt_right_hbmod) = S ((S (jt_index_hbmod)) * e)) /\\ exists fs_q_jt_hbmodright. d = fs_q_jt_hbmodright * S ((S (jt_index_hbmod)) * e) + (jt_right_hbmod))) -> (exists jt_left_hbmodmod jt_right_hbmodmod. (jt_left_hbmod)+(n)*jt_left_hbmodmod=(jt_right_hbmod)+(n)*jt_right_hbmodmod)))))", "specialize jordan_enumeration_reduce_primitive (n)", "specialize jordan_enumeration_reduce_primitive (k)", "specialize jordan_enumeration_reduce_primitive (E)", "specialize jordan_enumeration_reduce_primitive (F)", "specialize jordan_enumeration_reduce_primitive (G)", "specialize jordan_enumeration_reduce_primitive (H)", "specialize jordan_enumeration_reduce_primitive (v)", "specialize jordan_enumeration_reduce_primitive (b)", "specialize jordan_enumeration_reduce_primitive (c)", "apply jordan_enumeration_reduce_primitive", "exact hn", "exact hh", "exact hcomponents_right", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_witness", "cases hb_witness_witness_witness_right", "have hout : exists f g. ((((((exists fs_h_jt_coverentrycode. fs_h_jt_coverentrycode + S (f) = S ((S (v*x+x3)) * Q)) /\\ exists fs_q_jt_coverentrycode. P = fs_q_jt_coverentrycode * S ((S (v*x+x3)) * Q) + (f))) /\\ (((exists fs_h_jt_coverentryscale. fs_h_jt_coverentryscale + S (g) = S ((S (v*x+x3)) * T)) /\\ exists fs_q_jt_coverentryscale. R = fs_q_jt_coverentryscale * S ((S (v*x+x3)) * T) + (g))))) /\\ (((((forall jt_index_covercrtbound. (exists jt_gap_covercrtboundindex. jt_gap_covercrtboundindex+S (jt_index_covercrtbound)=(k)) -> exists jt_value_covercrtbound. ((((exists fs_h_jt_covercrtboundat. fs_h_jt_covercrtboundat + S (jt_value_covercrtbound) = S ((S (jt_index_covercrtbound)) * g)) /\\ exists fs_q_jt_covercrtboundat. f = fs_q_jt_covercrtboundat * S ((S (jt_index_covercrtbound)) * g) + (jt_value_covercrtbound))) /\\ (exists jt_gap_covercrtboundvalue. jt_gap_covercrtboundvalue+S (jt_value_covercrtbound)=(m*n)))) /\\ (((forall jt_index_covercrtleft jt_left_covercrtleft jt_right_covercrtleft. (exists jt_gap_covercrtleftindex. jt_gap_covercrtleftindex+S (jt_index_covercrtleft)=(k)) -> (((exists fs_h_jt_covercrtleftleft. fs_h_jt_covercrtleftleft + S (jt_left_covercrtleft) = S ((S (jt_index_covercrtleft)) * g)) /\\ exists fs_q_jt_covercrtleftleft. f = fs_q_jt_covercrtleftleft * S ((S (jt_index_covercrtleft)) * g) + (jt_left_covercrtleft))) -> (((exists fs_h_jt_covercrtleftright. fs_h_jt_covercrtleftright + S (jt_right_covercrtleft) = S ((S (jt_index_covercrtleft)) * x2)) /\\ exists fs_q_jt_covercrtleftright. x1 = fs_q_jt_covercrtleftright * S ((S (jt_index_covercrtleft)) * x2) + (jt_right_covercrtleft))) -> (exists jt_left_covercrtleftmod jt_right_covercrtleftmod. (jt_left_covercrtleft)+(m)*jt_left_covercrtleftmod=(jt_right_covercrtleft)+(m)*jt_right_covercrtleftmod)) /\\ (forall jt_index_covercrtright jt_left_covercrtright jt_right_covercrtright. (exists jt_gap_covercrtrightindex. jt_gap_covercrtrightindex+S (jt_index_covercrtright)=(k)) -> (((exists fs_h_jt_covercrtrightleft. fs_h_jt_covercrtrightleft + S (jt_left_covercrtright) = S ((S (jt_index_covercrtright)) * g)) /\\ exists fs_q_jt_covercrtrightleft. f = fs_q_jt_covercrtrightleft * S ((S (jt_index_covercrtright)) * g) + (jt_left_covercrtright))) -> (((exists fs_h_jt_covercrtrightright. fs_h_jt_covercrtrightright + S (jt_right_covercrtright) = S ((S (jt_index_covercrtright)) * x5)) /\\ exists fs_q_jt_covercrtrightright. x4 = fs_q_jt_covercrtrightright * S ((S (jt_index_covercrtright)) * x5) + (jt_right_covercrtright))) -> (exists jt_left_covercrtrightmod jt_right_covercrtrightmod. (jt_left_covercrtright)+(n)*jt_left_covercrtrightmod=(jt_right_covercrtright)+(n)*jt_right_covercrtrightmod)))))) /\\ (forall jt_divisor_coverprimitive. (exists jt_factor_coverprimitivemodulus. (m*n)=(jt_divisor_coverprimitive)*jt_factor_coverprimitivemodulus) -> (forall jt_index_coverprimitivecoordinates jt_value_coverprimitivecoordinates. (exists jt_gap_coverprimitivecoordinatesindex. jt_gap_coverprimitivecoordinatesindex+S (jt_index_coverprimitivecoordinates)=(k)) -> (((exists fs_h_jt_coverprimitivecoordinatesat. fs_h_jt_coverprimitivecoordinatesat + S (jt_value_coverprimitivecoordinates) = S ((S (jt_index_coverprimitivecoordinates)) * g)) /\\ exists fs_q_jt_coverprimitivecoordinatesat. f = fs_q_jt_coverprimitivecoordinatesat * S ((S (jt_index_coverprimitivecoordinates)) * g) + (jt_value_coverprimitivecoordinates))) -> (exists jt_factor_coverprimitivecoordinatesdivides. (jt_value_coverprimitivecoordinates)=(jt_divisor_coverprimitive)*jt_factor_coverprimitivecoordinatesdivides)) -> jt_divisor_coverprimitive=1))))", "specialize jordan_rectangle_crt_pair_value (m)", "specialize jordan_rectangle_crt_pair_value (n)", "specialize jordan_rectangle_crt_pair_value (k)", "specialize jordan_rectangle_crt_pair_value (A)", "specialize jordan_rectangle_crt_pair_value (B)", "specialize jordan_rectangle_crt_pair_value (C)", "specialize jordan_rectangle_crt_pair_value (D)", "specialize jordan_rectangle_crt_pair_value (u)", "specialize jordan_rectangle_crt_pair_value (E)", "specialize jordan_rectangle_crt_pair_value (F)", "specialize jordan_rectangle_crt_pair_value (G)", "specialize jordan_rectangle_crt_pair_value (H)", "specialize jordan_rectangle_crt_pair_value (v)", "specialize jordan_rectangle_crt_pair_value (P)", "specialize jordan_rectangle_crt_pair_value (Q)", "specialize jordan_rectangle_crt_pair_value (R)", "specialize jordan_rectangle_crt_pair_value (T)", "specialize jordan_rectangle_crt_pair_value (x)", "specialize jordan_rectangle_crt_pair_value (x3)", "specialize jordan_rectangle_crt_pair_value (x1)", "specialize jordan_rectangle_crt_pair_value (x2)", "specialize jordan_rectangle_crt_pair_value (x4)", "specialize jordan_rectangle_crt_pair_value (x5)", "apply jordan_rectangle_crt_pair_value", "exact hr", "exact ha_witness_witness_witness_left", "exact hb_witness_witness_witness_left", "exact ha_witness_witness_witness_right_left", "exact hb_witness_witness_witness_right_left", "cases hout", "cases hout_witness", "cases hout_witness_witness", "cases hout_witness_witness_right", "exists v*x+x3", "exists x6", "exists x7", "split", "specialize jordan_rectangle_flat_bound (u)", "specialize jordan_rectangle_flat_bound (v)", "specialize jordan_rectangle_flat_bound (x)", "specialize jordan_rectangle_flat_bound (x3)", "apply jordan_rectangle_flat_bound", "exact ha_witness_witness_witness_left", "exact hb_witness_witness_witness_left", "split", "exact hout_witness_witness_left", "specialize jordan_canonical_crt_tuple_unique (m)", "specialize jordan_canonical_crt_tuple_unique (n)", "specialize jordan_canonical_crt_tuple_unique (x1)", "specialize jordan_canonical_crt_tuple_unique (x2)", "specialize jordan_canonical_crt_tuple_unique (x4)", "specialize jordan_canonical_crt_tuple_unique (x5)", "specialize jordan_canonical_crt_tuple_unique (b)", "specialize jordan_canonical_crt_tuple_unique (c)", "specialize jordan_canonical_crt_tuple_unique (x6)", "specialize jordan_canonical_crt_tuple_unique (x7)", "specialize jordan_canonical_crt_tuple_unique (k)", "apply jordan_canonical_crt_tuple_unique", "exact hcop", "split", "exact hbound", "split", "exact ha_witness_witness_witness_right_right", "exact hb_witness_witness_witness_right_right", "exact hout_witness_witness_right_left" ], "script_sha256": "4f83da2ca76e716461c4ed03a99d0d7fd97a495dff51ce78bbd15f85b8e61a44", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "4f83da2ca76e716461c4ed03a99d0d7fd97a495dff51ce78bbd15f85b8e61a44", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "4cbf8e3e92b8d86002bdc9c90b51f74fe27a49f231d60bdb5b7858f2f56cd1f4" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v P Q R T b c. ~(m=0) -> ~(n=0) -> (forall jt_divisor_rectenumcop. (exists jt_factor_rectenumcopa. (m)=(jt_divisor_rectenumcop)*jt_factor_rectenumcopa) -> (exists jt_factor_rectenumcopb. (n)=(jt_divisor_rectenumcop)*jt_factor_rectenumcopb) -> jt_divisor_rectenumcop=1) -> (((forall jt_i_enumleft. (exists jt_gap_enumleftsoundindex. jt_gap_enumleftsoundindex+S (jt_i_enumleft)=(u)) -> exists jt_b_enumleft jt_c_enumleft. ((((((exists fs_h_jt_enumleftsoundcode. fs_h_jt_enumleftsoundcode + S (jt_b_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftsoundcode. A = fs_q_jt_enumleftsoundcode * S ((S (jt_i_enumleft)) * B) + (jt_b_enumleft))) /\\ (((exists fs_h_jt_enumleftsoundscale. fs_h_jt_enumleftsoundscale + S (jt_c_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftsoundscale. C = fs_q_jt_enumleftsoundscale * S ((S (jt_i_enumleft)) * D) + (jt_c_enumleft))))) /\\ (((forall jt_index_enumleftbound. (exists jt_gap_enumleftboundindex. jt_gap_enumleftboundindex+S (jt_index_enumleftbound)=(k)) -> exists jt_value_enumleftbound. ((((exists fs_h_jt_enumleftboundat. fs_h_jt_enumleftboundat + S (jt_value_enumleftbound) = S ((S (jt_index_enumleftbound)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftboundat. jt_b_enumleft = fs_q_jt_enumleftboundat * S ((S (jt_index_enumleftbound)) * jt_c_enumleft) + (jt_value_enumleftbound))) /\\ (exists jt_gap_enumleftboundvalue. jt_gap_enumleftboundvalue+S (jt_value_enumleftbound)=(m)))) /\\ (forall jt_divisor_enumleftprimitive. (exists jt_factor_enumleftprimitivemodulus. (m)=(jt_divisor_enumleftprimitive)*jt_factor_enumleftprimitivemodulus) -> (forall jt_index_enumleftprimitivecoordinates jt_value_enumleftprimitivecoordinates. (exists jt_gap_enumleftprimitivecoordinatesindex. jt_gap_enumleftprimitivecoordinatesindex+S (jt_index_enumleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumleftprimitivecoordinatesat. fs_h_jt_enumleftprimitivecoordinatesat + S (jt_value_enumleftprimitivecoordinates) = S ((S (jt_index_enumleftprimitivecoordinates)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftprimitivecoordinatesat. jt_b_enumleft = fs_q_jt_enumleftprimitivecoordinatesat * S ((S (jt_index_enumleftprimitivecoordinates)) * jt_c_enumleft) + (jt_value_enumleftprimitivecoordinates))) -> (exists jt_factor_enumleftprimitivecoordinatesdivides. (jt_value_enumleftprimitivecoordinates)=(jt_divisor_enumleftprimitive)*jt_factor_enumleftprimitivecoordinatesdivides)) -> jt_divisor_enumleftprimitive=1))))) /\\ (((forall jt_b_enumleft jt_c_enumleft. (forall jt_index_enumleftinputbound. (exists jt_gap_enumleftinputboundindex. jt_gap_enumleftinputboundindex+S (jt_index_enumleftinputbound)=(k)) -> exists jt_value_enumleftinputbound. ((((exists fs_h_jt_enumleftinputboundat. fs_h_jt_enumleftinputboundat + S (jt_value_enumleftinputbound) = S ((S (jt_index_enumleftinputbound)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftinputboundat. jt_b_enumleft = fs_q_jt_enumleftinputboundat * S ((S (jt_index_enumleftinputbound)) * jt_c_enumleft) + (jt_value_enumleftinputbound))) /\\ (exists jt_gap_enumleftinputboundvalue. jt_gap_enumleftinputboundvalue+S (jt_value_enumleftinputbound)=(m)))) -> (forall jt_divisor_enumleftinputprimitive. (exists jt_factor_enumleftinputprimitivemodulus. (m)=(jt_divisor_enumleftinputprimitive)*jt_factor_enumleftinputprimitivemodulus) -> (forall jt_index_enumleftinputprimitivecoordinates jt_value_enumleftinputprimitivecoordinates. (exists jt_gap_enumleftinputprimitivecoordinatesindex. jt_gap_enumleftinputprimitivecoordinatesindex+S (jt_index_enumleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumleftinputprimitivecoordinatesat. fs_h_jt_enumleftinputprimitivecoordinatesat + S (jt_value_enumleftinputprimitivecoordinates) = S ((S (jt_index_enumleftinputprimitivecoordinates)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftinputprimitivecoordinatesat. jt_b_enumleft = fs_q_jt_enumleftinputprimitivecoordinatesat * S ((S (jt_index_enumleftinputprimitivecoordinates)) * jt_c_enumleft) + (jt_value_enumleftinputprimitivecoordinates))) -> (exists jt_factor_enumleftinputprimitivecoordinatesdivides. (jt_value_enumleftinputprimitivecoordinates)=(jt_divisor_enumleftinputprimitive)*jt_factor_enumleftinputprimitivecoordinatesdivides)) -> jt_divisor_enumleftinputprimitive=1) -> exists jt_i_enumleft jt_d_enumleft jt_e_enumleft. ((exists jt_gap_enumleftcompleteindex. jt_gap_enumleftcompleteindex+S (jt_i_enumleft)=(u)) /\\ (((((((exists fs_h_jt_enumleftcompletecode. fs_h_jt_enumleftcompletecode + S (jt_d_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftcompletecode. A = fs_q_jt_enumleftcompletecode * S ((S (jt_i_enumleft)) * B) + (jt_d_enumleft))) /\\ (((exists fs_h_jt_enumleftcompletescale. fs_h_jt_enumleftcompletescale + S (jt_e_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftcompletescale. C = fs_q_jt_enumleftcompletescale * S ((S (jt_i_enumleft)) * D) + (jt_e_enumleft))))) /\\ (forall jt_index_enumleftrepresented jt_left_enumleftrepresented jt_right_enumleftrepresented. (exists jt_gap_enumleftrepresentedindex. jt_gap_enumleftrepresentedindex+S (jt_index_enumleftrepresented)=(k)) -> (((exists fs_h_jt_enumleftrepresentedleft. fs_h_jt_enumleftrepresentedleft + S (jt_left_enumleftrepresented) = S ((S (jt_index_enumleftrepresented)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftrepresentedleft. jt_b_enumleft = fs_q_jt_enumleftrepresentedleft * S ((S (jt_index_enumleftrepresented)) * jt_c_enumleft) + (jt_left_enumleftrepresented))) -> (((exists fs_h_jt_enumleftrepresentedright. fs_h_jt_enumleftrepresentedright + S (jt_right_enumleftrepresented) = S ((S (jt_index_enumleftrepresented)) * jt_e_enumleft)) /\\ exists fs_q_jt_enumleftrepresentedright. jt_d_enumleft = fs_q_jt_enumleftrepresentedright * S ((S (jt_index_enumleftrepresented)) * jt_e_enumleft) + (jt_right_enumleftrepresented))) -> jt_left_enumleftrepresented=jt_right_enumleftrepresented))))) /\\ (forall jt_i_enumleft jt_h_enumleft jt_b_enumleft jt_c_enumleft jt_d_enumleft jt_e_enumleft. (exists jt_gap_enumleftfirstindex. jt_gap_enumleftfirstindex+S (jt_i_enumleft)=(u)) -> (exists jt_gap_enumleftsecondindex. jt_gap_enumleftsecondindex+S (jt_h_enumleft)=(u)) -> (((((exists fs_h_jt_enumleftfirstcode. fs_h_jt_enumleftfirstcode + S (jt_b_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftfirstcode. A = fs_q_jt_enumleftfirstcode * S ((S (jt_i_enumleft)) * B) + (jt_b_enumleft))) /\\ (((exists fs_h_jt_enumleftfirstscale. fs_h_jt_enumleftfirstscale + S (jt_c_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftfirstscale. C = fs_q_jt_enumleftfirstscale * S ((S (jt_i_enumleft)) * D) + (jt_c_enumleft))))) -> (((((exists fs_h_jt_enumleftsecondcode. fs_h_jt_enumleftsecondcode + S (jt_d_enumleft) = S ((S (jt_h_enumleft)) * B)) /\\ exists fs_q_jt_enumleftsecondcode. A = fs_q_jt_enumleftsecondcode * S ((S (jt_h_enumleft)) * B) + (jt_d_enumleft))) /\\ (((exists fs_h_jt_enumleftsecondscale. fs_h_jt_enumleftsecondscale + S (jt_e_enumleft) = S ((S (jt_h_enumleft)) * D)) /\\ exists fs_q_jt_enumleftsecondscale. C = fs_q_jt_enumleftsecondscale * S ((S (jt_h_enumleft)) * D) + (jt_e_enumleft))))) -> (forall jt_index_enumleftsame jt_left_enumleftsame jt_right_enumleftsame. (exists jt_gap_enumleftsameindex. jt_gap_enumleftsameindex+S (jt_index_enumleftsame)=(k)) -> (((exists fs_h_jt_enumleftsameleft. fs_h_jt_enumleftsameleft + S (jt_left_enumleftsame) = S ((S (jt_index_enumleftsame)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftsameleft. jt_b_enumleft = fs_q_jt_enumleftsameleft * S ((S (jt_index_enumleftsame)) * jt_c_enumleft) + (jt_left_enumleftsame))) -> (((exists fs_h_jt_enumleftsameright. fs_h_jt_enumleftsameright + S (jt_right_enumleftsame) = S ((S (jt_index_enumleftsame)) * jt_e_enumleft)) /\\ exists fs_q_jt_enumleftsameright. jt_d_enumleft = fs_q_jt_enumleftsameright * S ((S (jt_index_enumleftsame)) * jt_e_enumleft) + (jt_right_enumleftsame))) -> jt_left_enumleftsame=jt_right_enumleftsame) -> jt_i_enumleft=jt_h_enumleft))))) -> (((forall jt_i_enumright. (exists jt_gap_enumrightsoundindex. jt_gap_enumrightsoundindex+S (jt_i_enumright)=(v)) -> exists jt_b_enumright jt_c_enumright. ((((((exists fs_h_jt_enumrightsoundcode. fs_h_jt_enumrightsoundcode + S (jt_b_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightsoundcode. E = fs_q_jt_enumrightsoundcode * S ((S (jt_i_enumright)) * F) + (jt_b_enumright))) /\\ (((exists fs_h_jt_enumrightsoundscale. fs_h_jt_enumrightsoundscale + S (jt_c_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightsoundscale. G = fs_q_jt_enumrightsoundscale * S ((S (jt_i_enumright)) * H) + (jt_c_enumright))))) /\\ (((forall jt_index_enumrightbound. (exists jt_gap_enumrightboundindex. jt_gap_enumrightboundindex+S (jt_index_enumrightbound)=(k)) -> exists jt_value_enumrightbound. ((((exists fs_h_jt_enumrightboundat. fs_h_jt_enumrightboundat + S (jt_value_enumrightbound) = S ((S (jt_index_enumrightbound)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightboundat. jt_b_enumright = fs_q_jt_enumrightboundat * S ((S (jt_index_enumrightbound)) * jt_c_enumright) + (jt_value_enumrightbound))) /\\ (exists jt_gap_enumrightboundvalue. jt_gap_enumrightboundvalue+S (jt_value_enumrightbound)=(n)))) /\\ (forall jt_divisor_enumrightprimitive. (exists jt_factor_enumrightprimitivemodulus. (n)=(jt_divisor_enumrightprimitive)*jt_factor_enumrightprimitivemodulus) -> (forall jt_index_enumrightprimitivecoordinates jt_value_enumrightprimitivecoordinates. (exists jt_gap_enumrightprimitivecoordinatesindex. jt_gap_enumrightprimitivecoordinatesindex+S (jt_index_enumrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrightprimitivecoordinatesat. fs_h_jt_enumrightprimitivecoordinatesat + S (jt_value_enumrightprimitivecoordinates) = S ((S (jt_index_enumrightprimitivecoordinates)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightprimitivecoordinatesat. jt_b_enumright = fs_q_jt_enumrightprimitivecoordinatesat * S ((S (jt_index_enumrightprimitivecoordinates)) * jt_c_enumright) + (jt_value_enumrightprimitivecoordinates))) -> (exists jt_factor_enumrightprimitivecoordinatesdivides. (jt_value_enumrightprimitivecoordinates)=(jt_divisor_enumrightprimitive)*jt_factor_enumrightprimitivecoordinatesdivides)) -> jt_divisor_enumrightprimitive=1))))) /\\ (((forall jt_b_enumright jt_c_enumright. (forall jt_index_enumrightinputbound. (exists jt_gap_enumrightinputboundindex. jt_gap_enumrightinputboundindex+S (jt_index_enumrightinputbound)=(k)) -> exists jt_value_enumrightinputbound. ((((exists fs_h_jt_enumrightinputboundat. fs_h_jt_enumrightinputboundat + S (jt_value_enumrightinputbound) = S ((S (jt_index_enumrightinputbound)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightinputboundat. jt_b_enumright = fs_q_jt_enumrightinputboundat * S ((S (jt_index_enumrightinputbound)) * jt_c_enumright) + (jt_value_enumrightinputbound))) /\\ (exists jt_gap_enumrightinputboundvalue. jt_gap_enumrightinputboundvalue+S (jt_value_enumrightinputbound)=(n)))) -> (forall jt_divisor_enumrightinputprimitive. (exists jt_factor_enumrightinputprimitivemodulus. (n)=(jt_divisor_enumrightinputprimitive)*jt_factor_enumrightinputprimitivemodulus) -> (forall jt_index_enumrightinputprimitivecoordinates jt_value_enumrightinputprimitivecoordinates. (exists jt_gap_enumrightinputprimitivecoordinatesindex. jt_gap_enumrightinputprimitivecoordinatesindex+S (jt_index_enumrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrightinputprimitivecoordinatesat. fs_h_jt_enumrightinputprimitivecoordinatesat + S (jt_value_enumrightinputprimitivecoordinates) = S ((S (jt_index_enumrightinputprimitivecoordinates)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightinputprimitivecoordinatesat. jt_b_enumright = fs_q_jt_enumrightinputprimitivecoordinatesat * S ((S (jt_index_enumrightinputprimitivecoordinates)) * jt_c_enumright) + (jt_value_enumrightinputprimitivecoordinates))) -> (exists jt_factor_enumrightinputprimitivecoordinatesdivides. (jt_value_enumrightinputprimitivecoordinates)=(jt_divisor_enumrightinputprimitive)*jt_factor_enumrightinputprimitivecoordinatesdivides)) -> jt_divisor_enumrightinputprimitive=1) -> exists jt_i_enumright jt_d_enumright jt_e_enumright. ((exists jt_gap_enumrightcompleteindex. jt_gap_enumrightcompleteindex+S (jt_i_enumright)=(v)) /\\ (((((((exists fs_h_jt_enumrightcompletecode. fs_h_jt_enumrightcompletecode + S (jt_d_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightcompletecode. E = fs_q_jt_enumrightcompletecode * S ((S (jt_i_enumright)) * F) + (jt_d_enumright))) /\\ (((exists fs_h_jt_enumrightcompletescale. fs_h_jt_enumrightcompletescale + S (jt_e_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightcompletescale. G = fs_q_jt_enumrightcompletescale * S ((S (jt_i_enumright)) * H) + (jt_e_enumright))))) /\\ (forall jt_index_enumrightrepresented jt_left_enumrightrepresented jt_right_enumrightrepresented. (exists jt_gap_enumrightrepresentedindex. jt_gap_enumrightrepresentedindex+S (jt_index_enumrightrepresented)=(k)) -> (((exists fs_h_jt_enumrightrepresentedleft. fs_h_jt_enumrightrepresentedleft + S (jt_left_enumrightrepresented) = S ((S (jt_index_enumrightrepresented)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightrepresentedleft. jt_b_enumright = fs_q_jt_enumrightrepresentedleft * S ((S (jt_index_enumrightrepresented)) * jt_c_enumright) + (jt_left_enumrightrepresented))) -> (((exists fs_h_jt_enumrightrepresentedright. fs_h_jt_enumrightrepresentedright + S (jt_right_enumrightrepresented) = S ((S (jt_index_enumrightrepresented)) * jt_e_enumright)) /\\ exists fs_q_jt_enumrightrepresentedright. jt_d_enumright = fs_q_jt_enumrightrepresentedright * S ((S (jt_index_enumrightrepresented)) * jt_e_enumright) + (jt_right_enumrightrepresented))) -> jt_left_enumrightrepresented=jt_right_enumrightrepresented))))) /\\ (forall jt_i_enumright jt_h_enumright jt_b_enumright jt_c_enumright jt_d_enumright jt_e_enumright. (exists jt_gap_enumrightfirstindex. jt_gap_enumrightfirstindex+S (jt_i_enumright)=(v)) -> (exists jt_gap_enumrightsecondindex. jt_gap_enumrightsecondindex+S (jt_h_enumright)=(v)) -> (((((exists fs_h_jt_enumrightfirstcode. fs_h_jt_enumrightfirstcode + S (jt_b_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightfirstcode. E = fs_q_jt_enumrightfirstcode * S ((S (jt_i_enumright)) * F) + (jt_b_enumright))) /\\ (((exists fs_h_jt_enumrightfirstscale. fs_h_jt_enumrightfirstscale + S (jt_c_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightfirstscale. G = fs_q_jt_enumrightfirstscale * S ((S (jt_i_enumright)) * H) + (jt_c_enumright))))) -> (((((exists fs_h_jt_enumrightsecondcode. fs_h_jt_enumrightsecondcode + S (jt_d_enumright) = S ((S (jt_h_enumright)) * F)) /\\ exists fs_q_jt_enumrightsecondcode. E = fs_q_jt_enumrightsecondcode * S ((S (jt_h_enumright)) * F) + (jt_d_enumright))) /\\ (((exists fs_h_jt_enumrightsecondscale. fs_h_jt_enumrightsecondscale + S (jt_e_enumright) = S ((S (jt_h_enumright)) * H)) /\\ exists fs_q_jt_enumrightsecondscale. G = fs_q_jt_enumrightsecondscale * S ((S (jt_h_enumright)) * H) + (jt_e_enumright))))) -> (forall jt_index_enumrightsame jt_left_enumrightsame jt_right_enumrightsame. (exists jt_gap_enumrightsameindex. jt_gap_enumrightsameindex+S (jt_index_enumrightsame)=(k)) -> (((exists fs_h_jt_enumrightsameleft. fs_h_jt_enumrightsameleft + S (jt_left_enumrightsame) = S ((S (jt_index_enumrightsame)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightsameleft. jt_b_enumright = fs_q_jt_enumrightsameleft * S ((S (jt_index_enumrightsame)) * jt_c_enumright) + (jt_left_enumrightsame))) -> (((exists fs_h_jt_enumrightsameright. fs_h_jt_enumrightsameright + S (jt_right_enumrightsame) = S ((S (jt_index_enumrightsame)) * jt_e_enumright)) /\\ exists fs_q_jt_enumrightsameright. jt_d_enumright = fs_q_jt_enumrightsameright * S ((S (jt_index_enumrightsame)) * jt_e_enumright) + (jt_right_enumrightsame))) -> jt_left_enumrightsame=jt_right_enumrightsame) -> jt_i_enumright=jt_h_enumright))))) -> (forall jt_index_enumrect. (exists jt_gap_enumrectindex. jt_gap_enumrectindex+S (jt_index_enumrect)=(u*v)) -> exists jt_row_enumrect jt_column_enumrect jt_b_enumrect jt_c_enumrect jt_d_enumrect jt_e_enumrect jt_f_enumrect jt_g_enumrect. ((exists jt_gap_enumrectrow. jt_gap_enumrectrow+S (jt_row_enumrect)=(u)) /\\ (((exists jt_gap_enumrectcolumn. jt_gap_enumrectcolumn+S (jt_column_enumrect)=(v)) /\\ (((jt_index_enumrect=(v)*jt_row_enumrect+jt_column_enumrect) /\\ (((((((exists fs_h_jt_enumrectleftcode. fs_h_jt_enumrectleftcode + S (jt_b_enumrect) = S ((S (jt_row_enumrect)) * B)) /\\ exists fs_q_jt_enumrectleftcode. A = fs_q_jt_enumrectleftcode * S ((S (jt_row_enumrect)) * B) + (jt_b_enumrect))) /\\ (((exists fs_h_jt_enumrectleftscale. fs_h_jt_enumrectleftscale + S (jt_c_enumrect) = S ((S (jt_row_enumrect)) * D)) /\\ exists fs_q_jt_enumrectleftscale. C = fs_q_jt_enumrectleftscale * S ((S (jt_row_enumrect)) * D) + (jt_c_enumrect))))) /\\ (((((((exists fs_h_jt_enumrectrightcode. fs_h_jt_enumrectrightcode + S (jt_d_enumrect) = S ((S (jt_column_enumrect)) * F)) /\\ exists fs_q_jt_enumrectrightcode. E = fs_q_jt_enumrectrightcode * S ((S (jt_column_enumrect)) * F) + (jt_d_enumrect))) /\\ (((exists fs_h_jt_enumrectrightscale. fs_h_jt_enumrectrightscale + S (jt_e_enumrect) = S ((S (jt_column_enumrect)) * H)) /\\ exists fs_q_jt_enumrectrightscale. G = fs_q_jt_enumrectrightscale * S ((S (jt_column_enumrect)) * H) + (jt_e_enumrect))))) /\\ (((((((exists fs_h_jt_enumrectoutputcode. fs_h_jt_enumrectoutputcode + S (jt_f_enumrect) = S ((S (jt_index_enumrect)) * Q)) /\\ exists fs_q_jt_enumrectoutputcode. P = fs_q_jt_enumrectoutputcode * S ((S (jt_index_enumrect)) * Q) + (jt_f_enumrect))) /\\ (((exists fs_h_jt_enumrectoutputscale. fs_h_jt_enumrectoutputscale + S (jt_g_enumrect) = S ((S (jt_index_enumrect)) * T)) /\\ exists fs_q_jt_enumrectoutputscale. R = fs_q_jt_enumrectoutputscale * S ((S (jt_index_enumrect)) * T) + (jt_g_enumrect))))) /\\ (((((forall jt_index_enumrectcrtbound. (exists jt_gap_enumrectcrtboundindex. jt_gap_enumrectcrtboundindex+S (jt_index_enumrectcrtbound)=(k)) -> exists jt_value_enumrectcrtbound. ((((exists fs_h_jt_enumrectcrtboundat. fs_h_jt_enumrectcrtboundat + S (jt_value_enumrectcrtbound) = S ((S (jt_index_enumrectcrtbound)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtboundat. jt_f_enumrect = fs_q_jt_enumrectcrtboundat * S ((S (jt_index_enumrectcrtbound)) * jt_g_enumrect) + (jt_value_enumrectcrtbound))) /\\ (exists jt_gap_enumrectcrtboundvalue. jt_gap_enumrectcrtboundvalue+S (jt_value_enumrectcrtbound)=(m*n)))) /\\ (((forall jt_index_enumrectcrtleft jt_left_enumrectcrtleft jt_right_enumrectcrtleft. (exists jt_gap_enumrectcrtleftindex. jt_gap_enumrectcrtleftindex+S (jt_index_enumrectcrtleft)=(k)) -> (((exists fs_h_jt_enumrectcrtleftleft. fs_h_jt_enumrectcrtleftleft + S (jt_left_enumrectcrtleft) = S ((S (jt_index_enumrectcrtleft)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtleftleft. jt_f_enumrect = fs_q_jt_enumrectcrtleftleft * S ((S (jt_index_enumrectcrtleft)) * jt_g_enumrect) + (jt_left_enumrectcrtleft))) -> (((exists fs_h_jt_enumrectcrtleftright. fs_h_jt_enumrectcrtleftright + S (jt_right_enumrectcrtleft) = S ((S (jt_index_enumrectcrtleft)) * jt_c_enumrect)) /\\ exists fs_q_jt_enumrectcrtleftright. jt_b_enumrect = fs_q_jt_enumrectcrtleftright * S ((S (jt_index_enumrectcrtleft)) * jt_c_enumrect) + (jt_right_enumrectcrtleft))) -> (exists jt_left_enumrectcrtleftmod jt_right_enumrectcrtleftmod. (jt_left_enumrectcrtleft)+(m)*jt_left_enumrectcrtleftmod=(jt_right_enumrectcrtleft)+(m)*jt_right_enumrectcrtleftmod)) /\\ (forall jt_index_enumrectcrtright jt_left_enumrectcrtright jt_right_enumrectcrtright. (exists jt_gap_enumrectcrtrightindex. jt_gap_enumrectcrtrightindex+S (jt_index_enumrectcrtright)=(k)) -> (((exists fs_h_jt_enumrectcrtrightleft. fs_h_jt_enumrectcrtrightleft + S (jt_left_enumrectcrtright) = S ((S (jt_index_enumrectcrtright)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtrightleft. jt_f_enumrect = fs_q_jt_enumrectcrtrightleft * S ((S (jt_index_enumrectcrtright)) * jt_g_enumrect) + (jt_left_enumrectcrtright))) -> (((exists fs_h_jt_enumrectcrtrightright. fs_h_jt_enumrectcrtrightright + S (jt_right_enumrectcrtright) = S ((S (jt_index_enumrectcrtright)) * jt_e_enumrect)) /\\ exists fs_q_jt_enumrectcrtrightright. jt_d_enumrect = fs_q_jt_enumrectcrtrightright * S ((S (jt_index_enumrectcrtright)) * jt_e_enumrect) + (jt_right_enumrectcrtright))) -> (exists jt_left_enumrectcrtrightmod jt_right_enumrectcrtrightmod. (jt_left_enumrectcrtright)+(n)*jt_left_enumrectcrtrightmod=(jt_right_enumrectcrtright)+(n)*jt_right_enumrectcrtrightmod)))))) /\\ (forall jt_divisor_enumrectprimitive. (exists jt_factor_enumrectprimitivemodulus. (m*n)=(jt_divisor_enumrectprimitive)*jt_factor_enumrectprimitivemodulus) -> (forall jt_index_enumrectprimitivecoordinates jt_value_enumrectprimitivecoordinates. (exists jt_gap_enumrectprimitivecoordinatesindex. jt_gap_enumrectprimitivecoordinatesindex+S (jt_index_enumrectprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrectprimitivecoordinatesat. fs_h_jt_enumrectprimitivecoordinatesat + S (jt_value_enumrectprimitivecoordinates) = S ((S (jt_index_enumrectprimitivecoordinates)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectprimitivecoordinatesat. jt_f_enumrect = fs_q_jt_enumrectprimitivecoordinatesat * S ((S (jt_index_enumrectprimitivecoordinates)) * jt_g_enumrect) + (jt_value_enumrectprimitivecoordinates))) -> (exists jt_factor_enumrectprimitivecoordinatesdivides. (jt_value_enumrectprimitivecoordinates)=(jt_divisor_enumrectprimitive)*jt_factor_enumrectprimitivecoordinatesdivides)) -> jt_divisor_enumrectprimitive=1))))))))))))))) -> (forall jt_index_coverbound. (exists jt_gap_coverboundindex. jt_gap_coverboundindex+S (jt_index_coverbound)=(k)) -> exists jt_value_coverbound. ((((exists fs_h_jt_coverboundat. fs_h_jt_coverboundat + S (jt_value_coverbound) = S ((S (jt_index_coverbound)) * c)) /\\ exists fs_q_jt_coverboundat. b = fs_q_jt_coverboundat * S ((S (jt_index_coverbound)) * c) + (jt_value_coverbound))) /\\ (exists jt_gap_coverboundvalue. jt_gap_coverboundvalue+S (jt_value_coverbound)=(m*n)))) -> (forall jt_divisor_coverprim. (exists jt_factor_coverprimmodulus. (m*n)=(jt_divisor_coverprim)*jt_factor_coverprimmodulus) -> (forall jt_index_coverprimcoordinates jt_value_coverprimcoordinates. (exists jt_gap_coverprimcoordinatesindex. jt_gap_coverprimcoordinatesindex+S (jt_index_coverprimcoordinates)=(k)) -> (((exists fs_h_jt_coverprimcoordinatesat. fs_h_jt_coverprimcoordinatesat + S (jt_value_coverprimcoordinates) = S ((S (jt_index_coverprimcoordinates)) * c)) /\\ exists fs_q_jt_coverprimcoordinatesat. b = fs_q_jt_coverprimcoordinatesat * S ((S (jt_index_coverprimcoordinates)) * c) + (jt_value_coverprimcoordinates))) -> (exists jt_factor_coverprimcoordinatesdivides. (jt_value_coverprimcoordinates)=(jt_divisor_coverprim)*jt_factor_coverprimcoordinatesdivides)) -> jt_divisor_coverprim=1) -> (exists jt_index_coverlisted jt_code_coverlisted jt_scale_coverlisted. ((exists jt_gap_coverlistedindex. jt_gap_coverlistedindex+S (jt_index_coverlisted)=(u*v)) /\\ (((((((exists fs_h_jt_coverlistedcode. fs_h_jt_coverlistedcode + S (jt_code_coverlisted) = S ((S (jt_index_coverlisted)) * Q)) /\\ exists fs_q_jt_coverlistedcode. P = fs_q_jt_coverlistedcode * S ((S (jt_index_coverlisted)) * Q) + (jt_code_coverlisted))) /\\ (((exists fs_h_jt_coverlistedscale. fs_h_jt_coverlistedscale + S (jt_scale_coverlisted) = S ((S (jt_index_coverlisted)) * T)) /\\ exists fs_q_jt_coverlistedscale. R = fs_q_jt_coverlistedscale * S ((S (jt_index_coverlisted)) * T) + (jt_scale_coverlisted))))) /\\ (forall jt_index_coverlistedequal jt_left_coverlistedequal jt_right_coverlistedequal. (exists jt_gap_coverlistedequalindex. jt_gap_coverlistedequalindex+S (jt_index_coverlistedequal)=(k)) -> (((exists fs_h_jt_coverlistedequalleft. fs_h_jt_coverlistedequalleft + S (jt_left_coverlistedequal) = S ((S (jt_index_coverlistedequal)) * c)) /\\ exists fs_q_jt_coverlistedequalleft. b = fs_q_jt_coverlistedequalleft * S ((S (jt_index_coverlistedequal)) * c) + (jt_left_coverlistedequal))) -> (((exists fs_h_jt_coverlistedequalright. fs_h_jt_coverlistedequalright + S (jt_right_coverlistedequal) = S ((S (jt_index_coverlistedequal)) * jt_scale_coverlisted)) /\\ exists fs_q_jt_coverlistedequalright. jt_code_coverlisted = fs_q_jt_coverlistedequalright * S ((S (jt_index_coverlistedequal)) * jt_scale_coverlisted) + (jt_right_coverlistedequal))) -> jt_left_coverlistedequal=jt_right_coverlistedequal)))))", "statement_sha256": "4cbf8e3e92b8d86002bdc9c90b51f74fe27a49f231d60bdb5b7858f2f56cd1f4", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every primitive product-modulus tuple reduces to a genuine source pair and is formally equal to its table output." }, { "admission_dependencies": [ "jordan_rectangle_crt_covers", "jordan_rectangle_crt_distinct" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 70, "body_proof_nodes": 167, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro hm", "intro hn", "intro hcop", "intro hl", "intro hh", "intro hr", "split", "intro p", "intro hp", "have hv : ∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. ∃ f. ∃ g. Lt(i,u) ∧ (Lt(j,v) ∧ (p = v · i + j ∧ (BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) ∧ (BetaAt(P,Q,p,f) ∧ BetaAt(R,T,p,g) ∧ (JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k) ∧ JordanPrimitiveTuple(m · n,f,g,k)))))))", "specialize hr (p)", "apply hr", "exact hp", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "exists x6", "exists x7", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "split", "have hc : JordanCanonicalTupleCRT(m,n,x2,x3,x4,x5,x6,x7,k)", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "cases hc", "exact hc_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "split", "intro b", "intro c", "intro hb", "intro hp", "specialize jordan_rectangle_crt_covers (m)", "specialize jordan_rectangle_crt_covers (n)", "specialize jordan_rectangle_crt_covers (k)", "specialize jordan_rectangle_crt_covers (A)", "specialize jordan_rectangle_crt_covers (B)", "specialize jordan_rectangle_crt_covers (C)", "specialize jordan_rectangle_crt_covers (D)", "specialize jordan_rectangle_crt_covers (u)", "specialize jordan_rectangle_crt_covers (E)", "specialize jordan_rectangle_crt_covers (F)", "specialize jordan_rectangle_crt_covers (G)", "specialize jordan_rectangle_crt_covers (H)", "specialize jordan_rectangle_crt_covers (v)", "specialize jordan_rectangle_crt_covers (P)", "specialize jordan_rectangle_crt_covers (Q)", "specialize jordan_rectangle_crt_covers (R)", "specialize jordan_rectangle_crt_covers (T)", "specialize jordan_rectangle_crt_covers (b)", "specialize jordan_rectangle_crt_covers (c)", "apply jordan_rectangle_crt_covers", "exact hm", "exact hn", "exact hcop", "exact hl", "exact hh", "exact hr", "exact hb", "exact hp", "intro i", "intro j", "intro b", "intro c", "intro d", "intro e", "intro hi", "intro hj", "intro he", "intro hf", "intro hsame", "specialize jordan_rectangle_crt_distinct (m)", "specialize jordan_rectangle_crt_distinct (n)", "specialize jordan_rectangle_crt_distinct (k)", "specialize jordan_rectangle_crt_distinct (A)", "specialize jordan_rectangle_crt_distinct (B)", "specialize jordan_rectangle_crt_distinct (C)", "specialize jordan_rectangle_crt_distinct (D)", "specialize jordan_rectangle_crt_distinct (u)", "specialize jordan_rectangle_crt_distinct (E)", "specialize jordan_rectangle_crt_distinct (F)", "specialize jordan_rectangle_crt_distinct (G)", "specialize jordan_rectangle_crt_distinct (H)", "specialize jordan_rectangle_crt_distinct (v)", "specialize jordan_rectangle_crt_distinct (P)", "specialize jordan_rectangle_crt_distinct (Q)", "specialize jordan_rectangle_crt_distinct (R)", "specialize jordan_rectangle_crt_distinct (T)", "specialize jordan_rectangle_crt_distinct (i)", "specialize jordan_rectangle_crt_distinct (j)", "specialize jordan_rectangle_crt_distinct (b)", "specialize jordan_rectangle_crt_distinct (c)", "specialize jordan_rectangle_crt_distinct (d)", "specialize jordan_rectangle_crt_distinct (e)", "apply jordan_rectangle_crt_distinct", "exact hl", "exact hh", "exact hr", "exact hi", "exact hj", "exact he", "exact hf", "exact hsame" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v) → JordanTupleEnumeration(k,m · n,P,Q,R,T,u · v)", "defined_statement_sha256": "3d6c029ee77820d22d3d718a7d26e59a5b3a6cb064a640c09b358dcd77ec0f3b", "definition_uses": { "ND0372": 1, "ND0374": 3, "ND0380": 2, "ND0381": 1, "PD0002": 2, "PD0005": 1, "PD0013": 6 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "5a8e6693884fa870726dac97097f177cd1ee5ebf5cadae1c5098a181782bf25e", "free_names": [], "script_definition_uses": { "ND0372": 1, "ND0380": 2, "PD0002": 2, "PD0013": 6 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro P" } ], [ { "kind": "text", "text": "intro Q" } ], [ { "kind": "text", "text": "intro R" } ], [ { "kind": "text", "text": "intro T" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ i. ∃ j. ∃ b. ∃ c. ∃ d. ∃ e. ∃ f. ∃ g. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (p = v · i + j ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,p,f)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(R,T,p,g)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,b,c,d,e,f,g,k)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m · n,f,g,k)" }, { "kind": "text", "text": "))))))" } ], [ { "kind": "text", "text": "specialize hr (p)" } ], [ { "kind": "text", "text": "apply hr" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "exists x6" } ], [ { "kind": "text", "text": "exists x7" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "have hc : " }, { "definition": "ND0380", "kind": "definition", "text": "JordanCanonicalTupleCRT(m,n,x2,x3,x4,x5,x6,x7,k)" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "exact hc_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (P)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (Q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (R)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (T)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (b)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_covers (c)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_covers" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro hf" } ], [ { "kind": "text", "text": "intro hsame" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (P)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (Q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (R)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (T)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (i)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (j)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (b)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (c)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (d)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_distinct (e)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_distinct" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "exact he" } ], [ { "kind": "text", "text": "exact hf" } ], [ { "kind": "text", "text": "exact hsame" } ] ], "statement_definition_uses": { "ND0374": 3, "ND0381": 1, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ P. ∀ Q. ∀ R. ∀ T. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m · n,P,Q,R,T,u · v)" } ] }, "dependencies": [ "jordan_rectangle_crt_covers", "jordan_rectangle_crt_distinct" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0048", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_rectangle_crt_enumeration", "original_ha_bundle_verified": true, "proof_bundle_node_id": 333, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro P", "intro Q", "intro R", "intro T", "intro hm", "intro hn", "intro hcop", "intro hl", "intro hh", "intro hr", "split", "intro p", "intro hp", "have hv : exists i j b c d e f g. ((exists jt_gap_soundrectvaluerow. jt_gap_soundrectvaluerow+S (i)=(u)) /\\ (((exists jt_gap_soundrectvaluecolumn. jt_gap_soundrectvaluecolumn+S (j)=(v)) /\\ (((p=(v)*(i)+(j)) /\\ (((((((exists fs_h_jt_soundrectvalueleftcode. fs_h_jt_soundrectvalueleftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_soundrectvalueleftcode. A = fs_q_jt_soundrectvalueleftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_soundrectvalueleftscale. fs_h_jt_soundrectvalueleftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_soundrectvalueleftscale. C = fs_q_jt_soundrectvalueleftscale * S ((S (i)) * D) + (c))))) /\\ (((((((exists fs_h_jt_soundrectvaluerightcode. fs_h_jt_soundrectvaluerightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_soundrectvaluerightcode. E = fs_q_jt_soundrectvaluerightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_soundrectvaluerightscale. fs_h_jt_soundrectvaluerightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_soundrectvaluerightscale. G = fs_q_jt_soundrectvaluerightscale * S ((S (j)) * H) + (e))))) /\\ (((((((exists fs_h_jt_soundrectvalueoutputcode. fs_h_jt_soundrectvalueoutputcode + S (f) = S ((S (p)) * Q)) /\\ exists fs_q_jt_soundrectvalueoutputcode. P = fs_q_jt_soundrectvalueoutputcode * S ((S (p)) * Q) + (f))) /\\ (((exists fs_h_jt_soundrectvalueoutputscale. fs_h_jt_soundrectvalueoutputscale + S (g) = S ((S (p)) * T)) /\\ exists fs_q_jt_soundrectvalueoutputscale. R = fs_q_jt_soundrectvalueoutputscale * S ((S (p)) * T) + (g))))) /\\ (((((forall jt_index_soundrectvaluecrtbound. (exists jt_gap_soundrectvaluecrtboundindex. jt_gap_soundrectvaluecrtboundindex+S (jt_index_soundrectvaluecrtbound)=(k)) -> exists jt_value_soundrectvaluecrtbound. ((((exists fs_h_jt_soundrectvaluecrtboundat. fs_h_jt_soundrectvaluecrtboundat + S (jt_value_soundrectvaluecrtbound) = S ((S (jt_index_soundrectvaluecrtbound)) * g)) /\\ exists fs_q_jt_soundrectvaluecrtboundat. f = fs_q_jt_soundrectvaluecrtboundat * S ((S (jt_index_soundrectvaluecrtbound)) * g) + (jt_value_soundrectvaluecrtbound))) /\\ (exists jt_gap_soundrectvaluecrtboundvalue. jt_gap_soundrectvaluecrtboundvalue+S (jt_value_soundrectvaluecrtbound)=(m*n)))) /\\ (((forall jt_index_soundrectvaluecrtleft jt_left_soundrectvaluecrtleft jt_right_soundrectvaluecrtleft. (exists jt_gap_soundrectvaluecrtleftindex. jt_gap_soundrectvaluecrtleftindex+S (jt_index_soundrectvaluecrtleft)=(k)) -> (((exists fs_h_jt_soundrectvaluecrtleftleft. fs_h_jt_soundrectvaluecrtleftleft + S (jt_left_soundrectvaluecrtleft) = S ((S (jt_index_soundrectvaluecrtleft)) * g)) /\\ exists fs_q_jt_soundrectvaluecrtleftleft. f = fs_q_jt_soundrectvaluecrtleftleft * S ((S (jt_index_soundrectvaluecrtleft)) * g) + (jt_left_soundrectvaluecrtleft))) -> (((exists fs_h_jt_soundrectvaluecrtleftright. fs_h_jt_soundrectvaluecrtleftright + S (jt_right_soundrectvaluecrtleft) = S ((S (jt_index_soundrectvaluecrtleft)) * c)) /\\ exists fs_q_jt_soundrectvaluecrtleftright. b = fs_q_jt_soundrectvaluecrtleftright * S ((S (jt_index_soundrectvaluecrtleft)) * c) + (jt_right_soundrectvaluecrtleft))) -> (exists jt_left_soundrectvaluecrtleftmod jt_right_soundrectvaluecrtleftmod. (jt_left_soundrectvaluecrtleft)+(m)*jt_left_soundrectvaluecrtleftmod=(jt_right_soundrectvaluecrtleft)+(m)*jt_right_soundrectvaluecrtleftmod)) /\\ (forall jt_index_soundrectvaluecrtright jt_left_soundrectvaluecrtright jt_right_soundrectvaluecrtright. (exists jt_gap_soundrectvaluecrtrightindex. jt_gap_soundrectvaluecrtrightindex+S (jt_index_soundrectvaluecrtright)=(k)) -> (((exists fs_h_jt_soundrectvaluecrtrightleft. fs_h_jt_soundrectvaluecrtrightleft + S (jt_left_soundrectvaluecrtright) = S ((S (jt_index_soundrectvaluecrtright)) * g)) /\\ exists fs_q_jt_soundrectvaluecrtrightleft. f = fs_q_jt_soundrectvaluecrtrightleft * S ((S (jt_index_soundrectvaluecrtright)) * g) + (jt_left_soundrectvaluecrtright))) -> (((exists fs_h_jt_soundrectvaluecrtrightright. fs_h_jt_soundrectvaluecrtrightright + S (jt_right_soundrectvaluecrtright) = S ((S (jt_index_soundrectvaluecrtright)) * e)) /\\ exists fs_q_jt_soundrectvaluecrtrightright. d = fs_q_jt_soundrectvaluecrtrightright * S ((S (jt_index_soundrectvaluecrtright)) * e) + (jt_right_soundrectvaluecrtright))) -> (exists jt_left_soundrectvaluecrtrightmod jt_right_soundrectvaluecrtrightmod. (jt_left_soundrectvaluecrtright)+(n)*jt_left_soundrectvaluecrtrightmod=(jt_right_soundrectvaluecrtright)+(n)*jt_right_soundrectvaluecrtrightmod)))))) /\\ (forall jt_divisor_soundrectvalueprimitive. (exists jt_factor_soundrectvalueprimitivemodulus. (m*n)=(jt_divisor_soundrectvalueprimitive)*jt_factor_soundrectvalueprimitivemodulus) -> (forall jt_index_soundrectvalueprimitivecoordinates jt_value_soundrectvalueprimitivecoordinates. (exists jt_gap_soundrectvalueprimitivecoordinatesindex. jt_gap_soundrectvalueprimitivecoordinatesindex+S (jt_index_soundrectvalueprimitivecoordinates)=(k)) -> (((exists fs_h_jt_soundrectvalueprimitivecoordinatesat. fs_h_jt_soundrectvalueprimitivecoordinatesat + S (jt_value_soundrectvalueprimitivecoordinates) = S ((S (jt_index_soundrectvalueprimitivecoordinates)) * g)) /\\ exists fs_q_jt_soundrectvalueprimitivecoordinatesat. f = fs_q_jt_soundrectvalueprimitivecoordinatesat * S ((S (jt_index_soundrectvalueprimitivecoordinates)) * g) + (jt_value_soundrectvalueprimitivecoordinates))) -> (exists jt_factor_soundrectvalueprimitivecoordinatesdivides. (jt_value_soundrectvalueprimitivecoordinates)=(jt_divisor_soundrectvalueprimitive)*jt_factor_soundrectvalueprimitivecoordinatesdivides)) -> jt_divisor_soundrectvalueprimitive=1))))))))))))))", "specialize hr (p)", "apply hr", "exact hp", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "cases hv_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right", "cases hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right", "exists x6", "exists x7", "split", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_left", "split", "have hc : ((forall jt_index_soundcrtbound. (exists jt_gap_soundcrtboundindex. jt_gap_soundcrtboundindex+S (jt_index_soundcrtbound)=(k)) -> exists jt_value_soundcrtbound. ((((exists fs_h_jt_soundcrtboundat. fs_h_jt_soundcrtboundat + S (jt_value_soundcrtbound) = S ((S (jt_index_soundcrtbound)) * x7)) /\\ exists fs_q_jt_soundcrtboundat. x6 = fs_q_jt_soundcrtboundat * S ((S (jt_index_soundcrtbound)) * x7) + (jt_value_soundcrtbound))) /\\ (exists jt_gap_soundcrtboundvalue. jt_gap_soundcrtboundvalue+S (jt_value_soundcrtbound)=(m*n)))) /\\ (((forall jt_index_soundcrtleft jt_left_soundcrtleft jt_right_soundcrtleft. (exists jt_gap_soundcrtleftindex. jt_gap_soundcrtleftindex+S (jt_index_soundcrtleft)=(k)) -> (((exists fs_h_jt_soundcrtleftleft. fs_h_jt_soundcrtleftleft + S (jt_left_soundcrtleft) = S ((S (jt_index_soundcrtleft)) * x7)) /\\ exists fs_q_jt_soundcrtleftleft. x6 = fs_q_jt_soundcrtleftleft * S ((S (jt_index_soundcrtleft)) * x7) + (jt_left_soundcrtleft))) -> (((exists fs_h_jt_soundcrtleftright. fs_h_jt_soundcrtleftright + S (jt_right_soundcrtleft) = S ((S (jt_index_soundcrtleft)) * x3)) /\\ exists fs_q_jt_soundcrtleftright. x2 = fs_q_jt_soundcrtleftright * S ((S (jt_index_soundcrtleft)) * x3) + (jt_right_soundcrtleft))) -> (exists jt_left_soundcrtleftmod jt_right_soundcrtleftmod. (jt_left_soundcrtleft)+(m)*jt_left_soundcrtleftmod=(jt_right_soundcrtleft)+(m)*jt_right_soundcrtleftmod)) /\\ (forall jt_index_soundcrtright jt_left_soundcrtright jt_right_soundcrtright. (exists jt_gap_soundcrtrightindex. jt_gap_soundcrtrightindex+S (jt_index_soundcrtright)=(k)) -> (((exists fs_h_jt_soundcrtrightleft. fs_h_jt_soundcrtrightleft + S (jt_left_soundcrtright) = S ((S (jt_index_soundcrtright)) * x7)) /\\ exists fs_q_jt_soundcrtrightleft. x6 = fs_q_jt_soundcrtrightleft * S ((S (jt_index_soundcrtright)) * x7) + (jt_left_soundcrtright))) -> (((exists fs_h_jt_soundcrtrightright. fs_h_jt_soundcrtrightright + S (jt_right_soundcrtright) = S ((S (jt_index_soundcrtright)) * x5)) /\\ exists fs_q_jt_soundcrtrightright. x4 = fs_q_jt_soundcrtrightright * S ((S (jt_index_soundcrtright)) * x5) + (jt_right_soundcrtright))) -> (exists jt_left_soundcrtrightmod jt_right_soundcrtrightmod. (jt_left_soundcrtright)+(n)*jt_left_soundcrtrightmod=(jt_right_soundcrtright)+(n)*jt_right_soundcrtrightmod)))))", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_left", "cases hc", "exact hc_left", "exact hv_witness_witness_witness_witness_witness_witness_witness_witness_right_right_right_right_right_right_right", "split", "intro b", "intro c", "intro hb", "intro hp", "specialize jordan_rectangle_crt_covers (m)", "specialize jordan_rectangle_crt_covers (n)", "specialize jordan_rectangle_crt_covers (k)", "specialize jordan_rectangle_crt_covers (A)", "specialize jordan_rectangle_crt_covers (B)", "specialize jordan_rectangle_crt_covers (C)", "specialize jordan_rectangle_crt_covers (D)", "specialize jordan_rectangle_crt_covers (u)", "specialize jordan_rectangle_crt_covers (E)", "specialize jordan_rectangle_crt_covers (F)", "specialize jordan_rectangle_crt_covers (G)", "specialize jordan_rectangle_crt_covers (H)", "specialize jordan_rectangle_crt_covers (v)", "specialize jordan_rectangle_crt_covers (P)", "specialize jordan_rectangle_crt_covers (Q)", "specialize jordan_rectangle_crt_covers (R)", "specialize jordan_rectangle_crt_covers (T)", "specialize jordan_rectangle_crt_covers (b)", "specialize jordan_rectangle_crt_covers (c)", "apply jordan_rectangle_crt_covers", "exact hm", "exact hn", "exact hcop", "exact hl", "exact hh", "exact hr", "exact hb", "exact hp", "intro i", "intro j", "intro b", "intro c", "intro d", "intro e", "intro hi", "intro hj", "intro he", "intro hf", "intro hsame", "specialize jordan_rectangle_crt_distinct (m)", "specialize jordan_rectangle_crt_distinct (n)", "specialize jordan_rectangle_crt_distinct (k)", "specialize jordan_rectangle_crt_distinct (A)", "specialize jordan_rectangle_crt_distinct (B)", "specialize jordan_rectangle_crt_distinct (C)", "specialize jordan_rectangle_crt_distinct (D)", "specialize jordan_rectangle_crt_distinct (u)", "specialize jordan_rectangle_crt_distinct (E)", "specialize jordan_rectangle_crt_distinct (F)", "specialize jordan_rectangle_crt_distinct (G)", "specialize jordan_rectangle_crt_distinct (H)", "specialize jordan_rectangle_crt_distinct (v)", "specialize jordan_rectangle_crt_distinct (P)", "specialize jordan_rectangle_crt_distinct (Q)", "specialize jordan_rectangle_crt_distinct (R)", "specialize jordan_rectangle_crt_distinct (T)", "specialize jordan_rectangle_crt_distinct (i)", "specialize jordan_rectangle_crt_distinct (j)", "specialize jordan_rectangle_crt_distinct (b)", "specialize jordan_rectangle_crt_distinct (c)", "specialize jordan_rectangle_crt_distinct (d)", "specialize jordan_rectangle_crt_distinct (e)", "apply jordan_rectangle_crt_distinct", "exact hl", "exact hh", "exact hr", "exact hi", "exact hj", "exact he", "exact hf", "exact hsame" ], "script_sha256": "a2ddb60e90042ba8f013576f70e8f05c69bf933ca231a1c81787e3623dd779c3", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "a2ddb60e90042ba8f013576f70e8f05c69bf933ca231a1c81787e3623dd779c3", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "5a8e6693884fa870726dac97097f177cd1ee5ebf5cadae1c5098a181782bf25e" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v P Q R T. ~(m=0) -> ~(n=0) -> (forall jt_divisor_rectenumcop. (exists jt_factor_rectenumcopa. (m)=(jt_divisor_rectenumcop)*jt_factor_rectenumcopa) -> (exists jt_factor_rectenumcopb. (n)=(jt_divisor_rectenumcop)*jt_factor_rectenumcopb) -> jt_divisor_rectenumcop=1) -> (((forall jt_i_enumleft. (exists jt_gap_enumleftsoundindex. jt_gap_enumleftsoundindex+S (jt_i_enumleft)=(u)) -> exists jt_b_enumleft jt_c_enumleft. ((((((exists fs_h_jt_enumleftsoundcode. fs_h_jt_enumleftsoundcode + S (jt_b_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftsoundcode. A = fs_q_jt_enumleftsoundcode * S ((S (jt_i_enumleft)) * B) + (jt_b_enumleft))) /\\ (((exists fs_h_jt_enumleftsoundscale. fs_h_jt_enumleftsoundscale + S (jt_c_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftsoundscale. C = fs_q_jt_enumleftsoundscale * S ((S (jt_i_enumleft)) * D) + (jt_c_enumleft))))) /\\ (((forall jt_index_enumleftbound. (exists jt_gap_enumleftboundindex. jt_gap_enumleftboundindex+S (jt_index_enumleftbound)=(k)) -> exists jt_value_enumleftbound. ((((exists fs_h_jt_enumleftboundat. fs_h_jt_enumleftboundat + S (jt_value_enumleftbound) = S ((S (jt_index_enumleftbound)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftboundat. jt_b_enumleft = fs_q_jt_enumleftboundat * S ((S (jt_index_enumleftbound)) * jt_c_enumleft) + (jt_value_enumleftbound))) /\\ (exists jt_gap_enumleftboundvalue. jt_gap_enumleftboundvalue+S (jt_value_enumleftbound)=(m)))) /\\ (forall jt_divisor_enumleftprimitive. (exists jt_factor_enumleftprimitivemodulus. (m)=(jt_divisor_enumleftprimitive)*jt_factor_enumleftprimitivemodulus) -> (forall jt_index_enumleftprimitivecoordinates jt_value_enumleftprimitivecoordinates. (exists jt_gap_enumleftprimitivecoordinatesindex. jt_gap_enumleftprimitivecoordinatesindex+S (jt_index_enumleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumleftprimitivecoordinatesat. fs_h_jt_enumleftprimitivecoordinatesat + S (jt_value_enumleftprimitivecoordinates) = S ((S (jt_index_enumleftprimitivecoordinates)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftprimitivecoordinatesat. jt_b_enumleft = fs_q_jt_enumleftprimitivecoordinatesat * S ((S (jt_index_enumleftprimitivecoordinates)) * jt_c_enumleft) + (jt_value_enumleftprimitivecoordinates))) -> (exists jt_factor_enumleftprimitivecoordinatesdivides. (jt_value_enumleftprimitivecoordinates)=(jt_divisor_enumleftprimitive)*jt_factor_enumleftprimitivecoordinatesdivides)) -> jt_divisor_enumleftprimitive=1))))) /\\ (((forall jt_b_enumleft jt_c_enumleft. (forall jt_index_enumleftinputbound. (exists jt_gap_enumleftinputboundindex. jt_gap_enumleftinputboundindex+S (jt_index_enumleftinputbound)=(k)) -> exists jt_value_enumleftinputbound. ((((exists fs_h_jt_enumleftinputboundat. fs_h_jt_enumleftinputboundat + S (jt_value_enumleftinputbound) = S ((S (jt_index_enumleftinputbound)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftinputboundat. jt_b_enumleft = fs_q_jt_enumleftinputboundat * S ((S (jt_index_enumleftinputbound)) * jt_c_enumleft) + (jt_value_enumleftinputbound))) /\\ (exists jt_gap_enumleftinputboundvalue. jt_gap_enumleftinputboundvalue+S (jt_value_enumleftinputbound)=(m)))) -> (forall jt_divisor_enumleftinputprimitive. (exists jt_factor_enumleftinputprimitivemodulus. (m)=(jt_divisor_enumleftinputprimitive)*jt_factor_enumleftinputprimitivemodulus) -> (forall jt_index_enumleftinputprimitivecoordinates jt_value_enumleftinputprimitivecoordinates. (exists jt_gap_enumleftinputprimitivecoordinatesindex. jt_gap_enumleftinputprimitivecoordinatesindex+S (jt_index_enumleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumleftinputprimitivecoordinatesat. fs_h_jt_enumleftinputprimitivecoordinatesat + S (jt_value_enumleftinputprimitivecoordinates) = S ((S (jt_index_enumleftinputprimitivecoordinates)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftinputprimitivecoordinatesat. jt_b_enumleft = fs_q_jt_enumleftinputprimitivecoordinatesat * S ((S (jt_index_enumleftinputprimitivecoordinates)) * jt_c_enumleft) + (jt_value_enumleftinputprimitivecoordinates))) -> (exists jt_factor_enumleftinputprimitivecoordinatesdivides. (jt_value_enumleftinputprimitivecoordinates)=(jt_divisor_enumleftinputprimitive)*jt_factor_enumleftinputprimitivecoordinatesdivides)) -> jt_divisor_enumleftinputprimitive=1) -> exists jt_i_enumleft jt_d_enumleft jt_e_enumleft. ((exists jt_gap_enumleftcompleteindex. jt_gap_enumleftcompleteindex+S (jt_i_enumleft)=(u)) /\\ (((((((exists fs_h_jt_enumleftcompletecode. fs_h_jt_enumleftcompletecode + S (jt_d_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftcompletecode. A = fs_q_jt_enumleftcompletecode * S ((S (jt_i_enumleft)) * B) + (jt_d_enumleft))) /\\ (((exists fs_h_jt_enumleftcompletescale. fs_h_jt_enumleftcompletescale + S (jt_e_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftcompletescale. C = fs_q_jt_enumleftcompletescale * S ((S (jt_i_enumleft)) * D) + (jt_e_enumleft))))) /\\ (forall jt_index_enumleftrepresented jt_left_enumleftrepresented jt_right_enumleftrepresented. (exists jt_gap_enumleftrepresentedindex. jt_gap_enumleftrepresentedindex+S (jt_index_enumleftrepresented)=(k)) -> (((exists fs_h_jt_enumleftrepresentedleft. fs_h_jt_enumleftrepresentedleft + S (jt_left_enumleftrepresented) = S ((S (jt_index_enumleftrepresented)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftrepresentedleft. jt_b_enumleft = fs_q_jt_enumleftrepresentedleft * S ((S (jt_index_enumleftrepresented)) * jt_c_enumleft) + (jt_left_enumleftrepresented))) -> (((exists fs_h_jt_enumleftrepresentedright. fs_h_jt_enumleftrepresentedright + S (jt_right_enumleftrepresented) = S ((S (jt_index_enumleftrepresented)) * jt_e_enumleft)) /\\ exists fs_q_jt_enumleftrepresentedright. jt_d_enumleft = fs_q_jt_enumleftrepresentedright * S ((S (jt_index_enumleftrepresented)) * jt_e_enumleft) + (jt_right_enumleftrepresented))) -> jt_left_enumleftrepresented=jt_right_enumleftrepresented))))) /\\ (forall jt_i_enumleft jt_h_enumleft jt_b_enumleft jt_c_enumleft jt_d_enumleft jt_e_enumleft. (exists jt_gap_enumleftfirstindex. jt_gap_enumleftfirstindex+S (jt_i_enumleft)=(u)) -> (exists jt_gap_enumleftsecondindex. jt_gap_enumleftsecondindex+S (jt_h_enumleft)=(u)) -> (((((exists fs_h_jt_enumleftfirstcode. fs_h_jt_enumleftfirstcode + S (jt_b_enumleft) = S ((S (jt_i_enumleft)) * B)) /\\ exists fs_q_jt_enumleftfirstcode. A = fs_q_jt_enumleftfirstcode * S ((S (jt_i_enumleft)) * B) + (jt_b_enumleft))) /\\ (((exists fs_h_jt_enumleftfirstscale. fs_h_jt_enumleftfirstscale + S (jt_c_enumleft) = S ((S (jt_i_enumleft)) * D)) /\\ exists fs_q_jt_enumleftfirstscale. C = fs_q_jt_enumleftfirstscale * S ((S (jt_i_enumleft)) * D) + (jt_c_enumleft))))) -> (((((exists fs_h_jt_enumleftsecondcode. fs_h_jt_enumleftsecondcode + S (jt_d_enumleft) = S ((S (jt_h_enumleft)) * B)) /\\ exists fs_q_jt_enumleftsecondcode. A = fs_q_jt_enumleftsecondcode * S ((S (jt_h_enumleft)) * B) + (jt_d_enumleft))) /\\ (((exists fs_h_jt_enumleftsecondscale. fs_h_jt_enumleftsecondscale + S (jt_e_enumleft) = S ((S (jt_h_enumleft)) * D)) /\\ exists fs_q_jt_enumleftsecondscale. C = fs_q_jt_enumleftsecondscale * S ((S (jt_h_enumleft)) * D) + (jt_e_enumleft))))) -> (forall jt_index_enumleftsame jt_left_enumleftsame jt_right_enumleftsame. (exists jt_gap_enumleftsameindex. jt_gap_enumleftsameindex+S (jt_index_enumleftsame)=(k)) -> (((exists fs_h_jt_enumleftsameleft. fs_h_jt_enumleftsameleft + S (jt_left_enumleftsame) = S ((S (jt_index_enumleftsame)) * jt_c_enumleft)) /\\ exists fs_q_jt_enumleftsameleft. jt_b_enumleft = fs_q_jt_enumleftsameleft * S ((S (jt_index_enumleftsame)) * jt_c_enumleft) + (jt_left_enumleftsame))) -> (((exists fs_h_jt_enumleftsameright. fs_h_jt_enumleftsameright + S (jt_right_enumleftsame) = S ((S (jt_index_enumleftsame)) * jt_e_enumleft)) /\\ exists fs_q_jt_enumleftsameright. jt_d_enumleft = fs_q_jt_enumleftsameright * S ((S (jt_index_enumleftsame)) * jt_e_enumleft) + (jt_right_enumleftsame))) -> jt_left_enumleftsame=jt_right_enumleftsame) -> jt_i_enumleft=jt_h_enumleft))))) -> (((forall jt_i_enumright. (exists jt_gap_enumrightsoundindex. jt_gap_enumrightsoundindex+S (jt_i_enumright)=(v)) -> exists jt_b_enumright jt_c_enumright. ((((((exists fs_h_jt_enumrightsoundcode. fs_h_jt_enumrightsoundcode + S (jt_b_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightsoundcode. E = fs_q_jt_enumrightsoundcode * S ((S (jt_i_enumright)) * F) + (jt_b_enumright))) /\\ (((exists fs_h_jt_enumrightsoundscale. fs_h_jt_enumrightsoundscale + S (jt_c_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightsoundscale. G = fs_q_jt_enumrightsoundscale * S ((S (jt_i_enumright)) * H) + (jt_c_enumright))))) /\\ (((forall jt_index_enumrightbound. (exists jt_gap_enumrightboundindex. jt_gap_enumrightboundindex+S (jt_index_enumrightbound)=(k)) -> exists jt_value_enumrightbound. ((((exists fs_h_jt_enumrightboundat. fs_h_jt_enumrightboundat + S (jt_value_enumrightbound) = S ((S (jt_index_enumrightbound)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightboundat. jt_b_enumright = fs_q_jt_enumrightboundat * S ((S (jt_index_enumrightbound)) * jt_c_enumright) + (jt_value_enumrightbound))) /\\ (exists jt_gap_enumrightboundvalue. jt_gap_enumrightboundvalue+S (jt_value_enumrightbound)=(n)))) /\\ (forall jt_divisor_enumrightprimitive. (exists jt_factor_enumrightprimitivemodulus. (n)=(jt_divisor_enumrightprimitive)*jt_factor_enumrightprimitivemodulus) -> (forall jt_index_enumrightprimitivecoordinates jt_value_enumrightprimitivecoordinates. (exists jt_gap_enumrightprimitivecoordinatesindex. jt_gap_enumrightprimitivecoordinatesindex+S (jt_index_enumrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrightprimitivecoordinatesat. fs_h_jt_enumrightprimitivecoordinatesat + S (jt_value_enumrightprimitivecoordinates) = S ((S (jt_index_enumrightprimitivecoordinates)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightprimitivecoordinatesat. jt_b_enumright = fs_q_jt_enumrightprimitivecoordinatesat * S ((S (jt_index_enumrightprimitivecoordinates)) * jt_c_enumright) + (jt_value_enumrightprimitivecoordinates))) -> (exists jt_factor_enumrightprimitivecoordinatesdivides. (jt_value_enumrightprimitivecoordinates)=(jt_divisor_enumrightprimitive)*jt_factor_enumrightprimitivecoordinatesdivides)) -> jt_divisor_enumrightprimitive=1))))) /\\ (((forall jt_b_enumright jt_c_enumright. (forall jt_index_enumrightinputbound. (exists jt_gap_enumrightinputboundindex. jt_gap_enumrightinputboundindex+S (jt_index_enumrightinputbound)=(k)) -> exists jt_value_enumrightinputbound. ((((exists fs_h_jt_enumrightinputboundat. fs_h_jt_enumrightinputboundat + S (jt_value_enumrightinputbound) = S ((S (jt_index_enumrightinputbound)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightinputboundat. jt_b_enumright = fs_q_jt_enumrightinputboundat * S ((S (jt_index_enumrightinputbound)) * jt_c_enumright) + (jt_value_enumrightinputbound))) /\\ (exists jt_gap_enumrightinputboundvalue. jt_gap_enumrightinputboundvalue+S (jt_value_enumrightinputbound)=(n)))) -> (forall jt_divisor_enumrightinputprimitive. (exists jt_factor_enumrightinputprimitivemodulus. (n)=(jt_divisor_enumrightinputprimitive)*jt_factor_enumrightinputprimitivemodulus) -> (forall jt_index_enumrightinputprimitivecoordinates jt_value_enumrightinputprimitivecoordinates. (exists jt_gap_enumrightinputprimitivecoordinatesindex. jt_gap_enumrightinputprimitivecoordinatesindex+S (jt_index_enumrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrightinputprimitivecoordinatesat. fs_h_jt_enumrightinputprimitivecoordinatesat + S (jt_value_enumrightinputprimitivecoordinates) = S ((S (jt_index_enumrightinputprimitivecoordinates)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightinputprimitivecoordinatesat. jt_b_enumright = fs_q_jt_enumrightinputprimitivecoordinatesat * S ((S (jt_index_enumrightinputprimitivecoordinates)) * jt_c_enumright) + (jt_value_enumrightinputprimitivecoordinates))) -> (exists jt_factor_enumrightinputprimitivecoordinatesdivides. (jt_value_enumrightinputprimitivecoordinates)=(jt_divisor_enumrightinputprimitive)*jt_factor_enumrightinputprimitivecoordinatesdivides)) -> jt_divisor_enumrightinputprimitive=1) -> exists jt_i_enumright jt_d_enumright jt_e_enumright. ((exists jt_gap_enumrightcompleteindex. jt_gap_enumrightcompleteindex+S (jt_i_enumright)=(v)) /\\ (((((((exists fs_h_jt_enumrightcompletecode. fs_h_jt_enumrightcompletecode + S (jt_d_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightcompletecode. E = fs_q_jt_enumrightcompletecode * S ((S (jt_i_enumright)) * F) + (jt_d_enumright))) /\\ (((exists fs_h_jt_enumrightcompletescale. fs_h_jt_enumrightcompletescale + S (jt_e_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightcompletescale. G = fs_q_jt_enumrightcompletescale * S ((S (jt_i_enumright)) * H) + (jt_e_enumright))))) /\\ (forall jt_index_enumrightrepresented jt_left_enumrightrepresented jt_right_enumrightrepresented. (exists jt_gap_enumrightrepresentedindex. jt_gap_enumrightrepresentedindex+S (jt_index_enumrightrepresented)=(k)) -> (((exists fs_h_jt_enumrightrepresentedleft. fs_h_jt_enumrightrepresentedleft + S (jt_left_enumrightrepresented) = S ((S (jt_index_enumrightrepresented)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightrepresentedleft. jt_b_enumright = fs_q_jt_enumrightrepresentedleft * S ((S (jt_index_enumrightrepresented)) * jt_c_enumright) + (jt_left_enumrightrepresented))) -> (((exists fs_h_jt_enumrightrepresentedright. fs_h_jt_enumrightrepresentedright + S (jt_right_enumrightrepresented) = S ((S (jt_index_enumrightrepresented)) * jt_e_enumright)) /\\ exists fs_q_jt_enumrightrepresentedright. jt_d_enumright = fs_q_jt_enumrightrepresentedright * S ((S (jt_index_enumrightrepresented)) * jt_e_enumright) + (jt_right_enumrightrepresented))) -> jt_left_enumrightrepresented=jt_right_enumrightrepresented))))) /\\ (forall jt_i_enumright jt_h_enumright jt_b_enumright jt_c_enumright jt_d_enumright jt_e_enumright. (exists jt_gap_enumrightfirstindex. jt_gap_enumrightfirstindex+S (jt_i_enumright)=(v)) -> (exists jt_gap_enumrightsecondindex. jt_gap_enumrightsecondindex+S (jt_h_enumright)=(v)) -> (((((exists fs_h_jt_enumrightfirstcode. fs_h_jt_enumrightfirstcode + S (jt_b_enumright) = S ((S (jt_i_enumright)) * F)) /\\ exists fs_q_jt_enumrightfirstcode. E = fs_q_jt_enumrightfirstcode * S ((S (jt_i_enumright)) * F) + (jt_b_enumright))) /\\ (((exists fs_h_jt_enumrightfirstscale. fs_h_jt_enumrightfirstscale + S (jt_c_enumright) = S ((S (jt_i_enumright)) * H)) /\\ exists fs_q_jt_enumrightfirstscale. G = fs_q_jt_enumrightfirstscale * S ((S (jt_i_enumright)) * H) + (jt_c_enumright))))) -> (((((exists fs_h_jt_enumrightsecondcode. fs_h_jt_enumrightsecondcode + S (jt_d_enumright) = S ((S (jt_h_enumright)) * F)) /\\ exists fs_q_jt_enumrightsecondcode. E = fs_q_jt_enumrightsecondcode * S ((S (jt_h_enumright)) * F) + (jt_d_enumright))) /\\ (((exists fs_h_jt_enumrightsecondscale. fs_h_jt_enumrightsecondscale + S (jt_e_enumright) = S ((S (jt_h_enumright)) * H)) /\\ exists fs_q_jt_enumrightsecondscale. G = fs_q_jt_enumrightsecondscale * S ((S (jt_h_enumright)) * H) + (jt_e_enumright))))) -> (forall jt_index_enumrightsame jt_left_enumrightsame jt_right_enumrightsame. (exists jt_gap_enumrightsameindex. jt_gap_enumrightsameindex+S (jt_index_enumrightsame)=(k)) -> (((exists fs_h_jt_enumrightsameleft. fs_h_jt_enumrightsameleft + S (jt_left_enumrightsame) = S ((S (jt_index_enumrightsame)) * jt_c_enumright)) /\\ exists fs_q_jt_enumrightsameleft. jt_b_enumright = fs_q_jt_enumrightsameleft * S ((S (jt_index_enumrightsame)) * jt_c_enumright) + (jt_left_enumrightsame))) -> (((exists fs_h_jt_enumrightsameright. fs_h_jt_enumrightsameright + S (jt_right_enumrightsame) = S ((S (jt_index_enumrightsame)) * jt_e_enumright)) /\\ exists fs_q_jt_enumrightsameright. jt_d_enumright = fs_q_jt_enumrightsameright * S ((S (jt_index_enumrightsame)) * jt_e_enumright) + (jt_right_enumrightsame))) -> jt_left_enumrightsame=jt_right_enumrightsame) -> jt_i_enumright=jt_h_enumright))))) -> (forall jt_index_enumrect. (exists jt_gap_enumrectindex. jt_gap_enumrectindex+S (jt_index_enumrect)=(u*v)) -> exists jt_row_enumrect jt_column_enumrect jt_b_enumrect jt_c_enumrect jt_d_enumrect jt_e_enumrect jt_f_enumrect jt_g_enumrect. ((exists jt_gap_enumrectrow. jt_gap_enumrectrow+S (jt_row_enumrect)=(u)) /\\ (((exists jt_gap_enumrectcolumn. jt_gap_enumrectcolumn+S (jt_column_enumrect)=(v)) /\\ (((jt_index_enumrect=(v)*jt_row_enumrect+jt_column_enumrect) /\\ (((((((exists fs_h_jt_enumrectleftcode. fs_h_jt_enumrectleftcode + S (jt_b_enumrect) = S ((S (jt_row_enumrect)) * B)) /\\ exists fs_q_jt_enumrectleftcode. A = fs_q_jt_enumrectleftcode * S ((S (jt_row_enumrect)) * B) + (jt_b_enumrect))) /\\ (((exists fs_h_jt_enumrectleftscale. fs_h_jt_enumrectleftscale + S (jt_c_enumrect) = S ((S (jt_row_enumrect)) * D)) /\\ exists fs_q_jt_enumrectleftscale. C = fs_q_jt_enumrectleftscale * S ((S (jt_row_enumrect)) * D) + (jt_c_enumrect))))) /\\ (((((((exists fs_h_jt_enumrectrightcode. fs_h_jt_enumrectrightcode + S (jt_d_enumrect) = S ((S (jt_column_enumrect)) * F)) /\\ exists fs_q_jt_enumrectrightcode. E = fs_q_jt_enumrectrightcode * S ((S (jt_column_enumrect)) * F) + (jt_d_enumrect))) /\\ (((exists fs_h_jt_enumrectrightscale. fs_h_jt_enumrectrightscale + S (jt_e_enumrect) = S ((S (jt_column_enumrect)) * H)) /\\ exists fs_q_jt_enumrectrightscale. G = fs_q_jt_enumrectrightscale * S ((S (jt_column_enumrect)) * H) + (jt_e_enumrect))))) /\\ (((((((exists fs_h_jt_enumrectoutputcode. fs_h_jt_enumrectoutputcode + S (jt_f_enumrect) = S ((S (jt_index_enumrect)) * Q)) /\\ exists fs_q_jt_enumrectoutputcode. P = fs_q_jt_enumrectoutputcode * S ((S (jt_index_enumrect)) * Q) + (jt_f_enumrect))) /\\ (((exists fs_h_jt_enumrectoutputscale. fs_h_jt_enumrectoutputscale + S (jt_g_enumrect) = S ((S (jt_index_enumrect)) * T)) /\\ exists fs_q_jt_enumrectoutputscale. R = fs_q_jt_enumrectoutputscale * S ((S (jt_index_enumrect)) * T) + (jt_g_enumrect))))) /\\ (((((forall jt_index_enumrectcrtbound. (exists jt_gap_enumrectcrtboundindex. jt_gap_enumrectcrtboundindex+S (jt_index_enumrectcrtbound)=(k)) -> exists jt_value_enumrectcrtbound. ((((exists fs_h_jt_enumrectcrtboundat. fs_h_jt_enumrectcrtboundat + S (jt_value_enumrectcrtbound) = S ((S (jt_index_enumrectcrtbound)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtboundat. jt_f_enumrect = fs_q_jt_enumrectcrtboundat * S ((S (jt_index_enumrectcrtbound)) * jt_g_enumrect) + (jt_value_enumrectcrtbound))) /\\ (exists jt_gap_enumrectcrtboundvalue. jt_gap_enumrectcrtboundvalue+S (jt_value_enumrectcrtbound)=(m*n)))) /\\ (((forall jt_index_enumrectcrtleft jt_left_enumrectcrtleft jt_right_enumrectcrtleft. (exists jt_gap_enumrectcrtleftindex. jt_gap_enumrectcrtleftindex+S (jt_index_enumrectcrtleft)=(k)) -> (((exists fs_h_jt_enumrectcrtleftleft. fs_h_jt_enumrectcrtleftleft + S (jt_left_enumrectcrtleft) = S ((S (jt_index_enumrectcrtleft)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtleftleft. jt_f_enumrect = fs_q_jt_enumrectcrtleftleft * S ((S (jt_index_enumrectcrtleft)) * jt_g_enumrect) + (jt_left_enumrectcrtleft))) -> (((exists fs_h_jt_enumrectcrtleftright. fs_h_jt_enumrectcrtleftright + S (jt_right_enumrectcrtleft) = S ((S (jt_index_enumrectcrtleft)) * jt_c_enumrect)) /\\ exists fs_q_jt_enumrectcrtleftright. jt_b_enumrect = fs_q_jt_enumrectcrtleftright * S ((S (jt_index_enumrectcrtleft)) * jt_c_enumrect) + (jt_right_enumrectcrtleft))) -> (exists jt_left_enumrectcrtleftmod jt_right_enumrectcrtleftmod. (jt_left_enumrectcrtleft)+(m)*jt_left_enumrectcrtleftmod=(jt_right_enumrectcrtleft)+(m)*jt_right_enumrectcrtleftmod)) /\\ (forall jt_index_enumrectcrtright jt_left_enumrectcrtright jt_right_enumrectcrtright. (exists jt_gap_enumrectcrtrightindex. jt_gap_enumrectcrtrightindex+S (jt_index_enumrectcrtright)=(k)) -> (((exists fs_h_jt_enumrectcrtrightleft. fs_h_jt_enumrectcrtrightleft + S (jt_left_enumrectcrtright) = S ((S (jt_index_enumrectcrtright)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectcrtrightleft. jt_f_enumrect = fs_q_jt_enumrectcrtrightleft * S ((S (jt_index_enumrectcrtright)) * jt_g_enumrect) + (jt_left_enumrectcrtright))) -> (((exists fs_h_jt_enumrectcrtrightright. fs_h_jt_enumrectcrtrightright + S (jt_right_enumrectcrtright) = S ((S (jt_index_enumrectcrtright)) * jt_e_enumrect)) /\\ exists fs_q_jt_enumrectcrtrightright. jt_d_enumrect = fs_q_jt_enumrectcrtrightright * S ((S (jt_index_enumrectcrtright)) * jt_e_enumrect) + (jt_right_enumrectcrtright))) -> (exists jt_left_enumrectcrtrightmod jt_right_enumrectcrtrightmod. (jt_left_enumrectcrtright)+(n)*jt_left_enumrectcrtrightmod=(jt_right_enumrectcrtright)+(n)*jt_right_enumrectcrtrightmod)))))) /\\ (forall jt_divisor_enumrectprimitive. (exists jt_factor_enumrectprimitivemodulus. (m*n)=(jt_divisor_enumrectprimitive)*jt_factor_enumrectprimitivemodulus) -> (forall jt_index_enumrectprimitivecoordinates jt_value_enumrectprimitivecoordinates. (exists jt_gap_enumrectprimitivecoordinatesindex. jt_gap_enumrectprimitivecoordinatesindex+S (jt_index_enumrectprimitivecoordinates)=(k)) -> (((exists fs_h_jt_enumrectprimitivecoordinatesat. fs_h_jt_enumrectprimitivecoordinatesat + S (jt_value_enumrectprimitivecoordinates) = S ((S (jt_index_enumrectprimitivecoordinates)) * jt_g_enumrect)) /\\ exists fs_q_jt_enumrectprimitivecoordinatesat. jt_f_enumrect = fs_q_jt_enumrectprimitivecoordinatesat * S ((S (jt_index_enumrectprimitivecoordinates)) * jt_g_enumrect) + (jt_value_enumrectprimitivecoordinates))) -> (exists jt_factor_enumrectprimitivecoordinatesdivides. (jt_value_enumrectprimitivecoordinates)=(jt_divisor_enumrectprimitive)*jt_factor_enumrectprimitivecoordinatesdivides)) -> jt_divisor_enumrectprimitive=1))))))))))))))) -> ((forall jt_i_productenum. (exists jt_gap_productenumsoundindex. jt_gap_productenumsoundindex+S (jt_i_productenum)=(u*v)) -> exists jt_b_productenum jt_c_productenum. ((((((exists fs_h_jt_productenumsoundcode. fs_h_jt_productenumsoundcode + S (jt_b_productenum) = S ((S (jt_i_productenum)) * Q)) /\\ exists fs_q_jt_productenumsoundcode. P = fs_q_jt_productenumsoundcode * S ((S (jt_i_productenum)) * Q) + (jt_b_productenum))) /\\ (((exists fs_h_jt_productenumsoundscale. fs_h_jt_productenumsoundscale + S (jt_c_productenum) = S ((S (jt_i_productenum)) * T)) /\\ exists fs_q_jt_productenumsoundscale. R = fs_q_jt_productenumsoundscale * S ((S (jt_i_productenum)) * T) + (jt_c_productenum))))) /\\ (((forall jt_index_productenumbound. (exists jt_gap_productenumboundindex. jt_gap_productenumboundindex+S (jt_index_productenumbound)=(k)) -> exists jt_value_productenumbound. ((((exists fs_h_jt_productenumboundat. fs_h_jt_productenumboundat + S (jt_value_productenumbound) = S ((S (jt_index_productenumbound)) * jt_c_productenum)) /\\ exists fs_q_jt_productenumboundat. jt_b_productenum = fs_q_jt_productenumboundat * S ((S (jt_index_productenumbound)) * jt_c_productenum) + (jt_value_productenumbound))) /\\ (exists jt_gap_productenumboundvalue. jt_gap_productenumboundvalue+S (jt_value_productenumbound)=(m*n)))) /\\ (forall jt_divisor_productenumprimitive. (exists jt_factor_productenumprimitivemodulus. (m*n)=(jt_divisor_productenumprimitive)*jt_factor_productenumprimitivemodulus) -> (forall jt_index_productenumprimitivecoordinates jt_value_productenumprimitivecoordinates. (exists jt_gap_productenumprimitivecoordinatesindex. jt_gap_productenumprimitivecoordinatesindex+S (jt_index_productenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_productenumprimitivecoordinatesat. fs_h_jt_productenumprimitivecoordinatesat + S (jt_value_productenumprimitivecoordinates) = S ((S (jt_index_productenumprimitivecoordinates)) * jt_c_productenum)) /\\ exists fs_q_jt_productenumprimitivecoordinatesat. jt_b_productenum = fs_q_jt_productenumprimitivecoordinatesat * S ((S (jt_index_productenumprimitivecoordinates)) * jt_c_productenum) + (jt_value_productenumprimitivecoordinates))) -> (exists jt_factor_productenumprimitivecoordinatesdivides. (jt_value_productenumprimitivecoordinates)=(jt_divisor_productenumprimitive)*jt_factor_productenumprimitivecoordinatesdivides)) -> jt_divisor_productenumprimitive=1))))) /\\ (((forall jt_b_productenum jt_c_productenum. (forall jt_index_productenuminputbound. (exists jt_gap_productenuminputboundindex. jt_gap_productenuminputboundindex+S (jt_index_productenuminputbound)=(k)) -> exists jt_value_productenuminputbound. ((((exists fs_h_jt_productenuminputboundat. fs_h_jt_productenuminputboundat + S (jt_value_productenuminputbound) = S ((S (jt_index_productenuminputbound)) * jt_c_productenum)) /\\ exists fs_q_jt_productenuminputboundat. jt_b_productenum = fs_q_jt_productenuminputboundat * S ((S (jt_index_productenuminputbound)) * jt_c_productenum) + (jt_value_productenuminputbound))) /\\ (exists jt_gap_productenuminputboundvalue. jt_gap_productenuminputboundvalue+S (jt_value_productenuminputbound)=(m*n)))) -> (forall jt_divisor_productenuminputprimitive. (exists jt_factor_productenuminputprimitivemodulus. (m*n)=(jt_divisor_productenuminputprimitive)*jt_factor_productenuminputprimitivemodulus) -> (forall jt_index_productenuminputprimitivecoordinates jt_value_productenuminputprimitivecoordinates. (exists jt_gap_productenuminputprimitivecoordinatesindex. jt_gap_productenuminputprimitivecoordinatesindex+S (jt_index_productenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_productenuminputprimitivecoordinatesat. fs_h_jt_productenuminputprimitivecoordinatesat + S (jt_value_productenuminputprimitivecoordinates) = S ((S (jt_index_productenuminputprimitivecoordinates)) * jt_c_productenum)) /\\ exists fs_q_jt_productenuminputprimitivecoordinatesat. jt_b_productenum = fs_q_jt_productenuminputprimitivecoordinatesat * S ((S (jt_index_productenuminputprimitivecoordinates)) * jt_c_productenum) + (jt_value_productenuminputprimitivecoordinates))) -> (exists jt_factor_productenuminputprimitivecoordinatesdivides. (jt_value_productenuminputprimitivecoordinates)=(jt_divisor_productenuminputprimitive)*jt_factor_productenuminputprimitivecoordinatesdivides)) -> jt_divisor_productenuminputprimitive=1) -> exists jt_i_productenum jt_d_productenum jt_e_productenum. ((exists jt_gap_productenumcompleteindex. jt_gap_productenumcompleteindex+S (jt_i_productenum)=(u*v)) /\\ (((((((exists fs_h_jt_productenumcompletecode. fs_h_jt_productenumcompletecode + S (jt_d_productenum) = S ((S (jt_i_productenum)) * Q)) /\\ exists fs_q_jt_productenumcompletecode. P = fs_q_jt_productenumcompletecode * S ((S (jt_i_productenum)) * Q) + (jt_d_productenum))) /\\ (((exists fs_h_jt_productenumcompletescale. fs_h_jt_productenumcompletescale + S (jt_e_productenum) = S ((S (jt_i_productenum)) * T)) /\\ exists fs_q_jt_productenumcompletescale. R = fs_q_jt_productenumcompletescale * S ((S (jt_i_productenum)) * T) + (jt_e_productenum))))) /\\ (forall jt_index_productenumrepresented jt_left_productenumrepresented jt_right_productenumrepresented. (exists jt_gap_productenumrepresentedindex. jt_gap_productenumrepresentedindex+S (jt_index_productenumrepresented)=(k)) -> (((exists fs_h_jt_productenumrepresentedleft. fs_h_jt_productenumrepresentedleft + S (jt_left_productenumrepresented) = S ((S (jt_index_productenumrepresented)) * jt_c_productenum)) /\\ exists fs_q_jt_productenumrepresentedleft. jt_b_productenum = fs_q_jt_productenumrepresentedleft * S ((S (jt_index_productenumrepresented)) * jt_c_productenum) + (jt_left_productenumrepresented))) -> (((exists fs_h_jt_productenumrepresentedright. fs_h_jt_productenumrepresentedright + S (jt_right_productenumrepresented) = S ((S (jt_index_productenumrepresented)) * jt_e_productenum)) /\\ exists fs_q_jt_productenumrepresentedright. jt_d_productenum = fs_q_jt_productenumrepresentedright * S ((S (jt_index_productenumrepresented)) * jt_e_productenum) + (jt_right_productenumrepresented))) -> jt_left_productenumrepresented=jt_right_productenumrepresented))))) /\\ (forall jt_i_productenum jt_h_productenum jt_b_productenum jt_c_productenum jt_d_productenum jt_e_productenum. (exists jt_gap_productenumfirstindex. jt_gap_productenumfirstindex+S (jt_i_productenum)=(u*v)) -> (exists jt_gap_productenumsecondindex. jt_gap_productenumsecondindex+S (jt_h_productenum)=(u*v)) -> (((((exists fs_h_jt_productenumfirstcode. fs_h_jt_productenumfirstcode + S (jt_b_productenum) = S ((S (jt_i_productenum)) * Q)) /\\ exists fs_q_jt_productenumfirstcode. P = fs_q_jt_productenumfirstcode * S ((S (jt_i_productenum)) * Q) + (jt_b_productenum))) /\\ (((exists fs_h_jt_productenumfirstscale. fs_h_jt_productenumfirstscale + S (jt_c_productenum) = S ((S (jt_i_productenum)) * T)) /\\ exists fs_q_jt_productenumfirstscale. R = fs_q_jt_productenumfirstscale * S ((S (jt_i_productenum)) * T) + (jt_c_productenum))))) -> (((((exists fs_h_jt_productenumsecondcode. fs_h_jt_productenumsecondcode + S (jt_d_productenum) = S ((S (jt_h_productenum)) * Q)) /\\ exists fs_q_jt_productenumsecondcode. P = fs_q_jt_productenumsecondcode * S ((S (jt_h_productenum)) * Q) + (jt_d_productenum))) /\\ (((exists fs_h_jt_productenumsecondscale. fs_h_jt_productenumsecondscale + S (jt_e_productenum) = S ((S (jt_h_productenum)) * T)) /\\ exists fs_q_jt_productenumsecondscale. R = fs_q_jt_productenumsecondscale * S ((S (jt_h_productenum)) * T) + (jt_e_productenum))))) -> (forall jt_index_productenumsame jt_left_productenumsame jt_right_productenumsame. (exists jt_gap_productenumsameindex. jt_gap_productenumsameindex+S (jt_index_productenumsame)=(k)) -> (((exists fs_h_jt_productenumsameleft. fs_h_jt_productenumsameleft + S (jt_left_productenumsame) = S ((S (jt_index_productenumsame)) * jt_c_productenum)) /\\ exists fs_q_jt_productenumsameleft. jt_b_productenum = fs_q_jt_productenumsameleft * S ((S (jt_index_productenumsame)) * jt_c_productenum) + (jt_left_productenumsame))) -> (((exists fs_h_jt_productenumsameright. fs_h_jt_productenumsameright + S (jt_right_productenumsame) = S ((S (jt_index_productenumsame)) * jt_e_productenum)) /\\ exists fs_q_jt_productenumsameright. jt_d_productenum = fs_q_jt_productenumsameright * S ((S (jt_index_productenumsame)) * jt_e_productenum) + (jt_right_productenumsame))) -> jt_left_productenumsame=jt_right_productenumsame) -> jt_i_productenum=jt_h_productenum))))", "statement_sha256": "5a8e6693884fa870726dac97097f177cd1ee5ebf5cadae1c5098a181782bf25e", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The constructed table is a duplicate-free exhaustive primitive tuple enumeration of literal length u*v." }, { "admission_dependencies": [ "jordan_rectangle_crt_exists", "le_refl", "jordan_rectangle_crt_enumeration" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 53, "body_proof_nodes": 90, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hm", "intro hn", "intro hcop", "intro hl", "intro hh", "have ht : ∀ q. Le(q,u · v) → ∃ x. ∃ y. ∃ z. ∃ i. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,q)", "specialize jordan_rectangle_crt_exists (m)", "specialize jordan_rectangle_crt_exists (n)", "specialize jordan_rectangle_crt_exists (k)", "specialize jordan_rectangle_crt_exists (A)", "specialize jordan_rectangle_crt_exists (B)", "specialize jordan_rectangle_crt_exists (C)", "specialize jordan_rectangle_crt_exists (D)", "specialize jordan_rectangle_crt_exists (u)", "specialize jordan_rectangle_crt_exists (E)", "specialize jordan_rectangle_crt_exists (F)", "specialize jordan_rectangle_crt_exists (G)", "specialize jordan_rectangle_crt_exists (H)", "specialize jordan_rectangle_crt_exists (v)", "apply jordan_rectangle_crt_exists", "exact hm", "exact hn", "exact hcop", "exact hl", "exact hh", "have hv : ∃ P. ∃ Q. ∃ R. ∃ T. JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v)", "specialize ht (u*v)", "apply ht", "specialize le_refl (u*v)", "apply le_refl", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "specialize jordan_rectangle_crt_enumeration (m)", "specialize jordan_rectangle_crt_enumeration (n)", "specialize jordan_rectangle_crt_enumeration (k)", "specialize jordan_rectangle_crt_enumeration (A)", "specialize jordan_rectangle_crt_enumeration (B)", "specialize jordan_rectangle_crt_enumeration (C)", "specialize jordan_rectangle_crt_enumeration (D)", "specialize jordan_rectangle_crt_enumeration (u)", "specialize jordan_rectangle_crt_enumeration (E)", "specialize jordan_rectangle_crt_enumeration (F)", "specialize jordan_rectangle_crt_enumeration (G)", "specialize jordan_rectangle_crt_enumeration (H)", "specialize jordan_rectangle_crt_enumeration (v)", "specialize jordan_rectangle_crt_enumeration (x)", "specialize jordan_rectangle_crt_enumeration (x1)", "specialize jordan_rectangle_crt_enumeration (x2)", "specialize jordan_rectangle_crt_enumeration (x3)", "apply jordan_rectangle_crt_enumeration", "exact hm", "exact hn", "exact hcop", "exact hl", "exact hh", "exact hv_witness_witness_witness_witness" ], "defined_statement": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ¬m = 0 → ¬n = 0 → Coprime(m,n) → JordanTupleEnumeration(k,m,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → ∃ x. ∃ y. ∃ z. ∃ i. JordanTupleEnumeration(k,m · n,x,y,z,i,u · v)", "defined_statement_sha256": "9b4b5ef02ee6304ecc14fd9cdc90b2124b909f5fcec4cd00792d7781d6d2a9b3", "definition_uses": { "ND0374": 3, "ND0381": 2, "PD0001": 1, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "8b3f625db3a4e9e10224e5a91a2506012b6fc4570e66af34a0701444f7bb48d5", "free_names": [], "script_definition_uses": { "ND0381": 2, "PD0001": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hh" } ], [ { "kind": "text", "text": "have ht : " }, { "kind": "text", "text": "∀ q. " }, { "definition": "PD0001", "kind": "definition", "text": "Le(q,u · v)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ i. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,x,y,z,i,q)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_exists (v)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_exists" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ P. ∃ Q. ∃ R. ∃ T. " }, { "definition": "ND0381", "kind": "definition", "text": "JordanRectangleCRT(m,n,k,A,B,C,D,u,E,F,G,H,v,P,Q,R,T,u · v)" } ], [ { "kind": "text", "text": "specialize ht (u*v)" } ], [ { "kind": "text", "text": "apply ht" } ], [ { "kind": "text", "text": "specialize le_refl (u*v)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "exists x3" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (m)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (n)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (k)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (A)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (B)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (C)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (D)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (u)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (E)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (F)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (G)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (H)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (v)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (x)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (x1)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (x2)" } ], [ { "kind": "text", "text": "specialize jordan_rectangle_crt_enumeration (x3)" } ], [ { "kind": "text", "text": "apply jordan_rectangle_crt_enumeration" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hh" } ], [ { "kind": "text", "text": "exact hv_witness_witness_witness_witness" } ] ], "statement_definition_uses": { "ND0374": 3, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ m. ∀ n. ∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ¬m = 0 → ¬n = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(m,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. ∃ i. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,m · n,x,y,z,i,u · v)" } ] }, "dependencies": [ "jordan_rectangle_crt_exists", "le_refl", "jordan_rectangle_crt_enumeration" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT0049", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_product_enumeration_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 334, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro m", "intro n", "intro k", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hm", "intro hn", "intro hcop", "intro hl", "intro hh", "have ht : forall q. (exists jt_gap_multablebound. jt_gap_multablebound+(q)=(u*v)) -> exists P Q R T. forall jt_index_multableprefix. (exists jt_gap_multableprefixindex. jt_gap_multableprefixindex+S (jt_index_multableprefix)=(q)) -> exists jt_row_multableprefix jt_column_multableprefix jt_b_multableprefix jt_c_multableprefix jt_d_multableprefix jt_e_multableprefix jt_f_multableprefix jt_g_multableprefix. ((exists jt_gap_multableprefixrow. jt_gap_multableprefixrow+S (jt_row_multableprefix)=(u)) /\\ (((exists jt_gap_multableprefixcolumn. jt_gap_multableprefixcolumn+S (jt_column_multableprefix)=(v)) /\\ (((jt_index_multableprefix=(v)*jt_row_multableprefix+jt_column_multableprefix) /\\ (((((((exists fs_h_jt_multableprefixleftcode. fs_h_jt_multableprefixleftcode + S (jt_b_multableprefix) = S ((S (jt_row_multableprefix)) * B)) /\\ exists fs_q_jt_multableprefixleftcode. A = fs_q_jt_multableprefixleftcode * S ((S (jt_row_multableprefix)) * B) + (jt_b_multableprefix))) /\\ (((exists fs_h_jt_multableprefixleftscale. fs_h_jt_multableprefixleftscale + S (jt_c_multableprefix) = S ((S (jt_row_multableprefix)) * D)) /\\ exists fs_q_jt_multableprefixleftscale. C = fs_q_jt_multableprefixleftscale * S ((S (jt_row_multableprefix)) * D) + (jt_c_multableprefix))))) /\\ (((((((exists fs_h_jt_multableprefixrightcode. fs_h_jt_multableprefixrightcode + S (jt_d_multableprefix) = S ((S (jt_column_multableprefix)) * F)) /\\ exists fs_q_jt_multableprefixrightcode. E = fs_q_jt_multableprefixrightcode * S ((S (jt_column_multableprefix)) * F) + (jt_d_multableprefix))) /\\ (((exists fs_h_jt_multableprefixrightscale. fs_h_jt_multableprefixrightscale + S (jt_e_multableprefix) = S ((S (jt_column_multableprefix)) * H)) /\\ exists fs_q_jt_multableprefixrightscale. G = fs_q_jt_multableprefixrightscale * S ((S (jt_column_multableprefix)) * H) + (jt_e_multableprefix))))) /\\ (((((((exists fs_h_jt_multableprefixoutputcode. fs_h_jt_multableprefixoutputcode + S (jt_f_multableprefix) = S ((S (jt_index_multableprefix)) * Q)) /\\ exists fs_q_jt_multableprefixoutputcode. P = fs_q_jt_multableprefixoutputcode * S ((S (jt_index_multableprefix)) * Q) + (jt_f_multableprefix))) /\\ (((exists fs_h_jt_multableprefixoutputscale. fs_h_jt_multableprefixoutputscale + S (jt_g_multableprefix) = S ((S (jt_index_multableprefix)) * T)) /\\ exists fs_q_jt_multableprefixoutputscale. R = fs_q_jt_multableprefixoutputscale * S ((S (jt_index_multableprefix)) * T) + (jt_g_multableprefix))))) /\\ (((((forall jt_index_multableprefixcrtbound. (exists jt_gap_multableprefixcrtboundindex. jt_gap_multableprefixcrtboundindex+S (jt_index_multableprefixcrtbound)=(k)) -> exists jt_value_multableprefixcrtbound. ((((exists fs_h_jt_multableprefixcrtboundat. fs_h_jt_multableprefixcrtboundat + S (jt_value_multableprefixcrtbound) = S ((S (jt_index_multableprefixcrtbound)) * jt_g_multableprefix)) /\\ exists fs_q_jt_multableprefixcrtboundat. jt_f_multableprefix = fs_q_jt_multableprefixcrtboundat * S ((S (jt_index_multableprefixcrtbound)) * jt_g_multableprefix) + (jt_value_multableprefixcrtbound))) /\\ (exists jt_gap_multableprefixcrtboundvalue. jt_gap_multableprefixcrtboundvalue+S (jt_value_multableprefixcrtbound)=(m*n)))) /\\ (((forall jt_index_multableprefixcrtleft jt_left_multableprefixcrtleft jt_right_multableprefixcrtleft. (exists jt_gap_multableprefixcrtleftindex. jt_gap_multableprefixcrtleftindex+S (jt_index_multableprefixcrtleft)=(k)) -> (((exists fs_h_jt_multableprefixcrtleftleft. fs_h_jt_multableprefixcrtleftleft + S (jt_left_multableprefixcrtleft) = S ((S (jt_index_multableprefixcrtleft)) * jt_g_multableprefix)) /\\ exists fs_q_jt_multableprefixcrtleftleft. jt_f_multableprefix = fs_q_jt_multableprefixcrtleftleft * S ((S (jt_index_multableprefixcrtleft)) * jt_g_multableprefix) + (jt_left_multableprefixcrtleft))) -> (((exists fs_h_jt_multableprefixcrtleftright. fs_h_jt_multableprefixcrtleftright + S (jt_right_multableprefixcrtleft) = S ((S (jt_index_multableprefixcrtleft)) * jt_c_multableprefix)) /\\ exists fs_q_jt_multableprefixcrtleftright. jt_b_multableprefix = fs_q_jt_multableprefixcrtleftright * S ((S (jt_index_multableprefixcrtleft)) * jt_c_multableprefix) + (jt_right_multableprefixcrtleft))) -> (exists jt_left_multableprefixcrtleftmod jt_right_multableprefixcrtleftmod. (jt_left_multableprefixcrtleft)+(m)*jt_left_multableprefixcrtleftmod=(jt_right_multableprefixcrtleft)+(m)*jt_right_multableprefixcrtleftmod)) /\\ (forall jt_index_multableprefixcrtright jt_left_multableprefixcrtright jt_right_multableprefixcrtright. (exists jt_gap_multableprefixcrtrightindex. jt_gap_multableprefixcrtrightindex+S (jt_index_multableprefixcrtright)=(k)) -> (((exists fs_h_jt_multableprefixcrtrightleft. fs_h_jt_multableprefixcrtrightleft + S (jt_left_multableprefixcrtright) = S ((S (jt_index_multableprefixcrtright)) * jt_g_multableprefix)) /\\ exists fs_q_jt_multableprefixcrtrightleft. jt_f_multableprefix = fs_q_jt_multableprefixcrtrightleft * S ((S (jt_index_multableprefixcrtright)) * jt_g_multableprefix) + (jt_left_multableprefixcrtright))) -> (((exists fs_h_jt_multableprefixcrtrightright. fs_h_jt_multableprefixcrtrightright + S (jt_right_multableprefixcrtright) = S ((S (jt_index_multableprefixcrtright)) * jt_e_multableprefix)) /\\ exists fs_q_jt_multableprefixcrtrightright. jt_d_multableprefix = fs_q_jt_multableprefixcrtrightright * S ((S (jt_index_multableprefixcrtright)) * jt_e_multableprefix) + (jt_right_multableprefixcrtright))) -> (exists jt_left_multableprefixcrtrightmod jt_right_multableprefixcrtrightmod. (jt_left_multableprefixcrtright)+(n)*jt_left_multableprefixcrtrightmod=(jt_right_multableprefixcrtright)+(n)*jt_right_multableprefixcrtrightmod)))))) /\\ (forall jt_divisor_multableprefixprimitive. (exists jt_factor_multableprefixprimitivemodulus. (m*n)=(jt_divisor_multableprefixprimitive)*jt_factor_multableprefixprimitivemodulus) -> (forall jt_index_multableprefixprimitivecoordinates jt_value_multableprefixprimitivecoordinates. (exists jt_gap_multableprefixprimitivecoordinatesindex. jt_gap_multableprefixprimitivecoordinatesindex+S (jt_index_multableprefixprimitivecoordinates)=(k)) -> (((exists fs_h_jt_multableprefixprimitivecoordinatesat. fs_h_jt_multableprefixprimitivecoordinatesat + S (jt_value_multableprefixprimitivecoordinates) = S ((S (jt_index_multableprefixprimitivecoordinates)) * jt_g_multableprefix)) /\\ exists fs_q_jt_multableprefixprimitivecoordinatesat. jt_f_multableprefix = fs_q_jt_multableprefixprimitivecoordinatesat * S ((S (jt_index_multableprefixprimitivecoordinates)) * jt_g_multableprefix) + (jt_value_multableprefixprimitivecoordinates))) -> (exists jt_factor_multableprefixprimitivecoordinatesdivides. (jt_value_multableprefixprimitivecoordinates)=(jt_divisor_multableprefixprimitive)*jt_factor_multableprefixprimitivecoordinatesdivides)) -> jt_divisor_multableprefixprimitive=1))))))))))))))", "specialize jordan_rectangle_crt_exists (m)", "specialize jordan_rectangle_crt_exists (n)", "specialize jordan_rectangle_crt_exists (k)", "specialize jordan_rectangle_crt_exists (A)", "specialize jordan_rectangle_crt_exists (B)", "specialize jordan_rectangle_crt_exists (C)", "specialize jordan_rectangle_crt_exists (D)", "specialize jordan_rectangle_crt_exists (u)", "specialize jordan_rectangle_crt_exists (E)", "specialize jordan_rectangle_crt_exists (F)", "specialize jordan_rectangle_crt_exists (G)", "specialize jordan_rectangle_crt_exists (H)", "specialize jordan_rectangle_crt_exists (v)", "apply jordan_rectangle_crt_exists", "exact hm", "exact hn", "exact hcop", "exact hl", "exact hh", "have hv : exists P Q R T. forall jt_index_mulenumtable. (exists jt_gap_mulenumtableindex. jt_gap_mulenumtableindex+S (jt_index_mulenumtable)=(u*v)) -> exists jt_row_mulenumtable jt_column_mulenumtable jt_b_mulenumtable jt_c_mulenumtable jt_d_mulenumtable jt_e_mulenumtable jt_f_mulenumtable jt_g_mulenumtable. ((exists jt_gap_mulenumtablerow. jt_gap_mulenumtablerow+S (jt_row_mulenumtable)=(u)) /\\ (((exists jt_gap_mulenumtablecolumn. jt_gap_mulenumtablecolumn+S (jt_column_mulenumtable)=(v)) /\\ (((jt_index_mulenumtable=(v)*jt_row_mulenumtable+jt_column_mulenumtable) /\\ (((((((exists fs_h_jt_mulenumtableleftcode. fs_h_jt_mulenumtableleftcode + S (jt_b_mulenumtable) = S ((S (jt_row_mulenumtable)) * B)) /\\ exists fs_q_jt_mulenumtableleftcode. A = fs_q_jt_mulenumtableleftcode * S ((S (jt_row_mulenumtable)) * B) + (jt_b_mulenumtable))) /\\ (((exists fs_h_jt_mulenumtableleftscale. fs_h_jt_mulenumtableleftscale + S (jt_c_mulenumtable) = S ((S (jt_row_mulenumtable)) * D)) /\\ exists fs_q_jt_mulenumtableleftscale. C = fs_q_jt_mulenumtableleftscale * S ((S (jt_row_mulenumtable)) * D) + (jt_c_mulenumtable))))) /\\ (((((((exists fs_h_jt_mulenumtablerightcode. fs_h_jt_mulenumtablerightcode + S (jt_d_mulenumtable) = S ((S (jt_column_mulenumtable)) * F)) /\\ exists fs_q_jt_mulenumtablerightcode. E = fs_q_jt_mulenumtablerightcode * S ((S (jt_column_mulenumtable)) * F) + (jt_d_mulenumtable))) /\\ (((exists fs_h_jt_mulenumtablerightscale. fs_h_jt_mulenumtablerightscale + S (jt_e_mulenumtable) = S ((S (jt_column_mulenumtable)) * H)) /\\ exists fs_q_jt_mulenumtablerightscale. G = fs_q_jt_mulenumtablerightscale * S ((S (jt_column_mulenumtable)) * H) + (jt_e_mulenumtable))))) /\\ (((((((exists fs_h_jt_mulenumtableoutputcode. fs_h_jt_mulenumtableoutputcode + S (jt_f_mulenumtable) = S ((S (jt_index_mulenumtable)) * Q)) /\\ exists fs_q_jt_mulenumtableoutputcode. P = fs_q_jt_mulenumtableoutputcode * S ((S (jt_index_mulenumtable)) * Q) + (jt_f_mulenumtable))) /\\ (((exists fs_h_jt_mulenumtableoutputscale. fs_h_jt_mulenumtableoutputscale + S (jt_g_mulenumtable) = S ((S (jt_index_mulenumtable)) * T)) /\\ exists fs_q_jt_mulenumtableoutputscale. R = fs_q_jt_mulenumtableoutputscale * S ((S (jt_index_mulenumtable)) * T) + (jt_g_mulenumtable))))) /\\ (((((forall jt_index_mulenumtablecrtbound. (exists jt_gap_mulenumtablecrtboundindex. jt_gap_mulenumtablecrtboundindex+S (jt_index_mulenumtablecrtbound)=(k)) -> exists jt_value_mulenumtablecrtbound. ((((exists fs_h_jt_mulenumtablecrtboundat. fs_h_jt_mulenumtablecrtboundat + S (jt_value_mulenumtablecrtbound) = S ((S (jt_index_mulenumtablecrtbound)) * jt_g_mulenumtable)) /\\ exists fs_q_jt_mulenumtablecrtboundat. jt_f_mulenumtable = fs_q_jt_mulenumtablecrtboundat * S ((S (jt_index_mulenumtablecrtbound)) * jt_g_mulenumtable) + (jt_value_mulenumtablecrtbound))) /\\ (exists jt_gap_mulenumtablecrtboundvalue. jt_gap_mulenumtablecrtboundvalue+S (jt_value_mulenumtablecrtbound)=(m*n)))) /\\ (((forall jt_index_mulenumtablecrtleft jt_left_mulenumtablecrtleft jt_right_mulenumtablecrtleft. (exists jt_gap_mulenumtablecrtleftindex. jt_gap_mulenumtablecrtleftindex+S (jt_index_mulenumtablecrtleft)=(k)) -> (((exists fs_h_jt_mulenumtablecrtleftleft. fs_h_jt_mulenumtablecrtleftleft + S (jt_left_mulenumtablecrtleft) = S ((S (jt_index_mulenumtablecrtleft)) * jt_g_mulenumtable)) /\\ exists fs_q_jt_mulenumtablecrtleftleft. jt_f_mulenumtable = fs_q_jt_mulenumtablecrtleftleft * S ((S (jt_index_mulenumtablecrtleft)) * jt_g_mulenumtable) + (jt_left_mulenumtablecrtleft))) -> (((exists fs_h_jt_mulenumtablecrtleftright. fs_h_jt_mulenumtablecrtleftright + S (jt_right_mulenumtablecrtleft) = S ((S (jt_index_mulenumtablecrtleft)) * jt_c_mulenumtable)) /\\ exists fs_q_jt_mulenumtablecrtleftright. jt_b_mulenumtable = fs_q_jt_mulenumtablecrtleftright * S ((S (jt_index_mulenumtablecrtleft)) * jt_c_mulenumtable) + (jt_right_mulenumtablecrtleft))) -> (exists jt_left_mulenumtablecrtleftmod jt_right_mulenumtablecrtleftmod. (jt_left_mulenumtablecrtleft)+(m)*jt_left_mulenumtablecrtleftmod=(jt_right_mulenumtablecrtleft)+(m)*jt_right_mulenumtablecrtleftmod)) /\\ (forall jt_index_mulenumtablecrtright jt_left_mulenumtablecrtright jt_right_mulenumtablecrtright. (exists jt_gap_mulenumtablecrtrightindex. jt_gap_mulenumtablecrtrightindex+S (jt_index_mulenumtablecrtright)=(k)) -> (((exists fs_h_jt_mulenumtablecrtrightleft. fs_h_jt_mulenumtablecrtrightleft + S (jt_left_mulenumtablecrtright) = S ((S (jt_index_mulenumtablecrtright)) * jt_g_mulenumtable)) /\\ exists fs_q_jt_mulenumtablecrtrightleft. jt_f_mulenumtable = fs_q_jt_mulenumtablecrtrightleft * S ((S (jt_index_mulenumtablecrtright)) * jt_g_mulenumtable) + (jt_left_mulenumtablecrtright))) -> (((exists fs_h_jt_mulenumtablecrtrightright. fs_h_jt_mulenumtablecrtrightright + S (jt_right_mulenumtablecrtright) = S ((S (jt_index_mulenumtablecrtright)) * jt_e_mulenumtable)) /\\ exists fs_q_jt_mulenumtablecrtrightright. jt_d_mulenumtable = fs_q_jt_mulenumtablecrtrightright * S ((S (jt_index_mulenumtablecrtright)) * jt_e_mulenumtable) + (jt_right_mulenumtablecrtright))) -> (exists jt_left_mulenumtablecrtrightmod jt_right_mulenumtablecrtrightmod. (jt_left_mulenumtablecrtright)+(n)*jt_left_mulenumtablecrtrightmod=(jt_right_mulenumtablecrtright)+(n)*jt_right_mulenumtablecrtrightmod)))))) /\\ (forall jt_divisor_mulenumtableprimitive. (exists jt_factor_mulenumtableprimitivemodulus. (m*n)=(jt_divisor_mulenumtableprimitive)*jt_factor_mulenumtableprimitivemodulus) -> (forall jt_index_mulenumtableprimitivecoordinates jt_value_mulenumtableprimitivecoordinates. (exists jt_gap_mulenumtableprimitivecoordinatesindex. jt_gap_mulenumtableprimitivecoordinatesindex+S (jt_index_mulenumtableprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumtableprimitivecoordinatesat. fs_h_jt_mulenumtableprimitivecoordinatesat + S (jt_value_mulenumtableprimitivecoordinates) = S ((S (jt_index_mulenumtableprimitivecoordinates)) * jt_g_mulenumtable)) /\\ exists fs_q_jt_mulenumtableprimitivecoordinatesat. jt_f_mulenumtable = fs_q_jt_mulenumtableprimitivecoordinatesat * S ((S (jt_index_mulenumtableprimitivecoordinates)) * jt_g_mulenumtable) + (jt_value_mulenumtableprimitivecoordinates))) -> (exists jt_factor_mulenumtableprimitivecoordinatesdivides. (jt_value_mulenumtableprimitivecoordinates)=(jt_divisor_mulenumtableprimitive)*jt_factor_mulenumtableprimitivecoordinatesdivides)) -> jt_divisor_mulenumtableprimitive=1))))))))))))))", "specialize ht (u*v)", "apply ht", "specialize le_refl (u*v)", "apply le_refl", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_witness", "exists x", "exists x1", "exists x2", "exists x3", "specialize jordan_rectangle_crt_enumeration (m)", "specialize jordan_rectangle_crt_enumeration (n)", "specialize jordan_rectangle_crt_enumeration (k)", "specialize jordan_rectangle_crt_enumeration (A)", "specialize jordan_rectangle_crt_enumeration (B)", "specialize jordan_rectangle_crt_enumeration (C)", "specialize jordan_rectangle_crt_enumeration (D)", "specialize jordan_rectangle_crt_enumeration (u)", "specialize jordan_rectangle_crt_enumeration (E)", "specialize jordan_rectangle_crt_enumeration (F)", "specialize jordan_rectangle_crt_enumeration (G)", "specialize jordan_rectangle_crt_enumeration (H)", "specialize jordan_rectangle_crt_enumeration (v)", "specialize jordan_rectangle_crt_enumeration (x)", "specialize jordan_rectangle_crt_enumeration (x1)", "specialize jordan_rectangle_crt_enumeration (x2)", "specialize jordan_rectangle_crt_enumeration (x3)", "apply jordan_rectangle_crt_enumeration", "exact hm", "exact hn", "exact hcop", "exact hl", "exact hh", "exact hv_witness_witness_witness_witness" ], "script_sha256": "4f6c81f6e33d66b8e7393cc1a752c353fe73e32e2117b482fcc1476797922551", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "4f6c81f6e33d66b8e7393cc1a752c353fe73e32e2117b482fcc1476797922551", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "8b3f625db3a4e9e10224e5a91a2506012b6fc4570e66af34a0701444f7bb48d5" } ], "stable_member": false, "statement": "forall m n k A B C D u E F G H v. ~(m=0) -> ~(n=0) -> (forall jt_divisor_mulenumcop. (exists jt_factor_mulenumcopa. (m)=(jt_divisor_mulenumcop)*jt_factor_mulenumcopa) -> (exists jt_factor_mulenumcopb. (n)=(jt_divisor_mulenumcop)*jt_factor_mulenumcopb) -> jt_divisor_mulenumcop=1) -> (((forall jt_i_mulenumleft. (exists jt_gap_mulenumleftsoundindex. jt_gap_mulenumleftsoundindex+S (jt_i_mulenumleft)=(u)) -> exists jt_b_mulenumleft jt_c_mulenumleft. ((((((exists fs_h_jt_mulenumleftsoundcode. fs_h_jt_mulenumleftsoundcode + S (jt_b_mulenumleft) = S ((S (jt_i_mulenumleft)) * B)) /\\ exists fs_q_jt_mulenumleftsoundcode. A = fs_q_jt_mulenumleftsoundcode * S ((S (jt_i_mulenumleft)) * B) + (jt_b_mulenumleft))) /\\ (((exists fs_h_jt_mulenumleftsoundscale. fs_h_jt_mulenumleftsoundscale + S (jt_c_mulenumleft) = S ((S (jt_i_mulenumleft)) * D)) /\\ exists fs_q_jt_mulenumleftsoundscale. C = fs_q_jt_mulenumleftsoundscale * S ((S (jt_i_mulenumleft)) * D) + (jt_c_mulenumleft))))) /\\ (((forall jt_index_mulenumleftbound. (exists jt_gap_mulenumleftboundindex. jt_gap_mulenumleftboundindex+S (jt_index_mulenumleftbound)=(k)) -> exists jt_value_mulenumleftbound. ((((exists fs_h_jt_mulenumleftboundat. fs_h_jt_mulenumleftboundat + S (jt_value_mulenumleftbound) = S ((S (jt_index_mulenumleftbound)) * jt_c_mulenumleft)) /\\ exists fs_q_jt_mulenumleftboundat. jt_b_mulenumleft = fs_q_jt_mulenumleftboundat * S ((S (jt_index_mulenumleftbound)) * jt_c_mulenumleft) + (jt_value_mulenumleftbound))) /\\ (exists jt_gap_mulenumleftboundvalue. jt_gap_mulenumleftboundvalue+S (jt_value_mulenumleftbound)=(m)))) /\\ (forall jt_divisor_mulenumleftprimitive. (exists jt_factor_mulenumleftprimitivemodulus. (m)=(jt_divisor_mulenumleftprimitive)*jt_factor_mulenumleftprimitivemodulus) -> (forall jt_index_mulenumleftprimitivecoordinates jt_value_mulenumleftprimitivecoordinates. (exists jt_gap_mulenumleftprimitivecoordinatesindex. jt_gap_mulenumleftprimitivecoordinatesindex+S (jt_index_mulenumleftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumleftprimitivecoordinatesat. fs_h_jt_mulenumleftprimitivecoordinatesat + S (jt_value_mulenumleftprimitivecoordinates) = S ((S (jt_index_mulenumleftprimitivecoordinates)) * jt_c_mulenumleft)) /\\ exists fs_q_jt_mulenumleftprimitivecoordinatesat. jt_b_mulenumleft = fs_q_jt_mulenumleftprimitivecoordinatesat * S ((S (jt_index_mulenumleftprimitivecoordinates)) * jt_c_mulenumleft) + (jt_value_mulenumleftprimitivecoordinates))) -> (exists jt_factor_mulenumleftprimitivecoordinatesdivides. (jt_value_mulenumleftprimitivecoordinates)=(jt_divisor_mulenumleftprimitive)*jt_factor_mulenumleftprimitivecoordinatesdivides)) -> jt_divisor_mulenumleftprimitive=1))))) /\\ (((forall jt_b_mulenumleft jt_c_mulenumleft. (forall jt_index_mulenumleftinputbound. (exists jt_gap_mulenumleftinputboundindex. jt_gap_mulenumleftinputboundindex+S (jt_index_mulenumleftinputbound)=(k)) -> exists jt_value_mulenumleftinputbound. ((((exists fs_h_jt_mulenumleftinputboundat. fs_h_jt_mulenumleftinputboundat + S (jt_value_mulenumleftinputbound) = S ((S (jt_index_mulenumleftinputbound)) * jt_c_mulenumleft)) /\\ exists fs_q_jt_mulenumleftinputboundat. jt_b_mulenumleft = fs_q_jt_mulenumleftinputboundat * S ((S (jt_index_mulenumleftinputbound)) * jt_c_mulenumleft) + (jt_value_mulenumleftinputbound))) /\\ (exists jt_gap_mulenumleftinputboundvalue. jt_gap_mulenumleftinputboundvalue+S (jt_value_mulenumleftinputbound)=(m)))) -> (forall jt_divisor_mulenumleftinputprimitive. (exists jt_factor_mulenumleftinputprimitivemodulus. (m)=(jt_divisor_mulenumleftinputprimitive)*jt_factor_mulenumleftinputprimitivemodulus) -> (forall jt_index_mulenumleftinputprimitivecoordinates jt_value_mulenumleftinputprimitivecoordinates. (exists jt_gap_mulenumleftinputprimitivecoordinatesindex. jt_gap_mulenumleftinputprimitivecoordinatesindex+S (jt_index_mulenumleftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumleftinputprimitivecoordinatesat. fs_h_jt_mulenumleftinputprimitivecoordinatesat + S (jt_value_mulenumleftinputprimitivecoordinates) = S ((S (jt_index_mulenumleftinputprimitivecoordinates)) * jt_c_mulenumleft)) /\\ exists fs_q_jt_mulenumleftinputprimitivecoordinatesat. jt_b_mulenumleft = fs_q_jt_mulenumleftinputprimitivecoordinatesat * S ((S (jt_index_mulenumleftinputprimitivecoordinates)) * jt_c_mulenumleft) + (jt_value_mulenumleftinputprimitivecoordinates))) -> (exists jt_factor_mulenumleftinputprimitivecoordinatesdivides. (jt_value_mulenumleftinputprimitivecoordinates)=(jt_divisor_mulenumleftinputprimitive)*jt_factor_mulenumleftinputprimitivecoordinatesdivides)) -> jt_divisor_mulenumleftinputprimitive=1) -> exists jt_i_mulenumleft jt_d_mulenumleft jt_e_mulenumleft. ((exists jt_gap_mulenumleftcompleteindex. jt_gap_mulenumleftcompleteindex+S (jt_i_mulenumleft)=(u)) /\\ (((((((exists fs_h_jt_mulenumleftcompletecode. fs_h_jt_mulenumleftcompletecode + S (jt_d_mulenumleft) = S ((S (jt_i_mulenumleft)) * B)) /\\ exists fs_q_jt_mulenumleftcompletecode. A = fs_q_jt_mulenumleftcompletecode * S ((S (jt_i_mulenumleft)) * B) + (jt_d_mulenumleft))) /\\ (((exists fs_h_jt_mulenumleftcompletescale. fs_h_jt_mulenumleftcompletescale + S (jt_e_mulenumleft) = S ((S (jt_i_mulenumleft)) * D)) /\\ exists fs_q_jt_mulenumleftcompletescale. C = fs_q_jt_mulenumleftcompletescale * S ((S (jt_i_mulenumleft)) * D) + (jt_e_mulenumleft))))) /\\ (forall jt_index_mulenumleftrepresented jt_left_mulenumleftrepresented jt_right_mulenumleftrepresented. (exists jt_gap_mulenumleftrepresentedindex. jt_gap_mulenumleftrepresentedindex+S (jt_index_mulenumleftrepresented)=(k)) -> (((exists fs_h_jt_mulenumleftrepresentedleft. fs_h_jt_mulenumleftrepresentedleft + S (jt_left_mulenumleftrepresented) = S ((S (jt_index_mulenumleftrepresented)) * jt_c_mulenumleft)) /\\ exists fs_q_jt_mulenumleftrepresentedleft. jt_b_mulenumleft = fs_q_jt_mulenumleftrepresentedleft * S ((S (jt_index_mulenumleftrepresented)) * jt_c_mulenumleft) + (jt_left_mulenumleftrepresented))) -> (((exists fs_h_jt_mulenumleftrepresentedright. fs_h_jt_mulenumleftrepresentedright + S (jt_right_mulenumleftrepresented) = S ((S (jt_index_mulenumleftrepresented)) * jt_e_mulenumleft)) /\\ exists fs_q_jt_mulenumleftrepresentedright. jt_d_mulenumleft = fs_q_jt_mulenumleftrepresentedright * S ((S (jt_index_mulenumleftrepresented)) * jt_e_mulenumleft) + (jt_right_mulenumleftrepresented))) -> jt_left_mulenumleftrepresented=jt_right_mulenumleftrepresented))))) /\\ (forall jt_i_mulenumleft jt_h_mulenumleft jt_b_mulenumleft jt_c_mulenumleft jt_d_mulenumleft jt_e_mulenumleft. (exists jt_gap_mulenumleftfirstindex. jt_gap_mulenumleftfirstindex+S (jt_i_mulenumleft)=(u)) -> (exists jt_gap_mulenumleftsecondindex. jt_gap_mulenumleftsecondindex+S (jt_h_mulenumleft)=(u)) -> (((((exists fs_h_jt_mulenumleftfirstcode. fs_h_jt_mulenumleftfirstcode + S (jt_b_mulenumleft) = S ((S (jt_i_mulenumleft)) * B)) /\\ exists fs_q_jt_mulenumleftfirstcode. A = fs_q_jt_mulenumleftfirstcode * S ((S (jt_i_mulenumleft)) * B) + (jt_b_mulenumleft))) /\\ (((exists fs_h_jt_mulenumleftfirstscale. fs_h_jt_mulenumleftfirstscale + S (jt_c_mulenumleft) = S ((S (jt_i_mulenumleft)) * D)) /\\ exists fs_q_jt_mulenumleftfirstscale. C = fs_q_jt_mulenumleftfirstscale * S ((S (jt_i_mulenumleft)) * D) + (jt_c_mulenumleft))))) -> (((((exists fs_h_jt_mulenumleftsecondcode. fs_h_jt_mulenumleftsecondcode + S (jt_d_mulenumleft) = S ((S (jt_h_mulenumleft)) * B)) /\\ exists fs_q_jt_mulenumleftsecondcode. A = fs_q_jt_mulenumleftsecondcode * S ((S (jt_h_mulenumleft)) * B) + (jt_d_mulenumleft))) /\\ (((exists fs_h_jt_mulenumleftsecondscale. fs_h_jt_mulenumleftsecondscale + S (jt_e_mulenumleft) = S ((S (jt_h_mulenumleft)) * D)) /\\ exists fs_q_jt_mulenumleftsecondscale. C = fs_q_jt_mulenumleftsecondscale * S ((S (jt_h_mulenumleft)) * D) + (jt_e_mulenumleft))))) -> (forall jt_index_mulenumleftsame jt_left_mulenumleftsame jt_right_mulenumleftsame. (exists jt_gap_mulenumleftsameindex. jt_gap_mulenumleftsameindex+S (jt_index_mulenumleftsame)=(k)) -> (((exists fs_h_jt_mulenumleftsameleft. fs_h_jt_mulenumleftsameleft + S (jt_left_mulenumleftsame) = S ((S (jt_index_mulenumleftsame)) * jt_c_mulenumleft)) /\\ exists fs_q_jt_mulenumleftsameleft. jt_b_mulenumleft = fs_q_jt_mulenumleftsameleft * S ((S (jt_index_mulenumleftsame)) * jt_c_mulenumleft) + (jt_left_mulenumleftsame))) -> (((exists fs_h_jt_mulenumleftsameright. fs_h_jt_mulenumleftsameright + S (jt_right_mulenumleftsame) = S ((S (jt_index_mulenumleftsame)) * jt_e_mulenumleft)) /\\ exists fs_q_jt_mulenumleftsameright. jt_d_mulenumleft = fs_q_jt_mulenumleftsameright * S ((S (jt_index_mulenumleftsame)) * jt_e_mulenumleft) + (jt_right_mulenumleftsame))) -> jt_left_mulenumleftsame=jt_right_mulenumleftsame) -> jt_i_mulenumleft=jt_h_mulenumleft))))) -> (((forall jt_i_mulenumright. (exists jt_gap_mulenumrightsoundindex. jt_gap_mulenumrightsoundindex+S (jt_i_mulenumright)=(v)) -> exists jt_b_mulenumright jt_c_mulenumright. ((((((exists fs_h_jt_mulenumrightsoundcode. fs_h_jt_mulenumrightsoundcode + S (jt_b_mulenumright) = S ((S (jt_i_mulenumright)) * F)) /\\ exists fs_q_jt_mulenumrightsoundcode. E = fs_q_jt_mulenumrightsoundcode * S ((S (jt_i_mulenumright)) * F) + (jt_b_mulenumright))) /\\ (((exists fs_h_jt_mulenumrightsoundscale. fs_h_jt_mulenumrightsoundscale + S (jt_c_mulenumright) = S ((S (jt_i_mulenumright)) * H)) /\\ exists fs_q_jt_mulenumrightsoundscale. G = fs_q_jt_mulenumrightsoundscale * S ((S (jt_i_mulenumright)) * H) + (jt_c_mulenumright))))) /\\ (((forall jt_index_mulenumrightbound. (exists jt_gap_mulenumrightboundindex. jt_gap_mulenumrightboundindex+S (jt_index_mulenumrightbound)=(k)) -> exists jt_value_mulenumrightbound. ((((exists fs_h_jt_mulenumrightboundat. fs_h_jt_mulenumrightboundat + S (jt_value_mulenumrightbound) = S ((S (jt_index_mulenumrightbound)) * jt_c_mulenumright)) /\\ exists fs_q_jt_mulenumrightboundat. jt_b_mulenumright = fs_q_jt_mulenumrightboundat * S ((S (jt_index_mulenumrightbound)) * jt_c_mulenumright) + (jt_value_mulenumrightbound))) /\\ (exists jt_gap_mulenumrightboundvalue. jt_gap_mulenumrightboundvalue+S (jt_value_mulenumrightbound)=(n)))) /\\ (forall jt_divisor_mulenumrightprimitive. (exists jt_factor_mulenumrightprimitivemodulus. (n)=(jt_divisor_mulenumrightprimitive)*jt_factor_mulenumrightprimitivemodulus) -> (forall jt_index_mulenumrightprimitivecoordinates jt_value_mulenumrightprimitivecoordinates. (exists jt_gap_mulenumrightprimitivecoordinatesindex. jt_gap_mulenumrightprimitivecoordinatesindex+S (jt_index_mulenumrightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumrightprimitivecoordinatesat. fs_h_jt_mulenumrightprimitivecoordinatesat + S (jt_value_mulenumrightprimitivecoordinates) = S ((S (jt_index_mulenumrightprimitivecoordinates)) * jt_c_mulenumright)) /\\ exists fs_q_jt_mulenumrightprimitivecoordinatesat. jt_b_mulenumright = fs_q_jt_mulenumrightprimitivecoordinatesat * S ((S (jt_index_mulenumrightprimitivecoordinates)) * jt_c_mulenumright) + (jt_value_mulenumrightprimitivecoordinates))) -> (exists jt_factor_mulenumrightprimitivecoordinatesdivides. (jt_value_mulenumrightprimitivecoordinates)=(jt_divisor_mulenumrightprimitive)*jt_factor_mulenumrightprimitivecoordinatesdivides)) -> jt_divisor_mulenumrightprimitive=1))))) /\\ (((forall jt_b_mulenumright jt_c_mulenumright. (forall jt_index_mulenumrightinputbound. (exists jt_gap_mulenumrightinputboundindex. jt_gap_mulenumrightinputboundindex+S (jt_index_mulenumrightinputbound)=(k)) -> exists jt_value_mulenumrightinputbound. ((((exists fs_h_jt_mulenumrightinputboundat. fs_h_jt_mulenumrightinputboundat + S (jt_value_mulenumrightinputbound) = S ((S (jt_index_mulenumrightinputbound)) * jt_c_mulenumright)) /\\ exists fs_q_jt_mulenumrightinputboundat. jt_b_mulenumright = fs_q_jt_mulenumrightinputboundat * S ((S (jt_index_mulenumrightinputbound)) * jt_c_mulenumright) + (jt_value_mulenumrightinputbound))) /\\ (exists jt_gap_mulenumrightinputboundvalue. jt_gap_mulenumrightinputboundvalue+S (jt_value_mulenumrightinputbound)=(n)))) -> (forall jt_divisor_mulenumrightinputprimitive. (exists jt_factor_mulenumrightinputprimitivemodulus. (n)=(jt_divisor_mulenumrightinputprimitive)*jt_factor_mulenumrightinputprimitivemodulus) -> (forall jt_index_mulenumrightinputprimitivecoordinates jt_value_mulenumrightinputprimitivecoordinates. (exists jt_gap_mulenumrightinputprimitivecoordinatesindex. jt_gap_mulenumrightinputprimitivecoordinatesindex+S (jt_index_mulenumrightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumrightinputprimitivecoordinatesat. fs_h_jt_mulenumrightinputprimitivecoordinatesat + S (jt_value_mulenumrightinputprimitivecoordinates) = S ((S (jt_index_mulenumrightinputprimitivecoordinates)) * jt_c_mulenumright)) /\\ exists fs_q_jt_mulenumrightinputprimitivecoordinatesat. jt_b_mulenumright = fs_q_jt_mulenumrightinputprimitivecoordinatesat * S ((S (jt_index_mulenumrightinputprimitivecoordinates)) * jt_c_mulenumright) + (jt_value_mulenumrightinputprimitivecoordinates))) -> (exists jt_factor_mulenumrightinputprimitivecoordinatesdivides. (jt_value_mulenumrightinputprimitivecoordinates)=(jt_divisor_mulenumrightinputprimitive)*jt_factor_mulenumrightinputprimitivecoordinatesdivides)) -> jt_divisor_mulenumrightinputprimitive=1) -> exists jt_i_mulenumright jt_d_mulenumright jt_e_mulenumright. ((exists jt_gap_mulenumrightcompleteindex. jt_gap_mulenumrightcompleteindex+S (jt_i_mulenumright)=(v)) /\\ (((((((exists fs_h_jt_mulenumrightcompletecode. fs_h_jt_mulenumrightcompletecode + S (jt_d_mulenumright) = S ((S (jt_i_mulenumright)) * F)) /\\ exists fs_q_jt_mulenumrightcompletecode. E = fs_q_jt_mulenumrightcompletecode * S ((S (jt_i_mulenumright)) * F) + (jt_d_mulenumright))) /\\ (((exists fs_h_jt_mulenumrightcompletescale. fs_h_jt_mulenumrightcompletescale + S (jt_e_mulenumright) = S ((S (jt_i_mulenumright)) * H)) /\\ exists fs_q_jt_mulenumrightcompletescale. G = fs_q_jt_mulenumrightcompletescale * S ((S (jt_i_mulenumright)) * H) + (jt_e_mulenumright))))) /\\ (forall jt_index_mulenumrightrepresented jt_left_mulenumrightrepresented jt_right_mulenumrightrepresented. (exists jt_gap_mulenumrightrepresentedindex. jt_gap_mulenumrightrepresentedindex+S (jt_index_mulenumrightrepresented)=(k)) -> (((exists fs_h_jt_mulenumrightrepresentedleft. fs_h_jt_mulenumrightrepresentedleft + S (jt_left_mulenumrightrepresented) = S ((S (jt_index_mulenumrightrepresented)) * jt_c_mulenumright)) /\\ exists fs_q_jt_mulenumrightrepresentedleft. jt_b_mulenumright = fs_q_jt_mulenumrightrepresentedleft * S ((S (jt_index_mulenumrightrepresented)) * jt_c_mulenumright) + (jt_left_mulenumrightrepresented))) -> (((exists fs_h_jt_mulenumrightrepresentedright. fs_h_jt_mulenumrightrepresentedright + S (jt_right_mulenumrightrepresented) = S ((S (jt_index_mulenumrightrepresented)) * jt_e_mulenumright)) /\\ exists fs_q_jt_mulenumrightrepresentedright. jt_d_mulenumright = fs_q_jt_mulenumrightrepresentedright * S ((S (jt_index_mulenumrightrepresented)) * jt_e_mulenumright) + (jt_right_mulenumrightrepresented))) -> jt_left_mulenumrightrepresented=jt_right_mulenumrightrepresented))))) /\\ (forall jt_i_mulenumright jt_h_mulenumright jt_b_mulenumright jt_c_mulenumright jt_d_mulenumright jt_e_mulenumright. (exists jt_gap_mulenumrightfirstindex. jt_gap_mulenumrightfirstindex+S (jt_i_mulenumright)=(v)) -> (exists jt_gap_mulenumrightsecondindex. jt_gap_mulenumrightsecondindex+S (jt_h_mulenumright)=(v)) -> (((((exists fs_h_jt_mulenumrightfirstcode. fs_h_jt_mulenumrightfirstcode + S (jt_b_mulenumright) = S ((S (jt_i_mulenumright)) * F)) /\\ exists fs_q_jt_mulenumrightfirstcode. E = fs_q_jt_mulenumrightfirstcode * S ((S (jt_i_mulenumright)) * F) + (jt_b_mulenumright))) /\\ (((exists fs_h_jt_mulenumrightfirstscale. fs_h_jt_mulenumrightfirstscale + S (jt_c_mulenumright) = S ((S (jt_i_mulenumright)) * H)) /\\ exists fs_q_jt_mulenumrightfirstscale. G = fs_q_jt_mulenumrightfirstscale * S ((S (jt_i_mulenumright)) * H) + (jt_c_mulenumright))))) -> (((((exists fs_h_jt_mulenumrightsecondcode. fs_h_jt_mulenumrightsecondcode + S (jt_d_mulenumright) = S ((S (jt_h_mulenumright)) * F)) /\\ exists fs_q_jt_mulenumrightsecondcode. E = fs_q_jt_mulenumrightsecondcode * S ((S (jt_h_mulenumright)) * F) + (jt_d_mulenumright))) /\\ (((exists fs_h_jt_mulenumrightsecondscale. fs_h_jt_mulenumrightsecondscale + S (jt_e_mulenumright) = S ((S (jt_h_mulenumright)) * H)) /\\ exists fs_q_jt_mulenumrightsecondscale. G = fs_q_jt_mulenumrightsecondscale * S ((S (jt_h_mulenumright)) * H) + (jt_e_mulenumright))))) -> (forall jt_index_mulenumrightsame jt_left_mulenumrightsame jt_right_mulenumrightsame. (exists jt_gap_mulenumrightsameindex. jt_gap_mulenumrightsameindex+S (jt_index_mulenumrightsame)=(k)) -> (((exists fs_h_jt_mulenumrightsameleft. fs_h_jt_mulenumrightsameleft + S (jt_left_mulenumrightsame) = S ((S (jt_index_mulenumrightsame)) * jt_c_mulenumright)) /\\ exists fs_q_jt_mulenumrightsameleft. jt_b_mulenumright = fs_q_jt_mulenumrightsameleft * S ((S (jt_index_mulenumrightsame)) * jt_c_mulenumright) + (jt_left_mulenumrightsame))) -> (((exists fs_h_jt_mulenumrightsameright. fs_h_jt_mulenumrightsameright + S (jt_right_mulenumrightsame) = S ((S (jt_index_mulenumrightsame)) * jt_e_mulenumright)) /\\ exists fs_q_jt_mulenumrightsameright. jt_d_mulenumright = fs_q_jt_mulenumrightsameright * S ((S (jt_index_mulenumrightsame)) * jt_e_mulenumright) + (jt_right_mulenumrightsame))) -> jt_left_mulenumrightsame=jt_right_mulenumrightsame) -> jt_i_mulenumright=jt_h_mulenumright))))) -> exists P Q R T. ((forall jt_i_mulenumoutput. (exists jt_gap_mulenumoutputsoundindex. jt_gap_mulenumoutputsoundindex+S (jt_i_mulenumoutput)=(u*v)) -> exists jt_b_mulenumoutput jt_c_mulenumoutput. ((((((exists fs_h_jt_mulenumoutputsoundcode. fs_h_jt_mulenumoutputsoundcode + S (jt_b_mulenumoutput) = S ((S (jt_i_mulenumoutput)) * Q)) /\\ exists fs_q_jt_mulenumoutputsoundcode. P = fs_q_jt_mulenumoutputsoundcode * S ((S (jt_i_mulenumoutput)) * Q) + (jt_b_mulenumoutput))) /\\ (((exists fs_h_jt_mulenumoutputsoundscale. fs_h_jt_mulenumoutputsoundscale + S (jt_c_mulenumoutput) = S ((S (jt_i_mulenumoutput)) * T)) /\\ exists fs_q_jt_mulenumoutputsoundscale. R = fs_q_jt_mulenumoutputsoundscale * S ((S (jt_i_mulenumoutput)) * T) + (jt_c_mulenumoutput))))) /\\ (((forall jt_index_mulenumoutputbound. (exists jt_gap_mulenumoutputboundindex. jt_gap_mulenumoutputboundindex+S (jt_index_mulenumoutputbound)=(k)) -> exists jt_value_mulenumoutputbound. ((((exists fs_h_jt_mulenumoutputboundat. fs_h_jt_mulenumoutputboundat + S (jt_value_mulenumoutputbound) = S ((S (jt_index_mulenumoutputbound)) * jt_c_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputboundat. jt_b_mulenumoutput = fs_q_jt_mulenumoutputboundat * S ((S (jt_index_mulenumoutputbound)) * jt_c_mulenumoutput) + (jt_value_mulenumoutputbound))) /\\ (exists jt_gap_mulenumoutputboundvalue. jt_gap_mulenumoutputboundvalue+S (jt_value_mulenumoutputbound)=(m*n)))) /\\ (forall jt_divisor_mulenumoutputprimitive. (exists jt_factor_mulenumoutputprimitivemodulus. (m*n)=(jt_divisor_mulenumoutputprimitive)*jt_factor_mulenumoutputprimitivemodulus) -> (forall jt_index_mulenumoutputprimitivecoordinates jt_value_mulenumoutputprimitivecoordinates. (exists jt_gap_mulenumoutputprimitivecoordinatesindex. jt_gap_mulenumoutputprimitivecoordinatesindex+S (jt_index_mulenumoutputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumoutputprimitivecoordinatesat. fs_h_jt_mulenumoutputprimitivecoordinatesat + S (jt_value_mulenumoutputprimitivecoordinates) = S ((S (jt_index_mulenumoutputprimitivecoordinates)) * jt_c_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputprimitivecoordinatesat. jt_b_mulenumoutput = fs_q_jt_mulenumoutputprimitivecoordinatesat * S ((S (jt_index_mulenumoutputprimitivecoordinates)) * jt_c_mulenumoutput) + (jt_value_mulenumoutputprimitivecoordinates))) -> (exists jt_factor_mulenumoutputprimitivecoordinatesdivides. (jt_value_mulenumoutputprimitivecoordinates)=(jt_divisor_mulenumoutputprimitive)*jt_factor_mulenumoutputprimitivecoordinatesdivides)) -> jt_divisor_mulenumoutputprimitive=1))))) /\\ (((forall jt_b_mulenumoutput jt_c_mulenumoutput. (forall jt_index_mulenumoutputinputbound. (exists jt_gap_mulenumoutputinputboundindex. jt_gap_mulenumoutputinputboundindex+S (jt_index_mulenumoutputinputbound)=(k)) -> exists jt_value_mulenumoutputinputbound. ((((exists fs_h_jt_mulenumoutputinputboundat. fs_h_jt_mulenumoutputinputboundat + S (jt_value_mulenumoutputinputbound) = S ((S (jt_index_mulenumoutputinputbound)) * jt_c_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputinputboundat. jt_b_mulenumoutput = fs_q_jt_mulenumoutputinputboundat * S ((S (jt_index_mulenumoutputinputbound)) * jt_c_mulenumoutput) + (jt_value_mulenumoutputinputbound))) /\\ (exists jt_gap_mulenumoutputinputboundvalue. jt_gap_mulenumoutputinputboundvalue+S (jt_value_mulenumoutputinputbound)=(m*n)))) -> (forall jt_divisor_mulenumoutputinputprimitive. (exists jt_factor_mulenumoutputinputprimitivemodulus. (m*n)=(jt_divisor_mulenumoutputinputprimitive)*jt_factor_mulenumoutputinputprimitivemodulus) -> (forall jt_index_mulenumoutputinputprimitivecoordinates jt_value_mulenumoutputinputprimitivecoordinates. (exists jt_gap_mulenumoutputinputprimitivecoordinatesindex. jt_gap_mulenumoutputinputprimitivecoordinatesindex+S (jt_index_mulenumoutputinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_mulenumoutputinputprimitivecoordinatesat. fs_h_jt_mulenumoutputinputprimitivecoordinatesat + S (jt_value_mulenumoutputinputprimitivecoordinates) = S ((S (jt_index_mulenumoutputinputprimitivecoordinates)) * jt_c_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputinputprimitivecoordinatesat. jt_b_mulenumoutput = fs_q_jt_mulenumoutputinputprimitivecoordinatesat * S ((S (jt_index_mulenumoutputinputprimitivecoordinates)) * jt_c_mulenumoutput) + (jt_value_mulenumoutputinputprimitivecoordinates))) -> (exists jt_factor_mulenumoutputinputprimitivecoordinatesdivides. (jt_value_mulenumoutputinputprimitivecoordinates)=(jt_divisor_mulenumoutputinputprimitive)*jt_factor_mulenumoutputinputprimitivecoordinatesdivides)) -> jt_divisor_mulenumoutputinputprimitive=1) -> exists jt_i_mulenumoutput jt_d_mulenumoutput jt_e_mulenumoutput. ((exists jt_gap_mulenumoutputcompleteindex. jt_gap_mulenumoutputcompleteindex+S (jt_i_mulenumoutput)=(u*v)) /\\ (((((((exists fs_h_jt_mulenumoutputcompletecode. fs_h_jt_mulenumoutputcompletecode + S (jt_d_mulenumoutput) = S ((S (jt_i_mulenumoutput)) * Q)) /\\ exists fs_q_jt_mulenumoutputcompletecode. P = fs_q_jt_mulenumoutputcompletecode * S ((S (jt_i_mulenumoutput)) * Q) + (jt_d_mulenumoutput))) /\\ (((exists fs_h_jt_mulenumoutputcompletescale. fs_h_jt_mulenumoutputcompletescale + S (jt_e_mulenumoutput) = S ((S (jt_i_mulenumoutput)) * T)) /\\ exists fs_q_jt_mulenumoutputcompletescale. R = fs_q_jt_mulenumoutputcompletescale * S ((S (jt_i_mulenumoutput)) * T) + (jt_e_mulenumoutput))))) /\\ (forall jt_index_mulenumoutputrepresented jt_left_mulenumoutputrepresented jt_right_mulenumoutputrepresented. (exists jt_gap_mulenumoutputrepresentedindex. jt_gap_mulenumoutputrepresentedindex+S (jt_index_mulenumoutputrepresented)=(k)) -> (((exists fs_h_jt_mulenumoutputrepresentedleft. fs_h_jt_mulenumoutputrepresentedleft + S (jt_left_mulenumoutputrepresented) = S ((S (jt_index_mulenumoutputrepresented)) * jt_c_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputrepresentedleft. jt_b_mulenumoutput = fs_q_jt_mulenumoutputrepresentedleft * S ((S (jt_index_mulenumoutputrepresented)) * jt_c_mulenumoutput) + (jt_left_mulenumoutputrepresented))) -> (((exists fs_h_jt_mulenumoutputrepresentedright. fs_h_jt_mulenumoutputrepresentedright + S (jt_right_mulenumoutputrepresented) = S ((S (jt_index_mulenumoutputrepresented)) * jt_e_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputrepresentedright. jt_d_mulenumoutput = fs_q_jt_mulenumoutputrepresentedright * S ((S (jt_index_mulenumoutputrepresented)) * jt_e_mulenumoutput) + (jt_right_mulenumoutputrepresented))) -> jt_left_mulenumoutputrepresented=jt_right_mulenumoutputrepresented))))) /\\ (forall jt_i_mulenumoutput jt_h_mulenumoutput jt_b_mulenumoutput jt_c_mulenumoutput jt_d_mulenumoutput jt_e_mulenumoutput. (exists jt_gap_mulenumoutputfirstindex. jt_gap_mulenumoutputfirstindex+S (jt_i_mulenumoutput)=(u*v)) -> (exists jt_gap_mulenumoutputsecondindex. jt_gap_mulenumoutputsecondindex+S (jt_h_mulenumoutput)=(u*v)) -> (((((exists fs_h_jt_mulenumoutputfirstcode. fs_h_jt_mulenumoutputfirstcode + S (jt_b_mulenumoutput) = S ((S (jt_i_mulenumoutput)) * Q)) /\\ exists fs_q_jt_mulenumoutputfirstcode. P = fs_q_jt_mulenumoutputfirstcode * S ((S (jt_i_mulenumoutput)) * Q) + (jt_b_mulenumoutput))) /\\ (((exists fs_h_jt_mulenumoutputfirstscale. fs_h_jt_mulenumoutputfirstscale + S (jt_c_mulenumoutput) = S ((S (jt_i_mulenumoutput)) * T)) /\\ exists fs_q_jt_mulenumoutputfirstscale. R = fs_q_jt_mulenumoutputfirstscale * S ((S (jt_i_mulenumoutput)) * T) + (jt_c_mulenumoutput))))) -> (((((exists fs_h_jt_mulenumoutputsecondcode. fs_h_jt_mulenumoutputsecondcode + S (jt_d_mulenumoutput) = S ((S (jt_h_mulenumoutput)) * Q)) /\\ exists fs_q_jt_mulenumoutputsecondcode. P = fs_q_jt_mulenumoutputsecondcode * S ((S (jt_h_mulenumoutput)) * Q) + (jt_d_mulenumoutput))) /\\ (((exists fs_h_jt_mulenumoutputsecondscale. fs_h_jt_mulenumoutputsecondscale + S (jt_e_mulenumoutput) = S ((S (jt_h_mulenumoutput)) * T)) /\\ exists fs_q_jt_mulenumoutputsecondscale. R = fs_q_jt_mulenumoutputsecondscale * S ((S (jt_h_mulenumoutput)) * T) + (jt_e_mulenumoutput))))) -> (forall jt_index_mulenumoutputsame jt_left_mulenumoutputsame jt_right_mulenumoutputsame. (exists jt_gap_mulenumoutputsameindex. jt_gap_mulenumoutputsameindex+S (jt_index_mulenumoutputsame)=(k)) -> (((exists fs_h_jt_mulenumoutputsameleft. fs_h_jt_mulenumoutputsameleft + S (jt_left_mulenumoutputsame) = S ((S (jt_index_mulenumoutputsame)) * jt_c_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputsameleft. jt_b_mulenumoutput = fs_q_jt_mulenumoutputsameleft * S ((S (jt_index_mulenumoutputsame)) * jt_c_mulenumoutput) + (jt_left_mulenumoutputsame))) -> (((exists fs_h_jt_mulenumoutputsameright. fs_h_jt_mulenumoutputsameright + S (jt_right_mulenumoutputsame) = S ((S (jt_index_mulenumoutputsame)) * jt_e_mulenumoutput)) /\\ exists fs_q_jt_mulenumoutputsameright. jt_d_mulenumoutput = fs_q_jt_mulenumoutputsameright * S ((S (jt_index_mulenumoutputsame)) * jt_e_mulenumoutput) + (jt_right_mulenumoutputsame))) -> jt_left_mulenumoutputsame=jt_right_mulenumoutputsame) -> jt_i_mulenumoutput=jt_h_mulenumoutput))))", "statement_sha256": "8b3f625db3a4e9e10224e5a91a2506012b6fc4570e66af34a0701444f7bb48d5", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Construct an actual product enumeration; no count-uniqueness or multiplicativity premise is used." }, { "admission_dependencies": [ "mul_ne_zero", "jordan_product_enumeration_exists" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 68, "body_proof_nodes": 140, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro a", "intro b", "intro u", "intro v", "intro hcop", "intro ha", "intro hb", "cases ha", "cases ha_right", "cases hb", "cases hb_right", "cases ha_right_right", "cases ha_right_right_witness", "cases ha_right_right_witness_witness", "cases ha_right_right_witness_witness_witness", "cases hb_right_right", "cases hb_right_right_witness", "cases hb_right_right_witness_witness", "cases hb_right_right_witness_witness_witness", "split", "exact ha_left", "split", "intro hz", "specialize mul_ne_zero (a)", "specialize mul_ne_zero (b)", "apply mul_ne_zero", "exact ha_right_left", "exact hb_right_left", "exact hz", "specialize jordan_product_enumeration_exists (a)", "specialize jordan_product_enumeration_exists (b)", "specialize jordan_product_enumeration_exists (k)", "specialize jordan_product_enumeration_exists (x)", "specialize jordan_product_enumeration_exists (x1)", "specialize jordan_product_enumeration_exists (x2)", "specialize jordan_product_enumeration_exists (x3)", "specialize jordan_product_enumeration_exists (u)", "specialize jordan_product_enumeration_exists (x4)", "specialize jordan_product_enumeration_exists (x5)", "specialize jordan_product_enumeration_exists (x6)", "specialize jordan_product_enumeration_exists (x7)", "specialize jordan_product_enumeration_exists (v)", "apply jordan_product_enumeration_exists", "exact ha_right_left", "exact hb_right_left", "exact hcop", "exact ha_right_right_witness_witness_witness_witness", "exact hb_right_right_witness_witness_witness_witness" ], "defined_statement": "∀ k. ∀ a. ∀ b. ∀ u. ∀ v. Coprime(a,b) → JordanTotient(k,a,u) → JordanTotient(k,b,v) → JordanTotient(k,a · b,u · v)", "defined_statement_sha256": "8a44525d1ec450eb99e4a0253ae9a413372d3b2356c1128e1bd8f9b31773e376", "definition_uses": { "ND0375": 3, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "f2c154c6ab76ff330735cca1784280f2d32b48777e27b76c27939e9061f06210", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "cases ha_right" } ], [ { "kind": "text", "text": "cases hb" } ], [ { "kind": "text", "text": "cases hb_right" } ], [ { "kind": "text", "text": "cases ha_right_right" } ], [ { "kind": "text", "text": "cases ha_right_right_witness" } ], [ { "kind": "text", "text": "cases ha_right_right_witness_witness" } ], [ { "kind": "text", "text": "cases ha_right_right_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hb_right_right" } ], [ { "kind": "text", "text": "cases hb_right_right_witness" } ], [ { "kind": "text", "text": "cases hb_right_right_witness_witness" } ], [ { "kind": "text", "text": "cases hb_right_right_witness_witness_witness" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact ha_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize mul_ne_zero (a)" } ], [ { "kind": "text", "text": "specialize mul_ne_zero (b)" } ], [ { "kind": "text", "text": "apply mul_ne_zero" } ], [ { "kind": "text", "text": "exact ha_right_left" } ], [ { "kind": "text", "text": "exact hb_right_left" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (a)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (b)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x1)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x2)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x3)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (u)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x4)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x5)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x6)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (x7)" } ], [ { "kind": "text", "text": "specialize jordan_product_enumeration_exists (v)" } ], [ { "kind": "text", "text": "apply jordan_product_enumeration_exists" } ], [ { "kind": "text", "text": "exact ha_right_left" } ], [ { "kind": "text", "text": "exact hb_right_left" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact ha_right_right_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "exact hb_right_right_witness_witness_witness_witness" } ] ], "statement_definition_uses": { "ND0375": 3, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ a. ∀ b. ∀ u. ∀ v. " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(a,b)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,b,v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a · b,u · v)" } ] }, "dependencies": [ "mul_ne_zero", "jordan_product_enumeration_exists" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT004A", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_coprime_product", "original_ha_bundle_verified": true, "proof_bundle_node_id": 335, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro a", "intro b", "intro u", "intro v", "intro hcop", "intro ha", "intro hb", "cases ha", "cases ha_right", "cases hb", "cases hb_right", "cases ha_right_right", "cases ha_right_right_witness", "cases ha_right_right_witness_witness", "cases ha_right_right_witness_witness_witness", "cases hb_right_right", "cases hb_right_right_witness", "cases hb_right_right_witness_witness", "cases hb_right_right_witness_witness_witness", "split", "exact ha_left", "split", "intro hz", "specialize mul_ne_zero (a)", "specialize mul_ne_zero (b)", "apply mul_ne_zero", "exact ha_right_left", "exact hb_right_left", "exact hz", "specialize jordan_product_enumeration_exists (a)", "specialize jordan_product_enumeration_exists (b)", "specialize jordan_product_enumeration_exists (k)", "specialize jordan_product_enumeration_exists (x)", "specialize jordan_product_enumeration_exists (x1)", "specialize jordan_product_enumeration_exists (x2)", "specialize jordan_product_enumeration_exists (x3)", "specialize jordan_product_enumeration_exists (u)", "specialize jordan_product_enumeration_exists (x4)", "specialize jordan_product_enumeration_exists (x5)", "specialize jordan_product_enumeration_exists (x6)", "specialize jordan_product_enumeration_exists (x7)", "specialize jordan_product_enumeration_exists (v)", "apply jordan_product_enumeration_exists", "exact ha_right_left", "exact hb_right_left", "exact hcop", "exact ha_right_right_witness_witness_witness_witness", "exact hb_right_right_witness_witness_witness_witness" ], "script_sha256": "b7f2811584bd7bc9bf8cecef78993b417ab753c71b5fa670904381cf14c2b3db", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "b7f2811584bd7bc9bf8cecef78993b417ab753c71b5fa670904381cf14c2b3db", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "f2c154c6ab76ff330735cca1784280f2d32b48777e27b76c27939e9061f06210" } ], "stable_member": false, "statement": "forall k a b u v. (forall jt_divisor_jmulcop. (exists jt_factor_jmulcopa. (a)=(jt_divisor_jmulcop)*jt_factor_jmulcopa) -> (exists jt_factor_jmulcopb. (b)=(jt_divisor_jmulcop)*jt_factor_jmulcopb) -> jt_divisor_jmulcop=1) -> (((~((k)=0)) /\\ (((~((a)=0)) /\\ (exists jt_codes_jmulleft jt_code_scale_jmulleft jt_scales_jmulleft jt_scale_scale_jmulleft. ((forall jt_i_jmulleftenum. (exists jt_gap_jmulleftenumsoundindex. jt_gap_jmulleftenumsoundindex+S (jt_i_jmulleftenum)=(u)) -> exists jt_b_jmulleftenum jt_c_jmulleftenum. ((((((exists fs_h_jt_jmulleftenumsoundcode. fs_h_jt_jmulleftenumsoundcode + S (jt_b_jmulleftenum) = S ((S (jt_i_jmulleftenum)) * jt_code_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumsoundcode. jt_codes_jmulleft = fs_q_jt_jmulleftenumsoundcode * S ((S (jt_i_jmulleftenum)) * jt_code_scale_jmulleft) + (jt_b_jmulleftenum))) /\\ (((exists fs_h_jt_jmulleftenumsoundscale. fs_h_jt_jmulleftenumsoundscale + S (jt_c_jmulleftenum) = S ((S (jt_i_jmulleftenum)) * jt_scale_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumsoundscale. jt_scales_jmulleft = fs_q_jt_jmulleftenumsoundscale * S ((S (jt_i_jmulleftenum)) * jt_scale_scale_jmulleft) + (jt_c_jmulleftenum))))) /\\ (((forall jt_index_jmulleftenumbound. (exists jt_gap_jmulleftenumboundindex. jt_gap_jmulleftenumboundindex+S (jt_index_jmulleftenumbound)=(k)) -> exists jt_value_jmulleftenumbound. ((((exists fs_h_jt_jmulleftenumboundat. fs_h_jt_jmulleftenumboundat + S (jt_value_jmulleftenumbound) = S ((S (jt_index_jmulleftenumbound)) * jt_c_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenumboundat. jt_b_jmulleftenum = fs_q_jt_jmulleftenumboundat * S ((S (jt_index_jmulleftenumbound)) * jt_c_jmulleftenum) + (jt_value_jmulleftenumbound))) /\\ (exists jt_gap_jmulleftenumboundvalue. jt_gap_jmulleftenumboundvalue+S (jt_value_jmulleftenumbound)=(a)))) /\\ (forall jt_divisor_jmulleftenumprimitive. (exists jt_factor_jmulleftenumprimitivemodulus. (a)=(jt_divisor_jmulleftenumprimitive)*jt_factor_jmulleftenumprimitivemodulus) -> (forall jt_index_jmulleftenumprimitivecoordinates jt_value_jmulleftenumprimitivecoordinates. (exists jt_gap_jmulleftenumprimitivecoordinatesindex. jt_gap_jmulleftenumprimitivecoordinatesindex+S (jt_index_jmulleftenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jmulleftenumprimitivecoordinatesat. fs_h_jt_jmulleftenumprimitivecoordinatesat + S (jt_value_jmulleftenumprimitivecoordinates) = S ((S (jt_index_jmulleftenumprimitivecoordinates)) * jt_c_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenumprimitivecoordinatesat. jt_b_jmulleftenum = fs_q_jt_jmulleftenumprimitivecoordinatesat * S ((S (jt_index_jmulleftenumprimitivecoordinates)) * jt_c_jmulleftenum) + (jt_value_jmulleftenumprimitivecoordinates))) -> (exists jt_factor_jmulleftenumprimitivecoordinatesdivides. (jt_value_jmulleftenumprimitivecoordinates)=(jt_divisor_jmulleftenumprimitive)*jt_factor_jmulleftenumprimitivecoordinatesdivides)) -> jt_divisor_jmulleftenumprimitive=1))))) /\\ (((forall jt_b_jmulleftenum jt_c_jmulleftenum. (forall jt_index_jmulleftenuminputbound. (exists jt_gap_jmulleftenuminputboundindex. jt_gap_jmulleftenuminputboundindex+S (jt_index_jmulleftenuminputbound)=(k)) -> exists jt_value_jmulleftenuminputbound. ((((exists fs_h_jt_jmulleftenuminputboundat. fs_h_jt_jmulleftenuminputboundat + S (jt_value_jmulleftenuminputbound) = S ((S (jt_index_jmulleftenuminputbound)) * jt_c_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenuminputboundat. jt_b_jmulleftenum = fs_q_jt_jmulleftenuminputboundat * S ((S (jt_index_jmulleftenuminputbound)) * jt_c_jmulleftenum) + (jt_value_jmulleftenuminputbound))) /\\ (exists jt_gap_jmulleftenuminputboundvalue. jt_gap_jmulleftenuminputboundvalue+S (jt_value_jmulleftenuminputbound)=(a)))) -> (forall jt_divisor_jmulleftenuminputprimitive. (exists jt_factor_jmulleftenuminputprimitivemodulus. (a)=(jt_divisor_jmulleftenuminputprimitive)*jt_factor_jmulleftenuminputprimitivemodulus) -> (forall jt_index_jmulleftenuminputprimitivecoordinates jt_value_jmulleftenuminputprimitivecoordinates. (exists jt_gap_jmulleftenuminputprimitivecoordinatesindex. jt_gap_jmulleftenuminputprimitivecoordinatesindex+S (jt_index_jmulleftenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jmulleftenuminputprimitivecoordinatesat. fs_h_jt_jmulleftenuminputprimitivecoordinatesat + S (jt_value_jmulleftenuminputprimitivecoordinates) = S ((S (jt_index_jmulleftenuminputprimitivecoordinates)) * jt_c_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenuminputprimitivecoordinatesat. jt_b_jmulleftenum = fs_q_jt_jmulleftenuminputprimitivecoordinatesat * S ((S (jt_index_jmulleftenuminputprimitivecoordinates)) * jt_c_jmulleftenum) + (jt_value_jmulleftenuminputprimitivecoordinates))) -> (exists jt_factor_jmulleftenuminputprimitivecoordinatesdivides. (jt_value_jmulleftenuminputprimitivecoordinates)=(jt_divisor_jmulleftenuminputprimitive)*jt_factor_jmulleftenuminputprimitivecoordinatesdivides)) -> jt_divisor_jmulleftenuminputprimitive=1) -> exists jt_i_jmulleftenum jt_d_jmulleftenum jt_e_jmulleftenum. ((exists jt_gap_jmulleftenumcompleteindex. jt_gap_jmulleftenumcompleteindex+S (jt_i_jmulleftenum)=(u)) /\\ (((((((exists fs_h_jt_jmulleftenumcompletecode. fs_h_jt_jmulleftenumcompletecode + S (jt_d_jmulleftenum) = S ((S (jt_i_jmulleftenum)) * jt_code_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumcompletecode. jt_codes_jmulleft = fs_q_jt_jmulleftenumcompletecode * S ((S (jt_i_jmulleftenum)) * jt_code_scale_jmulleft) + (jt_d_jmulleftenum))) /\\ (((exists fs_h_jt_jmulleftenumcompletescale. fs_h_jt_jmulleftenumcompletescale + S (jt_e_jmulleftenum) = S ((S (jt_i_jmulleftenum)) * jt_scale_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumcompletescale. jt_scales_jmulleft = fs_q_jt_jmulleftenumcompletescale * S ((S (jt_i_jmulleftenum)) * jt_scale_scale_jmulleft) + (jt_e_jmulleftenum))))) /\\ (forall jt_index_jmulleftenumrepresented jt_left_jmulleftenumrepresented jt_right_jmulleftenumrepresented. (exists jt_gap_jmulleftenumrepresentedindex. jt_gap_jmulleftenumrepresentedindex+S (jt_index_jmulleftenumrepresented)=(k)) -> (((exists fs_h_jt_jmulleftenumrepresentedleft. fs_h_jt_jmulleftenumrepresentedleft + S (jt_left_jmulleftenumrepresented) = S ((S (jt_index_jmulleftenumrepresented)) * jt_c_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenumrepresentedleft. jt_b_jmulleftenum = fs_q_jt_jmulleftenumrepresentedleft * S ((S (jt_index_jmulleftenumrepresented)) * jt_c_jmulleftenum) + (jt_left_jmulleftenumrepresented))) -> (((exists fs_h_jt_jmulleftenumrepresentedright. fs_h_jt_jmulleftenumrepresentedright + S (jt_right_jmulleftenumrepresented) = S ((S (jt_index_jmulleftenumrepresented)) * jt_e_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenumrepresentedright. jt_d_jmulleftenum = fs_q_jt_jmulleftenumrepresentedright * S ((S (jt_index_jmulleftenumrepresented)) * jt_e_jmulleftenum) + (jt_right_jmulleftenumrepresented))) -> jt_left_jmulleftenumrepresented=jt_right_jmulleftenumrepresented))))) /\\ (forall jt_i_jmulleftenum jt_h_jmulleftenum jt_b_jmulleftenum jt_c_jmulleftenum jt_d_jmulleftenum jt_e_jmulleftenum. (exists jt_gap_jmulleftenumfirstindex. jt_gap_jmulleftenumfirstindex+S (jt_i_jmulleftenum)=(u)) -> (exists jt_gap_jmulleftenumsecondindex. jt_gap_jmulleftenumsecondindex+S (jt_h_jmulleftenum)=(u)) -> (((((exists fs_h_jt_jmulleftenumfirstcode. fs_h_jt_jmulleftenumfirstcode + S (jt_b_jmulleftenum) = S ((S (jt_i_jmulleftenum)) * jt_code_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumfirstcode. jt_codes_jmulleft = fs_q_jt_jmulleftenumfirstcode * S ((S (jt_i_jmulleftenum)) * jt_code_scale_jmulleft) + (jt_b_jmulleftenum))) /\\ (((exists fs_h_jt_jmulleftenumfirstscale. fs_h_jt_jmulleftenumfirstscale + S (jt_c_jmulleftenum) = S ((S (jt_i_jmulleftenum)) * jt_scale_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumfirstscale. jt_scales_jmulleft = fs_q_jt_jmulleftenumfirstscale * S ((S (jt_i_jmulleftenum)) * jt_scale_scale_jmulleft) + (jt_c_jmulleftenum))))) -> (((((exists fs_h_jt_jmulleftenumsecondcode. fs_h_jt_jmulleftenumsecondcode + S (jt_d_jmulleftenum) = S ((S (jt_h_jmulleftenum)) * jt_code_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumsecondcode. jt_codes_jmulleft = fs_q_jt_jmulleftenumsecondcode * S ((S (jt_h_jmulleftenum)) * jt_code_scale_jmulleft) + (jt_d_jmulleftenum))) /\\ (((exists fs_h_jt_jmulleftenumsecondscale. fs_h_jt_jmulleftenumsecondscale + S (jt_e_jmulleftenum) = S ((S (jt_h_jmulleftenum)) * jt_scale_scale_jmulleft)) /\\ exists fs_q_jt_jmulleftenumsecondscale. jt_scales_jmulleft = fs_q_jt_jmulleftenumsecondscale * S ((S (jt_h_jmulleftenum)) * jt_scale_scale_jmulleft) + (jt_e_jmulleftenum))))) -> (forall jt_index_jmulleftenumsame jt_left_jmulleftenumsame jt_right_jmulleftenumsame. (exists jt_gap_jmulleftenumsameindex. jt_gap_jmulleftenumsameindex+S (jt_index_jmulleftenumsame)=(k)) -> (((exists fs_h_jt_jmulleftenumsameleft. fs_h_jt_jmulleftenumsameleft + S (jt_left_jmulleftenumsame) = S ((S (jt_index_jmulleftenumsame)) * jt_c_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenumsameleft. jt_b_jmulleftenum = fs_q_jt_jmulleftenumsameleft * S ((S (jt_index_jmulleftenumsame)) * jt_c_jmulleftenum) + (jt_left_jmulleftenumsame))) -> (((exists fs_h_jt_jmulleftenumsameright. fs_h_jt_jmulleftenumsameright + S (jt_right_jmulleftenumsame) = S ((S (jt_index_jmulleftenumsame)) * jt_e_jmulleftenum)) /\\ exists fs_q_jt_jmulleftenumsameright. jt_d_jmulleftenum = fs_q_jt_jmulleftenumsameright * S ((S (jt_index_jmulleftenumsame)) * jt_e_jmulleftenum) + (jt_right_jmulleftenumsame))) -> jt_left_jmulleftenumsame=jt_right_jmulleftenumsame) -> jt_i_jmulleftenum=jt_h_jmulleftenum))))))))) -> (((~((k)=0)) /\\ (((~((b)=0)) /\\ (exists jt_codes_jmulright jt_code_scale_jmulright jt_scales_jmulright jt_scale_scale_jmulright. ((forall jt_i_jmulrightenum. (exists jt_gap_jmulrightenumsoundindex. jt_gap_jmulrightenumsoundindex+S (jt_i_jmulrightenum)=(v)) -> exists jt_b_jmulrightenum jt_c_jmulrightenum. ((((((exists fs_h_jt_jmulrightenumsoundcode. fs_h_jt_jmulrightenumsoundcode + S (jt_b_jmulrightenum) = S ((S (jt_i_jmulrightenum)) * jt_code_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumsoundcode. jt_codes_jmulright = fs_q_jt_jmulrightenumsoundcode * S ((S (jt_i_jmulrightenum)) * jt_code_scale_jmulright) + (jt_b_jmulrightenum))) /\\ (((exists fs_h_jt_jmulrightenumsoundscale. fs_h_jt_jmulrightenumsoundscale + S (jt_c_jmulrightenum) = S ((S (jt_i_jmulrightenum)) * jt_scale_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumsoundscale. jt_scales_jmulright = fs_q_jt_jmulrightenumsoundscale * S ((S (jt_i_jmulrightenum)) * jt_scale_scale_jmulright) + (jt_c_jmulrightenum))))) /\\ (((forall jt_index_jmulrightenumbound. (exists jt_gap_jmulrightenumboundindex. jt_gap_jmulrightenumboundindex+S (jt_index_jmulrightenumbound)=(k)) -> exists jt_value_jmulrightenumbound. ((((exists fs_h_jt_jmulrightenumboundat. fs_h_jt_jmulrightenumboundat + S (jt_value_jmulrightenumbound) = S ((S (jt_index_jmulrightenumbound)) * jt_c_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenumboundat. jt_b_jmulrightenum = fs_q_jt_jmulrightenumboundat * S ((S (jt_index_jmulrightenumbound)) * jt_c_jmulrightenum) + (jt_value_jmulrightenumbound))) /\\ (exists jt_gap_jmulrightenumboundvalue. jt_gap_jmulrightenumboundvalue+S (jt_value_jmulrightenumbound)=(b)))) /\\ (forall jt_divisor_jmulrightenumprimitive. (exists jt_factor_jmulrightenumprimitivemodulus. (b)=(jt_divisor_jmulrightenumprimitive)*jt_factor_jmulrightenumprimitivemodulus) -> (forall jt_index_jmulrightenumprimitivecoordinates jt_value_jmulrightenumprimitivecoordinates. (exists jt_gap_jmulrightenumprimitivecoordinatesindex. jt_gap_jmulrightenumprimitivecoordinatesindex+S (jt_index_jmulrightenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jmulrightenumprimitivecoordinatesat. fs_h_jt_jmulrightenumprimitivecoordinatesat + S (jt_value_jmulrightenumprimitivecoordinates) = S ((S (jt_index_jmulrightenumprimitivecoordinates)) * jt_c_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenumprimitivecoordinatesat. jt_b_jmulrightenum = fs_q_jt_jmulrightenumprimitivecoordinatesat * S ((S (jt_index_jmulrightenumprimitivecoordinates)) * jt_c_jmulrightenum) + (jt_value_jmulrightenumprimitivecoordinates))) -> (exists jt_factor_jmulrightenumprimitivecoordinatesdivides. (jt_value_jmulrightenumprimitivecoordinates)=(jt_divisor_jmulrightenumprimitive)*jt_factor_jmulrightenumprimitivecoordinatesdivides)) -> jt_divisor_jmulrightenumprimitive=1))))) /\\ (((forall jt_b_jmulrightenum jt_c_jmulrightenum. (forall jt_index_jmulrightenuminputbound. (exists jt_gap_jmulrightenuminputboundindex. jt_gap_jmulrightenuminputboundindex+S (jt_index_jmulrightenuminputbound)=(k)) -> exists jt_value_jmulrightenuminputbound. ((((exists fs_h_jt_jmulrightenuminputboundat. fs_h_jt_jmulrightenuminputboundat + S (jt_value_jmulrightenuminputbound) = S ((S (jt_index_jmulrightenuminputbound)) * jt_c_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenuminputboundat. jt_b_jmulrightenum = fs_q_jt_jmulrightenuminputboundat * S ((S (jt_index_jmulrightenuminputbound)) * jt_c_jmulrightenum) + (jt_value_jmulrightenuminputbound))) /\\ (exists jt_gap_jmulrightenuminputboundvalue. jt_gap_jmulrightenuminputboundvalue+S (jt_value_jmulrightenuminputbound)=(b)))) -> (forall jt_divisor_jmulrightenuminputprimitive. (exists jt_factor_jmulrightenuminputprimitivemodulus. (b)=(jt_divisor_jmulrightenuminputprimitive)*jt_factor_jmulrightenuminputprimitivemodulus) -> (forall jt_index_jmulrightenuminputprimitivecoordinates jt_value_jmulrightenuminputprimitivecoordinates. (exists jt_gap_jmulrightenuminputprimitivecoordinatesindex. jt_gap_jmulrightenuminputprimitivecoordinatesindex+S (jt_index_jmulrightenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jmulrightenuminputprimitivecoordinatesat. fs_h_jt_jmulrightenuminputprimitivecoordinatesat + S (jt_value_jmulrightenuminputprimitivecoordinates) = S ((S (jt_index_jmulrightenuminputprimitivecoordinates)) * jt_c_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenuminputprimitivecoordinatesat. jt_b_jmulrightenum = fs_q_jt_jmulrightenuminputprimitivecoordinatesat * S ((S (jt_index_jmulrightenuminputprimitivecoordinates)) * jt_c_jmulrightenum) + (jt_value_jmulrightenuminputprimitivecoordinates))) -> (exists jt_factor_jmulrightenuminputprimitivecoordinatesdivides. (jt_value_jmulrightenuminputprimitivecoordinates)=(jt_divisor_jmulrightenuminputprimitive)*jt_factor_jmulrightenuminputprimitivecoordinatesdivides)) -> jt_divisor_jmulrightenuminputprimitive=1) -> exists jt_i_jmulrightenum jt_d_jmulrightenum jt_e_jmulrightenum. ((exists jt_gap_jmulrightenumcompleteindex. jt_gap_jmulrightenumcompleteindex+S (jt_i_jmulrightenum)=(v)) /\\ (((((((exists fs_h_jt_jmulrightenumcompletecode. fs_h_jt_jmulrightenumcompletecode + S (jt_d_jmulrightenum) = S ((S (jt_i_jmulrightenum)) * jt_code_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumcompletecode. jt_codes_jmulright = fs_q_jt_jmulrightenumcompletecode * S ((S (jt_i_jmulrightenum)) * jt_code_scale_jmulright) + (jt_d_jmulrightenum))) /\\ (((exists fs_h_jt_jmulrightenumcompletescale. fs_h_jt_jmulrightenumcompletescale + S (jt_e_jmulrightenum) = S ((S (jt_i_jmulrightenum)) * jt_scale_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumcompletescale. jt_scales_jmulright = fs_q_jt_jmulrightenumcompletescale * S ((S (jt_i_jmulrightenum)) * jt_scale_scale_jmulright) + (jt_e_jmulrightenum))))) /\\ (forall jt_index_jmulrightenumrepresented jt_left_jmulrightenumrepresented jt_right_jmulrightenumrepresented. (exists jt_gap_jmulrightenumrepresentedindex. jt_gap_jmulrightenumrepresentedindex+S (jt_index_jmulrightenumrepresented)=(k)) -> (((exists fs_h_jt_jmulrightenumrepresentedleft. fs_h_jt_jmulrightenumrepresentedleft + S (jt_left_jmulrightenumrepresented) = S ((S (jt_index_jmulrightenumrepresented)) * jt_c_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenumrepresentedleft. jt_b_jmulrightenum = fs_q_jt_jmulrightenumrepresentedleft * S ((S (jt_index_jmulrightenumrepresented)) * jt_c_jmulrightenum) + (jt_left_jmulrightenumrepresented))) -> (((exists fs_h_jt_jmulrightenumrepresentedright. fs_h_jt_jmulrightenumrepresentedright + S (jt_right_jmulrightenumrepresented) = S ((S (jt_index_jmulrightenumrepresented)) * jt_e_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenumrepresentedright. jt_d_jmulrightenum = fs_q_jt_jmulrightenumrepresentedright * S ((S (jt_index_jmulrightenumrepresented)) * jt_e_jmulrightenum) + (jt_right_jmulrightenumrepresented))) -> jt_left_jmulrightenumrepresented=jt_right_jmulrightenumrepresented))))) /\\ (forall jt_i_jmulrightenum jt_h_jmulrightenum jt_b_jmulrightenum jt_c_jmulrightenum jt_d_jmulrightenum jt_e_jmulrightenum. (exists jt_gap_jmulrightenumfirstindex. jt_gap_jmulrightenumfirstindex+S (jt_i_jmulrightenum)=(v)) -> (exists jt_gap_jmulrightenumsecondindex. jt_gap_jmulrightenumsecondindex+S (jt_h_jmulrightenum)=(v)) -> (((((exists fs_h_jt_jmulrightenumfirstcode. fs_h_jt_jmulrightenumfirstcode + S (jt_b_jmulrightenum) = S ((S (jt_i_jmulrightenum)) * jt_code_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumfirstcode. jt_codes_jmulright = fs_q_jt_jmulrightenumfirstcode * S ((S (jt_i_jmulrightenum)) * jt_code_scale_jmulright) + (jt_b_jmulrightenum))) /\\ (((exists fs_h_jt_jmulrightenumfirstscale. fs_h_jt_jmulrightenumfirstscale + S (jt_c_jmulrightenum) = S ((S (jt_i_jmulrightenum)) * jt_scale_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumfirstscale. jt_scales_jmulright = fs_q_jt_jmulrightenumfirstscale * S ((S (jt_i_jmulrightenum)) * jt_scale_scale_jmulright) + (jt_c_jmulrightenum))))) -> (((((exists fs_h_jt_jmulrightenumsecondcode. fs_h_jt_jmulrightenumsecondcode + S (jt_d_jmulrightenum) = S ((S (jt_h_jmulrightenum)) * jt_code_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumsecondcode. jt_codes_jmulright = fs_q_jt_jmulrightenumsecondcode * S ((S (jt_h_jmulrightenum)) * jt_code_scale_jmulright) + (jt_d_jmulrightenum))) /\\ (((exists fs_h_jt_jmulrightenumsecondscale. fs_h_jt_jmulrightenumsecondscale + S (jt_e_jmulrightenum) = S ((S (jt_h_jmulrightenum)) * jt_scale_scale_jmulright)) /\\ exists fs_q_jt_jmulrightenumsecondscale. jt_scales_jmulright = fs_q_jt_jmulrightenumsecondscale * S ((S (jt_h_jmulrightenum)) * jt_scale_scale_jmulright) + (jt_e_jmulrightenum))))) -> (forall jt_index_jmulrightenumsame jt_left_jmulrightenumsame jt_right_jmulrightenumsame. (exists jt_gap_jmulrightenumsameindex. jt_gap_jmulrightenumsameindex+S (jt_index_jmulrightenumsame)=(k)) -> (((exists fs_h_jt_jmulrightenumsameleft. fs_h_jt_jmulrightenumsameleft + S (jt_left_jmulrightenumsame) = S ((S (jt_index_jmulrightenumsame)) * jt_c_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenumsameleft. jt_b_jmulrightenum = fs_q_jt_jmulrightenumsameleft * S ((S (jt_index_jmulrightenumsame)) * jt_c_jmulrightenum) + (jt_left_jmulrightenumsame))) -> (((exists fs_h_jt_jmulrightenumsameright. fs_h_jt_jmulrightenumsameright + S (jt_right_jmulrightenumsame) = S ((S (jt_index_jmulrightenumsame)) * jt_e_jmulrightenum)) /\\ exists fs_q_jt_jmulrightenumsameright. jt_d_jmulrightenum = fs_q_jt_jmulrightenumsameright * S ((S (jt_index_jmulrightenumsame)) * jt_e_jmulrightenum) + (jt_right_jmulrightenumsame))) -> jt_left_jmulrightenumsame=jt_right_jmulrightenumsame) -> jt_i_jmulrightenum=jt_h_jmulrightenum))))))))) -> (((~((k)=0)) /\\ (((~((a*b)=0)) /\\ (exists jt_codes_jmulresult jt_code_scale_jmulresult jt_scales_jmulresult jt_scale_scale_jmulresult. ((forall jt_i_jmulresultenum. (exists jt_gap_jmulresultenumsoundindex. jt_gap_jmulresultenumsoundindex+S (jt_i_jmulresultenum)=(u*v)) -> exists jt_b_jmulresultenum jt_c_jmulresultenum. ((((((exists fs_h_jt_jmulresultenumsoundcode. fs_h_jt_jmulresultenumsoundcode + S (jt_b_jmulresultenum) = S ((S (jt_i_jmulresultenum)) * jt_code_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumsoundcode. jt_codes_jmulresult = fs_q_jt_jmulresultenumsoundcode * S ((S (jt_i_jmulresultenum)) * jt_code_scale_jmulresult) + (jt_b_jmulresultenum))) /\\ (((exists fs_h_jt_jmulresultenumsoundscale. fs_h_jt_jmulresultenumsoundscale + S (jt_c_jmulresultenum) = S ((S (jt_i_jmulresultenum)) * jt_scale_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumsoundscale. jt_scales_jmulresult = fs_q_jt_jmulresultenumsoundscale * S ((S (jt_i_jmulresultenum)) * jt_scale_scale_jmulresult) + (jt_c_jmulresultenum))))) /\\ (((forall jt_index_jmulresultenumbound. (exists jt_gap_jmulresultenumboundindex. jt_gap_jmulresultenumboundindex+S (jt_index_jmulresultenumbound)=(k)) -> exists jt_value_jmulresultenumbound. ((((exists fs_h_jt_jmulresultenumboundat. fs_h_jt_jmulresultenumboundat + S (jt_value_jmulresultenumbound) = S ((S (jt_index_jmulresultenumbound)) * jt_c_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenumboundat. jt_b_jmulresultenum = fs_q_jt_jmulresultenumboundat * S ((S (jt_index_jmulresultenumbound)) * jt_c_jmulresultenum) + (jt_value_jmulresultenumbound))) /\\ (exists jt_gap_jmulresultenumboundvalue. jt_gap_jmulresultenumboundvalue+S (jt_value_jmulresultenumbound)=(a*b)))) /\\ (forall jt_divisor_jmulresultenumprimitive. (exists jt_factor_jmulresultenumprimitivemodulus. (a*b)=(jt_divisor_jmulresultenumprimitive)*jt_factor_jmulresultenumprimitivemodulus) -> (forall jt_index_jmulresultenumprimitivecoordinates jt_value_jmulresultenumprimitivecoordinates. (exists jt_gap_jmulresultenumprimitivecoordinatesindex. jt_gap_jmulresultenumprimitivecoordinatesindex+S (jt_index_jmulresultenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jmulresultenumprimitivecoordinatesat. fs_h_jt_jmulresultenumprimitivecoordinatesat + S (jt_value_jmulresultenumprimitivecoordinates) = S ((S (jt_index_jmulresultenumprimitivecoordinates)) * jt_c_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenumprimitivecoordinatesat. jt_b_jmulresultenum = fs_q_jt_jmulresultenumprimitivecoordinatesat * S ((S (jt_index_jmulresultenumprimitivecoordinates)) * jt_c_jmulresultenum) + (jt_value_jmulresultenumprimitivecoordinates))) -> (exists jt_factor_jmulresultenumprimitivecoordinatesdivides. (jt_value_jmulresultenumprimitivecoordinates)=(jt_divisor_jmulresultenumprimitive)*jt_factor_jmulresultenumprimitivecoordinatesdivides)) -> jt_divisor_jmulresultenumprimitive=1))))) /\\ (((forall jt_b_jmulresultenum jt_c_jmulresultenum. (forall jt_index_jmulresultenuminputbound. (exists jt_gap_jmulresultenuminputboundindex. jt_gap_jmulresultenuminputboundindex+S (jt_index_jmulresultenuminputbound)=(k)) -> exists jt_value_jmulresultenuminputbound. ((((exists fs_h_jt_jmulresultenuminputboundat. fs_h_jt_jmulresultenuminputboundat + S (jt_value_jmulresultenuminputbound) = S ((S (jt_index_jmulresultenuminputbound)) * jt_c_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenuminputboundat. jt_b_jmulresultenum = fs_q_jt_jmulresultenuminputboundat * S ((S (jt_index_jmulresultenuminputbound)) * jt_c_jmulresultenum) + (jt_value_jmulresultenuminputbound))) /\\ (exists jt_gap_jmulresultenuminputboundvalue. jt_gap_jmulresultenuminputboundvalue+S (jt_value_jmulresultenuminputbound)=(a*b)))) -> (forall jt_divisor_jmulresultenuminputprimitive. (exists jt_factor_jmulresultenuminputprimitivemodulus. (a*b)=(jt_divisor_jmulresultenuminputprimitive)*jt_factor_jmulresultenuminputprimitivemodulus) -> (forall jt_index_jmulresultenuminputprimitivecoordinates jt_value_jmulresultenuminputprimitivecoordinates. (exists jt_gap_jmulresultenuminputprimitivecoordinatesindex. jt_gap_jmulresultenuminputprimitivecoordinatesindex+S (jt_index_jmulresultenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jmulresultenuminputprimitivecoordinatesat. fs_h_jt_jmulresultenuminputprimitivecoordinatesat + S (jt_value_jmulresultenuminputprimitivecoordinates) = S ((S (jt_index_jmulresultenuminputprimitivecoordinates)) * jt_c_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenuminputprimitivecoordinatesat. jt_b_jmulresultenum = fs_q_jt_jmulresultenuminputprimitivecoordinatesat * S ((S (jt_index_jmulresultenuminputprimitivecoordinates)) * jt_c_jmulresultenum) + (jt_value_jmulresultenuminputprimitivecoordinates))) -> (exists jt_factor_jmulresultenuminputprimitivecoordinatesdivides. (jt_value_jmulresultenuminputprimitivecoordinates)=(jt_divisor_jmulresultenuminputprimitive)*jt_factor_jmulresultenuminputprimitivecoordinatesdivides)) -> jt_divisor_jmulresultenuminputprimitive=1) -> exists jt_i_jmulresultenum jt_d_jmulresultenum jt_e_jmulresultenum. ((exists jt_gap_jmulresultenumcompleteindex. jt_gap_jmulresultenumcompleteindex+S (jt_i_jmulresultenum)=(u*v)) /\\ (((((((exists fs_h_jt_jmulresultenumcompletecode. fs_h_jt_jmulresultenumcompletecode + S (jt_d_jmulresultenum) = S ((S (jt_i_jmulresultenum)) * jt_code_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumcompletecode. jt_codes_jmulresult = fs_q_jt_jmulresultenumcompletecode * S ((S (jt_i_jmulresultenum)) * jt_code_scale_jmulresult) + (jt_d_jmulresultenum))) /\\ (((exists fs_h_jt_jmulresultenumcompletescale. fs_h_jt_jmulresultenumcompletescale + S (jt_e_jmulresultenum) = S ((S (jt_i_jmulresultenum)) * jt_scale_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumcompletescale. jt_scales_jmulresult = fs_q_jt_jmulresultenumcompletescale * S ((S (jt_i_jmulresultenum)) * jt_scale_scale_jmulresult) + (jt_e_jmulresultenum))))) /\\ (forall jt_index_jmulresultenumrepresented jt_left_jmulresultenumrepresented jt_right_jmulresultenumrepresented. (exists jt_gap_jmulresultenumrepresentedindex. jt_gap_jmulresultenumrepresentedindex+S (jt_index_jmulresultenumrepresented)=(k)) -> (((exists fs_h_jt_jmulresultenumrepresentedleft. fs_h_jt_jmulresultenumrepresentedleft + S (jt_left_jmulresultenumrepresented) = S ((S (jt_index_jmulresultenumrepresented)) * jt_c_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenumrepresentedleft. jt_b_jmulresultenum = fs_q_jt_jmulresultenumrepresentedleft * S ((S (jt_index_jmulresultenumrepresented)) * jt_c_jmulresultenum) + (jt_left_jmulresultenumrepresented))) -> (((exists fs_h_jt_jmulresultenumrepresentedright. fs_h_jt_jmulresultenumrepresentedright + S (jt_right_jmulresultenumrepresented) = S ((S (jt_index_jmulresultenumrepresented)) * jt_e_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenumrepresentedright. jt_d_jmulresultenum = fs_q_jt_jmulresultenumrepresentedright * S ((S (jt_index_jmulresultenumrepresented)) * jt_e_jmulresultenum) + (jt_right_jmulresultenumrepresented))) -> jt_left_jmulresultenumrepresented=jt_right_jmulresultenumrepresented))))) /\\ (forall jt_i_jmulresultenum jt_h_jmulresultenum jt_b_jmulresultenum jt_c_jmulresultenum jt_d_jmulresultenum jt_e_jmulresultenum. (exists jt_gap_jmulresultenumfirstindex. jt_gap_jmulresultenumfirstindex+S (jt_i_jmulresultenum)=(u*v)) -> (exists jt_gap_jmulresultenumsecondindex. jt_gap_jmulresultenumsecondindex+S (jt_h_jmulresultenum)=(u*v)) -> (((((exists fs_h_jt_jmulresultenumfirstcode. fs_h_jt_jmulresultenumfirstcode + S (jt_b_jmulresultenum) = S ((S (jt_i_jmulresultenum)) * jt_code_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumfirstcode. jt_codes_jmulresult = fs_q_jt_jmulresultenumfirstcode * S ((S (jt_i_jmulresultenum)) * jt_code_scale_jmulresult) + (jt_b_jmulresultenum))) /\\ (((exists fs_h_jt_jmulresultenumfirstscale. fs_h_jt_jmulresultenumfirstscale + S (jt_c_jmulresultenum) = S ((S (jt_i_jmulresultenum)) * jt_scale_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumfirstscale. jt_scales_jmulresult = fs_q_jt_jmulresultenumfirstscale * S ((S (jt_i_jmulresultenum)) * jt_scale_scale_jmulresult) + (jt_c_jmulresultenum))))) -> (((((exists fs_h_jt_jmulresultenumsecondcode. fs_h_jt_jmulresultenumsecondcode + S (jt_d_jmulresultenum) = S ((S (jt_h_jmulresultenum)) * jt_code_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumsecondcode. jt_codes_jmulresult = fs_q_jt_jmulresultenumsecondcode * S ((S (jt_h_jmulresultenum)) * jt_code_scale_jmulresult) + (jt_d_jmulresultenum))) /\\ (((exists fs_h_jt_jmulresultenumsecondscale. fs_h_jt_jmulresultenumsecondscale + S (jt_e_jmulresultenum) = S ((S (jt_h_jmulresultenum)) * jt_scale_scale_jmulresult)) /\\ exists fs_q_jt_jmulresultenumsecondscale. jt_scales_jmulresult = fs_q_jt_jmulresultenumsecondscale * S ((S (jt_h_jmulresultenum)) * jt_scale_scale_jmulresult) + (jt_e_jmulresultenum))))) -> (forall jt_index_jmulresultenumsame jt_left_jmulresultenumsame jt_right_jmulresultenumsame. (exists jt_gap_jmulresultenumsameindex. jt_gap_jmulresultenumsameindex+S (jt_index_jmulresultenumsame)=(k)) -> (((exists fs_h_jt_jmulresultenumsameleft. fs_h_jt_jmulresultenumsameleft + S (jt_left_jmulresultenumsame) = S ((S (jt_index_jmulresultenumsame)) * jt_c_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenumsameleft. jt_b_jmulresultenum = fs_q_jt_jmulresultenumsameleft * S ((S (jt_index_jmulresultenumsame)) * jt_c_jmulresultenum) + (jt_left_jmulresultenumsame))) -> (((exists fs_h_jt_jmulresultenumsameright. fs_h_jt_jmulresultenumsameright + S (jt_right_jmulresultenumsame) = S ((S (jt_index_jmulresultenumsame)) * jt_e_jmulresultenum)) /\\ exists fs_q_jt_jmulresultenumsameright. jt_d_jmulresultenum = fs_q_jt_jmulresultenumsameright * S ((S (jt_index_jmulresultenumsame)) * jt_e_jmulresultenum) + (jt_right_jmulresultenumsame))) -> jt_left_jmulresultenumsame=jt_right_jmulresultenumsame) -> jt_i_jmulresultenum=jt_h_jmulresultenum)))))))))", "statement_sha256": "f2c154c6ab76ff330735cca1784280f2d32b48777e27b76c27939e9061f06210", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Actual finite tuple-count Jordan values multiply at coprime moduli." }, { "admission_dependencies": [ "jordan_totient_exists", "jordan_totient_coprime_product" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 26, "body_proof_nodes": 46, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro a", "intro b", "intro hk", "intro ha", "intro hb", "intro hcop", "have hu : ∃ u. JordanTotient(k,a,u)", "specialize jordan_totient_exists (k)", "specialize jordan_totient_exists (a)", "apply jordan_totient_exists", "exact hk", "exact ha", "cases hu", "have hv : ∃ v. JordanTotient(k,b,v)", "specialize jordan_totient_exists (k)", "specialize jordan_totient_exists (b)", "apply jordan_totient_exists", "exact hk", "exact hb", "cases hv", "exists x", "exists x1", "exists x*x1", "split", "exact hu_witness", "split", "exact hv_witness", "split", "specialize jordan_totient_coprime_product (k)", "specialize jordan_totient_coprime_product (a)", "specialize jordan_totient_coprime_product (b)", "specialize jordan_totient_coprime_product (x)", "specialize jordan_totient_coprime_product (x1)", "apply jordan_totient_coprime_product", "exact hcop", "exact hu_witness", "exact hv_witness", "refl" ], "defined_statement": "∀ k. ∀ a. ∀ b. ¬k = 0 → ¬a = 0 → ¬b = 0 → Coprime(a,b) → ∃ x. ∃ y. ∃ z. JordanTotient(k,a,x) ∧ (JordanTotient(k,b,y) ∧ (JordanTotient(k,a · b,z) ∧ z = x · y))", "defined_statement_sha256": "6e95c9502537a6caf16aa50ba95cfa92680aec29140c33569dcdf1430148e36b", "definition_uses": { "ND0375": 5, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "5b4bbb844653f4befd6755a95510eaf54b8d8df88897b82fe25a530af55e981c", "free_names": [], "script_definition_uses": { "ND0375": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro hk" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "have hu : " }, { "kind": "text", "text": "∃ u. " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a,u)" } ], [ { "kind": "text", "text": "specialize jordan_totient_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_exists (a)" } ], [ { "kind": "text", "text": "apply jordan_totient_exists" } ], [ { "kind": "text", "text": "exact hk" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "cases hu" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ v. " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,b,v)" } ], [ { "kind": "text", "text": "specialize jordan_totient_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_exists (b)" } ], [ { "kind": "text", "text": "apply jordan_totient_exists" } ], [ { "kind": "text", "text": "exact hk" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "exists x*x1" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hu_witness" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (a)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (b)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (x)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (x1)" } ], [ { "kind": "text", "text": "apply jordan_totient_coprime_product" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact hu_witness" } ], [ { "kind": "text", "text": "exact hv_witness" } ], [ { "kind": "text", "text": "refl" } ] ], "statement_definition_uses": { "ND0375": 3, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ a. ∀ b. ¬k = 0 → ¬a = 0 → ¬b = 0 → " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(a,b)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∃ z. " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a,x)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,b,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a · b,z)" }, { "kind": "text", "text": " ∧ z = x · y))" } ] }, "dependencies": [ "jordan_totient_exists", "jordan_totient_coprime_product" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_candidate_theorems", "first_admitted_version": "v35", "id": "JT004B", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_multiplicativity_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 336, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro a", "intro b", "intro hk", "intro ha", "intro hb", "intro hcop", "have hu : exists u. ((~((k)=0)) /\\ (((~((a)=0)) /\\ (exists jt_codes_endpointleft jt_code_scale_endpointleft jt_scales_endpointleft jt_scale_scale_endpointleft. ((forall jt_i_endpointleftenum. (exists jt_gap_endpointleftenumsoundindex. jt_gap_endpointleftenumsoundindex+S (jt_i_endpointleftenum)=(u)) -> exists jt_b_endpointleftenum jt_c_endpointleftenum. ((((((exists fs_h_jt_endpointleftenumsoundcode. fs_h_jt_endpointleftenumsoundcode + S (jt_b_endpointleftenum) = S ((S (jt_i_endpointleftenum)) * jt_code_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumsoundcode. jt_codes_endpointleft = fs_q_jt_endpointleftenumsoundcode * S ((S (jt_i_endpointleftenum)) * jt_code_scale_endpointleft) + (jt_b_endpointleftenum))) /\\ (((exists fs_h_jt_endpointleftenumsoundscale. fs_h_jt_endpointleftenumsoundscale + S (jt_c_endpointleftenum) = S ((S (jt_i_endpointleftenum)) * jt_scale_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumsoundscale. jt_scales_endpointleft = fs_q_jt_endpointleftenumsoundscale * S ((S (jt_i_endpointleftenum)) * jt_scale_scale_endpointleft) + (jt_c_endpointleftenum))))) /\\ (((forall jt_index_endpointleftenumbound. (exists jt_gap_endpointleftenumboundindex. jt_gap_endpointleftenumboundindex+S (jt_index_endpointleftenumbound)=(k)) -> exists jt_value_endpointleftenumbound. ((((exists fs_h_jt_endpointleftenumboundat. fs_h_jt_endpointleftenumboundat + S (jt_value_endpointleftenumbound) = S ((S (jt_index_endpointleftenumbound)) * jt_c_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenumboundat. jt_b_endpointleftenum = fs_q_jt_endpointleftenumboundat * S ((S (jt_index_endpointleftenumbound)) * jt_c_endpointleftenum) + (jt_value_endpointleftenumbound))) /\\ (exists jt_gap_endpointleftenumboundvalue. jt_gap_endpointleftenumboundvalue+S (jt_value_endpointleftenumbound)=(a)))) /\\ (forall jt_divisor_endpointleftenumprimitive. (exists jt_factor_endpointleftenumprimitivemodulus. (a)=(jt_divisor_endpointleftenumprimitive)*jt_factor_endpointleftenumprimitivemodulus) -> (forall jt_index_endpointleftenumprimitivecoordinates jt_value_endpointleftenumprimitivecoordinates. (exists jt_gap_endpointleftenumprimitivecoordinatesindex. jt_gap_endpointleftenumprimitivecoordinatesindex+S (jt_index_endpointleftenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_endpointleftenumprimitivecoordinatesat. fs_h_jt_endpointleftenumprimitivecoordinatesat + S (jt_value_endpointleftenumprimitivecoordinates) = S ((S (jt_index_endpointleftenumprimitivecoordinates)) * jt_c_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenumprimitivecoordinatesat. jt_b_endpointleftenum = fs_q_jt_endpointleftenumprimitivecoordinatesat * S ((S (jt_index_endpointleftenumprimitivecoordinates)) * jt_c_endpointleftenum) + (jt_value_endpointleftenumprimitivecoordinates))) -> (exists jt_factor_endpointleftenumprimitivecoordinatesdivides. (jt_value_endpointleftenumprimitivecoordinates)=(jt_divisor_endpointleftenumprimitive)*jt_factor_endpointleftenumprimitivecoordinatesdivides)) -> jt_divisor_endpointleftenumprimitive=1))))) /\\ (((forall jt_b_endpointleftenum jt_c_endpointleftenum. (forall jt_index_endpointleftenuminputbound. (exists jt_gap_endpointleftenuminputboundindex. jt_gap_endpointleftenuminputboundindex+S (jt_index_endpointleftenuminputbound)=(k)) -> exists jt_value_endpointleftenuminputbound. ((((exists fs_h_jt_endpointleftenuminputboundat. fs_h_jt_endpointleftenuminputboundat + S (jt_value_endpointleftenuminputbound) = S ((S (jt_index_endpointleftenuminputbound)) * jt_c_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenuminputboundat. jt_b_endpointleftenum = fs_q_jt_endpointleftenuminputboundat * S ((S (jt_index_endpointleftenuminputbound)) * jt_c_endpointleftenum) + (jt_value_endpointleftenuminputbound))) /\\ (exists jt_gap_endpointleftenuminputboundvalue. jt_gap_endpointleftenuminputboundvalue+S (jt_value_endpointleftenuminputbound)=(a)))) -> (forall jt_divisor_endpointleftenuminputprimitive. (exists jt_factor_endpointleftenuminputprimitivemodulus. (a)=(jt_divisor_endpointleftenuminputprimitive)*jt_factor_endpointleftenuminputprimitivemodulus) -> (forall jt_index_endpointleftenuminputprimitivecoordinates jt_value_endpointleftenuminputprimitivecoordinates. (exists jt_gap_endpointleftenuminputprimitivecoordinatesindex. jt_gap_endpointleftenuminputprimitivecoordinatesindex+S (jt_index_endpointleftenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_endpointleftenuminputprimitivecoordinatesat. fs_h_jt_endpointleftenuminputprimitivecoordinatesat + S (jt_value_endpointleftenuminputprimitivecoordinates) = S ((S (jt_index_endpointleftenuminputprimitivecoordinates)) * jt_c_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenuminputprimitivecoordinatesat. jt_b_endpointleftenum = fs_q_jt_endpointleftenuminputprimitivecoordinatesat * S ((S (jt_index_endpointleftenuminputprimitivecoordinates)) * jt_c_endpointleftenum) + (jt_value_endpointleftenuminputprimitivecoordinates))) -> (exists jt_factor_endpointleftenuminputprimitivecoordinatesdivides. (jt_value_endpointleftenuminputprimitivecoordinates)=(jt_divisor_endpointleftenuminputprimitive)*jt_factor_endpointleftenuminputprimitivecoordinatesdivides)) -> jt_divisor_endpointleftenuminputprimitive=1) -> exists jt_i_endpointleftenum jt_d_endpointleftenum jt_e_endpointleftenum. ((exists jt_gap_endpointleftenumcompleteindex. jt_gap_endpointleftenumcompleteindex+S (jt_i_endpointleftenum)=(u)) /\\ (((((((exists fs_h_jt_endpointleftenumcompletecode. fs_h_jt_endpointleftenumcompletecode + S (jt_d_endpointleftenum) = S ((S (jt_i_endpointleftenum)) * jt_code_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumcompletecode. jt_codes_endpointleft = fs_q_jt_endpointleftenumcompletecode * S ((S (jt_i_endpointleftenum)) * jt_code_scale_endpointleft) + (jt_d_endpointleftenum))) /\\ (((exists fs_h_jt_endpointleftenumcompletescale. fs_h_jt_endpointleftenumcompletescale + S (jt_e_endpointleftenum) = S ((S (jt_i_endpointleftenum)) * jt_scale_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumcompletescale. jt_scales_endpointleft = fs_q_jt_endpointleftenumcompletescale * S ((S (jt_i_endpointleftenum)) * jt_scale_scale_endpointleft) + (jt_e_endpointleftenum))))) /\\ (forall jt_index_endpointleftenumrepresented jt_left_endpointleftenumrepresented jt_right_endpointleftenumrepresented. (exists jt_gap_endpointleftenumrepresentedindex. jt_gap_endpointleftenumrepresentedindex+S (jt_index_endpointleftenumrepresented)=(k)) -> (((exists fs_h_jt_endpointleftenumrepresentedleft. fs_h_jt_endpointleftenumrepresentedleft + S (jt_left_endpointleftenumrepresented) = S ((S (jt_index_endpointleftenumrepresented)) * jt_c_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenumrepresentedleft. jt_b_endpointleftenum = fs_q_jt_endpointleftenumrepresentedleft * S ((S (jt_index_endpointleftenumrepresented)) * jt_c_endpointleftenum) + (jt_left_endpointleftenumrepresented))) -> (((exists fs_h_jt_endpointleftenumrepresentedright. fs_h_jt_endpointleftenumrepresentedright + S (jt_right_endpointleftenumrepresented) = S ((S (jt_index_endpointleftenumrepresented)) * jt_e_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenumrepresentedright. jt_d_endpointleftenum = fs_q_jt_endpointleftenumrepresentedright * S ((S (jt_index_endpointleftenumrepresented)) * jt_e_endpointleftenum) + (jt_right_endpointleftenumrepresented))) -> jt_left_endpointleftenumrepresented=jt_right_endpointleftenumrepresented))))) /\\ (forall jt_i_endpointleftenum jt_h_endpointleftenum jt_b_endpointleftenum jt_c_endpointleftenum jt_d_endpointleftenum jt_e_endpointleftenum. (exists jt_gap_endpointleftenumfirstindex. jt_gap_endpointleftenumfirstindex+S (jt_i_endpointleftenum)=(u)) -> (exists jt_gap_endpointleftenumsecondindex. jt_gap_endpointleftenumsecondindex+S (jt_h_endpointleftenum)=(u)) -> (((((exists fs_h_jt_endpointleftenumfirstcode. fs_h_jt_endpointleftenumfirstcode + S (jt_b_endpointleftenum) = S ((S (jt_i_endpointleftenum)) * jt_code_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumfirstcode. jt_codes_endpointleft = fs_q_jt_endpointleftenumfirstcode * S ((S (jt_i_endpointleftenum)) * jt_code_scale_endpointleft) + (jt_b_endpointleftenum))) /\\ (((exists fs_h_jt_endpointleftenumfirstscale. fs_h_jt_endpointleftenumfirstscale + S (jt_c_endpointleftenum) = S ((S (jt_i_endpointleftenum)) * jt_scale_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumfirstscale. jt_scales_endpointleft = fs_q_jt_endpointleftenumfirstscale * S ((S (jt_i_endpointleftenum)) * jt_scale_scale_endpointleft) + (jt_c_endpointleftenum))))) -> (((((exists fs_h_jt_endpointleftenumsecondcode. fs_h_jt_endpointleftenumsecondcode + S (jt_d_endpointleftenum) = S ((S (jt_h_endpointleftenum)) * jt_code_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumsecondcode. jt_codes_endpointleft = fs_q_jt_endpointleftenumsecondcode * S ((S (jt_h_endpointleftenum)) * jt_code_scale_endpointleft) + (jt_d_endpointleftenum))) /\\ (((exists fs_h_jt_endpointleftenumsecondscale. fs_h_jt_endpointleftenumsecondscale + S (jt_e_endpointleftenum) = S ((S (jt_h_endpointleftenum)) * jt_scale_scale_endpointleft)) /\\ exists fs_q_jt_endpointleftenumsecondscale. jt_scales_endpointleft = fs_q_jt_endpointleftenumsecondscale * S ((S (jt_h_endpointleftenum)) * jt_scale_scale_endpointleft) + (jt_e_endpointleftenum))))) -> (forall jt_index_endpointleftenumsame jt_left_endpointleftenumsame jt_right_endpointleftenumsame. (exists jt_gap_endpointleftenumsameindex. jt_gap_endpointleftenumsameindex+S (jt_index_endpointleftenumsame)=(k)) -> (((exists fs_h_jt_endpointleftenumsameleft. fs_h_jt_endpointleftenumsameleft + S (jt_left_endpointleftenumsame) = S ((S (jt_index_endpointleftenumsame)) * jt_c_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenumsameleft. jt_b_endpointleftenum = fs_q_jt_endpointleftenumsameleft * S ((S (jt_index_endpointleftenumsame)) * jt_c_endpointleftenum) + (jt_left_endpointleftenumsame))) -> (((exists fs_h_jt_endpointleftenumsameright. fs_h_jt_endpointleftenumsameright + S (jt_right_endpointleftenumsame) = S ((S (jt_index_endpointleftenumsame)) * jt_e_endpointleftenum)) /\\ exists fs_q_jt_endpointleftenumsameright. jt_d_endpointleftenum = fs_q_jt_endpointleftenumsameright * S ((S (jt_index_endpointleftenumsame)) * jt_e_endpointleftenum) + (jt_right_endpointleftenumsame))) -> jt_left_endpointleftenumsame=jt_right_endpointleftenumsame) -> jt_i_endpointleftenum=jt_h_endpointleftenum))))))))", "specialize jordan_totient_exists (k)", "specialize jordan_totient_exists (a)", "apply jordan_totient_exists", "exact hk", "exact ha", "cases hu", "have hv : exists v. ((~((k)=0)) /\\ (((~((b)=0)) /\\ (exists jt_codes_endpointright jt_code_scale_endpointright jt_scales_endpointright jt_scale_scale_endpointright. ((forall jt_i_endpointrightenum. (exists jt_gap_endpointrightenumsoundindex. jt_gap_endpointrightenumsoundindex+S (jt_i_endpointrightenum)=(v)) -> exists jt_b_endpointrightenum jt_c_endpointrightenum. ((((((exists fs_h_jt_endpointrightenumsoundcode. fs_h_jt_endpointrightenumsoundcode + S (jt_b_endpointrightenum) = S ((S (jt_i_endpointrightenum)) * jt_code_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumsoundcode. jt_codes_endpointright = fs_q_jt_endpointrightenumsoundcode * S ((S (jt_i_endpointrightenum)) * jt_code_scale_endpointright) + (jt_b_endpointrightenum))) /\\ (((exists fs_h_jt_endpointrightenumsoundscale. fs_h_jt_endpointrightenumsoundscale + S (jt_c_endpointrightenum) = S ((S (jt_i_endpointrightenum)) * jt_scale_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumsoundscale. jt_scales_endpointright = fs_q_jt_endpointrightenumsoundscale * S ((S (jt_i_endpointrightenum)) * jt_scale_scale_endpointright) + (jt_c_endpointrightenum))))) /\\ (((forall jt_index_endpointrightenumbound. (exists jt_gap_endpointrightenumboundindex. jt_gap_endpointrightenumboundindex+S (jt_index_endpointrightenumbound)=(k)) -> exists jt_value_endpointrightenumbound. ((((exists fs_h_jt_endpointrightenumboundat. fs_h_jt_endpointrightenumboundat + S (jt_value_endpointrightenumbound) = S ((S (jt_index_endpointrightenumbound)) * jt_c_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenumboundat. jt_b_endpointrightenum = fs_q_jt_endpointrightenumboundat * S ((S (jt_index_endpointrightenumbound)) * jt_c_endpointrightenum) + (jt_value_endpointrightenumbound))) /\\ (exists jt_gap_endpointrightenumboundvalue. jt_gap_endpointrightenumboundvalue+S (jt_value_endpointrightenumbound)=(b)))) /\\ (forall jt_divisor_endpointrightenumprimitive. (exists jt_factor_endpointrightenumprimitivemodulus. (b)=(jt_divisor_endpointrightenumprimitive)*jt_factor_endpointrightenumprimitivemodulus) -> (forall jt_index_endpointrightenumprimitivecoordinates jt_value_endpointrightenumprimitivecoordinates. (exists jt_gap_endpointrightenumprimitivecoordinatesindex. jt_gap_endpointrightenumprimitivecoordinatesindex+S (jt_index_endpointrightenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_endpointrightenumprimitivecoordinatesat. fs_h_jt_endpointrightenumprimitivecoordinatesat + S (jt_value_endpointrightenumprimitivecoordinates) = S ((S (jt_index_endpointrightenumprimitivecoordinates)) * jt_c_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenumprimitivecoordinatesat. jt_b_endpointrightenum = fs_q_jt_endpointrightenumprimitivecoordinatesat * S ((S (jt_index_endpointrightenumprimitivecoordinates)) * jt_c_endpointrightenum) + (jt_value_endpointrightenumprimitivecoordinates))) -> (exists jt_factor_endpointrightenumprimitivecoordinatesdivides. (jt_value_endpointrightenumprimitivecoordinates)=(jt_divisor_endpointrightenumprimitive)*jt_factor_endpointrightenumprimitivecoordinatesdivides)) -> jt_divisor_endpointrightenumprimitive=1))))) /\\ (((forall jt_b_endpointrightenum jt_c_endpointrightenum. (forall jt_index_endpointrightenuminputbound. (exists jt_gap_endpointrightenuminputboundindex. jt_gap_endpointrightenuminputboundindex+S (jt_index_endpointrightenuminputbound)=(k)) -> exists jt_value_endpointrightenuminputbound. ((((exists fs_h_jt_endpointrightenuminputboundat. fs_h_jt_endpointrightenuminputboundat + S (jt_value_endpointrightenuminputbound) = S ((S (jt_index_endpointrightenuminputbound)) * jt_c_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenuminputboundat. jt_b_endpointrightenum = fs_q_jt_endpointrightenuminputboundat * S ((S (jt_index_endpointrightenuminputbound)) * jt_c_endpointrightenum) + (jt_value_endpointrightenuminputbound))) /\\ (exists jt_gap_endpointrightenuminputboundvalue. jt_gap_endpointrightenuminputboundvalue+S (jt_value_endpointrightenuminputbound)=(b)))) -> (forall jt_divisor_endpointrightenuminputprimitive. (exists jt_factor_endpointrightenuminputprimitivemodulus. (b)=(jt_divisor_endpointrightenuminputprimitive)*jt_factor_endpointrightenuminputprimitivemodulus) -> (forall jt_index_endpointrightenuminputprimitivecoordinates jt_value_endpointrightenuminputprimitivecoordinates. (exists jt_gap_endpointrightenuminputprimitivecoordinatesindex. jt_gap_endpointrightenuminputprimitivecoordinatesindex+S (jt_index_endpointrightenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_endpointrightenuminputprimitivecoordinatesat. fs_h_jt_endpointrightenuminputprimitivecoordinatesat + S (jt_value_endpointrightenuminputprimitivecoordinates) = S ((S (jt_index_endpointrightenuminputprimitivecoordinates)) * jt_c_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenuminputprimitivecoordinatesat. jt_b_endpointrightenum = fs_q_jt_endpointrightenuminputprimitivecoordinatesat * S ((S (jt_index_endpointrightenuminputprimitivecoordinates)) * jt_c_endpointrightenum) + (jt_value_endpointrightenuminputprimitivecoordinates))) -> (exists jt_factor_endpointrightenuminputprimitivecoordinatesdivides. (jt_value_endpointrightenuminputprimitivecoordinates)=(jt_divisor_endpointrightenuminputprimitive)*jt_factor_endpointrightenuminputprimitivecoordinatesdivides)) -> jt_divisor_endpointrightenuminputprimitive=1) -> exists jt_i_endpointrightenum jt_d_endpointrightenum jt_e_endpointrightenum. ((exists jt_gap_endpointrightenumcompleteindex. jt_gap_endpointrightenumcompleteindex+S (jt_i_endpointrightenum)=(v)) /\\ (((((((exists fs_h_jt_endpointrightenumcompletecode. fs_h_jt_endpointrightenumcompletecode + S (jt_d_endpointrightenum) = S ((S (jt_i_endpointrightenum)) * jt_code_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumcompletecode. jt_codes_endpointright = fs_q_jt_endpointrightenumcompletecode * S ((S (jt_i_endpointrightenum)) * jt_code_scale_endpointright) + (jt_d_endpointrightenum))) /\\ (((exists fs_h_jt_endpointrightenumcompletescale. fs_h_jt_endpointrightenumcompletescale + S (jt_e_endpointrightenum) = S ((S (jt_i_endpointrightenum)) * jt_scale_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumcompletescale. jt_scales_endpointright = fs_q_jt_endpointrightenumcompletescale * S ((S (jt_i_endpointrightenum)) * jt_scale_scale_endpointright) + (jt_e_endpointrightenum))))) /\\ (forall jt_index_endpointrightenumrepresented jt_left_endpointrightenumrepresented jt_right_endpointrightenumrepresented. (exists jt_gap_endpointrightenumrepresentedindex. jt_gap_endpointrightenumrepresentedindex+S (jt_index_endpointrightenumrepresented)=(k)) -> (((exists fs_h_jt_endpointrightenumrepresentedleft. fs_h_jt_endpointrightenumrepresentedleft + S (jt_left_endpointrightenumrepresented) = S ((S (jt_index_endpointrightenumrepresented)) * jt_c_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenumrepresentedleft. jt_b_endpointrightenum = fs_q_jt_endpointrightenumrepresentedleft * S ((S (jt_index_endpointrightenumrepresented)) * jt_c_endpointrightenum) + (jt_left_endpointrightenumrepresented))) -> (((exists fs_h_jt_endpointrightenumrepresentedright. fs_h_jt_endpointrightenumrepresentedright + S (jt_right_endpointrightenumrepresented) = S ((S (jt_index_endpointrightenumrepresented)) * jt_e_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenumrepresentedright. jt_d_endpointrightenum = fs_q_jt_endpointrightenumrepresentedright * S ((S (jt_index_endpointrightenumrepresented)) * jt_e_endpointrightenum) + (jt_right_endpointrightenumrepresented))) -> jt_left_endpointrightenumrepresented=jt_right_endpointrightenumrepresented))))) /\\ (forall jt_i_endpointrightenum jt_h_endpointrightenum jt_b_endpointrightenum jt_c_endpointrightenum jt_d_endpointrightenum jt_e_endpointrightenum. (exists jt_gap_endpointrightenumfirstindex. jt_gap_endpointrightenumfirstindex+S (jt_i_endpointrightenum)=(v)) -> (exists jt_gap_endpointrightenumsecondindex. jt_gap_endpointrightenumsecondindex+S (jt_h_endpointrightenum)=(v)) -> (((((exists fs_h_jt_endpointrightenumfirstcode. fs_h_jt_endpointrightenumfirstcode + S (jt_b_endpointrightenum) = S ((S (jt_i_endpointrightenum)) * jt_code_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumfirstcode. jt_codes_endpointright = fs_q_jt_endpointrightenumfirstcode * S ((S (jt_i_endpointrightenum)) * jt_code_scale_endpointright) + (jt_b_endpointrightenum))) /\\ (((exists fs_h_jt_endpointrightenumfirstscale. fs_h_jt_endpointrightenumfirstscale + S (jt_c_endpointrightenum) = S ((S (jt_i_endpointrightenum)) * jt_scale_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumfirstscale. jt_scales_endpointright = fs_q_jt_endpointrightenumfirstscale * S ((S (jt_i_endpointrightenum)) * jt_scale_scale_endpointright) + (jt_c_endpointrightenum))))) -> (((((exists fs_h_jt_endpointrightenumsecondcode. fs_h_jt_endpointrightenumsecondcode + S (jt_d_endpointrightenum) = S ((S (jt_h_endpointrightenum)) * jt_code_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumsecondcode. jt_codes_endpointright = fs_q_jt_endpointrightenumsecondcode * S ((S (jt_h_endpointrightenum)) * jt_code_scale_endpointright) + (jt_d_endpointrightenum))) /\\ (((exists fs_h_jt_endpointrightenumsecondscale. fs_h_jt_endpointrightenumsecondscale + S (jt_e_endpointrightenum) = S ((S (jt_h_endpointrightenum)) * jt_scale_scale_endpointright)) /\\ exists fs_q_jt_endpointrightenumsecondscale. jt_scales_endpointright = fs_q_jt_endpointrightenumsecondscale * S ((S (jt_h_endpointrightenum)) * jt_scale_scale_endpointright) + (jt_e_endpointrightenum))))) -> (forall jt_index_endpointrightenumsame jt_left_endpointrightenumsame jt_right_endpointrightenumsame. (exists jt_gap_endpointrightenumsameindex. jt_gap_endpointrightenumsameindex+S (jt_index_endpointrightenumsame)=(k)) -> (((exists fs_h_jt_endpointrightenumsameleft. fs_h_jt_endpointrightenumsameleft + S (jt_left_endpointrightenumsame) = S ((S (jt_index_endpointrightenumsame)) * jt_c_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenumsameleft. jt_b_endpointrightenum = fs_q_jt_endpointrightenumsameleft * S ((S (jt_index_endpointrightenumsame)) * jt_c_endpointrightenum) + (jt_left_endpointrightenumsame))) -> (((exists fs_h_jt_endpointrightenumsameright. fs_h_jt_endpointrightenumsameright + S (jt_right_endpointrightenumsame) = S ((S (jt_index_endpointrightenumsame)) * jt_e_endpointrightenum)) /\\ exists fs_q_jt_endpointrightenumsameright. jt_d_endpointrightenum = fs_q_jt_endpointrightenumsameright * S ((S (jt_index_endpointrightenumsame)) * jt_e_endpointrightenum) + (jt_right_endpointrightenumsame))) -> jt_left_endpointrightenumsame=jt_right_endpointrightenumsame) -> jt_i_endpointrightenum=jt_h_endpointrightenum))))))))", "specialize jordan_totient_exists (k)", "specialize jordan_totient_exists (b)", "apply jordan_totient_exists", "exact hk", "exact hb", "cases hv", "exists x", "exists x1", "exists x*x1", "split", "exact hu_witness", "split", "exact hv_witness", "split", "specialize jordan_totient_coprime_product (k)", "specialize jordan_totient_coprime_product (a)", "specialize jordan_totient_coprime_product (b)", "specialize jordan_totient_coprime_product (x)", "specialize jordan_totient_coprime_product (x1)", "apply jordan_totient_coprime_product", "exact hcop", "exact hu_witness", "exact hv_witness", "refl" ], "script_sha256": "4ca53a50fb0414baf90456aebdef1d5e1eb2510583cac3b871aa328fd22d9d92", "source_filename": "jordan_multiplicativity_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_candidate_theorems", "script_sha256": "4ca53a50fb0414baf90456aebdef1d5e1eb2510583cac3b871aa328fd22d9d92", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_candidate", "source_sha256": "aeff3b3adb320e30388290654fc88beea3ccbe9c84adba543b47e741c5a11b86", "statement_sha256": "5b4bbb844653f4befd6755a95510eaf54b8d8df88897b82fe25a530af55e981c" } ], "stable_member": false, "statement": "forall k a b. ~(k=0) -> ~(a=0) -> ~(b=0) -> (forall jt_divisor_jendpointcop. (exists jt_factor_jendpointcopa. (a)=(jt_divisor_jendpointcop)*jt_factor_jendpointcopa) -> (exists jt_factor_jendpointcopb. (b)=(jt_divisor_jendpointcop)*jt_factor_jendpointcopb) -> jt_divisor_jendpointcop=1) -> exists u v w. ((((~((k)=0)) /\\ (((~((a)=0)) /\\ (exists jt_codes_jendpointleft jt_code_scale_jendpointleft jt_scales_jendpointleft jt_scale_scale_jendpointleft. ((forall jt_i_jendpointleftenum. (exists jt_gap_jendpointleftenumsoundindex. jt_gap_jendpointleftenumsoundindex+S (jt_i_jendpointleftenum)=(u)) -> exists jt_b_jendpointleftenum jt_c_jendpointleftenum. ((((((exists fs_h_jt_jendpointleftenumsoundcode. fs_h_jt_jendpointleftenumsoundcode + S (jt_b_jendpointleftenum) = S ((S (jt_i_jendpointleftenum)) * jt_code_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumsoundcode. jt_codes_jendpointleft = fs_q_jt_jendpointleftenumsoundcode * S ((S (jt_i_jendpointleftenum)) * jt_code_scale_jendpointleft) + (jt_b_jendpointleftenum))) /\\ (((exists fs_h_jt_jendpointleftenumsoundscale. fs_h_jt_jendpointleftenumsoundscale + S (jt_c_jendpointleftenum) = S ((S (jt_i_jendpointleftenum)) * jt_scale_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumsoundscale. jt_scales_jendpointleft = fs_q_jt_jendpointleftenumsoundscale * S ((S (jt_i_jendpointleftenum)) * jt_scale_scale_jendpointleft) + (jt_c_jendpointleftenum))))) /\\ (((forall jt_index_jendpointleftenumbound. (exists jt_gap_jendpointleftenumboundindex. jt_gap_jendpointleftenumboundindex+S (jt_index_jendpointleftenumbound)=(k)) -> exists jt_value_jendpointleftenumbound. ((((exists fs_h_jt_jendpointleftenumboundat. fs_h_jt_jendpointleftenumboundat + S (jt_value_jendpointleftenumbound) = S ((S (jt_index_jendpointleftenumbound)) * jt_c_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenumboundat. jt_b_jendpointleftenum = fs_q_jt_jendpointleftenumboundat * S ((S (jt_index_jendpointleftenumbound)) * jt_c_jendpointleftenum) + (jt_value_jendpointleftenumbound))) /\\ (exists jt_gap_jendpointleftenumboundvalue. jt_gap_jendpointleftenumboundvalue+S (jt_value_jendpointleftenumbound)=(a)))) /\\ (forall jt_divisor_jendpointleftenumprimitive. (exists jt_factor_jendpointleftenumprimitivemodulus. (a)=(jt_divisor_jendpointleftenumprimitive)*jt_factor_jendpointleftenumprimitivemodulus) -> (forall jt_index_jendpointleftenumprimitivecoordinates jt_value_jendpointleftenumprimitivecoordinates. (exists jt_gap_jendpointleftenumprimitivecoordinatesindex. jt_gap_jendpointleftenumprimitivecoordinatesindex+S (jt_index_jendpointleftenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jendpointleftenumprimitivecoordinatesat. fs_h_jt_jendpointleftenumprimitivecoordinatesat + S (jt_value_jendpointleftenumprimitivecoordinates) = S ((S (jt_index_jendpointleftenumprimitivecoordinates)) * jt_c_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenumprimitivecoordinatesat. jt_b_jendpointleftenum = fs_q_jt_jendpointleftenumprimitivecoordinatesat * S ((S (jt_index_jendpointleftenumprimitivecoordinates)) * jt_c_jendpointleftenum) + (jt_value_jendpointleftenumprimitivecoordinates))) -> (exists jt_factor_jendpointleftenumprimitivecoordinatesdivides. (jt_value_jendpointleftenumprimitivecoordinates)=(jt_divisor_jendpointleftenumprimitive)*jt_factor_jendpointleftenumprimitivecoordinatesdivides)) -> jt_divisor_jendpointleftenumprimitive=1))))) /\\ (((forall jt_b_jendpointleftenum jt_c_jendpointleftenum. (forall jt_index_jendpointleftenuminputbound. (exists jt_gap_jendpointleftenuminputboundindex. jt_gap_jendpointleftenuminputboundindex+S (jt_index_jendpointleftenuminputbound)=(k)) -> exists jt_value_jendpointleftenuminputbound. ((((exists fs_h_jt_jendpointleftenuminputboundat. fs_h_jt_jendpointleftenuminputboundat + S (jt_value_jendpointleftenuminputbound) = S ((S (jt_index_jendpointleftenuminputbound)) * jt_c_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenuminputboundat. jt_b_jendpointleftenum = fs_q_jt_jendpointleftenuminputboundat * S ((S (jt_index_jendpointleftenuminputbound)) * jt_c_jendpointleftenum) + (jt_value_jendpointleftenuminputbound))) /\\ (exists jt_gap_jendpointleftenuminputboundvalue. jt_gap_jendpointleftenuminputboundvalue+S (jt_value_jendpointleftenuminputbound)=(a)))) -> (forall jt_divisor_jendpointleftenuminputprimitive. (exists jt_factor_jendpointleftenuminputprimitivemodulus. (a)=(jt_divisor_jendpointleftenuminputprimitive)*jt_factor_jendpointleftenuminputprimitivemodulus) -> (forall jt_index_jendpointleftenuminputprimitivecoordinates jt_value_jendpointleftenuminputprimitivecoordinates. (exists jt_gap_jendpointleftenuminputprimitivecoordinatesindex. jt_gap_jendpointleftenuminputprimitivecoordinatesindex+S (jt_index_jendpointleftenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jendpointleftenuminputprimitivecoordinatesat. fs_h_jt_jendpointleftenuminputprimitivecoordinatesat + S (jt_value_jendpointleftenuminputprimitivecoordinates) = S ((S (jt_index_jendpointleftenuminputprimitivecoordinates)) * jt_c_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenuminputprimitivecoordinatesat. jt_b_jendpointleftenum = fs_q_jt_jendpointleftenuminputprimitivecoordinatesat * S ((S (jt_index_jendpointleftenuminputprimitivecoordinates)) * jt_c_jendpointleftenum) + (jt_value_jendpointleftenuminputprimitivecoordinates))) -> (exists jt_factor_jendpointleftenuminputprimitivecoordinatesdivides. (jt_value_jendpointleftenuminputprimitivecoordinates)=(jt_divisor_jendpointleftenuminputprimitive)*jt_factor_jendpointleftenuminputprimitivecoordinatesdivides)) -> jt_divisor_jendpointleftenuminputprimitive=1) -> exists jt_i_jendpointleftenum jt_d_jendpointleftenum jt_e_jendpointleftenum. ((exists jt_gap_jendpointleftenumcompleteindex. jt_gap_jendpointleftenumcompleteindex+S (jt_i_jendpointleftenum)=(u)) /\\ (((((((exists fs_h_jt_jendpointleftenumcompletecode. fs_h_jt_jendpointleftenumcompletecode + S (jt_d_jendpointleftenum) = S ((S (jt_i_jendpointleftenum)) * jt_code_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumcompletecode. jt_codes_jendpointleft = fs_q_jt_jendpointleftenumcompletecode * S ((S (jt_i_jendpointleftenum)) * jt_code_scale_jendpointleft) + (jt_d_jendpointleftenum))) /\\ (((exists fs_h_jt_jendpointleftenumcompletescale. fs_h_jt_jendpointleftenumcompletescale + S (jt_e_jendpointleftenum) = S ((S (jt_i_jendpointleftenum)) * jt_scale_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumcompletescale. jt_scales_jendpointleft = fs_q_jt_jendpointleftenumcompletescale * S ((S (jt_i_jendpointleftenum)) * jt_scale_scale_jendpointleft) + (jt_e_jendpointleftenum))))) /\\ (forall jt_index_jendpointleftenumrepresented jt_left_jendpointleftenumrepresented jt_right_jendpointleftenumrepresented. (exists jt_gap_jendpointleftenumrepresentedindex. jt_gap_jendpointleftenumrepresentedindex+S (jt_index_jendpointleftenumrepresented)=(k)) -> (((exists fs_h_jt_jendpointleftenumrepresentedleft. fs_h_jt_jendpointleftenumrepresentedleft + S (jt_left_jendpointleftenumrepresented) = S ((S (jt_index_jendpointleftenumrepresented)) * jt_c_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenumrepresentedleft. jt_b_jendpointleftenum = fs_q_jt_jendpointleftenumrepresentedleft * S ((S (jt_index_jendpointleftenumrepresented)) * jt_c_jendpointleftenum) + (jt_left_jendpointleftenumrepresented))) -> (((exists fs_h_jt_jendpointleftenumrepresentedright. fs_h_jt_jendpointleftenumrepresentedright + S (jt_right_jendpointleftenumrepresented) = S ((S (jt_index_jendpointleftenumrepresented)) * jt_e_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenumrepresentedright. jt_d_jendpointleftenum = fs_q_jt_jendpointleftenumrepresentedright * S ((S (jt_index_jendpointleftenumrepresented)) * jt_e_jendpointleftenum) + (jt_right_jendpointleftenumrepresented))) -> jt_left_jendpointleftenumrepresented=jt_right_jendpointleftenumrepresented))))) /\\ (forall jt_i_jendpointleftenum jt_h_jendpointleftenum jt_b_jendpointleftenum jt_c_jendpointleftenum jt_d_jendpointleftenum jt_e_jendpointleftenum. (exists jt_gap_jendpointleftenumfirstindex. jt_gap_jendpointleftenumfirstindex+S (jt_i_jendpointleftenum)=(u)) -> (exists jt_gap_jendpointleftenumsecondindex. jt_gap_jendpointleftenumsecondindex+S (jt_h_jendpointleftenum)=(u)) -> (((((exists fs_h_jt_jendpointleftenumfirstcode. fs_h_jt_jendpointleftenumfirstcode + S (jt_b_jendpointleftenum) = S ((S (jt_i_jendpointleftenum)) * jt_code_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumfirstcode. jt_codes_jendpointleft = fs_q_jt_jendpointleftenumfirstcode * S ((S (jt_i_jendpointleftenum)) * jt_code_scale_jendpointleft) + (jt_b_jendpointleftenum))) /\\ (((exists fs_h_jt_jendpointleftenumfirstscale. fs_h_jt_jendpointleftenumfirstscale + S (jt_c_jendpointleftenum) = S ((S (jt_i_jendpointleftenum)) * jt_scale_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumfirstscale. jt_scales_jendpointleft = fs_q_jt_jendpointleftenumfirstscale * S ((S (jt_i_jendpointleftenum)) * jt_scale_scale_jendpointleft) + (jt_c_jendpointleftenum))))) -> (((((exists fs_h_jt_jendpointleftenumsecondcode. fs_h_jt_jendpointleftenumsecondcode + S (jt_d_jendpointleftenum) = S ((S (jt_h_jendpointleftenum)) * jt_code_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumsecondcode. jt_codes_jendpointleft = fs_q_jt_jendpointleftenumsecondcode * S ((S (jt_h_jendpointleftenum)) * jt_code_scale_jendpointleft) + (jt_d_jendpointleftenum))) /\\ (((exists fs_h_jt_jendpointleftenumsecondscale. fs_h_jt_jendpointleftenumsecondscale + S (jt_e_jendpointleftenum) = S ((S (jt_h_jendpointleftenum)) * jt_scale_scale_jendpointleft)) /\\ exists fs_q_jt_jendpointleftenumsecondscale. jt_scales_jendpointleft = fs_q_jt_jendpointleftenumsecondscale * S ((S (jt_h_jendpointleftenum)) * jt_scale_scale_jendpointleft) + (jt_e_jendpointleftenum))))) -> (forall jt_index_jendpointleftenumsame jt_left_jendpointleftenumsame jt_right_jendpointleftenumsame. (exists jt_gap_jendpointleftenumsameindex. jt_gap_jendpointleftenumsameindex+S (jt_index_jendpointleftenumsame)=(k)) -> (((exists fs_h_jt_jendpointleftenumsameleft. fs_h_jt_jendpointleftenumsameleft + S (jt_left_jendpointleftenumsame) = S ((S (jt_index_jendpointleftenumsame)) * jt_c_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenumsameleft. jt_b_jendpointleftenum = fs_q_jt_jendpointleftenumsameleft * S ((S (jt_index_jendpointleftenumsame)) * jt_c_jendpointleftenum) + (jt_left_jendpointleftenumsame))) -> (((exists fs_h_jt_jendpointleftenumsameright. fs_h_jt_jendpointleftenumsameright + S (jt_right_jendpointleftenumsame) = S ((S (jt_index_jendpointleftenumsame)) * jt_e_jendpointleftenum)) /\\ exists fs_q_jt_jendpointleftenumsameright. jt_d_jendpointleftenum = fs_q_jt_jendpointleftenumsameright * S ((S (jt_index_jendpointleftenumsame)) * jt_e_jendpointleftenum) + (jt_right_jendpointleftenumsame))) -> jt_left_jendpointleftenumsame=jt_right_jendpointleftenumsame) -> jt_i_jendpointleftenum=jt_h_jendpointleftenum))))))))) /\\ (((((~((k)=0)) /\\ (((~((b)=0)) /\\ (exists jt_codes_jendpointright jt_code_scale_jendpointright jt_scales_jendpointright jt_scale_scale_jendpointright. ((forall jt_i_jendpointrightenum. (exists jt_gap_jendpointrightenumsoundindex. jt_gap_jendpointrightenumsoundindex+S (jt_i_jendpointrightenum)=(v)) -> exists jt_b_jendpointrightenum jt_c_jendpointrightenum. ((((((exists fs_h_jt_jendpointrightenumsoundcode. fs_h_jt_jendpointrightenumsoundcode + S (jt_b_jendpointrightenum) = S ((S (jt_i_jendpointrightenum)) * jt_code_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumsoundcode. jt_codes_jendpointright = fs_q_jt_jendpointrightenumsoundcode * S ((S (jt_i_jendpointrightenum)) * jt_code_scale_jendpointright) + (jt_b_jendpointrightenum))) /\\ (((exists fs_h_jt_jendpointrightenumsoundscale. fs_h_jt_jendpointrightenumsoundscale + S (jt_c_jendpointrightenum) = S ((S (jt_i_jendpointrightenum)) * jt_scale_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumsoundscale. jt_scales_jendpointright = fs_q_jt_jendpointrightenumsoundscale * S ((S (jt_i_jendpointrightenum)) * jt_scale_scale_jendpointright) + (jt_c_jendpointrightenum))))) /\\ (((forall jt_index_jendpointrightenumbound. (exists jt_gap_jendpointrightenumboundindex. jt_gap_jendpointrightenumboundindex+S (jt_index_jendpointrightenumbound)=(k)) -> exists jt_value_jendpointrightenumbound. ((((exists fs_h_jt_jendpointrightenumboundat. fs_h_jt_jendpointrightenumboundat + S (jt_value_jendpointrightenumbound) = S ((S (jt_index_jendpointrightenumbound)) * jt_c_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenumboundat. jt_b_jendpointrightenum = fs_q_jt_jendpointrightenumboundat * S ((S (jt_index_jendpointrightenumbound)) * jt_c_jendpointrightenum) + (jt_value_jendpointrightenumbound))) /\\ (exists jt_gap_jendpointrightenumboundvalue. jt_gap_jendpointrightenumboundvalue+S (jt_value_jendpointrightenumbound)=(b)))) /\\ (forall jt_divisor_jendpointrightenumprimitive. (exists jt_factor_jendpointrightenumprimitivemodulus. (b)=(jt_divisor_jendpointrightenumprimitive)*jt_factor_jendpointrightenumprimitivemodulus) -> (forall jt_index_jendpointrightenumprimitivecoordinates jt_value_jendpointrightenumprimitivecoordinates. (exists jt_gap_jendpointrightenumprimitivecoordinatesindex. jt_gap_jendpointrightenumprimitivecoordinatesindex+S (jt_index_jendpointrightenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jendpointrightenumprimitivecoordinatesat. fs_h_jt_jendpointrightenumprimitivecoordinatesat + S (jt_value_jendpointrightenumprimitivecoordinates) = S ((S (jt_index_jendpointrightenumprimitivecoordinates)) * jt_c_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenumprimitivecoordinatesat. jt_b_jendpointrightenum = fs_q_jt_jendpointrightenumprimitivecoordinatesat * S ((S (jt_index_jendpointrightenumprimitivecoordinates)) * jt_c_jendpointrightenum) + (jt_value_jendpointrightenumprimitivecoordinates))) -> (exists jt_factor_jendpointrightenumprimitivecoordinatesdivides. (jt_value_jendpointrightenumprimitivecoordinates)=(jt_divisor_jendpointrightenumprimitive)*jt_factor_jendpointrightenumprimitivecoordinatesdivides)) -> jt_divisor_jendpointrightenumprimitive=1))))) /\\ (((forall jt_b_jendpointrightenum jt_c_jendpointrightenum. (forall jt_index_jendpointrightenuminputbound. (exists jt_gap_jendpointrightenuminputboundindex. jt_gap_jendpointrightenuminputboundindex+S (jt_index_jendpointrightenuminputbound)=(k)) -> exists jt_value_jendpointrightenuminputbound. ((((exists fs_h_jt_jendpointrightenuminputboundat. fs_h_jt_jendpointrightenuminputboundat + S (jt_value_jendpointrightenuminputbound) = S ((S (jt_index_jendpointrightenuminputbound)) * jt_c_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenuminputboundat. jt_b_jendpointrightenum = fs_q_jt_jendpointrightenuminputboundat * S ((S (jt_index_jendpointrightenuminputbound)) * jt_c_jendpointrightenum) + (jt_value_jendpointrightenuminputbound))) /\\ (exists jt_gap_jendpointrightenuminputboundvalue. jt_gap_jendpointrightenuminputboundvalue+S (jt_value_jendpointrightenuminputbound)=(b)))) -> (forall jt_divisor_jendpointrightenuminputprimitive. (exists jt_factor_jendpointrightenuminputprimitivemodulus. (b)=(jt_divisor_jendpointrightenuminputprimitive)*jt_factor_jendpointrightenuminputprimitivemodulus) -> (forall jt_index_jendpointrightenuminputprimitivecoordinates jt_value_jendpointrightenuminputprimitivecoordinates. (exists jt_gap_jendpointrightenuminputprimitivecoordinatesindex. jt_gap_jendpointrightenuminputprimitivecoordinatesindex+S (jt_index_jendpointrightenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jendpointrightenuminputprimitivecoordinatesat. fs_h_jt_jendpointrightenuminputprimitivecoordinatesat + S (jt_value_jendpointrightenuminputprimitivecoordinates) = S ((S (jt_index_jendpointrightenuminputprimitivecoordinates)) * jt_c_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenuminputprimitivecoordinatesat. jt_b_jendpointrightenum = fs_q_jt_jendpointrightenuminputprimitivecoordinatesat * S ((S (jt_index_jendpointrightenuminputprimitivecoordinates)) * jt_c_jendpointrightenum) + (jt_value_jendpointrightenuminputprimitivecoordinates))) -> (exists jt_factor_jendpointrightenuminputprimitivecoordinatesdivides. (jt_value_jendpointrightenuminputprimitivecoordinates)=(jt_divisor_jendpointrightenuminputprimitive)*jt_factor_jendpointrightenuminputprimitivecoordinatesdivides)) -> jt_divisor_jendpointrightenuminputprimitive=1) -> exists jt_i_jendpointrightenum jt_d_jendpointrightenum jt_e_jendpointrightenum. ((exists jt_gap_jendpointrightenumcompleteindex. jt_gap_jendpointrightenumcompleteindex+S (jt_i_jendpointrightenum)=(v)) /\\ (((((((exists fs_h_jt_jendpointrightenumcompletecode. fs_h_jt_jendpointrightenumcompletecode + S (jt_d_jendpointrightenum) = S ((S (jt_i_jendpointrightenum)) * jt_code_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumcompletecode. jt_codes_jendpointright = fs_q_jt_jendpointrightenumcompletecode * S ((S (jt_i_jendpointrightenum)) * jt_code_scale_jendpointright) + (jt_d_jendpointrightenum))) /\\ (((exists fs_h_jt_jendpointrightenumcompletescale. fs_h_jt_jendpointrightenumcompletescale + S (jt_e_jendpointrightenum) = S ((S (jt_i_jendpointrightenum)) * jt_scale_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumcompletescale. jt_scales_jendpointright = fs_q_jt_jendpointrightenumcompletescale * S ((S (jt_i_jendpointrightenum)) * jt_scale_scale_jendpointright) + (jt_e_jendpointrightenum))))) /\\ (forall jt_index_jendpointrightenumrepresented jt_left_jendpointrightenumrepresented jt_right_jendpointrightenumrepresented. (exists jt_gap_jendpointrightenumrepresentedindex. jt_gap_jendpointrightenumrepresentedindex+S (jt_index_jendpointrightenumrepresented)=(k)) -> (((exists fs_h_jt_jendpointrightenumrepresentedleft. fs_h_jt_jendpointrightenumrepresentedleft + S (jt_left_jendpointrightenumrepresented) = S ((S (jt_index_jendpointrightenumrepresented)) * jt_c_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenumrepresentedleft. jt_b_jendpointrightenum = fs_q_jt_jendpointrightenumrepresentedleft * S ((S (jt_index_jendpointrightenumrepresented)) * jt_c_jendpointrightenum) + (jt_left_jendpointrightenumrepresented))) -> (((exists fs_h_jt_jendpointrightenumrepresentedright. fs_h_jt_jendpointrightenumrepresentedright + S (jt_right_jendpointrightenumrepresented) = S ((S (jt_index_jendpointrightenumrepresented)) * jt_e_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenumrepresentedright. jt_d_jendpointrightenum = fs_q_jt_jendpointrightenumrepresentedright * S ((S (jt_index_jendpointrightenumrepresented)) * jt_e_jendpointrightenum) + (jt_right_jendpointrightenumrepresented))) -> jt_left_jendpointrightenumrepresented=jt_right_jendpointrightenumrepresented))))) /\\ (forall jt_i_jendpointrightenum jt_h_jendpointrightenum jt_b_jendpointrightenum jt_c_jendpointrightenum jt_d_jendpointrightenum jt_e_jendpointrightenum. (exists jt_gap_jendpointrightenumfirstindex. jt_gap_jendpointrightenumfirstindex+S (jt_i_jendpointrightenum)=(v)) -> (exists jt_gap_jendpointrightenumsecondindex. jt_gap_jendpointrightenumsecondindex+S (jt_h_jendpointrightenum)=(v)) -> (((((exists fs_h_jt_jendpointrightenumfirstcode. fs_h_jt_jendpointrightenumfirstcode + S (jt_b_jendpointrightenum) = S ((S (jt_i_jendpointrightenum)) * jt_code_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumfirstcode. jt_codes_jendpointright = fs_q_jt_jendpointrightenumfirstcode * S ((S (jt_i_jendpointrightenum)) * jt_code_scale_jendpointright) + (jt_b_jendpointrightenum))) /\\ (((exists fs_h_jt_jendpointrightenumfirstscale. fs_h_jt_jendpointrightenumfirstscale + S (jt_c_jendpointrightenum) = S ((S (jt_i_jendpointrightenum)) * jt_scale_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumfirstscale. jt_scales_jendpointright = fs_q_jt_jendpointrightenumfirstscale * S ((S (jt_i_jendpointrightenum)) * jt_scale_scale_jendpointright) + (jt_c_jendpointrightenum))))) -> (((((exists fs_h_jt_jendpointrightenumsecondcode. fs_h_jt_jendpointrightenumsecondcode + S (jt_d_jendpointrightenum) = S ((S (jt_h_jendpointrightenum)) * jt_code_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumsecondcode. jt_codes_jendpointright = fs_q_jt_jendpointrightenumsecondcode * S ((S (jt_h_jendpointrightenum)) * jt_code_scale_jendpointright) + (jt_d_jendpointrightenum))) /\\ (((exists fs_h_jt_jendpointrightenumsecondscale. fs_h_jt_jendpointrightenumsecondscale + S (jt_e_jendpointrightenum) = S ((S (jt_h_jendpointrightenum)) * jt_scale_scale_jendpointright)) /\\ exists fs_q_jt_jendpointrightenumsecondscale. jt_scales_jendpointright = fs_q_jt_jendpointrightenumsecondscale * S ((S (jt_h_jendpointrightenum)) * jt_scale_scale_jendpointright) + (jt_e_jendpointrightenum))))) -> (forall jt_index_jendpointrightenumsame jt_left_jendpointrightenumsame jt_right_jendpointrightenumsame. (exists jt_gap_jendpointrightenumsameindex. jt_gap_jendpointrightenumsameindex+S (jt_index_jendpointrightenumsame)=(k)) -> (((exists fs_h_jt_jendpointrightenumsameleft. fs_h_jt_jendpointrightenumsameleft + S (jt_left_jendpointrightenumsame) = S ((S (jt_index_jendpointrightenumsame)) * jt_c_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenumsameleft. jt_b_jendpointrightenum = fs_q_jt_jendpointrightenumsameleft * S ((S (jt_index_jendpointrightenumsame)) * jt_c_jendpointrightenum) + (jt_left_jendpointrightenumsame))) -> (((exists fs_h_jt_jendpointrightenumsameright. fs_h_jt_jendpointrightenumsameright + S (jt_right_jendpointrightenumsame) = S ((S (jt_index_jendpointrightenumsame)) * jt_e_jendpointrightenum)) /\\ exists fs_q_jt_jendpointrightenumsameright. jt_d_jendpointrightenum = fs_q_jt_jendpointrightenumsameright * S ((S (jt_index_jendpointrightenumsame)) * jt_e_jendpointrightenum) + (jt_right_jendpointrightenumsame))) -> jt_left_jendpointrightenumsame=jt_right_jendpointrightenumsame) -> jt_i_jendpointrightenum=jt_h_jendpointrightenum))))))))) /\\ (((((~((k)=0)) /\\ (((~((a*b)=0)) /\\ (exists jt_codes_jendpointproduct jt_code_scale_jendpointproduct jt_scales_jendpointproduct jt_scale_scale_jendpointproduct. ((forall jt_i_jendpointproductenum. (exists jt_gap_jendpointproductenumsoundindex. jt_gap_jendpointproductenumsoundindex+S (jt_i_jendpointproductenum)=(w)) -> exists jt_b_jendpointproductenum jt_c_jendpointproductenum. ((((((exists fs_h_jt_jendpointproductenumsoundcode. fs_h_jt_jendpointproductenumsoundcode + S (jt_b_jendpointproductenum) = S ((S (jt_i_jendpointproductenum)) * jt_code_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumsoundcode. jt_codes_jendpointproduct = fs_q_jt_jendpointproductenumsoundcode * S ((S (jt_i_jendpointproductenum)) * jt_code_scale_jendpointproduct) + (jt_b_jendpointproductenum))) /\\ (((exists fs_h_jt_jendpointproductenumsoundscale. fs_h_jt_jendpointproductenumsoundscale + S (jt_c_jendpointproductenum) = S ((S (jt_i_jendpointproductenum)) * jt_scale_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumsoundscale. jt_scales_jendpointproduct = fs_q_jt_jendpointproductenumsoundscale * S ((S (jt_i_jendpointproductenum)) * jt_scale_scale_jendpointproduct) + (jt_c_jendpointproductenum))))) /\\ (((forall jt_index_jendpointproductenumbound. (exists jt_gap_jendpointproductenumboundindex. jt_gap_jendpointproductenumboundindex+S (jt_index_jendpointproductenumbound)=(k)) -> exists jt_value_jendpointproductenumbound. ((((exists fs_h_jt_jendpointproductenumboundat. fs_h_jt_jendpointproductenumboundat + S (jt_value_jendpointproductenumbound) = S ((S (jt_index_jendpointproductenumbound)) * jt_c_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenumboundat. jt_b_jendpointproductenum = fs_q_jt_jendpointproductenumboundat * S ((S (jt_index_jendpointproductenumbound)) * jt_c_jendpointproductenum) + (jt_value_jendpointproductenumbound))) /\\ (exists jt_gap_jendpointproductenumboundvalue. jt_gap_jendpointproductenumboundvalue+S (jt_value_jendpointproductenumbound)=(a*b)))) /\\ (forall jt_divisor_jendpointproductenumprimitive. (exists jt_factor_jendpointproductenumprimitivemodulus. (a*b)=(jt_divisor_jendpointproductenumprimitive)*jt_factor_jendpointproductenumprimitivemodulus) -> (forall jt_index_jendpointproductenumprimitivecoordinates jt_value_jendpointproductenumprimitivecoordinates. (exists jt_gap_jendpointproductenumprimitivecoordinatesindex. jt_gap_jendpointproductenumprimitivecoordinatesindex+S (jt_index_jendpointproductenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jendpointproductenumprimitivecoordinatesat. fs_h_jt_jendpointproductenumprimitivecoordinatesat + S (jt_value_jendpointproductenumprimitivecoordinates) = S ((S (jt_index_jendpointproductenumprimitivecoordinates)) * jt_c_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenumprimitivecoordinatesat. jt_b_jendpointproductenum = fs_q_jt_jendpointproductenumprimitivecoordinatesat * S ((S (jt_index_jendpointproductenumprimitivecoordinates)) * jt_c_jendpointproductenum) + (jt_value_jendpointproductenumprimitivecoordinates))) -> (exists jt_factor_jendpointproductenumprimitivecoordinatesdivides. (jt_value_jendpointproductenumprimitivecoordinates)=(jt_divisor_jendpointproductenumprimitive)*jt_factor_jendpointproductenumprimitivecoordinatesdivides)) -> jt_divisor_jendpointproductenumprimitive=1))))) /\\ (((forall jt_b_jendpointproductenum jt_c_jendpointproductenum. (forall jt_index_jendpointproductenuminputbound. (exists jt_gap_jendpointproductenuminputboundindex. jt_gap_jendpointproductenuminputboundindex+S (jt_index_jendpointproductenuminputbound)=(k)) -> exists jt_value_jendpointproductenuminputbound. ((((exists fs_h_jt_jendpointproductenuminputboundat. fs_h_jt_jendpointproductenuminputboundat + S (jt_value_jendpointproductenuminputbound) = S ((S (jt_index_jendpointproductenuminputbound)) * jt_c_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenuminputboundat. jt_b_jendpointproductenum = fs_q_jt_jendpointproductenuminputboundat * S ((S (jt_index_jendpointproductenuminputbound)) * jt_c_jendpointproductenum) + (jt_value_jendpointproductenuminputbound))) /\\ (exists jt_gap_jendpointproductenuminputboundvalue. jt_gap_jendpointproductenuminputboundvalue+S (jt_value_jendpointproductenuminputbound)=(a*b)))) -> (forall jt_divisor_jendpointproductenuminputprimitive. (exists jt_factor_jendpointproductenuminputprimitivemodulus. (a*b)=(jt_divisor_jendpointproductenuminputprimitive)*jt_factor_jendpointproductenuminputprimitivemodulus) -> (forall jt_index_jendpointproductenuminputprimitivecoordinates jt_value_jendpointproductenuminputprimitivecoordinates. (exists jt_gap_jendpointproductenuminputprimitivecoordinatesindex. jt_gap_jendpointproductenuminputprimitivecoordinatesindex+S (jt_index_jendpointproductenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_jendpointproductenuminputprimitivecoordinatesat. fs_h_jt_jendpointproductenuminputprimitivecoordinatesat + S (jt_value_jendpointproductenuminputprimitivecoordinates) = S ((S (jt_index_jendpointproductenuminputprimitivecoordinates)) * jt_c_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenuminputprimitivecoordinatesat. jt_b_jendpointproductenum = fs_q_jt_jendpointproductenuminputprimitivecoordinatesat * S ((S (jt_index_jendpointproductenuminputprimitivecoordinates)) * jt_c_jendpointproductenum) + (jt_value_jendpointproductenuminputprimitivecoordinates))) -> (exists jt_factor_jendpointproductenuminputprimitivecoordinatesdivides. (jt_value_jendpointproductenuminputprimitivecoordinates)=(jt_divisor_jendpointproductenuminputprimitive)*jt_factor_jendpointproductenuminputprimitivecoordinatesdivides)) -> jt_divisor_jendpointproductenuminputprimitive=1) -> exists jt_i_jendpointproductenum jt_d_jendpointproductenum jt_e_jendpointproductenum. ((exists jt_gap_jendpointproductenumcompleteindex. jt_gap_jendpointproductenumcompleteindex+S (jt_i_jendpointproductenum)=(w)) /\\ (((((((exists fs_h_jt_jendpointproductenumcompletecode. fs_h_jt_jendpointproductenumcompletecode + S (jt_d_jendpointproductenum) = S ((S (jt_i_jendpointproductenum)) * jt_code_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumcompletecode. jt_codes_jendpointproduct = fs_q_jt_jendpointproductenumcompletecode * S ((S (jt_i_jendpointproductenum)) * jt_code_scale_jendpointproduct) + (jt_d_jendpointproductenum))) /\\ (((exists fs_h_jt_jendpointproductenumcompletescale. fs_h_jt_jendpointproductenumcompletescale + S (jt_e_jendpointproductenum) = S ((S (jt_i_jendpointproductenum)) * jt_scale_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumcompletescale. jt_scales_jendpointproduct = fs_q_jt_jendpointproductenumcompletescale * S ((S (jt_i_jendpointproductenum)) * jt_scale_scale_jendpointproduct) + (jt_e_jendpointproductenum))))) /\\ (forall jt_index_jendpointproductenumrepresented jt_left_jendpointproductenumrepresented jt_right_jendpointproductenumrepresented. (exists jt_gap_jendpointproductenumrepresentedindex. jt_gap_jendpointproductenumrepresentedindex+S (jt_index_jendpointproductenumrepresented)=(k)) -> (((exists fs_h_jt_jendpointproductenumrepresentedleft. fs_h_jt_jendpointproductenumrepresentedleft + S (jt_left_jendpointproductenumrepresented) = S ((S (jt_index_jendpointproductenumrepresented)) * jt_c_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenumrepresentedleft. jt_b_jendpointproductenum = fs_q_jt_jendpointproductenumrepresentedleft * S ((S (jt_index_jendpointproductenumrepresented)) * jt_c_jendpointproductenum) + (jt_left_jendpointproductenumrepresented))) -> (((exists fs_h_jt_jendpointproductenumrepresentedright. fs_h_jt_jendpointproductenumrepresentedright + S (jt_right_jendpointproductenumrepresented) = S ((S (jt_index_jendpointproductenumrepresented)) * jt_e_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenumrepresentedright. jt_d_jendpointproductenum = fs_q_jt_jendpointproductenumrepresentedright * S ((S (jt_index_jendpointproductenumrepresented)) * jt_e_jendpointproductenum) + (jt_right_jendpointproductenumrepresented))) -> jt_left_jendpointproductenumrepresented=jt_right_jendpointproductenumrepresented))))) /\\ (forall jt_i_jendpointproductenum jt_h_jendpointproductenum jt_b_jendpointproductenum jt_c_jendpointproductenum jt_d_jendpointproductenum jt_e_jendpointproductenum. (exists jt_gap_jendpointproductenumfirstindex. jt_gap_jendpointproductenumfirstindex+S (jt_i_jendpointproductenum)=(w)) -> (exists jt_gap_jendpointproductenumsecondindex. jt_gap_jendpointproductenumsecondindex+S (jt_h_jendpointproductenum)=(w)) -> (((((exists fs_h_jt_jendpointproductenumfirstcode. fs_h_jt_jendpointproductenumfirstcode + S (jt_b_jendpointproductenum) = S ((S (jt_i_jendpointproductenum)) * jt_code_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumfirstcode. jt_codes_jendpointproduct = fs_q_jt_jendpointproductenumfirstcode * S ((S (jt_i_jendpointproductenum)) * jt_code_scale_jendpointproduct) + (jt_b_jendpointproductenum))) /\\ (((exists fs_h_jt_jendpointproductenumfirstscale. fs_h_jt_jendpointproductenumfirstscale + S (jt_c_jendpointproductenum) = S ((S (jt_i_jendpointproductenum)) * jt_scale_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumfirstscale. jt_scales_jendpointproduct = fs_q_jt_jendpointproductenumfirstscale * S ((S (jt_i_jendpointproductenum)) * jt_scale_scale_jendpointproduct) + (jt_c_jendpointproductenum))))) -> (((((exists fs_h_jt_jendpointproductenumsecondcode. fs_h_jt_jendpointproductenumsecondcode + S (jt_d_jendpointproductenum) = S ((S (jt_h_jendpointproductenum)) * jt_code_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumsecondcode. jt_codes_jendpointproduct = fs_q_jt_jendpointproductenumsecondcode * S ((S (jt_h_jendpointproductenum)) * jt_code_scale_jendpointproduct) + (jt_d_jendpointproductenum))) /\\ (((exists fs_h_jt_jendpointproductenumsecondscale. fs_h_jt_jendpointproductenumsecondscale + S (jt_e_jendpointproductenum) = S ((S (jt_h_jendpointproductenum)) * jt_scale_scale_jendpointproduct)) /\\ exists fs_q_jt_jendpointproductenumsecondscale. jt_scales_jendpointproduct = fs_q_jt_jendpointproductenumsecondscale * S ((S (jt_h_jendpointproductenum)) * jt_scale_scale_jendpointproduct) + (jt_e_jendpointproductenum))))) -> (forall jt_index_jendpointproductenumsame jt_left_jendpointproductenumsame jt_right_jendpointproductenumsame. (exists jt_gap_jendpointproductenumsameindex. jt_gap_jendpointproductenumsameindex+S (jt_index_jendpointproductenumsame)=(k)) -> (((exists fs_h_jt_jendpointproductenumsameleft. fs_h_jt_jendpointproductenumsameleft + S (jt_left_jendpointproductenumsame) = S ((S (jt_index_jendpointproductenumsame)) * jt_c_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenumsameleft. jt_b_jendpointproductenum = fs_q_jt_jendpointproductenumsameleft * S ((S (jt_index_jendpointproductenumsame)) * jt_c_jendpointproductenum) + (jt_left_jendpointproductenumsame))) -> (((exists fs_h_jt_jendpointproductenumsameright. fs_h_jt_jendpointproductenumsameright + S (jt_right_jendpointproductenumsame) = S ((S (jt_index_jendpointproductenumsame)) * jt_e_jendpointproductenum)) /\\ exists fs_q_jt_jendpointproductenumsameright. jt_d_jendpointproductenum = fs_q_jt_jendpointproductenumsameright * S ((S (jt_index_jendpointproductenumsame)) * jt_e_jendpointproductenum) + (jt_right_jendpointproductenumsame))) -> jt_left_jendpointproductenumsame=jt_right_jendpointproductenumsame) -> jt_i_jendpointproductenum=jt_h_jendpointproductenum))))))))) /\\ (w=u*v))))))", "statement_sha256": "5b4bbb844653f4befd6755a95510eaf54b8d8df88897b82fe25a530af55e981c", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The requested constructive G008 coprime multiplicativity endpoint with all three actual counts supplied." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 39, "body_proof_nodes": 104, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro i", "intro j", "intro b", "intro c", "intro d", "intro e", "intro hl", "intro hr", "intro he", "intro a0", "intro a1", "intro a2", "intro a3", "intro hnewl", "intro hnewr", "cases hl", "cases hr", "cases hnewl", "cases hnewr", "have heq_b : b=a0", "specialize beta_at_unique (A)", "specialize beta_at_unique (B)", "specialize beta_at_unique (i)", "specialize beta_at_unique (b)", "specialize beta_at_unique (a0)", "apply beta_at_unique", "exact hl_left", "exact hnewl_left", "have heq_c : c=a1", "specialize beta_at_unique (C)", "specialize beta_at_unique (D)", "specialize beta_at_unique (i)", "specialize beta_at_unique (c)", "specialize beta_at_unique (a1)", "apply beta_at_unique", "exact hl_right", "exact hnewl_right", "have heq_d : d=a2", "specialize beta_at_unique (E)", "specialize beta_at_unique (F)", "specialize beta_at_unique (j)", "specialize beta_at_unique (d)", "specialize beta_at_unique (a2)", "apply beta_at_unique", "exact hr_left", "exact hnewr_left", "have heq_e : e=a3", "specialize beta_at_unique (G)", "specialize beta_at_unique (H)", "specialize beta_at_unique (j)", "specialize beta_at_unique (e)", "specialize beta_at_unique (a3)", "apply beta_at_unique", "exact hr_right", "exact hnewr_right", "rewrite heq_b at he", "rewrite heq_c at he", "rewrite heq_c at he", "rewrite heq_d at he", "rewrite heq_e at he", "rewrite heq_e at he", "exact he" ], "defined_statement": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ i. ∀ j. ∀ b. ∀ c. ∀ d. ∀ e. BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) → BetaAt(E,F,j,d) ∧ BetaAt(G,H,j,e) → IntegerVectorZero(b,c,d,e,k) → ∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) → BetaAt(E,F,j,z) ∧ BetaAt(G,H,j,n) → IntegerVectorZero(x,y,z,n,k)", "defined_statement_sha256": "a6e4530c26dc5527e334bda50dce901551ac24d3dce7c522f231c0e2eca02bf3", "definition_uses": { "ND0121": 2, "PD0013": 8 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "7e9dba769c3fc1220484f0ab4bae5ceb5370b9903cc76538cf799da44809e604", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro he" } ], [ { "kind": "text", "text": "intro a0" } ], [ { "kind": "text", "text": "intro a1" } ], [ { "kind": "text", "text": "intro a2" } ], [ { "kind": "text", "text": "intro a3" } ], [ { "kind": "text", "text": "intro hnewl" } ], [ { "kind": "text", "text": "intro hnewr" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hr" } ], [ { "kind": "text", "text": "cases hnewl" } ], [ { "kind": "text", "text": "cases hnewr" } ], [ { "kind": "text", "text": "have heq_b : b=a0" } ], [ { "kind": "text", "text": "specialize beta_at_unique (A)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (B)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a0)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hl_left" } ], [ { "kind": "text", "text": "exact hnewl_left" } ], [ { "kind": "text", "text": "have heq_c : c=a1" } ], [ { "kind": "text", "text": "specialize beta_at_unique (C)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (D)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a1)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hl_right" } ], [ { "kind": "text", "text": "exact hnewl_right" } ], [ { "kind": "text", "text": "have heq_d : d=a2" } ], [ { "kind": "text", "text": "specialize beta_at_unique (E)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (F)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (d)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a2)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hr_left" } ], [ { "kind": "text", "text": "exact hnewr_left" } ], [ { "kind": "text", "text": "have heq_e : e=a3" } ], [ { "kind": "text", "text": "specialize beta_at_unique (G)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (H)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (e)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a3)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hr_right" } ], [ { "kind": "text", "text": "exact hnewr_right" } ], [ { "kind": "text", "text": "rewrite heq_b at he" } ], [ { "kind": "text", "text": "rewrite heq_c at he" } ], [ { "kind": "text", "text": "rewrite heq_c at he" } ], [ { "kind": "text", "text": "rewrite heq_d at he" } ], [ { "kind": "text", "text": "rewrite heq_e at he" } ], [ { "kind": "text", "text": "rewrite heq_e at he" } ], [ { "kind": "text", "text": "exact he" } ] ], "statement_definition_uses": { "ND0121": 2, "PD0013": 8 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ i. ∀ j. ∀ b. ∀ c. ∀ d. ∀ e. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,d)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,e)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" }, { "kind": "text", "text": " → ∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT004C", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_position_match_from_entries", "original_ha_bundle_verified": true, "proof_bundle_node_id": 337, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro i", "intro j", "intro b", "intro c", "intro d", "intro e", "intro hl", "intro hr", "intro he", "intro a0", "intro a1", "intro a2", "intro a3", "intro hnewl", "intro hnewr", "cases hl", "cases hr", "cases hnewl", "cases hnewr", "have heq_b : b=a0", "specialize beta_at_unique (A)", "specialize beta_at_unique (B)", "specialize beta_at_unique (i)", "specialize beta_at_unique (b)", "specialize beta_at_unique (a0)", "apply beta_at_unique", "exact hl_left", "exact hnewl_left", "have heq_c : c=a1", "specialize beta_at_unique (C)", "specialize beta_at_unique (D)", "specialize beta_at_unique (i)", "specialize beta_at_unique (c)", "specialize beta_at_unique (a1)", "apply beta_at_unique", "exact hl_right", "exact hnewl_right", "have heq_d : d=a2", "specialize beta_at_unique (E)", "specialize beta_at_unique (F)", "specialize beta_at_unique (j)", "specialize beta_at_unique (d)", "specialize beta_at_unique (a2)", "apply beta_at_unique", "exact hr_left", "exact hnewr_left", "have heq_e : e=a3", "specialize beta_at_unique (G)", "specialize beta_at_unique (H)", "specialize beta_at_unique (j)", "specialize beta_at_unique (e)", "specialize beta_at_unique (a3)", "apply beta_at_unique", "exact hr_right", "exact hnewr_right", "rewrite heq_b at he", "rewrite heq_c at he", "rewrite heq_c at he", "rewrite heq_d at he", "rewrite heq_e at he", "rewrite heq_e at he", "exact he" ], "script_sha256": "d360947d36373a16c5ba6e248e124eae813db33b96f1caf57f3f8e98d2f941b6", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "d360947d36373a16c5ba6e248e124eae813db33b96f1caf57f3f8e98d2f941b6", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "7e9dba769c3fc1220484f0ab4bae5ceb5370b9903cc76538cf799da44809e604" } ], "stable_member": false, "statement": "forall k A B C D E F G H i j b c d e. (((((exists fs_h_jt_position_leftcode. fs_h_jt_position_leftcode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_position_leftcode. A = fs_q_jt_position_leftcode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_position_leftscale. fs_h_jt_position_leftscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_position_leftscale. C = fs_q_jt_position_leftscale * S ((S (i)) * D) + (c))))) -> (((((exists fs_h_jt_position_rightcode. fs_h_jt_position_rightcode + S (d) = S ((S (j)) * F)) /\\ exists fs_q_jt_position_rightcode. E = fs_q_jt_position_rightcode * S ((S (j)) * F) + (d))) /\\ (((exists fs_h_jt_position_rightscale. fs_h_jt_position_rightscale + S (e) = S ((S (j)) * H)) /\\ exists fs_q_jt_position_rightscale. G = fs_q_jt_position_rightscale * S ((S (j)) * H) + (e))))) -> (forall jt_index_position_equal jt_left_position_equal jt_right_position_equal. (exists jt_gap_position_equalindex. jt_gap_position_equalindex+S (jt_index_position_equal)=(k)) -> (((exists fs_h_jt_position_equalleft. fs_h_jt_position_equalleft + S (jt_left_position_equal) = S ((S (jt_index_position_equal)) * c)) /\\ exists fs_q_jt_position_equalleft. b = fs_q_jt_position_equalleft * S ((S (jt_index_position_equal)) * c) + (jt_left_position_equal))) -> (((exists fs_h_jt_position_equalright. fs_h_jt_position_equalright + S (jt_right_position_equal) = S ((S (jt_index_position_equal)) * e)) /\\ exists fs_q_jt_position_equalright. d = fs_q_jt_position_equalright * S ((S (jt_index_position_equal)) * e) + (jt_right_position_equal))) -> jt_left_position_equal=jt_right_position_equal) -> (forall jt_b_position_result jt_c_position_result jt_d_position_result jt_e_position_result. (((((exists fs_h_jt_position_resultleftcode. fs_h_jt_position_resultleftcode + S (jt_b_position_result) = S ((S (i)) * B)) /\\ exists fs_q_jt_position_resultleftcode. A = fs_q_jt_position_resultleftcode * S ((S (i)) * B) + (jt_b_position_result))) /\\ (((exists fs_h_jt_position_resultleftscale. fs_h_jt_position_resultleftscale + S (jt_c_position_result) = S ((S (i)) * D)) /\\ exists fs_q_jt_position_resultleftscale. C = fs_q_jt_position_resultleftscale * S ((S (i)) * D) + (jt_c_position_result))))) -> (((((exists fs_h_jt_position_resultrightcode. fs_h_jt_position_resultrightcode + S (jt_d_position_result) = S ((S (j)) * F)) /\\ exists fs_q_jt_position_resultrightcode. E = fs_q_jt_position_resultrightcode * S ((S (j)) * F) + (jt_d_position_result))) /\\ (((exists fs_h_jt_position_resultrightscale. fs_h_jt_position_resultrightscale + S (jt_e_position_result) = S ((S (j)) * H)) /\\ exists fs_q_jt_position_resultrightscale. G = fs_q_jt_position_resultrightscale * S ((S (j)) * H) + (jt_e_position_result))))) -> (forall jt_index_position_resultequal jt_left_position_resultequal jt_right_position_resultequal. (exists jt_gap_position_resultequalindex. jt_gap_position_resultequalindex+S (jt_index_position_resultequal)=(k)) -> (((exists fs_h_jt_position_resultequalleft. fs_h_jt_position_resultequalleft + S (jt_left_position_resultequal) = S ((S (jt_index_position_resultequal)) * jt_c_position_result)) /\\ exists fs_q_jt_position_resultequalleft. jt_b_position_result = fs_q_jt_position_resultequalleft * S ((S (jt_index_position_resultequal)) * jt_c_position_result) + (jt_left_position_resultequal))) -> (((exists fs_h_jt_position_resultequalright. fs_h_jt_position_resultequalright + S (jt_right_position_resultequal) = S ((S (jt_index_position_resultequal)) * jt_e_position_result)) /\\ exists fs_q_jt_position_resultequalright. jt_d_position_result = fs_q_jt_position_resultequalright * S ((S (jt_index_position_resultequal)) * jt_e_position_result) + (jt_right_position_resultequal))) -> jt_left_position_resultequal=jt_right_position_resultequal))", "statement_sha256": "7e9dba769c3fc1220484f0ab4bae5ceb5370b9903cc76538cf799da44809e604", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Actual outer beta functionality transports chosen tuple equality to every decoding at the same two positions." }, { "admission_dependencies": [ "jordan_enumeration_complete", "jordan_enumeration_position_match_from_entries" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 44, "body_proof_nodes": 82, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro i", "intro hl", "intro hr", "intro hi", "cases hl", "have hv : ∃ b. ∃ c. BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaPrefixInto(b,c,k,n) ∧ JordanPrimitiveTuple(n,b,c,k))", "specialize hl_left (i)", "apply hl_left", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_right", "have hw : JordanTupleListed(x,x1,k,E,F,G,H,v)", "specialize jordan_enumeration_complete (k)", "specialize jordan_enumeration_complete (n)", "specialize jordan_enumeration_complete (E)", "specialize jordan_enumeration_complete (F)", "specialize jordan_enumeration_complete (G)", "specialize jordan_enumeration_complete (H)", "specialize jordan_enumeration_complete (v)", "specialize jordan_enumeration_complete (x)", "specialize jordan_enumeration_complete (x1)", "apply jordan_enumeration_complete", "exact hr", "exact hv_witness_witness_right_left", "exact hv_witness_witness_right_right", "cases hw", "cases hw_witness", "cases hw_witness_witness", "cases hw_witness_witness_witness", "cases hw_witness_witness_witness_right", "exists x2", "split", "exact hw_witness_witness_witness_left", "specialize jordan_enumeration_position_match_from_entries (k)", "specialize jordan_enumeration_position_match_from_entries (A)", "specialize jordan_enumeration_position_match_from_entries (B)", "specialize jordan_enumeration_position_match_from_entries (C)", "specialize jordan_enumeration_position_match_from_entries (D)", "specialize jordan_enumeration_position_match_from_entries (E)", "specialize jordan_enumeration_position_match_from_entries (F)", "specialize jordan_enumeration_position_match_from_entries (G)", "specialize jordan_enumeration_position_match_from_entries (H)", "specialize jordan_enumeration_position_match_from_entries (i)", "specialize jordan_enumeration_position_match_from_entries (x2)", "specialize jordan_enumeration_position_match_from_entries (x)", "specialize jordan_enumeration_position_match_from_entries (x1)", "specialize jordan_enumeration_position_match_from_entries (x3)", "specialize jordan_enumeration_position_match_from_entries (x4)", "apply jordan_enumeration_position_match_from_entries", "exact hv_witness_witness_left", "exact hw_witness_witness_witness_right_left", "exact hw_witness_witness_witness_right_right" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ i. JordanTupleEnumeration(k,n,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → Lt(i,u) → ∃ x. Lt(x,v) ∧ (∀ y. ∀ z. ∀ m. ∀ j. BetaAt(A,B,i,y) ∧ BetaAt(C,D,i,z) → BetaAt(E,F,x,m) ∧ BetaAt(G,H,x,j) → IntegerVectorZero(y,z,m,j,k))", "defined_statement_sha256": "73499dfca57795f809475b563810ed97c96ec6e7f5db1fea784e790ae074c6ad", "definition_uses": { "ND0121": 1, "ND0262": 1, "ND0372": 1, "ND0374": 2, "ND0376": 1, "PD0002": 2, "PD0013": 6 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "ad48f138ed36432f448b3ec94079040c891340610738aaee3988dfcb9c2d1230", "free_names": [], "script_definition_uses": { "ND0262": 1, "ND0372": 1, "ND0376": 1, "PD0013": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ b. ∃ c. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hl_left (i)" } ], [ { "kind": "text", "text": "apply hl_left" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness" } ], [ { "kind": "text", "text": "cases hv_witness_witness_right" } ], [ { "kind": "text", "text": "have hw : " }, { "definition": "ND0376", "kind": "definition", "text": "JordanTupleListed(x,x1,k,E,F,G,H,v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_complete (x1)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_complete" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hv_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_witness_right_right" } ], [ { "kind": "text", "text": "cases hw" } ], [ { "kind": "text", "text": "cases hw_witness" } ], [ { "kind": "text", "text": "cases hw_witness_witness" } ], [ { "kind": "text", "text": "cases hw_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hw_witness_witness_witness_right" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hw_witness_witness_witness_left" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (i)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (x2)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (x1)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (x3)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_from_entries (x4)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_position_match_from_entries" } ], [ { "kind": "text", "text": "exact hv_witness_witness_left" } ], [ { "kind": "text", "text": "exact hw_witness_witness_witness_right_left" } ], [ { "kind": "text", "text": "exact hw_witness_witness_witness_right_right" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0374": 2, "PD0002": 2, "PD0013": 4 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ i. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,u)" }, { "kind": "text", "text": " → ∃ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,v)" }, { "kind": "text", "text": " ∧ (∀ y. ∀ z. ∀ m. ∀ j. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,z)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,x,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,x,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(y,z,m,j,k)" }, { "kind": "text", "text": ")" } ] }, "dependencies": [ "jordan_enumeration_complete", "jordan_enumeration_position_match_from_entries" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT004D", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_position_match_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 338, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro i", "intro hl", "intro hr", "intro hi", "cases hl", "have hv : exists b c. ((((((exists fs_h_jt_chosen_sourceentrycode. fs_h_jt_chosen_sourceentrycode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_chosen_sourceentrycode. A = fs_q_jt_chosen_sourceentrycode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_chosen_sourceentryscale. fs_h_jt_chosen_sourceentryscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_chosen_sourceentryscale. C = fs_q_jt_chosen_sourceentryscale * S ((S (i)) * D) + (c))))) /\\ (((forall jt_index_chosen_sourcebound. (exists jt_gap_chosen_sourceboundindex. jt_gap_chosen_sourceboundindex+S (jt_index_chosen_sourcebound)=(k)) -> exists jt_value_chosen_sourcebound. ((((exists fs_h_jt_chosen_sourceboundat. fs_h_jt_chosen_sourceboundat + S (jt_value_chosen_sourcebound) = S ((S (jt_index_chosen_sourcebound)) * c)) /\\ exists fs_q_jt_chosen_sourceboundat. b = fs_q_jt_chosen_sourceboundat * S ((S (jt_index_chosen_sourcebound)) * c) + (jt_value_chosen_sourcebound))) /\\ (exists jt_gap_chosen_sourceboundvalue. jt_gap_chosen_sourceboundvalue+S (jt_value_chosen_sourcebound)=(n)))) /\\ (forall jt_divisor_chosen_sourceprimitive. (exists jt_factor_chosen_sourceprimitivemodulus. (n)=(jt_divisor_chosen_sourceprimitive)*jt_factor_chosen_sourceprimitivemodulus) -> (forall jt_index_chosen_sourceprimitivecoordinates jt_value_chosen_sourceprimitivecoordinates. (exists jt_gap_chosen_sourceprimitivecoordinatesindex. jt_gap_chosen_sourceprimitivecoordinatesindex+S (jt_index_chosen_sourceprimitivecoordinates)=(k)) -> (((exists fs_h_jt_chosen_sourceprimitivecoordinatesat. fs_h_jt_chosen_sourceprimitivecoordinatesat + S (jt_value_chosen_sourceprimitivecoordinates) = S ((S (jt_index_chosen_sourceprimitivecoordinates)) * c)) /\\ exists fs_q_jt_chosen_sourceprimitivecoordinatesat. b = fs_q_jt_chosen_sourceprimitivecoordinatesat * S ((S (jt_index_chosen_sourceprimitivecoordinates)) * c) + (jt_value_chosen_sourceprimitivecoordinates))) -> (exists jt_factor_chosen_sourceprimitivecoordinatesdivides. (jt_value_chosen_sourceprimitivecoordinates)=(jt_divisor_chosen_sourceprimitive)*jt_factor_chosen_sourceprimitivecoordinatesdivides)) -> jt_divisor_chosen_sourceprimitive=1))))", "specialize hl_left (i)", "apply hl_left", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_witness", "cases hv_witness_witness_right", "have hw : exists jt_index_chosen_target jt_code_chosen_target jt_scale_chosen_target. ((exists jt_gap_chosen_targetindex. jt_gap_chosen_targetindex+S (jt_index_chosen_target)=(v)) /\\ (((((((exists fs_h_jt_chosen_targetcode. fs_h_jt_chosen_targetcode + S (jt_code_chosen_target) = S ((S (jt_index_chosen_target)) * F)) /\\ exists fs_q_jt_chosen_targetcode. E = fs_q_jt_chosen_targetcode * S ((S (jt_index_chosen_target)) * F) + (jt_code_chosen_target))) /\\ (((exists fs_h_jt_chosen_targetscale. fs_h_jt_chosen_targetscale + S (jt_scale_chosen_target) = S ((S (jt_index_chosen_target)) * H)) /\\ exists fs_q_jt_chosen_targetscale. G = fs_q_jt_chosen_targetscale * S ((S (jt_index_chosen_target)) * H) + (jt_scale_chosen_target))))) /\\ (forall jt_index_chosen_targetequal jt_left_chosen_targetequal jt_right_chosen_targetequal. (exists jt_gap_chosen_targetequalindex. jt_gap_chosen_targetequalindex+S (jt_index_chosen_targetequal)=(k)) -> (((exists fs_h_jt_chosen_targetequalleft. fs_h_jt_chosen_targetequalleft + S (jt_left_chosen_targetequal) = S ((S (jt_index_chosen_targetequal)) * x1)) /\\ exists fs_q_jt_chosen_targetequalleft. x = fs_q_jt_chosen_targetequalleft * S ((S (jt_index_chosen_targetequal)) * x1) + (jt_left_chosen_targetequal))) -> (((exists fs_h_jt_chosen_targetequalright. fs_h_jt_chosen_targetequalright + S (jt_right_chosen_targetequal) = S ((S (jt_index_chosen_targetequal)) * jt_scale_chosen_target)) /\\ exists fs_q_jt_chosen_targetequalright. jt_code_chosen_target = fs_q_jt_chosen_targetequalright * S ((S (jt_index_chosen_targetequal)) * jt_scale_chosen_target) + (jt_right_chosen_targetequal))) -> jt_left_chosen_targetequal=jt_right_chosen_targetequal))))", "specialize jordan_enumeration_complete (k)", "specialize jordan_enumeration_complete (n)", "specialize jordan_enumeration_complete (E)", "specialize jordan_enumeration_complete (F)", "specialize jordan_enumeration_complete (G)", "specialize jordan_enumeration_complete (H)", "specialize jordan_enumeration_complete (v)", "specialize jordan_enumeration_complete (x)", "specialize jordan_enumeration_complete (x1)", "apply jordan_enumeration_complete", "exact hr", "exact hv_witness_witness_right_left", "exact hv_witness_witness_right_right", "cases hw", "cases hw_witness", "cases hw_witness_witness", "cases hw_witness_witness_witness", "cases hw_witness_witness_witness_right", "exists x2", "split", "exact hw_witness_witness_witness_left", "specialize jordan_enumeration_position_match_from_entries (k)", "specialize jordan_enumeration_position_match_from_entries (A)", "specialize jordan_enumeration_position_match_from_entries (B)", "specialize jordan_enumeration_position_match_from_entries (C)", "specialize jordan_enumeration_position_match_from_entries (D)", "specialize jordan_enumeration_position_match_from_entries (E)", "specialize jordan_enumeration_position_match_from_entries (F)", "specialize jordan_enumeration_position_match_from_entries (G)", "specialize jordan_enumeration_position_match_from_entries (H)", "specialize jordan_enumeration_position_match_from_entries (i)", "specialize jordan_enumeration_position_match_from_entries (x2)", "specialize jordan_enumeration_position_match_from_entries (x)", "specialize jordan_enumeration_position_match_from_entries (x1)", "specialize jordan_enumeration_position_match_from_entries (x3)", "specialize jordan_enumeration_position_match_from_entries (x4)", "apply jordan_enumeration_position_match_from_entries", "exact hv_witness_witness_left", "exact hw_witness_witness_witness_right_left", "exact hw_witness_witness_witness_right_right" ], "script_sha256": "a2ecd5cd05adacbfd3115a9bc5fac6357d87b41dd276d70d0f160b57188221f5", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "a2ecd5cd05adacbfd3115a9bc5fac6357d87b41dd276d70d0f160b57188221f5", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "ad48f138ed36432f448b3ec94079040c891340610738aaee3988dfcb9c2d1230" } ], "stable_member": false, "statement": "forall k n A B C D u E F G H v i. (((forall jt_i_source_enum. (exists jt_gap_source_enumsoundindex. jt_gap_source_enumsoundindex+S (jt_i_source_enum)=(u)) -> exists jt_b_source_enum jt_c_source_enum. ((((((exists fs_h_jt_source_enumsoundcode. fs_h_jt_source_enumsoundcode + S (jt_b_source_enum) = S ((S (jt_i_source_enum)) * B)) /\\ exists fs_q_jt_source_enumsoundcode. A = fs_q_jt_source_enumsoundcode * S ((S (jt_i_source_enum)) * B) + (jt_b_source_enum))) /\\ (((exists fs_h_jt_source_enumsoundscale. fs_h_jt_source_enumsoundscale + S (jt_c_source_enum) = S ((S (jt_i_source_enum)) * D)) /\\ exists fs_q_jt_source_enumsoundscale. C = fs_q_jt_source_enumsoundscale * S ((S (jt_i_source_enum)) * D) + (jt_c_source_enum))))) /\\ (((forall jt_index_source_enumbound. (exists jt_gap_source_enumboundindex. jt_gap_source_enumboundindex+S (jt_index_source_enumbound)=(k)) -> exists jt_value_source_enumbound. ((((exists fs_h_jt_source_enumboundat. fs_h_jt_source_enumboundat + S (jt_value_source_enumbound) = S ((S (jt_index_source_enumbound)) * jt_c_source_enum)) /\\ exists fs_q_jt_source_enumboundat. jt_b_source_enum = fs_q_jt_source_enumboundat * S ((S (jt_index_source_enumbound)) * jt_c_source_enum) + (jt_value_source_enumbound))) /\\ (exists jt_gap_source_enumboundvalue. jt_gap_source_enumboundvalue+S (jt_value_source_enumbound)=(n)))) /\\ (forall jt_divisor_source_enumprimitive. (exists jt_factor_source_enumprimitivemodulus. (n)=(jt_divisor_source_enumprimitive)*jt_factor_source_enumprimitivemodulus) -> (forall jt_index_source_enumprimitivecoordinates jt_value_source_enumprimitivecoordinates. (exists jt_gap_source_enumprimitivecoordinatesindex. jt_gap_source_enumprimitivecoordinatesindex+S (jt_index_source_enumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_source_enumprimitivecoordinatesat. fs_h_jt_source_enumprimitivecoordinatesat + S (jt_value_source_enumprimitivecoordinates) = S ((S (jt_index_source_enumprimitivecoordinates)) * jt_c_source_enum)) /\\ exists fs_q_jt_source_enumprimitivecoordinatesat. jt_b_source_enum = fs_q_jt_source_enumprimitivecoordinatesat * S ((S (jt_index_source_enumprimitivecoordinates)) * jt_c_source_enum) + (jt_value_source_enumprimitivecoordinates))) -> (exists jt_factor_source_enumprimitivecoordinatesdivides. (jt_value_source_enumprimitivecoordinates)=(jt_divisor_source_enumprimitive)*jt_factor_source_enumprimitivecoordinatesdivides)) -> jt_divisor_source_enumprimitive=1))))) /\\ (((forall jt_b_source_enum jt_c_source_enum. (forall jt_index_source_enuminputbound. (exists jt_gap_source_enuminputboundindex. jt_gap_source_enuminputboundindex+S (jt_index_source_enuminputbound)=(k)) -> exists jt_value_source_enuminputbound. ((((exists fs_h_jt_source_enuminputboundat. fs_h_jt_source_enuminputboundat + S (jt_value_source_enuminputbound) = S ((S (jt_index_source_enuminputbound)) * jt_c_source_enum)) /\\ exists fs_q_jt_source_enuminputboundat. jt_b_source_enum = fs_q_jt_source_enuminputboundat * S ((S (jt_index_source_enuminputbound)) * jt_c_source_enum) + (jt_value_source_enuminputbound))) /\\ (exists jt_gap_source_enuminputboundvalue. jt_gap_source_enuminputboundvalue+S (jt_value_source_enuminputbound)=(n)))) -> (forall jt_divisor_source_enuminputprimitive. (exists jt_factor_source_enuminputprimitivemodulus. (n)=(jt_divisor_source_enuminputprimitive)*jt_factor_source_enuminputprimitivemodulus) -> (forall jt_index_source_enuminputprimitivecoordinates jt_value_source_enuminputprimitivecoordinates. (exists jt_gap_source_enuminputprimitivecoordinatesindex. jt_gap_source_enuminputprimitivecoordinatesindex+S (jt_index_source_enuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_source_enuminputprimitivecoordinatesat. fs_h_jt_source_enuminputprimitivecoordinatesat + S (jt_value_source_enuminputprimitivecoordinates) = S ((S (jt_index_source_enuminputprimitivecoordinates)) * jt_c_source_enum)) /\\ exists fs_q_jt_source_enuminputprimitivecoordinatesat. jt_b_source_enum = fs_q_jt_source_enuminputprimitivecoordinatesat * S ((S (jt_index_source_enuminputprimitivecoordinates)) * jt_c_source_enum) + (jt_value_source_enuminputprimitivecoordinates))) -> (exists jt_factor_source_enuminputprimitivecoordinatesdivides. (jt_value_source_enuminputprimitivecoordinates)=(jt_divisor_source_enuminputprimitive)*jt_factor_source_enuminputprimitivecoordinatesdivides)) -> jt_divisor_source_enuminputprimitive=1) -> exists jt_i_source_enum jt_d_source_enum jt_e_source_enum. ((exists jt_gap_source_enumcompleteindex. jt_gap_source_enumcompleteindex+S (jt_i_source_enum)=(u)) /\\ (((((((exists fs_h_jt_source_enumcompletecode. fs_h_jt_source_enumcompletecode + S (jt_d_source_enum) = S ((S (jt_i_source_enum)) * B)) /\\ exists fs_q_jt_source_enumcompletecode. A = fs_q_jt_source_enumcompletecode * S ((S (jt_i_source_enum)) * B) + (jt_d_source_enum))) /\\ (((exists fs_h_jt_source_enumcompletescale. fs_h_jt_source_enumcompletescale + S (jt_e_source_enum) = S ((S (jt_i_source_enum)) * D)) /\\ exists fs_q_jt_source_enumcompletescale. C = fs_q_jt_source_enumcompletescale * S ((S (jt_i_source_enum)) * D) + (jt_e_source_enum))))) /\\ (forall jt_index_source_enumrepresented jt_left_source_enumrepresented jt_right_source_enumrepresented. (exists jt_gap_source_enumrepresentedindex. jt_gap_source_enumrepresentedindex+S (jt_index_source_enumrepresented)=(k)) -> (((exists fs_h_jt_source_enumrepresentedleft. fs_h_jt_source_enumrepresentedleft + S (jt_left_source_enumrepresented) = S ((S (jt_index_source_enumrepresented)) * jt_c_source_enum)) /\\ exists fs_q_jt_source_enumrepresentedleft. jt_b_source_enum = fs_q_jt_source_enumrepresentedleft * S ((S (jt_index_source_enumrepresented)) * jt_c_source_enum) + (jt_left_source_enumrepresented))) -> (((exists fs_h_jt_source_enumrepresentedright. fs_h_jt_source_enumrepresentedright + S (jt_right_source_enumrepresented) = S ((S (jt_index_source_enumrepresented)) * jt_e_source_enum)) /\\ exists fs_q_jt_source_enumrepresentedright. jt_d_source_enum = fs_q_jt_source_enumrepresentedright * S ((S (jt_index_source_enumrepresented)) * jt_e_source_enum) + (jt_right_source_enumrepresented))) -> jt_left_source_enumrepresented=jt_right_source_enumrepresented))))) /\\ (forall jt_i_source_enum jt_h_source_enum jt_b_source_enum jt_c_source_enum jt_d_source_enum jt_e_source_enum. (exists jt_gap_source_enumfirstindex. jt_gap_source_enumfirstindex+S (jt_i_source_enum)=(u)) -> (exists jt_gap_source_enumsecondindex. jt_gap_source_enumsecondindex+S (jt_h_source_enum)=(u)) -> (((((exists fs_h_jt_source_enumfirstcode. fs_h_jt_source_enumfirstcode + S (jt_b_source_enum) = S ((S (jt_i_source_enum)) * B)) /\\ exists fs_q_jt_source_enumfirstcode. A = fs_q_jt_source_enumfirstcode * S ((S (jt_i_source_enum)) * B) + (jt_b_source_enum))) /\\ (((exists fs_h_jt_source_enumfirstscale. fs_h_jt_source_enumfirstscale + S (jt_c_source_enum) = S ((S (jt_i_source_enum)) * D)) /\\ exists fs_q_jt_source_enumfirstscale. C = fs_q_jt_source_enumfirstscale * S ((S (jt_i_source_enum)) * D) + (jt_c_source_enum))))) -> (((((exists fs_h_jt_source_enumsecondcode. fs_h_jt_source_enumsecondcode + S (jt_d_source_enum) = S ((S (jt_h_source_enum)) * B)) /\\ exists fs_q_jt_source_enumsecondcode. A = fs_q_jt_source_enumsecondcode * S ((S (jt_h_source_enum)) * B) + (jt_d_source_enum))) /\\ (((exists fs_h_jt_source_enumsecondscale. fs_h_jt_source_enumsecondscale + S (jt_e_source_enum) = S ((S (jt_h_source_enum)) * D)) /\\ exists fs_q_jt_source_enumsecondscale. C = fs_q_jt_source_enumsecondscale * S ((S (jt_h_source_enum)) * D) + (jt_e_source_enum))))) -> (forall jt_index_source_enumsame jt_left_source_enumsame jt_right_source_enumsame. (exists jt_gap_source_enumsameindex. jt_gap_source_enumsameindex+S (jt_index_source_enumsame)=(k)) -> (((exists fs_h_jt_source_enumsameleft. fs_h_jt_source_enumsameleft + S (jt_left_source_enumsame) = S ((S (jt_index_source_enumsame)) * jt_c_source_enum)) /\\ exists fs_q_jt_source_enumsameleft. jt_b_source_enum = fs_q_jt_source_enumsameleft * S ((S (jt_index_source_enumsame)) * jt_c_source_enum) + (jt_left_source_enumsame))) -> (((exists fs_h_jt_source_enumsameright. fs_h_jt_source_enumsameright + S (jt_right_source_enumsame) = S ((S (jt_index_source_enumsame)) * jt_e_source_enum)) /\\ exists fs_q_jt_source_enumsameright. jt_d_source_enum = fs_q_jt_source_enumsameright * S ((S (jt_index_source_enumsame)) * jt_e_source_enum) + (jt_right_source_enumsame))) -> jt_left_source_enumsame=jt_right_source_enumsame) -> jt_i_source_enum=jt_h_source_enum))))) -> (((forall jt_i_target_enum. (exists jt_gap_target_enumsoundindex. jt_gap_target_enumsoundindex+S (jt_i_target_enum)=(v)) -> exists jt_b_target_enum jt_c_target_enum. ((((((exists fs_h_jt_target_enumsoundcode. fs_h_jt_target_enumsoundcode + S (jt_b_target_enum) = S ((S (jt_i_target_enum)) * F)) /\\ exists fs_q_jt_target_enumsoundcode. E = fs_q_jt_target_enumsoundcode * S ((S (jt_i_target_enum)) * F) + (jt_b_target_enum))) /\\ (((exists fs_h_jt_target_enumsoundscale. fs_h_jt_target_enumsoundscale + S (jt_c_target_enum) = S ((S (jt_i_target_enum)) * H)) /\\ exists fs_q_jt_target_enumsoundscale. G = fs_q_jt_target_enumsoundscale * S ((S (jt_i_target_enum)) * H) + (jt_c_target_enum))))) /\\ (((forall jt_index_target_enumbound. (exists jt_gap_target_enumboundindex. jt_gap_target_enumboundindex+S (jt_index_target_enumbound)=(k)) -> exists jt_value_target_enumbound. ((((exists fs_h_jt_target_enumboundat. fs_h_jt_target_enumboundat + S (jt_value_target_enumbound) = S ((S (jt_index_target_enumbound)) * jt_c_target_enum)) /\\ exists fs_q_jt_target_enumboundat. jt_b_target_enum = fs_q_jt_target_enumboundat * S ((S (jt_index_target_enumbound)) * jt_c_target_enum) + (jt_value_target_enumbound))) /\\ (exists jt_gap_target_enumboundvalue. jt_gap_target_enumboundvalue+S (jt_value_target_enumbound)=(n)))) /\\ (forall jt_divisor_target_enumprimitive. (exists jt_factor_target_enumprimitivemodulus. (n)=(jt_divisor_target_enumprimitive)*jt_factor_target_enumprimitivemodulus) -> (forall jt_index_target_enumprimitivecoordinates jt_value_target_enumprimitivecoordinates. (exists jt_gap_target_enumprimitivecoordinatesindex. jt_gap_target_enumprimitivecoordinatesindex+S (jt_index_target_enumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_target_enumprimitivecoordinatesat. fs_h_jt_target_enumprimitivecoordinatesat + S (jt_value_target_enumprimitivecoordinates) = S ((S (jt_index_target_enumprimitivecoordinates)) * jt_c_target_enum)) /\\ exists fs_q_jt_target_enumprimitivecoordinatesat. jt_b_target_enum = fs_q_jt_target_enumprimitivecoordinatesat * S ((S (jt_index_target_enumprimitivecoordinates)) * jt_c_target_enum) + (jt_value_target_enumprimitivecoordinates))) -> (exists jt_factor_target_enumprimitivecoordinatesdivides. (jt_value_target_enumprimitivecoordinates)=(jt_divisor_target_enumprimitive)*jt_factor_target_enumprimitivecoordinatesdivides)) -> jt_divisor_target_enumprimitive=1))))) /\\ (((forall jt_b_target_enum jt_c_target_enum. (forall jt_index_target_enuminputbound. (exists jt_gap_target_enuminputboundindex. jt_gap_target_enuminputboundindex+S (jt_index_target_enuminputbound)=(k)) -> exists jt_value_target_enuminputbound. ((((exists fs_h_jt_target_enuminputboundat. fs_h_jt_target_enuminputboundat + S (jt_value_target_enuminputbound) = S ((S (jt_index_target_enuminputbound)) * jt_c_target_enum)) /\\ exists fs_q_jt_target_enuminputboundat. jt_b_target_enum = fs_q_jt_target_enuminputboundat * S ((S (jt_index_target_enuminputbound)) * jt_c_target_enum) + (jt_value_target_enuminputbound))) /\\ (exists jt_gap_target_enuminputboundvalue. jt_gap_target_enuminputboundvalue+S (jt_value_target_enuminputbound)=(n)))) -> (forall jt_divisor_target_enuminputprimitive. (exists jt_factor_target_enuminputprimitivemodulus. (n)=(jt_divisor_target_enuminputprimitive)*jt_factor_target_enuminputprimitivemodulus) -> (forall jt_index_target_enuminputprimitivecoordinates jt_value_target_enuminputprimitivecoordinates. (exists jt_gap_target_enuminputprimitivecoordinatesindex. jt_gap_target_enuminputprimitivecoordinatesindex+S (jt_index_target_enuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_target_enuminputprimitivecoordinatesat. fs_h_jt_target_enuminputprimitivecoordinatesat + S (jt_value_target_enuminputprimitivecoordinates) = S ((S (jt_index_target_enuminputprimitivecoordinates)) * jt_c_target_enum)) /\\ exists fs_q_jt_target_enuminputprimitivecoordinatesat. jt_b_target_enum = fs_q_jt_target_enuminputprimitivecoordinatesat * S ((S (jt_index_target_enuminputprimitivecoordinates)) * jt_c_target_enum) + (jt_value_target_enuminputprimitivecoordinates))) -> (exists jt_factor_target_enuminputprimitivecoordinatesdivides. (jt_value_target_enuminputprimitivecoordinates)=(jt_divisor_target_enuminputprimitive)*jt_factor_target_enuminputprimitivecoordinatesdivides)) -> jt_divisor_target_enuminputprimitive=1) -> exists jt_i_target_enum jt_d_target_enum jt_e_target_enum. ((exists jt_gap_target_enumcompleteindex. jt_gap_target_enumcompleteindex+S (jt_i_target_enum)=(v)) /\\ (((((((exists fs_h_jt_target_enumcompletecode. fs_h_jt_target_enumcompletecode + S (jt_d_target_enum) = S ((S (jt_i_target_enum)) * F)) /\\ exists fs_q_jt_target_enumcompletecode. E = fs_q_jt_target_enumcompletecode * S ((S (jt_i_target_enum)) * F) + (jt_d_target_enum))) /\\ (((exists fs_h_jt_target_enumcompletescale. fs_h_jt_target_enumcompletescale + S (jt_e_target_enum) = S ((S (jt_i_target_enum)) * H)) /\\ exists fs_q_jt_target_enumcompletescale. G = fs_q_jt_target_enumcompletescale * S ((S (jt_i_target_enum)) * H) + (jt_e_target_enum))))) /\\ (forall jt_index_target_enumrepresented jt_left_target_enumrepresented jt_right_target_enumrepresented. (exists jt_gap_target_enumrepresentedindex. jt_gap_target_enumrepresentedindex+S (jt_index_target_enumrepresented)=(k)) -> (((exists fs_h_jt_target_enumrepresentedleft. fs_h_jt_target_enumrepresentedleft + S (jt_left_target_enumrepresented) = S ((S (jt_index_target_enumrepresented)) * jt_c_target_enum)) /\\ exists fs_q_jt_target_enumrepresentedleft. jt_b_target_enum = fs_q_jt_target_enumrepresentedleft * S ((S (jt_index_target_enumrepresented)) * jt_c_target_enum) + (jt_left_target_enumrepresented))) -> (((exists fs_h_jt_target_enumrepresentedright. fs_h_jt_target_enumrepresentedright + S (jt_right_target_enumrepresented) = S ((S (jt_index_target_enumrepresented)) * jt_e_target_enum)) /\\ exists fs_q_jt_target_enumrepresentedright. jt_d_target_enum = fs_q_jt_target_enumrepresentedright * S ((S (jt_index_target_enumrepresented)) * jt_e_target_enum) + (jt_right_target_enumrepresented))) -> jt_left_target_enumrepresented=jt_right_target_enumrepresented))))) /\\ (forall jt_i_target_enum jt_h_target_enum jt_b_target_enum jt_c_target_enum jt_d_target_enum jt_e_target_enum. (exists jt_gap_target_enumfirstindex. jt_gap_target_enumfirstindex+S (jt_i_target_enum)=(v)) -> (exists jt_gap_target_enumsecondindex. jt_gap_target_enumsecondindex+S (jt_h_target_enum)=(v)) -> (((((exists fs_h_jt_target_enumfirstcode. fs_h_jt_target_enumfirstcode + S (jt_b_target_enum) = S ((S (jt_i_target_enum)) * F)) /\\ exists fs_q_jt_target_enumfirstcode. E = fs_q_jt_target_enumfirstcode * S ((S (jt_i_target_enum)) * F) + (jt_b_target_enum))) /\\ (((exists fs_h_jt_target_enumfirstscale. fs_h_jt_target_enumfirstscale + S (jt_c_target_enum) = S ((S (jt_i_target_enum)) * H)) /\\ exists fs_q_jt_target_enumfirstscale. G = fs_q_jt_target_enumfirstscale * S ((S (jt_i_target_enum)) * H) + (jt_c_target_enum))))) -> (((((exists fs_h_jt_target_enumsecondcode. fs_h_jt_target_enumsecondcode + S (jt_d_target_enum) = S ((S (jt_h_target_enum)) * F)) /\\ exists fs_q_jt_target_enumsecondcode. E = fs_q_jt_target_enumsecondcode * S ((S (jt_h_target_enum)) * F) + (jt_d_target_enum))) /\\ (((exists fs_h_jt_target_enumsecondscale. fs_h_jt_target_enumsecondscale + S (jt_e_target_enum) = S ((S (jt_h_target_enum)) * H)) /\\ exists fs_q_jt_target_enumsecondscale. G = fs_q_jt_target_enumsecondscale * S ((S (jt_h_target_enum)) * H) + (jt_e_target_enum))))) -> (forall jt_index_target_enumsame jt_left_target_enumsame jt_right_target_enumsame. (exists jt_gap_target_enumsameindex. jt_gap_target_enumsameindex+S (jt_index_target_enumsame)=(k)) -> (((exists fs_h_jt_target_enumsameleft. fs_h_jt_target_enumsameleft + S (jt_left_target_enumsame) = S ((S (jt_index_target_enumsame)) * jt_c_target_enum)) /\\ exists fs_q_jt_target_enumsameleft. jt_b_target_enum = fs_q_jt_target_enumsameleft * S ((S (jt_index_target_enumsame)) * jt_c_target_enum) + (jt_left_target_enumsame))) -> (((exists fs_h_jt_target_enumsameright. fs_h_jt_target_enumsameright + S (jt_right_target_enumsame) = S ((S (jt_index_target_enumsame)) * jt_e_target_enum)) /\\ exists fs_q_jt_target_enumsameright. jt_d_target_enum = fs_q_jt_target_enumsameright * S ((S (jt_index_target_enumsame)) * jt_e_target_enum) + (jt_right_target_enumsame))) -> jt_left_target_enumsame=jt_right_target_enumsame) -> jt_i_target_enum=jt_h_target_enum))))) -> (exists jt_gap_source_index. jt_gap_source_index+S (i)=(u)) -> (exists j. ((exists jt_gap_chosen_image_bound. jt_gap_chosen_image_bound+S (j)=(v)) /\\ (forall jt_b_chosen_image_match jt_c_chosen_image_match jt_d_chosen_image_match jt_e_chosen_image_match. (((((exists fs_h_jt_chosen_image_matchleftcode. fs_h_jt_chosen_image_matchleftcode + S (jt_b_chosen_image_match) = S ((S (i)) * B)) /\\ exists fs_q_jt_chosen_image_matchleftcode. A = fs_q_jt_chosen_image_matchleftcode * S ((S (i)) * B) + (jt_b_chosen_image_match))) /\\ (((exists fs_h_jt_chosen_image_matchleftscale. fs_h_jt_chosen_image_matchleftscale + S (jt_c_chosen_image_match) = S ((S (i)) * D)) /\\ exists fs_q_jt_chosen_image_matchleftscale. C = fs_q_jt_chosen_image_matchleftscale * S ((S (i)) * D) + (jt_c_chosen_image_match))))) -> (((((exists fs_h_jt_chosen_image_matchrightcode. fs_h_jt_chosen_image_matchrightcode + S (jt_d_chosen_image_match) = S ((S (j)) * F)) /\\ exists fs_q_jt_chosen_image_matchrightcode. E = fs_q_jt_chosen_image_matchrightcode * S ((S (j)) * F) + (jt_d_chosen_image_match))) /\\ (((exists fs_h_jt_chosen_image_matchrightscale. fs_h_jt_chosen_image_matchrightscale + S (jt_e_chosen_image_match) = S ((S (j)) * H)) /\\ exists fs_q_jt_chosen_image_matchrightscale. G = fs_q_jt_chosen_image_matchrightscale * S ((S (j)) * H) + (jt_e_chosen_image_match))))) -> (forall jt_index_chosen_image_matchequal jt_left_chosen_image_matchequal jt_right_chosen_image_matchequal. (exists jt_gap_chosen_image_matchequalindex. jt_gap_chosen_image_matchequalindex+S (jt_index_chosen_image_matchequal)=(k)) -> (((exists fs_h_jt_chosen_image_matchequalleft. fs_h_jt_chosen_image_matchequalleft + S (jt_left_chosen_image_matchequal) = S ((S (jt_index_chosen_image_matchequal)) * jt_c_chosen_image_match)) /\\ exists fs_q_jt_chosen_image_matchequalleft. jt_b_chosen_image_match = fs_q_jt_chosen_image_matchequalleft * S ((S (jt_index_chosen_image_matchequal)) * jt_c_chosen_image_match) + (jt_left_chosen_image_matchequal))) -> (((exists fs_h_jt_chosen_image_matchequalright. fs_h_jt_chosen_image_matchequalright + S (jt_right_chosen_image_matchequal) = S ((S (jt_index_chosen_image_matchequal)) * jt_e_chosen_image_match)) /\\ exists fs_q_jt_chosen_image_matchequalright. jt_d_chosen_image_match = fs_q_jt_chosen_image_matchequalright * S ((S (jt_index_chosen_image_matchequal)) * jt_e_chosen_image_match) + (jt_right_chosen_image_matchequal))) -> jt_left_chosen_image_matchequal=jt_right_chosen_image_matchequal))))", "statement_sha256": "ad48f138ed36432f448b3ec94079040c891340610738aaee3988dfcb9c2d1230", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every actual source position has a bounded target position representing precisely the same coordinate tuple." }, { "admission_dependencies": [ "lt_not_le", "zero_le" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 25, "body_proof_nodes": 34, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro Z", "intro W", "intro v", "intro i", "intro hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "defined_statement": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ Z. ∀ W. ∀ v. ∀ x. Lt(x,0) → ∃ y. BetaAt(Z,W,x,y) ∧ (Lt(y,v) ∧ (∀ z. ∀ n. ∀ m. ∀ i. BetaAt(A,B,x,z) ∧ BetaAt(C,D,x,n) → BetaAt(E,F,y,m) ∧ BetaAt(G,H,y,i) → IntegerVectorZero(z,n,m,i,k)))", "defined_statement_sha256": "4c0b2eec9d81cca5aed34bb4b53eb927082caf415b95b3140c0afee12d0fb4aa", "definition_uses": { "ND0121": 1, "PD0002": 2, "PD0013": 5 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "cf279f7c9c53d1de712ce7f599d16a7149f817c0bbb9c5ca5939ad1188aca5a7", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro Z" } ], [ { "kind": "text", "text": "intro W" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize lt_not_le (i)" } ], [ { "kind": "text", "text": "specialize lt_not_le (0)" } ], [ { "kind": "text", "text": "apply lt_not_le" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "specialize zero_le (i)" } ], [ { "kind": "text", "text": "apply zero_le" } ] ], "statement_definition_uses": { "ND0121": 1, "PD0002": 2, "PD0013": 5 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ Z. ∀ W. ∀ v. ∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,0)" }, { "kind": "text", "text": " → ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,x,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,v)" }, { "kind": "text", "text": " ∧ (∀ z. ∀ n. ∀ m. ∀ i. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,n)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,y,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,y,i)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(z,n,m,i,k)" }, { "kind": "text", "text": "))" } ] }, "dependencies": [ "lt_not_le", "zero_le" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT004E", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_index_map_empty", "original_ha_bundle_verified": true, "proof_bundle_node_id": 339, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro Z", "intro W", "intro v", "intro i", "intro hi", "exfalso", "specialize lt_not_le (i)", "specialize lt_not_le (0)", "apply lt_not_le", "exact hi", "specialize zero_le (i)", "apply zero_le" ], "script_sha256": "41ce22239b09f748cff2e1f03758177d5008e811ce4074f84c29cae37829de64", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "41ce22239b09f748cff2e1f03758177d5008e811ce4074f84c29cae37829de64", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "cf279f7c9c53d1de712ce7f599d16a7149f817c0bbb9c5ca5939ad1188aca5a7" } ], "stable_member": false, "statement": "forall k A B C D E F G H Z W v. (forall jt_index_empty_map. (exists jt_gap_empty_mapindex. jt_gap_empty_mapindex+S (jt_index_empty_map)=(0)) -> exists jt_image_empty_map. ((((exists fs_h_jt_empty_mapat. fs_h_jt_empty_mapat + S (jt_image_empty_map) = S ((S (jt_index_empty_map)) * W)) /\\ exists fs_q_jt_empty_mapat. Z = fs_q_jt_empty_mapat * S ((S (jt_index_empty_map)) * W) + (jt_image_empty_map))) /\\ (((exists jt_gap_empty_mapbound. jt_gap_empty_mapbound+S (jt_image_empty_map)=(v)) /\\ (forall jt_b_empty_mapmatch jt_c_empty_mapmatch jt_d_empty_mapmatch jt_e_empty_mapmatch. (((((exists fs_h_jt_empty_mapmatchleftcode. fs_h_jt_empty_mapmatchleftcode + S (jt_b_empty_mapmatch) = S ((S (jt_index_empty_map)) * B)) /\\ exists fs_q_jt_empty_mapmatchleftcode. A = fs_q_jt_empty_mapmatchleftcode * S ((S (jt_index_empty_map)) * B) + (jt_b_empty_mapmatch))) /\\ (((exists fs_h_jt_empty_mapmatchleftscale. fs_h_jt_empty_mapmatchleftscale + S (jt_c_empty_mapmatch) = S ((S (jt_index_empty_map)) * D)) /\\ exists fs_q_jt_empty_mapmatchleftscale. C = fs_q_jt_empty_mapmatchleftscale * S ((S (jt_index_empty_map)) * D) + (jt_c_empty_mapmatch))))) -> (((((exists fs_h_jt_empty_mapmatchrightcode. fs_h_jt_empty_mapmatchrightcode + S (jt_d_empty_mapmatch) = S ((S (jt_image_empty_map)) * F)) /\\ exists fs_q_jt_empty_mapmatchrightcode. E = fs_q_jt_empty_mapmatchrightcode * S ((S (jt_image_empty_map)) * F) + (jt_d_empty_mapmatch))) /\\ (((exists fs_h_jt_empty_mapmatchrightscale. fs_h_jt_empty_mapmatchrightscale + S (jt_e_empty_mapmatch) = S ((S (jt_image_empty_map)) * H)) /\\ exists fs_q_jt_empty_mapmatchrightscale. G = fs_q_jt_empty_mapmatchrightscale * S ((S (jt_image_empty_map)) * H) + (jt_e_empty_mapmatch))))) -> (forall jt_index_empty_mapmatchequal jt_left_empty_mapmatchequal jt_right_empty_mapmatchequal. (exists jt_gap_empty_mapmatchequalindex. jt_gap_empty_mapmatchequalindex+S (jt_index_empty_mapmatchequal)=(k)) -> (((exists fs_h_jt_empty_mapmatchequalleft. fs_h_jt_empty_mapmatchequalleft + S (jt_left_empty_mapmatchequal) = S ((S (jt_index_empty_mapmatchequal)) * jt_c_empty_mapmatch)) /\\ exists fs_q_jt_empty_mapmatchequalleft. jt_b_empty_mapmatch = fs_q_jt_empty_mapmatchequalleft * S ((S (jt_index_empty_mapmatchequal)) * jt_c_empty_mapmatch) + (jt_left_empty_mapmatchequal))) -> (((exists fs_h_jt_empty_mapmatchequalright. fs_h_jt_empty_mapmatchequalright + S (jt_right_empty_mapmatchequal) = S ((S (jt_index_empty_mapmatchequal)) * jt_e_empty_mapmatch)) /\\ exists fs_q_jt_empty_mapmatchequalright. jt_d_empty_mapmatch = fs_q_jt_empty_mapmatchequalright * S ((S (jt_index_empty_mapmatchequal)) * jt_e_empty_mapmatch) + (jt_right_empty_mapmatchequal))) -> jt_left_empty_mapmatchequal=jt_right_empty_mapmatchequal))))))", "statement_sha256": "cf279f7c9c53d1de712ce7f599d16a7149f817c0bbb9c5ca5939ad1188aca5a7", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The zero-length index map is vacuous, independently of target length." }, { "admission_dependencies": [ "beta_prefix_extend", "finite_lt_succ_eq_or_lt" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 35, "body_proof_nodes": 85, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro Z", "intro W", "intro q", "intro v", "intro j", "intro hm", "intro hj", "intro hmatch", "have hext : ∃ P. ∃ Q. BetaAt(P,Q,q,j) ∧ BetaPrefixEqual(Z,W,P,Q,q)", "specialize beta_prefix_extend (q)", "specialize beta_prefix_extend (Z)", "specialize beta_prefix_extend (W)", "specialize beta_prefix_extend (j)", "apply beta_prefix_extend", "cases hext", "cases hext_witness", "cases hext_witness_witness", "exists x", "exists x1", "intro i", "intro hi", "have hc : i = q ∨ Lt(i,q)", "specialize finite_lt_succ_eq_or_lt (q)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hc", "exists j", "split", "rewrite hc_left", "rewrite hc_left", "exact hext_witness_witness_left", "split", "exact hj", "rewrite hc_left", "rewrite hc_left", "rewrite hc_left", "rewrite hc_left", "exact hmatch", "have hv : ∃ r. BetaAt(Z,W,i,r) ∧ (Lt(r,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) → BetaAt(E,F,r,z) ∧ BetaAt(G,H,r,n) → IntegerVectorZero(x,y,z,n,k)))", "specialize hm (i)", "apply hm", "exact hc_right", "cases hv", "cases hv_witness", "cases hv_witness_right", "exists x2", "split", "specialize hext_witness_witness_right (i)", "specialize hext_witness_witness_right (x2)", "apply hext_witness_witness_right", "exact hc_right", "exact hv_witness_left", "split", "exact hv_witness_right_left", "exact hv_witness_right_right" ], "defined_statement": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ Z. ∀ W. ∀ q. ∀ v. ∀ j. (∀ x. Lt(x,q) → ∃ y. BetaAt(Z,W,x,y) ∧ (Lt(y,v) ∧ (∀ z. ∀ n. ∀ m. ∀ i. BetaAt(A,B,x,z) ∧ BetaAt(C,D,x,n) → BetaAt(E,F,y,m) ∧ BetaAt(G,H,y,i) → IntegerVectorZero(z,n,m,i,k)))) → Lt(j,v) → (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,q,x) ∧ BetaAt(C,D,q,y) → BetaAt(E,F,j,z) ∧ BetaAt(G,H,j,n) → IntegerVectorZero(x,y,z,n,k)) → ∃ x. ∃ y. ∀ z. Lt(z,S q) → ∃ n. BetaAt(x,y,z,n) ∧ (Lt(n,v) ∧ (∀ m. ∀ i. ∀ u. ∀ w. BetaAt(A,B,z,m) ∧ BetaAt(C,D,z,i) → BetaAt(E,F,n,u) ∧ BetaAt(G,H,n,w) → IntegerVectorZero(m,i,u,w,k)))", "defined_statement_sha256": "06dc9d9efb64a9d5fec2779155421325608ec14aa53723baced1438b05b15b59", "definition_uses": { "ND0121": 4, "ND0263": 1, "PD0002": 7, "PD0013": 20 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "1452a194f9a4d8552aff8f4093912eb4616f67ad27f4e1001c2d389974c5040a", "free_names": [], "script_definition_uses": { "ND0121": 1, "ND0263": 1, "PD0002": 2, "PD0013": 6 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro Z" } ], [ { "kind": "text", "text": "intro W" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "intro hmatch" } ], [ { "kind": "text", "text": "have hext : " }, { "kind": "text", "text": "∃ P. ∃ Q. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(P,Q,q,j)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0263", "kind": "definition", "text": "BetaPrefixEqual(Z,W,P,Q,q)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (q)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (Z)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (W)" } ], [ { "kind": "text", "text": "specialize beta_prefix_extend (j)" } ], [ { "kind": "text", "text": "apply beta_prefix_extend" } ], [ { "kind": "text", "text": "cases hext" } ], [ { "kind": "text", "text": "cases hext_witness" } ], [ { "kind": "text", "text": "cases hext_witness_witness" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "exists x1" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "have hc : " }, { "kind": "text", "text": "i = q ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,q)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (q)" } ], [ { "kind": "text", "text": "specialize finite_lt_succ_eq_or_lt (i)" } ], [ { "kind": "text", "text": "apply finite_lt_succ_eq_or_lt" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "exists j" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "exact hext_witness_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "rewrite hc_left" } ], [ { "kind": "text", "text": "exact hmatch" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ r. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,i,r)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(r,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,r,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,r,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize hm (i)" } ], [ { "kind": "text", "text": "apply hm" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_right" } ], [ { "kind": "text", "text": "exists x2" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize hext_witness_witness_right (i)" } ], [ { "kind": "text", "text": "specialize hext_witness_witness_right (x2)" } ], [ { "kind": "text", "text": "apply hext_witness_witness_right" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "exact hv_witness_left" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_right_right" } ] ], "statement_definition_uses": { "ND0121": 3, "PD0002": 5, "PD0013": 14 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ Z. ∀ W. ∀ q. ∀ v. ∀ j. (∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,q)" }, { "kind": "text", "text": " → ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,x,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,v)" }, { "kind": "text", "text": " ∧ (∀ z. ∀ n. ∀ m. ∀ i. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,n)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,y,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,y,i)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(z,n,m,i,k)" }, { "kind": "text", "text": "))) → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " → (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,q,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,q,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": ") → ∃ x. ∃ y. ∀ z. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,S q)" }, { "kind": "text", "text": " → ∃ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(x,y,z,n)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(n,v)" }, { "kind": "text", "text": " ∧ (∀ m. ∀ i. ∀ u. ∀ w. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,z,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,z,i)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,n,u)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,n,w)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(m,i,u,w,k)" }, { "kind": "text", "text": "))" } ] }, "dependencies": [ "beta_prefix_extend", "finite_lt_succ_eq_or_lt" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT004F", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_index_map_append", "original_ha_bundle_verified": true, "proof_bundle_node_id": 340, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro Z", "intro W", "intro q", "intro v", "intro j", "intro hm", "intro hj", "intro hmatch", "have hext : exists P Q. ((((exists fs_h_jt_append_image. fs_h_jt_append_image + S (j) = S ((S (q)) * Q)) /\\ exists fs_q_jt_append_image. P = fs_q_jt_append_image * S ((S (q)) * Q) + (j))) /\\ (forall jt_index_append_preserves jt_value_append_preserves. (exists jt_gap_append_preservesindex. jt_gap_append_preservesindex+S (jt_index_append_preserves)=(q)) -> (((exists fs_h_jt_append_preservesold. fs_h_jt_append_preservesold + S (jt_value_append_preserves) = S ((S (jt_index_append_preserves)) * W)) /\\ exists fs_q_jt_append_preservesold. Z = fs_q_jt_append_preservesold * S ((S (jt_index_append_preserves)) * W) + (jt_value_append_preserves))) -> (((exists fs_h_jt_append_preservesnew. fs_h_jt_append_preservesnew + S (jt_value_append_preserves) = S ((S (jt_index_append_preserves)) * Q)) /\\ exists fs_q_jt_append_preservesnew. P = fs_q_jt_append_preservesnew * S ((S (jt_index_append_preserves)) * Q) + (jt_value_append_preserves)))))", "specialize beta_prefix_extend (q)", "specialize beta_prefix_extend (Z)", "specialize beta_prefix_extend (W)", "specialize beta_prefix_extend (j)", "apply beta_prefix_extend", "cases hext", "cases hext_witness", "cases hext_witness_witness", "exists x", "exists x1", "intro i", "intro hi", "have hc : i=q \\/ (exists jt_gap_append_old_index. jt_gap_append_old_index+S (i)=(q))", "specialize finite_lt_succ_eq_or_lt (q)", "specialize finite_lt_succ_eq_or_lt (i)", "apply finite_lt_succ_eq_or_lt", "exact hi", "cases hc", "exists j", "split", "rewrite hc_left", "rewrite hc_left", "exact hext_witness_witness_left", "split", "exact hj", "rewrite hc_left", "rewrite hc_left", "rewrite hc_left", "rewrite hc_left", "exact hmatch", "have hv : exists r. ((((exists fs_h_jt_append_previousat. fs_h_jt_append_previousat + S (r) = S ((S (i)) * W)) /\\ exists fs_q_jt_append_previousat. Z = fs_q_jt_append_previousat * S ((S (i)) * W) + (r))) /\\ (((exists jt_gap_append_previousbound. jt_gap_append_previousbound+S (r)=(v)) /\\ (forall jt_b_append_previousmatch jt_c_append_previousmatch jt_d_append_previousmatch jt_e_append_previousmatch. (((((exists fs_h_jt_append_previousmatchleftcode. fs_h_jt_append_previousmatchleftcode + S (jt_b_append_previousmatch) = S ((S (i)) * B)) /\\ exists fs_q_jt_append_previousmatchleftcode. A = fs_q_jt_append_previousmatchleftcode * S ((S (i)) * B) + (jt_b_append_previousmatch))) /\\ (((exists fs_h_jt_append_previousmatchleftscale. fs_h_jt_append_previousmatchleftscale + S (jt_c_append_previousmatch) = S ((S (i)) * D)) /\\ exists fs_q_jt_append_previousmatchleftscale. C = fs_q_jt_append_previousmatchleftscale * S ((S (i)) * D) + (jt_c_append_previousmatch))))) -> (((((exists fs_h_jt_append_previousmatchrightcode. fs_h_jt_append_previousmatchrightcode + S (jt_d_append_previousmatch) = S ((S (r)) * F)) /\\ exists fs_q_jt_append_previousmatchrightcode. E = fs_q_jt_append_previousmatchrightcode * S ((S (r)) * F) + (jt_d_append_previousmatch))) /\\ (((exists fs_h_jt_append_previousmatchrightscale. fs_h_jt_append_previousmatchrightscale + S (jt_e_append_previousmatch) = S ((S (r)) * H)) /\\ exists fs_q_jt_append_previousmatchrightscale. G = fs_q_jt_append_previousmatchrightscale * S ((S (r)) * H) + (jt_e_append_previousmatch))))) -> (forall jt_index_append_previousmatchequal jt_left_append_previousmatchequal jt_right_append_previousmatchequal. (exists jt_gap_append_previousmatchequalindex. jt_gap_append_previousmatchequalindex+S (jt_index_append_previousmatchequal)=(k)) -> (((exists fs_h_jt_append_previousmatchequalleft. fs_h_jt_append_previousmatchequalleft + S (jt_left_append_previousmatchequal) = S ((S (jt_index_append_previousmatchequal)) * jt_c_append_previousmatch)) /\\ exists fs_q_jt_append_previousmatchequalleft. jt_b_append_previousmatch = fs_q_jt_append_previousmatchequalleft * S ((S (jt_index_append_previousmatchequal)) * jt_c_append_previousmatch) + (jt_left_append_previousmatchequal))) -> (((exists fs_h_jt_append_previousmatchequalright. fs_h_jt_append_previousmatchequalright + S (jt_right_append_previousmatchequal) = S ((S (jt_index_append_previousmatchequal)) * jt_e_append_previousmatch)) /\\ exists fs_q_jt_append_previousmatchequalright. jt_d_append_previousmatch = fs_q_jt_append_previousmatchequalright * S ((S (jt_index_append_previousmatchequal)) * jt_e_append_previousmatch) + (jt_right_append_previousmatchequal))) -> jt_left_append_previousmatchequal=jt_right_append_previousmatchequal)))))", "specialize hm (i)", "apply hm", "exact hc_right", "cases hv", "cases hv_witness", "cases hv_witness_right", "exists x2", "split", "specialize hext_witness_witness_right (i)", "specialize hext_witness_witness_right (x2)", "apply hext_witness_witness_right", "exact hc_right", "exact hv_witness_left", "split", "exact hv_witness_right_left", "exact hv_witness_right_right" ], "script_sha256": "c9b1199cbb483325c0a622fdd75a16bc7070e8a861f7d5b051cdfb54cec40642", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "c9b1199cbb483325c0a622fdd75a16bc7070e8a861f7d5b051cdfb54cec40642", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "1452a194f9a4d8552aff8f4093912eb4616f67ad27f4e1001c2d389974c5040a" } ], "stable_member": false, "statement": "forall k A B C D E F G H Z W q v j. (forall jt_index_append_old. (exists jt_gap_append_oldindex. jt_gap_append_oldindex+S (jt_index_append_old)=(q)) -> exists jt_image_append_old. ((((exists fs_h_jt_append_oldat. fs_h_jt_append_oldat + S (jt_image_append_old) = S ((S (jt_index_append_old)) * W)) /\\ exists fs_q_jt_append_oldat. Z = fs_q_jt_append_oldat * S ((S (jt_index_append_old)) * W) + (jt_image_append_old))) /\\ (((exists jt_gap_append_oldbound. jt_gap_append_oldbound+S (jt_image_append_old)=(v)) /\\ (forall jt_b_append_oldmatch jt_c_append_oldmatch jt_d_append_oldmatch jt_e_append_oldmatch. (((((exists fs_h_jt_append_oldmatchleftcode. fs_h_jt_append_oldmatchleftcode + S (jt_b_append_oldmatch) = S ((S (jt_index_append_old)) * B)) /\\ exists fs_q_jt_append_oldmatchleftcode. A = fs_q_jt_append_oldmatchleftcode * S ((S (jt_index_append_old)) * B) + (jt_b_append_oldmatch))) /\\ (((exists fs_h_jt_append_oldmatchleftscale. fs_h_jt_append_oldmatchleftscale + S (jt_c_append_oldmatch) = S ((S (jt_index_append_old)) * D)) /\\ exists fs_q_jt_append_oldmatchleftscale. C = fs_q_jt_append_oldmatchleftscale * S ((S (jt_index_append_old)) * D) + (jt_c_append_oldmatch))))) -> (((((exists fs_h_jt_append_oldmatchrightcode. fs_h_jt_append_oldmatchrightcode + S (jt_d_append_oldmatch) = S ((S (jt_image_append_old)) * F)) /\\ exists fs_q_jt_append_oldmatchrightcode. E = fs_q_jt_append_oldmatchrightcode * S ((S (jt_image_append_old)) * F) + (jt_d_append_oldmatch))) /\\ (((exists fs_h_jt_append_oldmatchrightscale. fs_h_jt_append_oldmatchrightscale + S (jt_e_append_oldmatch) = S ((S (jt_image_append_old)) * H)) /\\ exists fs_q_jt_append_oldmatchrightscale. G = fs_q_jt_append_oldmatchrightscale * S ((S (jt_image_append_old)) * H) + (jt_e_append_oldmatch))))) -> (forall jt_index_append_oldmatchequal jt_left_append_oldmatchequal jt_right_append_oldmatchequal. (exists jt_gap_append_oldmatchequalindex. jt_gap_append_oldmatchequalindex+S (jt_index_append_oldmatchequal)=(k)) -> (((exists fs_h_jt_append_oldmatchequalleft. fs_h_jt_append_oldmatchequalleft + S (jt_left_append_oldmatchequal) = S ((S (jt_index_append_oldmatchequal)) * jt_c_append_oldmatch)) /\\ exists fs_q_jt_append_oldmatchequalleft. jt_b_append_oldmatch = fs_q_jt_append_oldmatchequalleft * S ((S (jt_index_append_oldmatchequal)) * jt_c_append_oldmatch) + (jt_left_append_oldmatchequal))) -> (((exists fs_h_jt_append_oldmatchequalright. fs_h_jt_append_oldmatchequalright + S (jt_right_append_oldmatchequal) = S ((S (jt_index_append_oldmatchequal)) * jt_e_append_oldmatch)) /\\ exists fs_q_jt_append_oldmatchequalright. jt_d_append_oldmatch = fs_q_jt_append_oldmatchequalright * S ((S (jt_index_append_oldmatchequal)) * jt_e_append_oldmatch) + (jt_right_append_oldmatchequal))) -> jt_left_append_oldmatchequal=jt_right_append_oldmatchequal)))))) -> (exists jt_gap_append_bound. jt_gap_append_bound+S (j)=(v)) -> (forall jt_b_append_match jt_c_append_match jt_d_append_match jt_e_append_match. (((((exists fs_h_jt_append_matchleftcode. fs_h_jt_append_matchleftcode + S (jt_b_append_match) = S ((S (q)) * B)) /\\ exists fs_q_jt_append_matchleftcode. A = fs_q_jt_append_matchleftcode * S ((S (q)) * B) + (jt_b_append_match))) /\\ (((exists fs_h_jt_append_matchleftscale. fs_h_jt_append_matchleftscale + S (jt_c_append_match) = S ((S (q)) * D)) /\\ exists fs_q_jt_append_matchleftscale. C = fs_q_jt_append_matchleftscale * S ((S (q)) * D) + (jt_c_append_match))))) -> (((((exists fs_h_jt_append_matchrightcode. fs_h_jt_append_matchrightcode + S (jt_d_append_match) = S ((S (j)) * F)) /\\ exists fs_q_jt_append_matchrightcode. E = fs_q_jt_append_matchrightcode * S ((S (j)) * F) + (jt_d_append_match))) /\\ (((exists fs_h_jt_append_matchrightscale. fs_h_jt_append_matchrightscale + S (jt_e_append_match) = S ((S (j)) * H)) /\\ exists fs_q_jt_append_matchrightscale. G = fs_q_jt_append_matchrightscale * S ((S (j)) * H) + (jt_e_append_match))))) -> (forall jt_index_append_matchequal jt_left_append_matchequal jt_right_append_matchequal. (exists jt_gap_append_matchequalindex. jt_gap_append_matchequalindex+S (jt_index_append_matchequal)=(k)) -> (((exists fs_h_jt_append_matchequalleft. fs_h_jt_append_matchequalleft + S (jt_left_append_matchequal) = S ((S (jt_index_append_matchequal)) * jt_c_append_match)) /\\ exists fs_q_jt_append_matchequalleft. jt_b_append_match = fs_q_jt_append_matchequalleft * S ((S (jt_index_append_matchequal)) * jt_c_append_match) + (jt_left_append_matchequal))) -> (((exists fs_h_jt_append_matchequalright. fs_h_jt_append_matchequalright + S (jt_right_append_matchequal) = S ((S (jt_index_append_matchequal)) * jt_e_append_match)) /\\ exists fs_q_jt_append_matchequalright. jt_d_append_match = fs_q_jt_append_matchequalright * S ((S (jt_index_append_matchequal)) * jt_e_append_match) + (jt_right_append_matchequal))) -> jt_left_append_matchequal=jt_right_append_matchequal)) -> (exists P Q. forall jt_index_append_result. (exists jt_gap_append_resultindex. jt_gap_append_resultindex+S (jt_index_append_result)=(S q)) -> exists jt_image_append_result. ((((exists fs_h_jt_append_resultat. fs_h_jt_append_resultat + S (jt_image_append_result) = S ((S (jt_index_append_result)) * Q)) /\\ exists fs_q_jt_append_resultat. P = fs_q_jt_append_resultat * S ((S (jt_index_append_result)) * Q) + (jt_image_append_result))) /\\ (((exists jt_gap_append_resultbound. jt_gap_append_resultbound+S (jt_image_append_result)=(v)) /\\ (forall jt_b_append_resultmatch jt_c_append_resultmatch jt_d_append_resultmatch jt_e_append_resultmatch. (((((exists fs_h_jt_append_resultmatchleftcode. fs_h_jt_append_resultmatchleftcode + S (jt_b_append_resultmatch) = S ((S (jt_index_append_result)) * B)) /\\ exists fs_q_jt_append_resultmatchleftcode. A = fs_q_jt_append_resultmatchleftcode * S ((S (jt_index_append_result)) * B) + (jt_b_append_resultmatch))) /\\ (((exists fs_h_jt_append_resultmatchleftscale. fs_h_jt_append_resultmatchleftscale + S (jt_c_append_resultmatch) = S ((S (jt_index_append_result)) * D)) /\\ exists fs_q_jt_append_resultmatchleftscale. C = fs_q_jt_append_resultmatchleftscale * S ((S (jt_index_append_result)) * D) + (jt_c_append_resultmatch))))) -> (((((exists fs_h_jt_append_resultmatchrightcode. fs_h_jt_append_resultmatchrightcode + S (jt_d_append_resultmatch) = S ((S (jt_image_append_result)) * F)) /\\ exists fs_q_jt_append_resultmatchrightcode. E = fs_q_jt_append_resultmatchrightcode * S ((S (jt_image_append_result)) * F) + (jt_d_append_resultmatch))) /\\ (((exists fs_h_jt_append_resultmatchrightscale. fs_h_jt_append_resultmatchrightscale + S (jt_e_append_resultmatch) = S ((S (jt_image_append_result)) * H)) /\\ exists fs_q_jt_append_resultmatchrightscale. G = fs_q_jt_append_resultmatchrightscale * S ((S (jt_image_append_result)) * H) + (jt_e_append_resultmatch))))) -> (forall jt_index_append_resultmatchequal jt_left_append_resultmatchequal jt_right_append_resultmatchequal. (exists jt_gap_append_resultmatchequalindex. jt_gap_append_resultmatchequalindex+S (jt_index_append_resultmatchequal)=(k)) -> (((exists fs_h_jt_append_resultmatchequalleft. fs_h_jt_append_resultmatchequalleft + S (jt_left_append_resultmatchequal) = S ((S (jt_index_append_resultmatchequal)) * jt_c_append_resultmatch)) /\\ exists fs_q_jt_append_resultmatchequalleft. jt_b_append_resultmatch = fs_q_jt_append_resultmatchequalleft * S ((S (jt_index_append_resultmatchequal)) * jt_c_append_resultmatch) + (jt_left_append_resultmatchequal))) -> (((exists fs_h_jt_append_resultmatchequalright. fs_h_jt_append_resultmatchequalright + S (jt_right_append_resultmatchequal) = S ((S (jt_index_append_resultmatchequal)) * jt_e_append_resultmatch)) /\\ exists fs_q_jt_append_resultmatchequalright. jt_d_append_resultmatch = fs_q_jt_append_resultmatchequalright * S ((S (jt_index_append_resultmatchequal)) * jt_e_append_resultmatch) + (jt_right_append_resultmatchequal))) -> jt_left_append_resultmatchequal=jt_right_append_resultmatchequal))))))", "statement_sha256": "1452a194f9a4d8552aff8f4093912eb4616f67ad27f4e1001c2d389974c5040a", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Append one genuine matched target index and preserve all previous mapped positions by beta extension." }, { "admission_dependencies": [ "jordan_enumeration_index_map_empty", "le_trans", "le_succ_self", "jordan_enumeration_position_match_exists", "jordan_enumeration_index_map_append" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 44, "body_proof_nodes": 127, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "induction q", "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "intro hq", "exists 0", "exists 0", "specialize jordan_enumeration_index_map_empty (k)", "specialize jordan_enumeration_index_map_empty (A)", "specialize jordan_enumeration_index_map_empty (B)", "specialize jordan_enumeration_index_map_empty (C)", "specialize jordan_enumeration_index_map_empty (D)", "specialize jordan_enumeration_index_map_empty (E)", "specialize jordan_enumeration_index_map_empty (F)", "specialize jordan_enumeration_index_map_empty (G)", "specialize jordan_enumeration_index_map_empty (H)", "specialize jordan_enumeration_index_map_empty (0)", "specialize jordan_enumeration_index_map_empty (0)", "specialize jordan_enumeration_index_map_empty (v)", "apply jordan_enumeration_index_map_empty", "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "intro hq", "have hp : ∃ Z. ∃ W. ∀ jt_index_induction_prefix. Lt(jt_index_induction_prefix,q) → ∃ x. BetaAt(Z,W,jt_index_induction_prefix,x) ∧ (Lt(x,v) ∧ (∀ y. ∀ z. ∀ n. ∀ m. BetaAt(A,B,jt_index_induction_prefix,y) ∧ BetaAt(C,D,jt_index_induction_prefix,z) → BetaAt(E,F,x,n) ∧ BetaAt(G,H,x,m) → IntegerVectorZero(y,z,n,m,k)))", "specialize IH (k)", "specialize IH (n)", "specialize IH (A)", "specialize IH (B)", "specialize IH (C)", "specialize IH (D)", "specialize IH (u)", "specialize IH (E)", "specialize IH (F)", "specialize IH (G)", "specialize IH (H)", "specialize IH (v)", "apply IH", "exact hl", "exact hr", "specialize le_trans (q)", "specialize le_trans (S q)", "specialize le_trans (u)", "apply le_trans", "specialize le_succ_self (q)", "apply le_succ_self", "exact hq", "cases hp", "cases hp_witness", "have hm : ∃ j. Lt(j,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,q,x) ∧ BetaAt(C,D,q,y) → BetaAt(E,F,j,z) ∧ BetaAt(G,H,j,n) → IntegerVectorZero(x,y,z,n,k))", "specialize jordan_enumeration_position_match_exists (k)", "specialize jordan_enumeration_position_match_exists (n)", "specialize jordan_enumeration_position_match_exists (A)", "specialize jordan_enumeration_position_match_exists (B)", "specialize jordan_enumeration_position_match_exists (C)", "specialize jordan_enumeration_position_match_exists (D)", "specialize jordan_enumeration_position_match_exists (u)", "specialize jordan_enumeration_position_match_exists (E)", "specialize jordan_enumeration_position_match_exists (F)", "specialize jordan_enumeration_position_match_exists (G)", "specialize jordan_enumeration_position_match_exists (H)", "specialize jordan_enumeration_position_match_exists (v)", "specialize jordan_enumeration_position_match_exists (q)", "apply jordan_enumeration_position_match_exists", "exact hl", "exact hr", "exact hq", "cases hm", "cases hm_witness", "specialize jordan_enumeration_index_map_append (k)", "specialize jordan_enumeration_index_map_append (A)", "specialize jordan_enumeration_index_map_append (B)", "specialize jordan_enumeration_index_map_append (C)", "specialize jordan_enumeration_index_map_append (D)", "specialize jordan_enumeration_index_map_append (E)", "specialize jordan_enumeration_index_map_append (F)", "specialize jordan_enumeration_index_map_append (G)", "specialize jordan_enumeration_index_map_append (H)", "specialize jordan_enumeration_index_map_append (x)", "specialize jordan_enumeration_index_map_append (x1)", "specialize jordan_enumeration_index_map_append (q)", "specialize jordan_enumeration_index_map_append (v)", "specialize jordan_enumeration_index_map_append (x2)", "apply jordan_enumeration_index_map_append", "exact hp_witness_witness", "exact hm_witness_left", "exact hm_witness_right" ], "defined_statement": "∀ q. ∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. JordanTupleEnumeration(k,n,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → Le(q,u) → ∃ x. ∃ y. ∀ z. Lt(z,q) → ∃ m. BetaAt(x,y,z,m) ∧ (Lt(m,v) ∧ (∀ i. ∀ j. ∀ w. ∀ x0. BetaAt(A,B,z,i) ∧ BetaAt(C,D,z,j) → BetaAt(E,F,m,w) ∧ BetaAt(G,H,m,x0) → IntegerVectorZero(i,j,w,x0,k)))", "defined_statement_sha256": "9227697ce02bfddfce0d8658aedda70169e6d24aff6fb32a2cd834a79201104c", "definition_uses": { "ND0121": 3, "ND0374": 2, "PD0001": 1, "PD0002": 5, "PD0013": 14 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "08c2140fa4220b50407d0f59cc64df65d06678ec9be0f0752f534a8e6af93647", "free_names": [], "script_definition_uses": { "ND0121": 2, "PD0002": 3, "PD0013": 9 }, "script_parts": [ [ { "kind": "text", "text": "induction q" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (0)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (0)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_empty (v)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_index_map_empty" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hq" } ], [ { "kind": "text", "text": "have hp : " }, { "kind": "text", "text": "∃ Z. ∃ W. ∀ jt_index_induction_prefix. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(jt_index_induction_prefix,q)" }, { "kind": "text", "text": " → ∃ x. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,jt_index_induction_prefix,x)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,v)" }, { "kind": "text", "text": " ∧ (∀ y. ∀ z. ∀ n. ∀ m. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,jt_index_induction_prefix,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,jt_index_induction_prefix,z)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,x,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,x,m)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(y,z,n,m,k)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize IH (k)" } ], [ { "kind": "text", "text": "specialize IH (n)" } ], [ { "kind": "text", "text": "specialize IH (A)" } ], [ { "kind": "text", "text": "specialize IH (B)" } ], [ { "kind": "text", "text": "specialize IH (C)" } ], [ { "kind": "text", "text": "specialize IH (D)" } ], [ { "kind": "text", "text": "specialize IH (u)" } ], [ { "kind": "text", "text": "specialize IH (E)" } ], [ { "kind": "text", "text": "specialize IH (F)" } ], [ { "kind": "text", "text": "specialize IH (G)" } ], [ { "kind": "text", "text": "specialize IH (H)" } ], [ { "kind": "text", "text": "specialize IH (v)" } ], [ { "kind": "text", "text": "apply IH" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "specialize le_trans (q)" } ], [ { "kind": "text", "text": "specialize le_trans (S q)" } ], [ { "kind": "text", "text": "specialize le_trans (u)" } ], [ { "kind": "text", "text": "apply le_trans" } ], [ { "kind": "text", "text": "specialize le_succ_self (q)" } ], [ { "kind": "text", "text": "apply le_succ_self" } ], [ { "kind": "text", "text": "exact hq" } ], [ { "kind": "text", "text": "cases hp" } ], [ { "kind": "text", "text": "cases hp_witness" } ], [ { "kind": "text", "text": "have hm : " }, { "kind": "text", "text": "∃ j. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,q,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,q,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_position_match_exists (q)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_position_match_exists" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hq" } ], [ { "kind": "text", "text": "cases hm" } ], [ { "kind": "text", "text": "cases hm_witness" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (x1)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (q)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_append (x2)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_index_map_append" } ], [ { "kind": "text", "text": "exact hp_witness_witness" } ], [ { "kind": "text", "text": "exact hm_witness_left" } ], [ { "kind": "text", "text": "exact hm_witness_right" } ] ], "statement_definition_uses": { "ND0121": 1, "ND0374": 2, "PD0001": 1, "PD0002": 2, "PD0013": 5 }, "statement_parts": [ { "kind": "text", "text": "∀ q. ∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0001", "kind": "definition", "text": "Le(q,u)" }, { "kind": "text", "text": " → ∃ x. ∃ y. ∀ z. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,q)" }, { "kind": "text", "text": " → ∃ m. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(x,y,z,m)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(m,v)" }, { "kind": "text", "text": " ∧ (∀ i. ∀ j. ∀ w. ∀ x0. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,z,i)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,z,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,m,w)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,m,x0)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(i,j,w,x0,k)" }, { "kind": "text", "text": "))" } ] }, "dependencies": [ "jordan_enumeration_index_map_empty", "le_trans", "le_succ_self", "jordan_enumeration_position_match_exists", "jordan_enumeration_index_map_append" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT0050", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_index_map_exists", "original_ha_bundle_verified": true, "proof_bundle_node_id": 341, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "induction q", "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "intro hq", "exists 0", "exists 0", "specialize jordan_enumeration_index_map_empty (k)", "specialize jordan_enumeration_index_map_empty (A)", "specialize jordan_enumeration_index_map_empty (B)", "specialize jordan_enumeration_index_map_empty (C)", "specialize jordan_enumeration_index_map_empty (D)", "specialize jordan_enumeration_index_map_empty (E)", "specialize jordan_enumeration_index_map_empty (F)", "specialize jordan_enumeration_index_map_empty (G)", "specialize jordan_enumeration_index_map_empty (H)", "specialize jordan_enumeration_index_map_empty (0)", "specialize jordan_enumeration_index_map_empty (0)", "specialize jordan_enumeration_index_map_empty (v)", "apply jordan_enumeration_index_map_empty", "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "intro hq", "have hp : exists Z W. forall jt_index_induction_prefix. (exists jt_gap_induction_prefixindex. jt_gap_induction_prefixindex+S (jt_index_induction_prefix)=(q)) -> exists jt_image_induction_prefix. ((((exists fs_h_jt_induction_prefixat. fs_h_jt_induction_prefixat + S (jt_image_induction_prefix) = S ((S (jt_index_induction_prefix)) * W)) /\\ exists fs_q_jt_induction_prefixat. Z = fs_q_jt_induction_prefixat * S ((S (jt_index_induction_prefix)) * W) + (jt_image_induction_prefix))) /\\ (((exists jt_gap_induction_prefixbound. jt_gap_induction_prefixbound+S (jt_image_induction_prefix)=(v)) /\\ (forall jt_b_induction_prefixmatch jt_c_induction_prefixmatch jt_d_induction_prefixmatch jt_e_induction_prefixmatch. (((((exists fs_h_jt_induction_prefixmatchleftcode. fs_h_jt_induction_prefixmatchleftcode + S (jt_b_induction_prefixmatch) = S ((S (jt_index_induction_prefix)) * B)) /\\ exists fs_q_jt_induction_prefixmatchleftcode. A = fs_q_jt_induction_prefixmatchleftcode * S ((S (jt_index_induction_prefix)) * B) + (jt_b_induction_prefixmatch))) /\\ (((exists fs_h_jt_induction_prefixmatchleftscale. fs_h_jt_induction_prefixmatchleftscale + S (jt_c_induction_prefixmatch) = S ((S (jt_index_induction_prefix)) * D)) /\\ exists fs_q_jt_induction_prefixmatchleftscale. C = fs_q_jt_induction_prefixmatchleftscale * S ((S (jt_index_induction_prefix)) * D) + (jt_c_induction_prefixmatch))))) -> (((((exists fs_h_jt_induction_prefixmatchrightcode. fs_h_jt_induction_prefixmatchrightcode + S (jt_d_induction_prefixmatch) = S ((S (jt_image_induction_prefix)) * F)) /\\ exists fs_q_jt_induction_prefixmatchrightcode. E = fs_q_jt_induction_prefixmatchrightcode * S ((S (jt_image_induction_prefix)) * F) + (jt_d_induction_prefixmatch))) /\\ (((exists fs_h_jt_induction_prefixmatchrightscale. fs_h_jt_induction_prefixmatchrightscale + S (jt_e_induction_prefixmatch) = S ((S (jt_image_induction_prefix)) * H)) /\\ exists fs_q_jt_induction_prefixmatchrightscale. G = fs_q_jt_induction_prefixmatchrightscale * S ((S (jt_image_induction_prefix)) * H) + (jt_e_induction_prefixmatch))))) -> (forall jt_index_induction_prefixmatchequal jt_left_induction_prefixmatchequal jt_right_induction_prefixmatchequal. (exists jt_gap_induction_prefixmatchequalindex. jt_gap_induction_prefixmatchequalindex+S (jt_index_induction_prefixmatchequal)=(k)) -> (((exists fs_h_jt_induction_prefixmatchequalleft. fs_h_jt_induction_prefixmatchequalleft + S (jt_left_induction_prefixmatchequal) = S ((S (jt_index_induction_prefixmatchequal)) * jt_c_induction_prefixmatch)) /\\ exists fs_q_jt_induction_prefixmatchequalleft. jt_b_induction_prefixmatch = fs_q_jt_induction_prefixmatchequalleft * S ((S (jt_index_induction_prefixmatchequal)) * jt_c_induction_prefixmatch) + (jt_left_induction_prefixmatchequal))) -> (((exists fs_h_jt_induction_prefixmatchequalright. fs_h_jt_induction_prefixmatchequalright + S (jt_right_induction_prefixmatchequal) = S ((S (jt_index_induction_prefixmatchequal)) * jt_e_induction_prefixmatch)) /\\ exists fs_q_jt_induction_prefixmatchequalright. jt_d_induction_prefixmatch = fs_q_jt_induction_prefixmatchequalright * S ((S (jt_index_induction_prefixmatchequal)) * jt_e_induction_prefixmatch) + (jt_right_induction_prefixmatchequal))) -> jt_left_induction_prefixmatchequal=jt_right_induction_prefixmatchequal)))))", "specialize IH (k)", "specialize IH (n)", "specialize IH (A)", "specialize IH (B)", "specialize IH (C)", "specialize IH (D)", "specialize IH (u)", "specialize IH (E)", "specialize IH (F)", "specialize IH (G)", "specialize IH (H)", "specialize IH (v)", "apply IH", "exact hl", "exact hr", "specialize le_trans (q)", "specialize le_trans (S q)", "specialize le_trans (u)", "apply le_trans", "specialize le_succ_self (q)", "apply le_succ_self", "exact hq", "cases hp", "cases hp_witness", "have hm : exists j. ((exists jt_gap_induction_image_bound. jt_gap_induction_image_bound+S (j)=(v)) /\\ (forall jt_b_induction_image_match jt_c_induction_image_match jt_d_induction_image_match jt_e_induction_image_match. (((((exists fs_h_jt_induction_image_matchleftcode. fs_h_jt_induction_image_matchleftcode + S (jt_b_induction_image_match) = S ((S (q)) * B)) /\\ exists fs_q_jt_induction_image_matchleftcode. A = fs_q_jt_induction_image_matchleftcode * S ((S (q)) * B) + (jt_b_induction_image_match))) /\\ (((exists fs_h_jt_induction_image_matchleftscale. fs_h_jt_induction_image_matchleftscale + S (jt_c_induction_image_match) = S ((S (q)) * D)) /\\ exists fs_q_jt_induction_image_matchleftscale. C = fs_q_jt_induction_image_matchleftscale * S ((S (q)) * D) + (jt_c_induction_image_match))))) -> (((((exists fs_h_jt_induction_image_matchrightcode. fs_h_jt_induction_image_matchrightcode + S (jt_d_induction_image_match) = S ((S (j)) * F)) /\\ exists fs_q_jt_induction_image_matchrightcode. E = fs_q_jt_induction_image_matchrightcode * S ((S (j)) * F) + (jt_d_induction_image_match))) /\\ (((exists fs_h_jt_induction_image_matchrightscale. fs_h_jt_induction_image_matchrightscale + S (jt_e_induction_image_match) = S ((S (j)) * H)) /\\ exists fs_q_jt_induction_image_matchrightscale. G = fs_q_jt_induction_image_matchrightscale * S ((S (j)) * H) + (jt_e_induction_image_match))))) -> (forall jt_index_induction_image_matchequal jt_left_induction_image_matchequal jt_right_induction_image_matchequal. (exists jt_gap_induction_image_matchequalindex. jt_gap_induction_image_matchequalindex+S (jt_index_induction_image_matchequal)=(k)) -> (((exists fs_h_jt_induction_image_matchequalleft. fs_h_jt_induction_image_matchequalleft + S (jt_left_induction_image_matchequal) = S ((S (jt_index_induction_image_matchequal)) * jt_c_induction_image_match)) /\\ exists fs_q_jt_induction_image_matchequalleft. jt_b_induction_image_match = fs_q_jt_induction_image_matchequalleft * S ((S (jt_index_induction_image_matchequal)) * jt_c_induction_image_match) + (jt_left_induction_image_matchequal))) -> (((exists fs_h_jt_induction_image_matchequalright. fs_h_jt_induction_image_matchequalright + S (jt_right_induction_image_matchequal) = S ((S (jt_index_induction_image_matchequal)) * jt_e_induction_image_match)) /\\ exists fs_q_jt_induction_image_matchequalright. jt_d_induction_image_match = fs_q_jt_induction_image_matchequalright * S ((S (jt_index_induction_image_matchequal)) * jt_e_induction_image_match) + (jt_right_induction_image_matchequal))) -> jt_left_induction_image_matchequal=jt_right_induction_image_matchequal)))", "specialize jordan_enumeration_position_match_exists (k)", "specialize jordan_enumeration_position_match_exists (n)", "specialize jordan_enumeration_position_match_exists (A)", "specialize jordan_enumeration_position_match_exists (B)", "specialize jordan_enumeration_position_match_exists (C)", "specialize jordan_enumeration_position_match_exists (D)", "specialize jordan_enumeration_position_match_exists (u)", "specialize jordan_enumeration_position_match_exists (E)", "specialize jordan_enumeration_position_match_exists (F)", "specialize jordan_enumeration_position_match_exists (G)", "specialize jordan_enumeration_position_match_exists (H)", "specialize jordan_enumeration_position_match_exists (v)", "specialize jordan_enumeration_position_match_exists (q)", "apply jordan_enumeration_position_match_exists", "exact hl", "exact hr", "exact hq", "cases hm", "cases hm_witness", "specialize jordan_enumeration_index_map_append (k)", "specialize jordan_enumeration_index_map_append (A)", "specialize jordan_enumeration_index_map_append (B)", "specialize jordan_enumeration_index_map_append (C)", "specialize jordan_enumeration_index_map_append (D)", "specialize jordan_enumeration_index_map_append (E)", "specialize jordan_enumeration_index_map_append (F)", "specialize jordan_enumeration_index_map_append (G)", "specialize jordan_enumeration_index_map_append (H)", "specialize jordan_enumeration_index_map_append (x)", "specialize jordan_enumeration_index_map_append (x1)", "specialize jordan_enumeration_index_map_append (q)", "specialize jordan_enumeration_index_map_append (v)", "specialize jordan_enumeration_index_map_append (x2)", "apply jordan_enumeration_index_map_append", "exact hp_witness_witness", "exact hm_witness_left", "exact hm_witness_right" ], "script_sha256": "dc2555bdd63251f212300598962d1fde8e176d1f9d859de8be05060f2a44558c", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "dc2555bdd63251f212300598962d1fde8e176d1f9d859de8be05060f2a44558c", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "08c2140fa4220b50407d0f59cc64df65d06678ec9be0f0752f534a8e6af93647" } ], "stable_member": false, "statement": "forall q k n A B C D u E F G H v. (((forall jt_i_exists_source. (exists jt_gap_exists_sourcesoundindex. jt_gap_exists_sourcesoundindex+S (jt_i_exists_source)=(u)) -> exists jt_b_exists_source jt_c_exists_source. ((((((exists fs_h_jt_exists_sourcesoundcode. fs_h_jt_exists_sourcesoundcode + S (jt_b_exists_source) = S ((S (jt_i_exists_source)) * B)) /\\ exists fs_q_jt_exists_sourcesoundcode. A = fs_q_jt_exists_sourcesoundcode * S ((S (jt_i_exists_source)) * B) + (jt_b_exists_source))) /\\ (((exists fs_h_jt_exists_sourcesoundscale. fs_h_jt_exists_sourcesoundscale + S (jt_c_exists_source) = S ((S (jt_i_exists_source)) * D)) /\\ exists fs_q_jt_exists_sourcesoundscale. C = fs_q_jt_exists_sourcesoundscale * S ((S (jt_i_exists_source)) * D) + (jt_c_exists_source))))) /\\ (((forall jt_index_exists_sourcebound. (exists jt_gap_exists_sourceboundindex. jt_gap_exists_sourceboundindex+S (jt_index_exists_sourcebound)=(k)) -> exists jt_value_exists_sourcebound. ((((exists fs_h_jt_exists_sourceboundat. fs_h_jt_exists_sourceboundat + S (jt_value_exists_sourcebound) = S ((S (jt_index_exists_sourcebound)) * jt_c_exists_source)) /\\ exists fs_q_jt_exists_sourceboundat. jt_b_exists_source = fs_q_jt_exists_sourceboundat * S ((S (jt_index_exists_sourcebound)) * jt_c_exists_source) + (jt_value_exists_sourcebound))) /\\ (exists jt_gap_exists_sourceboundvalue. jt_gap_exists_sourceboundvalue+S (jt_value_exists_sourcebound)=(n)))) /\\ (forall jt_divisor_exists_sourceprimitive. (exists jt_factor_exists_sourceprimitivemodulus. (n)=(jt_divisor_exists_sourceprimitive)*jt_factor_exists_sourceprimitivemodulus) -> (forall jt_index_exists_sourceprimitivecoordinates jt_value_exists_sourceprimitivecoordinates. (exists jt_gap_exists_sourceprimitivecoordinatesindex. jt_gap_exists_sourceprimitivecoordinatesindex+S (jt_index_exists_sourceprimitivecoordinates)=(k)) -> (((exists fs_h_jt_exists_sourceprimitivecoordinatesat. fs_h_jt_exists_sourceprimitivecoordinatesat + S (jt_value_exists_sourceprimitivecoordinates) = S ((S (jt_index_exists_sourceprimitivecoordinates)) * jt_c_exists_source)) /\\ exists fs_q_jt_exists_sourceprimitivecoordinatesat. jt_b_exists_source = fs_q_jt_exists_sourceprimitivecoordinatesat * S ((S (jt_index_exists_sourceprimitivecoordinates)) * jt_c_exists_source) + (jt_value_exists_sourceprimitivecoordinates))) -> (exists jt_factor_exists_sourceprimitivecoordinatesdivides. (jt_value_exists_sourceprimitivecoordinates)=(jt_divisor_exists_sourceprimitive)*jt_factor_exists_sourceprimitivecoordinatesdivides)) -> jt_divisor_exists_sourceprimitive=1))))) /\\ (((forall jt_b_exists_source jt_c_exists_source. (forall jt_index_exists_sourceinputbound. (exists jt_gap_exists_sourceinputboundindex. jt_gap_exists_sourceinputboundindex+S (jt_index_exists_sourceinputbound)=(k)) -> exists jt_value_exists_sourceinputbound. ((((exists fs_h_jt_exists_sourceinputboundat. fs_h_jt_exists_sourceinputboundat + S (jt_value_exists_sourceinputbound) = S ((S (jt_index_exists_sourceinputbound)) * jt_c_exists_source)) /\\ exists fs_q_jt_exists_sourceinputboundat. jt_b_exists_source = fs_q_jt_exists_sourceinputboundat * S ((S (jt_index_exists_sourceinputbound)) * jt_c_exists_source) + (jt_value_exists_sourceinputbound))) /\\ (exists jt_gap_exists_sourceinputboundvalue. jt_gap_exists_sourceinputboundvalue+S (jt_value_exists_sourceinputbound)=(n)))) -> (forall jt_divisor_exists_sourceinputprimitive. (exists jt_factor_exists_sourceinputprimitivemodulus. (n)=(jt_divisor_exists_sourceinputprimitive)*jt_factor_exists_sourceinputprimitivemodulus) -> (forall jt_index_exists_sourceinputprimitivecoordinates jt_value_exists_sourceinputprimitivecoordinates. (exists jt_gap_exists_sourceinputprimitivecoordinatesindex. jt_gap_exists_sourceinputprimitivecoordinatesindex+S (jt_index_exists_sourceinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_exists_sourceinputprimitivecoordinatesat. fs_h_jt_exists_sourceinputprimitivecoordinatesat + S (jt_value_exists_sourceinputprimitivecoordinates) = S ((S (jt_index_exists_sourceinputprimitivecoordinates)) * jt_c_exists_source)) /\\ exists fs_q_jt_exists_sourceinputprimitivecoordinatesat. jt_b_exists_source = fs_q_jt_exists_sourceinputprimitivecoordinatesat * S ((S (jt_index_exists_sourceinputprimitivecoordinates)) * jt_c_exists_source) + (jt_value_exists_sourceinputprimitivecoordinates))) -> (exists jt_factor_exists_sourceinputprimitivecoordinatesdivides. (jt_value_exists_sourceinputprimitivecoordinates)=(jt_divisor_exists_sourceinputprimitive)*jt_factor_exists_sourceinputprimitivecoordinatesdivides)) -> jt_divisor_exists_sourceinputprimitive=1) -> exists jt_i_exists_source jt_d_exists_source jt_e_exists_source. ((exists jt_gap_exists_sourcecompleteindex. jt_gap_exists_sourcecompleteindex+S (jt_i_exists_source)=(u)) /\\ (((((((exists fs_h_jt_exists_sourcecompletecode. fs_h_jt_exists_sourcecompletecode + S (jt_d_exists_source) = S ((S (jt_i_exists_source)) * B)) /\\ exists fs_q_jt_exists_sourcecompletecode. A = fs_q_jt_exists_sourcecompletecode * S ((S (jt_i_exists_source)) * B) + (jt_d_exists_source))) /\\ (((exists fs_h_jt_exists_sourcecompletescale. fs_h_jt_exists_sourcecompletescale + S (jt_e_exists_source) = S ((S (jt_i_exists_source)) * D)) /\\ exists fs_q_jt_exists_sourcecompletescale. C = fs_q_jt_exists_sourcecompletescale * S ((S (jt_i_exists_source)) * D) + (jt_e_exists_source))))) /\\ (forall jt_index_exists_sourcerepresented jt_left_exists_sourcerepresented jt_right_exists_sourcerepresented. (exists jt_gap_exists_sourcerepresentedindex. jt_gap_exists_sourcerepresentedindex+S (jt_index_exists_sourcerepresented)=(k)) -> (((exists fs_h_jt_exists_sourcerepresentedleft. fs_h_jt_exists_sourcerepresentedleft + S (jt_left_exists_sourcerepresented) = S ((S (jt_index_exists_sourcerepresented)) * jt_c_exists_source)) /\\ exists fs_q_jt_exists_sourcerepresentedleft. jt_b_exists_source = fs_q_jt_exists_sourcerepresentedleft * S ((S (jt_index_exists_sourcerepresented)) * jt_c_exists_source) + (jt_left_exists_sourcerepresented))) -> (((exists fs_h_jt_exists_sourcerepresentedright. fs_h_jt_exists_sourcerepresentedright + S (jt_right_exists_sourcerepresented) = S ((S (jt_index_exists_sourcerepresented)) * jt_e_exists_source)) /\\ exists fs_q_jt_exists_sourcerepresentedright. jt_d_exists_source = fs_q_jt_exists_sourcerepresentedright * S ((S (jt_index_exists_sourcerepresented)) * jt_e_exists_source) + (jt_right_exists_sourcerepresented))) -> jt_left_exists_sourcerepresented=jt_right_exists_sourcerepresented))))) /\\ (forall jt_i_exists_source jt_h_exists_source jt_b_exists_source jt_c_exists_source jt_d_exists_source jt_e_exists_source. (exists jt_gap_exists_sourcefirstindex. jt_gap_exists_sourcefirstindex+S (jt_i_exists_source)=(u)) -> (exists jt_gap_exists_sourcesecondindex. jt_gap_exists_sourcesecondindex+S (jt_h_exists_source)=(u)) -> (((((exists fs_h_jt_exists_sourcefirstcode. fs_h_jt_exists_sourcefirstcode + S (jt_b_exists_source) = S ((S (jt_i_exists_source)) * B)) /\\ exists fs_q_jt_exists_sourcefirstcode. A = fs_q_jt_exists_sourcefirstcode * S ((S (jt_i_exists_source)) * B) + (jt_b_exists_source))) /\\ (((exists fs_h_jt_exists_sourcefirstscale. fs_h_jt_exists_sourcefirstscale + S (jt_c_exists_source) = S ((S (jt_i_exists_source)) * D)) /\\ exists fs_q_jt_exists_sourcefirstscale. C = fs_q_jt_exists_sourcefirstscale * S ((S (jt_i_exists_source)) * D) + (jt_c_exists_source))))) -> (((((exists fs_h_jt_exists_sourcesecondcode. fs_h_jt_exists_sourcesecondcode + S (jt_d_exists_source) = S ((S (jt_h_exists_source)) * B)) /\\ exists fs_q_jt_exists_sourcesecondcode. A = fs_q_jt_exists_sourcesecondcode * S ((S (jt_h_exists_source)) * B) + (jt_d_exists_source))) /\\ (((exists fs_h_jt_exists_sourcesecondscale. fs_h_jt_exists_sourcesecondscale + S (jt_e_exists_source) = S ((S (jt_h_exists_source)) * D)) /\\ exists fs_q_jt_exists_sourcesecondscale. C = fs_q_jt_exists_sourcesecondscale * S ((S (jt_h_exists_source)) * D) + (jt_e_exists_source))))) -> (forall jt_index_exists_sourcesame jt_left_exists_sourcesame jt_right_exists_sourcesame. (exists jt_gap_exists_sourcesameindex. jt_gap_exists_sourcesameindex+S (jt_index_exists_sourcesame)=(k)) -> (((exists fs_h_jt_exists_sourcesameleft. fs_h_jt_exists_sourcesameleft + S (jt_left_exists_sourcesame) = S ((S (jt_index_exists_sourcesame)) * jt_c_exists_source)) /\\ exists fs_q_jt_exists_sourcesameleft. jt_b_exists_source = fs_q_jt_exists_sourcesameleft * S ((S (jt_index_exists_sourcesame)) * jt_c_exists_source) + (jt_left_exists_sourcesame))) -> (((exists fs_h_jt_exists_sourcesameright. fs_h_jt_exists_sourcesameright + S (jt_right_exists_sourcesame) = S ((S (jt_index_exists_sourcesame)) * jt_e_exists_source)) /\\ exists fs_q_jt_exists_sourcesameright. jt_d_exists_source = fs_q_jt_exists_sourcesameright * S ((S (jt_index_exists_sourcesame)) * jt_e_exists_source) + (jt_right_exists_sourcesame))) -> jt_left_exists_sourcesame=jt_right_exists_sourcesame) -> jt_i_exists_source=jt_h_exists_source))))) -> (((forall jt_i_exists_target. (exists jt_gap_exists_targetsoundindex. jt_gap_exists_targetsoundindex+S (jt_i_exists_target)=(v)) -> exists jt_b_exists_target jt_c_exists_target. ((((((exists fs_h_jt_exists_targetsoundcode. fs_h_jt_exists_targetsoundcode + S (jt_b_exists_target) = S ((S (jt_i_exists_target)) * F)) /\\ exists fs_q_jt_exists_targetsoundcode. E = fs_q_jt_exists_targetsoundcode * S ((S (jt_i_exists_target)) * F) + (jt_b_exists_target))) /\\ (((exists fs_h_jt_exists_targetsoundscale. fs_h_jt_exists_targetsoundscale + S (jt_c_exists_target) = S ((S (jt_i_exists_target)) * H)) /\\ exists fs_q_jt_exists_targetsoundscale. G = fs_q_jt_exists_targetsoundscale * S ((S (jt_i_exists_target)) * H) + (jt_c_exists_target))))) /\\ (((forall jt_index_exists_targetbound. (exists jt_gap_exists_targetboundindex. jt_gap_exists_targetboundindex+S (jt_index_exists_targetbound)=(k)) -> exists jt_value_exists_targetbound. ((((exists fs_h_jt_exists_targetboundat. fs_h_jt_exists_targetboundat + S (jt_value_exists_targetbound) = S ((S (jt_index_exists_targetbound)) * jt_c_exists_target)) /\\ exists fs_q_jt_exists_targetboundat. jt_b_exists_target = fs_q_jt_exists_targetboundat * S ((S (jt_index_exists_targetbound)) * jt_c_exists_target) + (jt_value_exists_targetbound))) /\\ (exists jt_gap_exists_targetboundvalue. jt_gap_exists_targetboundvalue+S (jt_value_exists_targetbound)=(n)))) /\\ (forall jt_divisor_exists_targetprimitive. (exists jt_factor_exists_targetprimitivemodulus. (n)=(jt_divisor_exists_targetprimitive)*jt_factor_exists_targetprimitivemodulus) -> (forall jt_index_exists_targetprimitivecoordinates jt_value_exists_targetprimitivecoordinates. (exists jt_gap_exists_targetprimitivecoordinatesindex. jt_gap_exists_targetprimitivecoordinatesindex+S (jt_index_exists_targetprimitivecoordinates)=(k)) -> (((exists fs_h_jt_exists_targetprimitivecoordinatesat. fs_h_jt_exists_targetprimitivecoordinatesat + S (jt_value_exists_targetprimitivecoordinates) = S ((S (jt_index_exists_targetprimitivecoordinates)) * jt_c_exists_target)) /\\ exists fs_q_jt_exists_targetprimitivecoordinatesat. jt_b_exists_target = fs_q_jt_exists_targetprimitivecoordinatesat * S ((S (jt_index_exists_targetprimitivecoordinates)) * jt_c_exists_target) + (jt_value_exists_targetprimitivecoordinates))) -> (exists jt_factor_exists_targetprimitivecoordinatesdivides. (jt_value_exists_targetprimitivecoordinates)=(jt_divisor_exists_targetprimitive)*jt_factor_exists_targetprimitivecoordinatesdivides)) -> jt_divisor_exists_targetprimitive=1))))) /\\ (((forall jt_b_exists_target jt_c_exists_target. (forall jt_index_exists_targetinputbound. (exists jt_gap_exists_targetinputboundindex. jt_gap_exists_targetinputboundindex+S (jt_index_exists_targetinputbound)=(k)) -> exists jt_value_exists_targetinputbound. ((((exists fs_h_jt_exists_targetinputboundat. fs_h_jt_exists_targetinputboundat + S (jt_value_exists_targetinputbound) = S ((S (jt_index_exists_targetinputbound)) * jt_c_exists_target)) /\\ exists fs_q_jt_exists_targetinputboundat. jt_b_exists_target = fs_q_jt_exists_targetinputboundat * S ((S (jt_index_exists_targetinputbound)) * jt_c_exists_target) + (jt_value_exists_targetinputbound))) /\\ (exists jt_gap_exists_targetinputboundvalue. jt_gap_exists_targetinputboundvalue+S (jt_value_exists_targetinputbound)=(n)))) -> (forall jt_divisor_exists_targetinputprimitive. (exists jt_factor_exists_targetinputprimitivemodulus. (n)=(jt_divisor_exists_targetinputprimitive)*jt_factor_exists_targetinputprimitivemodulus) -> (forall jt_index_exists_targetinputprimitivecoordinates jt_value_exists_targetinputprimitivecoordinates. (exists jt_gap_exists_targetinputprimitivecoordinatesindex. jt_gap_exists_targetinputprimitivecoordinatesindex+S (jt_index_exists_targetinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_exists_targetinputprimitivecoordinatesat. fs_h_jt_exists_targetinputprimitivecoordinatesat + S (jt_value_exists_targetinputprimitivecoordinates) = S ((S (jt_index_exists_targetinputprimitivecoordinates)) * jt_c_exists_target)) /\\ exists fs_q_jt_exists_targetinputprimitivecoordinatesat. jt_b_exists_target = fs_q_jt_exists_targetinputprimitivecoordinatesat * S ((S (jt_index_exists_targetinputprimitivecoordinates)) * jt_c_exists_target) + (jt_value_exists_targetinputprimitivecoordinates))) -> (exists jt_factor_exists_targetinputprimitivecoordinatesdivides. (jt_value_exists_targetinputprimitivecoordinates)=(jt_divisor_exists_targetinputprimitive)*jt_factor_exists_targetinputprimitivecoordinatesdivides)) -> jt_divisor_exists_targetinputprimitive=1) -> exists jt_i_exists_target jt_d_exists_target jt_e_exists_target. ((exists jt_gap_exists_targetcompleteindex. jt_gap_exists_targetcompleteindex+S (jt_i_exists_target)=(v)) /\\ (((((((exists fs_h_jt_exists_targetcompletecode. fs_h_jt_exists_targetcompletecode + S (jt_d_exists_target) = S ((S (jt_i_exists_target)) * F)) /\\ exists fs_q_jt_exists_targetcompletecode. E = fs_q_jt_exists_targetcompletecode * S ((S (jt_i_exists_target)) * F) + (jt_d_exists_target))) /\\ (((exists fs_h_jt_exists_targetcompletescale. fs_h_jt_exists_targetcompletescale + S (jt_e_exists_target) = S ((S (jt_i_exists_target)) * H)) /\\ exists fs_q_jt_exists_targetcompletescale. G = fs_q_jt_exists_targetcompletescale * S ((S (jt_i_exists_target)) * H) + (jt_e_exists_target))))) /\\ (forall jt_index_exists_targetrepresented jt_left_exists_targetrepresented jt_right_exists_targetrepresented. (exists jt_gap_exists_targetrepresentedindex. jt_gap_exists_targetrepresentedindex+S (jt_index_exists_targetrepresented)=(k)) -> (((exists fs_h_jt_exists_targetrepresentedleft. fs_h_jt_exists_targetrepresentedleft + S (jt_left_exists_targetrepresented) = S ((S (jt_index_exists_targetrepresented)) * jt_c_exists_target)) /\\ exists fs_q_jt_exists_targetrepresentedleft. jt_b_exists_target = fs_q_jt_exists_targetrepresentedleft * S ((S (jt_index_exists_targetrepresented)) * jt_c_exists_target) + (jt_left_exists_targetrepresented))) -> (((exists fs_h_jt_exists_targetrepresentedright. fs_h_jt_exists_targetrepresentedright + S (jt_right_exists_targetrepresented) = S ((S (jt_index_exists_targetrepresented)) * jt_e_exists_target)) /\\ exists fs_q_jt_exists_targetrepresentedright. jt_d_exists_target = fs_q_jt_exists_targetrepresentedright * S ((S (jt_index_exists_targetrepresented)) * jt_e_exists_target) + (jt_right_exists_targetrepresented))) -> jt_left_exists_targetrepresented=jt_right_exists_targetrepresented))))) /\\ (forall jt_i_exists_target jt_h_exists_target jt_b_exists_target jt_c_exists_target jt_d_exists_target jt_e_exists_target. (exists jt_gap_exists_targetfirstindex. jt_gap_exists_targetfirstindex+S (jt_i_exists_target)=(v)) -> (exists jt_gap_exists_targetsecondindex. jt_gap_exists_targetsecondindex+S (jt_h_exists_target)=(v)) -> (((((exists fs_h_jt_exists_targetfirstcode. fs_h_jt_exists_targetfirstcode + S (jt_b_exists_target) = S ((S (jt_i_exists_target)) * F)) /\\ exists fs_q_jt_exists_targetfirstcode. E = fs_q_jt_exists_targetfirstcode * S ((S (jt_i_exists_target)) * F) + (jt_b_exists_target))) /\\ (((exists fs_h_jt_exists_targetfirstscale. fs_h_jt_exists_targetfirstscale + S (jt_c_exists_target) = S ((S (jt_i_exists_target)) * H)) /\\ exists fs_q_jt_exists_targetfirstscale. G = fs_q_jt_exists_targetfirstscale * S ((S (jt_i_exists_target)) * H) + (jt_c_exists_target))))) -> (((((exists fs_h_jt_exists_targetsecondcode. fs_h_jt_exists_targetsecondcode + S (jt_d_exists_target) = S ((S (jt_h_exists_target)) * F)) /\\ exists fs_q_jt_exists_targetsecondcode. E = fs_q_jt_exists_targetsecondcode * S ((S (jt_h_exists_target)) * F) + (jt_d_exists_target))) /\\ (((exists fs_h_jt_exists_targetsecondscale. fs_h_jt_exists_targetsecondscale + S (jt_e_exists_target) = S ((S (jt_h_exists_target)) * H)) /\\ exists fs_q_jt_exists_targetsecondscale. G = fs_q_jt_exists_targetsecondscale * S ((S (jt_h_exists_target)) * H) + (jt_e_exists_target))))) -> (forall jt_index_exists_targetsame jt_left_exists_targetsame jt_right_exists_targetsame. (exists jt_gap_exists_targetsameindex. jt_gap_exists_targetsameindex+S (jt_index_exists_targetsame)=(k)) -> (((exists fs_h_jt_exists_targetsameleft. fs_h_jt_exists_targetsameleft + S (jt_left_exists_targetsame) = S ((S (jt_index_exists_targetsame)) * jt_c_exists_target)) /\\ exists fs_q_jt_exists_targetsameleft. jt_b_exists_target = fs_q_jt_exists_targetsameleft * S ((S (jt_index_exists_targetsame)) * jt_c_exists_target) + (jt_left_exists_targetsame))) -> (((exists fs_h_jt_exists_targetsameright. fs_h_jt_exists_targetsameright + S (jt_right_exists_targetsame) = S ((S (jt_index_exists_targetsame)) * jt_e_exists_target)) /\\ exists fs_q_jt_exists_targetsameright. jt_d_exists_target = fs_q_jt_exists_targetsameright * S ((S (jt_index_exists_targetsame)) * jt_e_exists_target) + (jt_right_exists_targetsame))) -> jt_left_exists_targetsame=jt_right_exists_targetsame) -> jt_i_exists_target=jt_h_exists_target))))) -> (exists jt_gap_exists_bound. jt_gap_exists_bound+(q)=(u)) -> (exists Z W. forall jt_index_exists_map. (exists jt_gap_exists_mapindex. jt_gap_exists_mapindex+S (jt_index_exists_map)=(q)) -> exists jt_image_exists_map. ((((exists fs_h_jt_exists_mapat. fs_h_jt_exists_mapat + S (jt_image_exists_map) = S ((S (jt_index_exists_map)) * W)) /\\ exists fs_q_jt_exists_mapat. Z = fs_q_jt_exists_mapat * S ((S (jt_index_exists_map)) * W) + (jt_image_exists_map))) /\\ (((exists jt_gap_exists_mapbound. jt_gap_exists_mapbound+S (jt_image_exists_map)=(v)) /\\ (forall jt_b_exists_mapmatch jt_c_exists_mapmatch jt_d_exists_mapmatch jt_e_exists_mapmatch. (((((exists fs_h_jt_exists_mapmatchleftcode. fs_h_jt_exists_mapmatchleftcode + S (jt_b_exists_mapmatch) = S ((S (jt_index_exists_map)) * B)) /\\ exists fs_q_jt_exists_mapmatchleftcode. A = fs_q_jt_exists_mapmatchleftcode * S ((S (jt_index_exists_map)) * B) + (jt_b_exists_mapmatch))) /\\ (((exists fs_h_jt_exists_mapmatchleftscale. fs_h_jt_exists_mapmatchleftscale + S (jt_c_exists_mapmatch) = S ((S (jt_index_exists_map)) * D)) /\\ exists fs_q_jt_exists_mapmatchleftscale. C = fs_q_jt_exists_mapmatchleftscale * S ((S (jt_index_exists_map)) * D) + (jt_c_exists_mapmatch))))) -> (((((exists fs_h_jt_exists_mapmatchrightcode. fs_h_jt_exists_mapmatchrightcode + S (jt_d_exists_mapmatch) = S ((S (jt_image_exists_map)) * F)) /\\ exists fs_q_jt_exists_mapmatchrightcode. E = fs_q_jt_exists_mapmatchrightcode * S ((S (jt_image_exists_map)) * F) + (jt_d_exists_mapmatch))) /\\ (((exists fs_h_jt_exists_mapmatchrightscale. fs_h_jt_exists_mapmatchrightscale + S (jt_e_exists_mapmatch) = S ((S (jt_image_exists_map)) * H)) /\\ exists fs_q_jt_exists_mapmatchrightscale. G = fs_q_jt_exists_mapmatchrightscale * S ((S (jt_image_exists_map)) * H) + (jt_e_exists_mapmatch))))) -> (forall jt_index_exists_mapmatchequal jt_left_exists_mapmatchequal jt_right_exists_mapmatchequal. (exists jt_gap_exists_mapmatchequalindex. jt_gap_exists_mapmatchequalindex+S (jt_index_exists_mapmatchequal)=(k)) -> (((exists fs_h_jt_exists_mapmatchequalleft. fs_h_jt_exists_mapmatchequalleft + S (jt_left_exists_mapmatchequal) = S ((S (jt_index_exists_mapmatchequal)) * jt_c_exists_mapmatch)) /\\ exists fs_q_jt_exists_mapmatchequalleft. jt_b_exists_mapmatch = fs_q_jt_exists_mapmatchequalleft * S ((S (jt_index_exists_mapmatchequal)) * jt_c_exists_mapmatch) + (jt_left_exists_mapmatchequal))) -> (((exists fs_h_jt_exists_mapmatchequalright. fs_h_jt_exists_mapmatchequalright + S (jt_right_exists_mapmatchequal) = S ((S (jt_index_exists_mapmatchequal)) * jt_e_exists_mapmatch)) /\\ exists fs_q_jt_exists_mapmatchequalright. jt_d_exists_mapmatch = fs_q_jt_exists_mapmatchequalright * S ((S (jt_index_exists_mapmatchequal)) * jt_e_exists_mapmatch) + (jt_right_exists_mapmatchequal))) -> jt_left_exists_mapmatchequal=jt_right_exists_mapmatchequal))))))", "statement_sha256": "08c2140fa4220b50407d0f59cc64df65d06678ec9be0f0752f534a8e6af93647", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Finite induction constructs an actual beta map for every source prefix, with no finite-choice axiom." }, { "admission_dependencies": [ "beta_at_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 33, "body_proof_nodes": 91, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro Z", "intro W", "intro q", "intro v", "intro i", "intro j", "intro hm", "intro hi", "intro hat", "have hv : ∃ r. BetaAt(Z,W,i,r) ∧ (Lt(r,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) → BetaAt(E,F,r,z) ∧ BetaAt(G,H,r,n) → IntegerVectorZero(x,y,z,n,k)))", "specialize hm (i)", "apply hm", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_right", "have he : x=j", "specialize beta_at_unique (Z)", "specialize beta_at_unique (W)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (j)", "apply beta_at_unique", "exact hv_witness_left", "exact hat", "split", "rewrite he at hv_witness_right_left", "exact hv_witness_right_left", "rewrite he at hv_witness_right_right", "rewrite he at hv_witness_right_right", "rewrite he at hv_witness_right_right", "rewrite he at hv_witness_right_right", "exact hv_witness_right_right" ], "defined_statement": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ Z. ∀ W. ∀ q. ∀ v. ∀ i. ∀ j. (∀ x. Lt(x,q) → ∃ y. BetaAt(Z,W,x,y) ∧ (Lt(y,v) ∧ (∀ z. ∀ n. ∀ m. ∀ u. BetaAt(A,B,x,z) ∧ BetaAt(C,D,x,n) → BetaAt(E,F,y,m) ∧ BetaAt(G,H,y,u) → IntegerVectorZero(z,n,m,u,k)))) → Lt(i,q) → BetaAt(Z,W,i,j) → Lt(j,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) → BetaAt(E,F,j,z) ∧ BetaAt(G,H,j,n) → IntegerVectorZero(x,y,z,n,k))", "defined_statement_sha256": "09c2f0d9bec47a2c14196b9ce5458938004d48983f3ed8998886abbf66a91ece", "definition_uses": { "ND0121": 3, "PD0002": 5, "PD0013": 15 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "52e88d2ceee12522fcdddf484364f737a9bb1d888e03b4adccd42d8dd8644f91", "free_names": [], "script_definition_uses": { "ND0121": 1, "PD0002": 1, "PD0013": 5 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro Z" } ], [ { "kind": "text", "text": "intro W" } ], [ { "kind": "text", "text": "intro q" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hat" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ r. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,i,r)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(r,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,r,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,r,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize hm (i)" } ], [ { "kind": "text", "text": "apply hm" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_right" } ], [ { "kind": "text", "text": "have he : x=j" } ], [ { "kind": "text", "text": "specialize beta_at_unique (Z)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (W)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (j)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact hv_witness_left" } ], [ { "kind": "text", "text": "exact hat" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "rewrite he at hv_witness_right_left" } ], [ { "kind": "text", "text": "exact hv_witness_right_left" } ], [ { "kind": "text", "text": "rewrite he at hv_witness_right_right" } ], [ { "kind": "text", "text": "rewrite he at hv_witness_right_right" } ], [ { "kind": "text", "text": "rewrite he at hv_witness_right_right" } ], [ { "kind": "text", "text": "rewrite he at hv_witness_right_right" } ], [ { "kind": "text", "text": "exact hv_witness_right_right" } ] ], "statement_definition_uses": { "ND0121": 2, "PD0002": 4, "PD0013": 10 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ A. ∀ B. ∀ C. ∀ D. ∀ E. ∀ F. ∀ G. ∀ H. ∀ Z. ∀ W. ∀ q. ∀ v. ∀ i. ∀ j. (∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,q)" }, { "kind": "text", "text": " → ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,x,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,v)" }, { "kind": "text", "text": " ∧ (∀ z. ∀ n. ∀ m. ∀ u. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,n)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,y,m)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,y,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(z,n,m,u,k)" }, { "kind": "text", "text": "))) → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,q)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,i,j)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": ")" } ] }, "dependencies": [ "beta_at_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT0051", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_index_map_entry", "original_ha_bundle_verified": true, "proof_bundle_node_id": 342, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro A", "intro B", "intro C", "intro D", "intro E", "intro F", "intro G", "intro H", "intro Z", "intro W", "intro q", "intro v", "intro i", "intro j", "intro hm", "intro hi", "intro hat", "have hv : exists r. ((((exists fs_h_jt_actual_mapat. fs_h_jt_actual_mapat + S (r) = S ((S (i)) * W)) /\\ exists fs_q_jt_actual_mapat. Z = fs_q_jt_actual_mapat * S ((S (i)) * W) + (r))) /\\ (((exists jt_gap_actual_mapbound. jt_gap_actual_mapbound+S (r)=(v)) /\\ (forall jt_b_actual_mapmatch jt_c_actual_mapmatch jt_d_actual_mapmatch jt_e_actual_mapmatch. (((((exists fs_h_jt_actual_mapmatchleftcode. fs_h_jt_actual_mapmatchleftcode + S (jt_b_actual_mapmatch) = S ((S (i)) * B)) /\\ exists fs_q_jt_actual_mapmatchleftcode. A = fs_q_jt_actual_mapmatchleftcode * S ((S (i)) * B) + (jt_b_actual_mapmatch))) /\\ (((exists fs_h_jt_actual_mapmatchleftscale. fs_h_jt_actual_mapmatchleftscale + S (jt_c_actual_mapmatch) = S ((S (i)) * D)) /\\ exists fs_q_jt_actual_mapmatchleftscale. C = fs_q_jt_actual_mapmatchleftscale * S ((S (i)) * D) + (jt_c_actual_mapmatch))))) -> (((((exists fs_h_jt_actual_mapmatchrightcode. fs_h_jt_actual_mapmatchrightcode + S (jt_d_actual_mapmatch) = S ((S (r)) * F)) /\\ exists fs_q_jt_actual_mapmatchrightcode. E = fs_q_jt_actual_mapmatchrightcode * S ((S (r)) * F) + (jt_d_actual_mapmatch))) /\\ (((exists fs_h_jt_actual_mapmatchrightscale. fs_h_jt_actual_mapmatchrightscale + S (jt_e_actual_mapmatch) = S ((S (r)) * H)) /\\ exists fs_q_jt_actual_mapmatchrightscale. G = fs_q_jt_actual_mapmatchrightscale * S ((S (r)) * H) + (jt_e_actual_mapmatch))))) -> (forall jt_index_actual_mapmatchequal jt_left_actual_mapmatchequal jt_right_actual_mapmatchequal. (exists jt_gap_actual_mapmatchequalindex. jt_gap_actual_mapmatchequalindex+S (jt_index_actual_mapmatchequal)=(k)) -> (((exists fs_h_jt_actual_mapmatchequalleft. fs_h_jt_actual_mapmatchequalleft + S (jt_left_actual_mapmatchequal) = S ((S (jt_index_actual_mapmatchequal)) * jt_c_actual_mapmatch)) /\\ exists fs_q_jt_actual_mapmatchequalleft. jt_b_actual_mapmatch = fs_q_jt_actual_mapmatchequalleft * S ((S (jt_index_actual_mapmatchequal)) * jt_c_actual_mapmatch) + (jt_left_actual_mapmatchequal))) -> (((exists fs_h_jt_actual_mapmatchequalright. fs_h_jt_actual_mapmatchequalright + S (jt_right_actual_mapmatchequal) = S ((S (jt_index_actual_mapmatchequal)) * jt_e_actual_mapmatch)) /\\ exists fs_q_jt_actual_mapmatchequalright. jt_d_actual_mapmatch = fs_q_jt_actual_mapmatchequalright * S ((S (jt_index_actual_mapmatchequal)) * jt_e_actual_mapmatch) + (jt_right_actual_mapmatchequal))) -> jt_left_actual_mapmatchequal=jt_right_actual_mapmatchequal)))))", "specialize hm (i)", "apply hm", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_right", "have he : x=j", "specialize beta_at_unique (Z)", "specialize beta_at_unique (W)", "specialize beta_at_unique (i)", "specialize beta_at_unique (x)", "specialize beta_at_unique (j)", "apply beta_at_unique", "exact hv_witness_left", "exact hat", "split", "rewrite he at hv_witness_right_left", "exact hv_witness_right_left", "rewrite he at hv_witness_right_right", "rewrite he at hv_witness_right_right", "rewrite he at hv_witness_right_right", "rewrite he at hv_witness_right_right", "exact hv_witness_right_right" ], "script_sha256": "5156a9e3e1c8b06b74b92c4211de424c918321d7caddb38324481a3dc11635a0", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "5156a9e3e1c8b06b74b92c4211de424c918321d7caddb38324481a3dc11635a0", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "52e88d2ceee12522fcdddf484364f737a9bb1d888e03b4adccd42d8dd8644f91" } ], "stable_member": false, "statement": "forall k A B C D E F G H Z W q v i j. (forall jt_index_actual_given_map. (exists jt_gap_actual_given_mapindex. jt_gap_actual_given_mapindex+S (jt_index_actual_given_map)=(q)) -> exists jt_image_actual_given_map. ((((exists fs_h_jt_actual_given_mapat. fs_h_jt_actual_given_mapat + S (jt_image_actual_given_map) = S ((S (jt_index_actual_given_map)) * W)) /\\ exists fs_q_jt_actual_given_mapat. Z = fs_q_jt_actual_given_mapat * S ((S (jt_index_actual_given_map)) * W) + (jt_image_actual_given_map))) /\\ (((exists jt_gap_actual_given_mapbound. jt_gap_actual_given_mapbound+S (jt_image_actual_given_map)=(v)) /\\ (forall jt_b_actual_given_mapmatch jt_c_actual_given_mapmatch jt_d_actual_given_mapmatch jt_e_actual_given_mapmatch. (((((exists fs_h_jt_actual_given_mapmatchleftcode. fs_h_jt_actual_given_mapmatchleftcode + S (jt_b_actual_given_mapmatch) = S ((S (jt_index_actual_given_map)) * B)) /\\ exists fs_q_jt_actual_given_mapmatchleftcode. A = fs_q_jt_actual_given_mapmatchleftcode * S ((S (jt_index_actual_given_map)) * B) + (jt_b_actual_given_mapmatch))) /\\ (((exists fs_h_jt_actual_given_mapmatchleftscale. fs_h_jt_actual_given_mapmatchleftscale + S (jt_c_actual_given_mapmatch) = S ((S (jt_index_actual_given_map)) * D)) /\\ exists fs_q_jt_actual_given_mapmatchleftscale. C = fs_q_jt_actual_given_mapmatchleftscale * S ((S (jt_index_actual_given_map)) * D) + (jt_c_actual_given_mapmatch))))) -> (((((exists fs_h_jt_actual_given_mapmatchrightcode. fs_h_jt_actual_given_mapmatchrightcode + S (jt_d_actual_given_mapmatch) = S ((S (jt_image_actual_given_map)) * F)) /\\ exists fs_q_jt_actual_given_mapmatchrightcode. E = fs_q_jt_actual_given_mapmatchrightcode * S ((S (jt_image_actual_given_map)) * F) + (jt_d_actual_given_mapmatch))) /\\ (((exists fs_h_jt_actual_given_mapmatchrightscale. fs_h_jt_actual_given_mapmatchrightscale + S (jt_e_actual_given_mapmatch) = S ((S (jt_image_actual_given_map)) * H)) /\\ exists fs_q_jt_actual_given_mapmatchrightscale. G = fs_q_jt_actual_given_mapmatchrightscale * S ((S (jt_image_actual_given_map)) * H) + (jt_e_actual_given_mapmatch))))) -> (forall jt_index_actual_given_mapmatchequal jt_left_actual_given_mapmatchequal jt_right_actual_given_mapmatchequal. (exists jt_gap_actual_given_mapmatchequalindex. jt_gap_actual_given_mapmatchequalindex+S (jt_index_actual_given_mapmatchequal)=(k)) -> (((exists fs_h_jt_actual_given_mapmatchequalleft. fs_h_jt_actual_given_mapmatchequalleft + S (jt_left_actual_given_mapmatchequal) = S ((S (jt_index_actual_given_mapmatchequal)) * jt_c_actual_given_mapmatch)) /\\ exists fs_q_jt_actual_given_mapmatchequalleft. jt_b_actual_given_mapmatch = fs_q_jt_actual_given_mapmatchequalleft * S ((S (jt_index_actual_given_mapmatchequal)) * jt_c_actual_given_mapmatch) + (jt_left_actual_given_mapmatchequal))) -> (((exists fs_h_jt_actual_given_mapmatchequalright. fs_h_jt_actual_given_mapmatchequalright + S (jt_right_actual_given_mapmatchequal) = S ((S (jt_index_actual_given_mapmatchequal)) * jt_e_actual_given_mapmatch)) /\\ exists fs_q_jt_actual_given_mapmatchequalright. jt_d_actual_given_mapmatch = fs_q_jt_actual_given_mapmatchequalright * S ((S (jt_index_actual_given_mapmatchequal)) * jt_e_actual_given_mapmatch) + (jt_right_actual_given_mapmatchequal))) -> jt_left_actual_given_mapmatchequal=jt_right_actual_given_mapmatchequal)))))) -> (exists jt_gap_actual_index. jt_gap_actual_index+S (i)=(q)) -> (((exists fs_h_jt_actual_value. fs_h_jt_actual_value + S (j) = S ((S (i)) * W)) /\\ exists fs_q_jt_actual_value. Z = fs_q_jt_actual_value * S ((S (i)) * W) + (j))) -> (((exists jt_gap_actual_result_bound. jt_gap_actual_result_bound+S (j)=(v)) /\\ (forall jt_b_actual_result_match jt_c_actual_result_match jt_d_actual_result_match jt_e_actual_result_match. (((((exists fs_h_jt_actual_result_matchleftcode. fs_h_jt_actual_result_matchleftcode + S (jt_b_actual_result_match) = S ((S (i)) * B)) /\\ exists fs_q_jt_actual_result_matchleftcode. A = fs_q_jt_actual_result_matchleftcode * S ((S (i)) * B) + (jt_b_actual_result_match))) /\\ (((exists fs_h_jt_actual_result_matchleftscale. fs_h_jt_actual_result_matchleftscale + S (jt_c_actual_result_match) = S ((S (i)) * D)) /\\ exists fs_q_jt_actual_result_matchleftscale. C = fs_q_jt_actual_result_matchleftscale * S ((S (i)) * D) + (jt_c_actual_result_match))))) -> (((((exists fs_h_jt_actual_result_matchrightcode. fs_h_jt_actual_result_matchrightcode + S (jt_d_actual_result_match) = S ((S (j)) * F)) /\\ exists fs_q_jt_actual_result_matchrightcode. E = fs_q_jt_actual_result_matchrightcode * S ((S (j)) * F) + (jt_d_actual_result_match))) /\\ (((exists fs_h_jt_actual_result_matchrightscale. fs_h_jt_actual_result_matchrightscale + S (jt_e_actual_result_match) = S ((S (j)) * H)) /\\ exists fs_q_jt_actual_result_matchrightscale. G = fs_q_jt_actual_result_matchrightscale * S ((S (j)) * H) + (jt_e_actual_result_match))))) -> (forall jt_index_actual_result_matchequal jt_left_actual_result_matchequal jt_right_actual_result_matchequal. (exists jt_gap_actual_result_matchequalindex. jt_gap_actual_result_matchequalindex+S (jt_index_actual_result_matchequal)=(k)) -> (((exists fs_h_jt_actual_result_matchequalleft. fs_h_jt_actual_result_matchequalleft + S (jt_left_actual_result_matchequal) = S ((S (jt_index_actual_result_matchequal)) * jt_c_actual_result_match)) /\\ exists fs_q_jt_actual_result_matchequalleft. jt_b_actual_result_match = fs_q_jt_actual_result_matchequalleft * S ((S (jt_index_actual_result_matchequal)) * jt_c_actual_result_match) + (jt_left_actual_result_matchequal))) -> (((exists fs_h_jt_actual_result_matchequalright. fs_h_jt_actual_result_matchequalright + S (jt_right_actual_result_matchequal) = S ((S (jt_index_actual_result_matchequal)) * jt_e_actual_result_match)) /\\ exists fs_q_jt_actual_result_matchequalright. jt_d_actual_result_match = fs_q_jt_actual_result_matchequalright * S ((S (jt_index_actual_result_matchequal)) * jt_e_actual_result_match) + (jt_right_actual_result_matchequal))) -> jt_left_actual_result_matchequal=jt_right_actual_result_matchequal))))", "statement_sha256": "52e88d2ceee12522fcdddf484364f737a9bb1d888e03b4adccd42d8dd8644f91", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every actual decoded map value is bounded and matches the represented source tuple." }, { "admission_dependencies": [ "jordan_enumeration_index_map_entry", "jordan_tuple_equal_symm", "jordan_tuple_equal_trans", "jordan_enumeration_distinct" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 64, "body_proof_nodes": 199, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro Z", "intro W", "intro hl", "intro hr", "intro hm", "split", "intro i", "intro hi", "have hv : ∃ j. BetaAt(Z,W,i,j) ∧ (Lt(j,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) → BetaAt(E,F,j,z) ∧ BetaAt(G,H,j,n) → IntegerVectorZero(x,y,z,n,k)))", "specialize hm (i)", "apply hm", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_right", "exists x", "split", "exact hv_witness_left", "exact hv_witness_right_left", "intro i", "intro j", "intro r", "intro hi", "intro hj", "intro hat", "intro hbt", "have hmi : Lt(r,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,i,x) ∧ BetaAt(C,D,i,y) → BetaAt(E,F,r,z) ∧ BetaAt(G,H,r,n) → IntegerVectorZero(x,y,z,n,k))", "specialize jordan_enumeration_index_map_entry (k)", "specialize jordan_enumeration_index_map_entry (A)", "specialize jordan_enumeration_index_map_entry (B)", "specialize jordan_enumeration_index_map_entry (C)", "specialize jordan_enumeration_index_map_entry (D)", "specialize jordan_enumeration_index_map_entry (E)", "specialize jordan_enumeration_index_map_entry (F)", "specialize jordan_enumeration_index_map_entry (G)", "specialize jordan_enumeration_index_map_entry (H)", "specialize jordan_enumeration_index_map_entry (Z)", "specialize jordan_enumeration_index_map_entry (W)", "specialize jordan_enumeration_index_map_entry (u)", "specialize jordan_enumeration_index_map_entry (v)", "specialize jordan_enumeration_index_map_entry (i)", "specialize jordan_enumeration_index_map_entry (r)", "apply jordan_enumeration_index_map_entry", "exact hm", "exact hi", "exact hat", "have hmj : Lt(r,v) ∧ (∀ x. ∀ y. ∀ z. ∀ n. BetaAt(A,B,j,x) ∧ BetaAt(C,D,j,y) → BetaAt(E,F,r,z) ∧ BetaAt(G,H,r,n) → IntegerVectorZero(x,y,z,n,k))", "specialize jordan_enumeration_index_map_entry (k)", "specialize jordan_enumeration_index_map_entry (A)", "specialize jordan_enumeration_index_map_entry (B)", "specialize jordan_enumeration_index_map_entry (C)", "specialize jordan_enumeration_index_map_entry (D)", "specialize jordan_enumeration_index_map_entry (E)", "specialize jordan_enumeration_index_map_entry (F)", "specialize jordan_enumeration_index_map_entry (G)", "specialize jordan_enumeration_index_map_entry (H)", "specialize jordan_enumeration_index_map_entry (Z)", "specialize jordan_enumeration_index_map_entry (W)", "specialize jordan_enumeration_index_map_entry (u)", "specialize jordan_enumeration_index_map_entry (v)", "specialize jordan_enumeration_index_map_entry (j)", "specialize jordan_enumeration_index_map_entry (r)", "apply jordan_enumeration_index_map_entry", "exact hm", "exact hj", "exact hbt", "cases hmi", "cases hmj", "cases hl", "cases hr", "have ht : ∃ b. ∃ c. BetaAt(E,F,r,b) ∧ BetaAt(G,H,r,c) ∧ (BetaPrefixInto(b,c,k,n) ∧ JordanPrimitiveTuple(n,b,c,k))", "specialize hr_left (r)", "apply hr_left", "exact hmi_left", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_right", "have ha : ∃ b. ∃ c. BetaAt(A,B,i,b) ∧ BetaAt(C,D,i,c) ∧ (BetaPrefixInto(b,c,k,n) ∧ JordanPrimitiveTuple(n,b,c,k))", "specialize hl_left (i)", "apply hl_left", "exact hi", "cases ha", "cases ha_witness", "cases ha_witness_witness", "cases ha_witness_witness_right", "have hb : ∃ b. ∃ c. BetaAt(A,B,j,b) ∧ BetaAt(C,D,j,c) ∧ (BetaPrefixInto(b,c,k,n) ∧ JordanPrimitiveTuple(n,b,c,k))", "specialize hl_left (j)", "apply hl_left", "exact hj", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_right", "have hea : IntegerVectorZero(x2,x3,x,x1,k)", "specialize hmi_right (x2)", "specialize hmi_right (x3)", "specialize hmi_right (x)", "specialize hmi_right (x1)", "apply hmi_right", "exact ha_witness_witness_left", "exact ht_witness_witness_left", "have heb : IntegerVectorZero(x4,x5,x,x1,k)", "specialize hmj_right (x4)", "specialize hmj_right (x5)", "specialize hmj_right (x)", "specialize hmj_right (x1)", "apply hmj_right", "exact hb_witness_witness_left", "exact ht_witness_witness_left", "have hrev : IntegerVectorZero(x,x1,x4,x5,k)", "specialize jordan_tuple_equal_symm (x4)", "specialize jordan_tuple_equal_symm (x5)", "specialize jordan_tuple_equal_symm (x)", "specialize jordan_tuple_equal_symm (x1)", "specialize jordan_tuple_equal_symm (k)", "apply jordan_tuple_equal_symm", "exact heb", "have heq : IntegerVectorZero(x2,x3,x4,x5,k)", "specialize jordan_tuple_equal_trans (x2)", "specialize jordan_tuple_equal_trans (x3)", "specialize jordan_tuple_equal_trans (x)", "specialize jordan_tuple_equal_trans (x1)", "specialize jordan_tuple_equal_trans (x4)", "specialize jordan_tuple_equal_trans (x5)", "specialize jordan_tuple_equal_trans (k)", "apply jordan_tuple_equal_trans", "exact hea", "exact hrev", "specialize jordan_enumeration_distinct (k)", "specialize jordan_enumeration_distinct (n)", "specialize jordan_enumeration_distinct (A)", "specialize jordan_enumeration_distinct (B)", "specialize jordan_enumeration_distinct (C)", "specialize jordan_enumeration_distinct (D)", "specialize jordan_enumeration_distinct (u)", "specialize jordan_enumeration_distinct (i)", "specialize jordan_enumeration_distinct (j)", "specialize jordan_enumeration_distinct (x2)", "specialize jordan_enumeration_distinct (x3)", "specialize jordan_enumeration_distinct (x4)", "specialize jordan_enumeration_distinct (x5)", "apply jordan_enumeration_distinct", "exact hl", "exact hi", "exact hj", "exact ha_witness_witness_left", "exact hb_witness_witness_left", "exact heq" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ Z. ∀ W. JordanTupleEnumeration(k,n,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → (∀ x. Lt(x,u) → ∃ y. BetaAt(Z,W,x,y) ∧ (Lt(y,v) ∧ (∀ z. ∀ m. ∀ i. ∀ j. BetaAt(A,B,x,z) ∧ BetaAt(C,D,x,m) → BetaAt(E,F,y,i) ∧ BetaAt(G,H,y,j) → IntegerVectorZero(z,m,i,j,k)))) → FiniteMatrixSelector(Z,W,u,v)", "defined_statement_sha256": "ef379470a6288e8c62b2a07e1851de430539d72e6b42fc7f68259072e97a36ad", "definition_uses": { "ND0113": 1, "ND0121": 8, "ND0262": 3, "ND0372": 3, "ND0374": 2, "PD0002": 5, "PD0013": 24 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "ee21ed551d2ea2469e34b393ffd50c729e41d07c3a92f11f10d8f7e66540050c", "free_names": [], "script_definition_uses": { "ND0121": 7, "ND0262": 3, "ND0372": 3, "PD0002": 3, "PD0013": 19 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro Z" } ], [ { "kind": "text", "text": "intro W" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "have hv : " }, { "kind": "text", "text": "∃ j. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,i,j)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(j,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,j,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,j,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize hm (i)" } ], [ { "kind": "text", "text": "apply hm" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hv" } ], [ { "kind": "text", "text": "cases hv_witness" } ], [ { "kind": "text", "text": "cases hv_witness_right" } ], [ { "kind": "text", "text": "exists x" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hv_witness_left" } ], [ { "kind": "text", "text": "exact hv_witness_right_left" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro r" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "intro hj" } ], [ { "kind": "text", "text": "intro hat" } ], [ { "kind": "text", "text": "intro hbt" } ], [ { "kind": "text", "text": "have hmi : " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(r,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,r,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,r,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (Z)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (W)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (i)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (r)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_index_map_entry" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hat" } ], [ { "kind": "text", "text": "have hmj : " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(r,v)" }, { "kind": "text", "text": " ∧ (∀ x. ∀ y. ∀ z. ∀ n. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,j,x)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,j,y)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,r,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,r,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,y,z,n,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (Z)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (W)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (j)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_entry (r)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_index_map_entry" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "exact hbt" } ], [ { "kind": "text", "text": "cases hmi" } ], [ { "kind": "text", "text": "cases hmj" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hr" } ], [ { "kind": "text", "text": "have ht : " }, { "kind": "text", "text": "∃ b. ∃ c. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,r,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,r,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hr_left (r)" } ], [ { "kind": "text", "text": "apply hr_left" } ], [ { "kind": "text", "text": "exact hmi_left" } ], [ { "kind": "text", "text": "cases ht" } ], [ { "kind": "text", "text": "cases ht_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness" } ], [ { "kind": "text", "text": "cases ht_witness_witness_right" } ], [ { "kind": "text", "text": "have ha : " }, { "kind": "text", "text": "∃ b. ∃ c. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,i,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,i,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hl_left (i)" } ], [ { "kind": "text", "text": "apply hl_left" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases ha" } ], [ { "kind": "text", "text": "cases ha_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness" } ], [ { "kind": "text", "text": "cases ha_witness_witness_right" } ], [ { "kind": "text", "text": "have hb : " }, { "kind": "text", "text": "∃ b. ∃ c. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,j,b)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,j,c)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": ")" } ], [ { "kind": "text", "text": "specialize hl_left (j)" } ], [ { "kind": "text", "text": "apply hl_left" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "cases hb" } ], [ { "kind": "text", "text": "cases hb_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness" } ], [ { "kind": "text", "text": "cases hb_witness_witness_right" } ], [ { "kind": "text", "text": "have hea : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x2,x3,x,x1,k)" } ], [ { "kind": "text", "text": "specialize hmi_right (x2)" } ], [ { "kind": "text", "text": "specialize hmi_right (x3)" } ], [ { "kind": "text", "text": "specialize hmi_right (x)" } ], [ { "kind": "text", "text": "specialize hmi_right (x1)" } ], [ { "kind": "text", "text": "apply hmi_right" } ], [ { "kind": "text", "text": "exact ha_witness_witness_left" } ], [ { "kind": "text", "text": "exact ht_witness_witness_left" } ], [ { "kind": "text", "text": "have heb : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x4,x5,x,x1,k)" } ], [ { "kind": "text", "text": "specialize hmj_right (x4)" } ], [ { "kind": "text", "text": "specialize hmj_right (x5)" } ], [ { "kind": "text", "text": "specialize hmj_right (x)" } ], [ { "kind": "text", "text": "specialize hmj_right (x1)" } ], [ { "kind": "text", "text": "apply hmj_right" } ], [ { "kind": "text", "text": "exact hb_witness_witness_left" } ], [ { "kind": "text", "text": "exact ht_witness_witness_left" } ], [ { "kind": "text", "text": "have hrev : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x,x1,x4,x5,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x5)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_symm (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_symm" } ], [ { "kind": "text", "text": "exact heb" } ], [ { "kind": "text", "text": "have heq : " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(x2,x3,x4,x5,k)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x2)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x3)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x1)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x4)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (x5)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_equal_trans (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_equal_trans" } ], [ { "kind": "text", "text": "exact hea" } ], [ { "kind": "text", "text": "exact hrev" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (i)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (j)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x2)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x3)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x4)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_distinct (x5)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_distinct" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "exact hj" } ], [ { "kind": "text", "text": "exact ha_witness_witness_left" } ], [ { "kind": "text", "text": "exact hb_witness_witness_left" } ], [ { "kind": "text", "text": "exact heq" } ] ], "statement_definition_uses": { "ND0113": 1, "ND0121": 1, "ND0374": 2, "PD0002": 2, "PD0013": 5 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. ∀ Z. ∀ W. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → (∀ x. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,u)" }, { "kind": "text", "text": " → ∃ y. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,x,y)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(y,v)" }, { "kind": "text", "text": " ∧ (∀ z. ∀ m. ∀ i. ∀ j. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,x,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,x,m)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,y,i)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,y,j)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(z,m,i,j,k)" }, { "kind": "text", "text": "))) → " }, { "definition": "ND0113", "kind": "definition", "text": "FiniteMatrixSelector(Z,W,u,v)" } ] }, "dependencies": [ "jordan_enumeration_index_map_entry", "jordan_tuple_equal_symm", "jordan_tuple_equal_trans", "jordan_enumeration_distinct" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT0052", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_index_map_bounded_injective", "original_ha_bundle_verified": true, "proof_bundle_node_id": 343, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro Z", "intro W", "intro hl", "intro hr", "intro hm", "split", "intro i", "intro hi", "have hv : exists j. ((((exists fs_h_jt_bounded_mapat. fs_h_jt_bounded_mapat + S (j) = S ((S (i)) * W)) /\\ exists fs_q_jt_bounded_mapat. Z = fs_q_jt_bounded_mapat * S ((S (i)) * W) + (j))) /\\ (((exists jt_gap_bounded_mapbound. jt_gap_bounded_mapbound+S (j)=(v)) /\\ (forall jt_b_bounded_mapmatch jt_c_bounded_mapmatch jt_d_bounded_mapmatch jt_e_bounded_mapmatch. (((((exists fs_h_jt_bounded_mapmatchleftcode. fs_h_jt_bounded_mapmatchleftcode + S (jt_b_bounded_mapmatch) = S ((S (i)) * B)) /\\ exists fs_q_jt_bounded_mapmatchleftcode. A = fs_q_jt_bounded_mapmatchleftcode * S ((S (i)) * B) + (jt_b_bounded_mapmatch))) /\\ (((exists fs_h_jt_bounded_mapmatchleftscale. fs_h_jt_bounded_mapmatchleftscale + S (jt_c_bounded_mapmatch) = S ((S (i)) * D)) /\\ exists fs_q_jt_bounded_mapmatchleftscale. C = fs_q_jt_bounded_mapmatchleftscale * S ((S (i)) * D) + (jt_c_bounded_mapmatch))))) -> (((((exists fs_h_jt_bounded_mapmatchrightcode. fs_h_jt_bounded_mapmatchrightcode + S (jt_d_bounded_mapmatch) = S ((S (j)) * F)) /\\ exists fs_q_jt_bounded_mapmatchrightcode. E = fs_q_jt_bounded_mapmatchrightcode * S ((S (j)) * F) + (jt_d_bounded_mapmatch))) /\\ (((exists fs_h_jt_bounded_mapmatchrightscale. fs_h_jt_bounded_mapmatchrightscale + S (jt_e_bounded_mapmatch) = S ((S (j)) * H)) /\\ exists fs_q_jt_bounded_mapmatchrightscale. G = fs_q_jt_bounded_mapmatchrightscale * S ((S (j)) * H) + (jt_e_bounded_mapmatch))))) -> (forall jt_index_bounded_mapmatchequal jt_left_bounded_mapmatchequal jt_right_bounded_mapmatchequal. (exists jt_gap_bounded_mapmatchequalindex. jt_gap_bounded_mapmatchequalindex+S (jt_index_bounded_mapmatchequal)=(k)) -> (((exists fs_h_jt_bounded_mapmatchequalleft. fs_h_jt_bounded_mapmatchequalleft + S (jt_left_bounded_mapmatchequal) = S ((S (jt_index_bounded_mapmatchequal)) * jt_c_bounded_mapmatch)) /\\ exists fs_q_jt_bounded_mapmatchequalleft. jt_b_bounded_mapmatch = fs_q_jt_bounded_mapmatchequalleft * S ((S (jt_index_bounded_mapmatchequal)) * jt_c_bounded_mapmatch) + (jt_left_bounded_mapmatchequal))) -> (((exists fs_h_jt_bounded_mapmatchequalright. fs_h_jt_bounded_mapmatchequalright + S (jt_right_bounded_mapmatchequal) = S ((S (jt_index_bounded_mapmatchequal)) * jt_e_bounded_mapmatch)) /\\ exists fs_q_jt_bounded_mapmatchequalright. jt_d_bounded_mapmatch = fs_q_jt_bounded_mapmatchequalright * S ((S (jt_index_bounded_mapmatchequal)) * jt_e_bounded_mapmatch) + (jt_right_bounded_mapmatchequal))) -> jt_left_bounded_mapmatchequal=jt_right_bounded_mapmatchequal)))))", "specialize hm (i)", "apply hm", "exact hi", "cases hv", "cases hv_witness", "cases hv_witness_right", "exists x", "split", "exact hv_witness_left", "exact hv_witness_right_left", "intro i", "intro j", "intro r", "intro hi", "intro hj", "intro hat", "intro hbt", "have hmi : ((exists jt_gap_hmibound. jt_gap_hmibound+S (r)=(v)) /\\ (forall jt_b_hmimatch jt_c_hmimatch jt_d_hmimatch jt_e_hmimatch. (((((exists fs_h_jt_hmimatchleftcode. fs_h_jt_hmimatchleftcode + S (jt_b_hmimatch) = S ((S (i)) * B)) /\\ exists fs_q_jt_hmimatchleftcode. A = fs_q_jt_hmimatchleftcode * S ((S (i)) * B) + (jt_b_hmimatch))) /\\ (((exists fs_h_jt_hmimatchleftscale. fs_h_jt_hmimatchleftscale + S (jt_c_hmimatch) = S ((S (i)) * D)) /\\ exists fs_q_jt_hmimatchleftscale. C = fs_q_jt_hmimatchleftscale * S ((S (i)) * D) + (jt_c_hmimatch))))) -> (((((exists fs_h_jt_hmimatchrightcode. fs_h_jt_hmimatchrightcode + S (jt_d_hmimatch) = S ((S (r)) * F)) /\\ exists fs_q_jt_hmimatchrightcode. E = fs_q_jt_hmimatchrightcode * S ((S (r)) * F) + (jt_d_hmimatch))) /\\ (((exists fs_h_jt_hmimatchrightscale. fs_h_jt_hmimatchrightscale + S (jt_e_hmimatch) = S ((S (r)) * H)) /\\ exists fs_q_jt_hmimatchrightscale. G = fs_q_jt_hmimatchrightscale * S ((S (r)) * H) + (jt_e_hmimatch))))) -> (forall jt_index_hmimatchequal jt_left_hmimatchequal jt_right_hmimatchequal. (exists jt_gap_hmimatchequalindex. jt_gap_hmimatchequalindex+S (jt_index_hmimatchequal)=(k)) -> (((exists fs_h_jt_hmimatchequalleft. fs_h_jt_hmimatchequalleft + S (jt_left_hmimatchequal) = S ((S (jt_index_hmimatchequal)) * jt_c_hmimatch)) /\\ exists fs_q_jt_hmimatchequalleft. jt_b_hmimatch = fs_q_jt_hmimatchequalleft * S ((S (jt_index_hmimatchequal)) * jt_c_hmimatch) + (jt_left_hmimatchequal))) -> (((exists fs_h_jt_hmimatchequalright. fs_h_jt_hmimatchequalright + S (jt_right_hmimatchequal) = S ((S (jt_index_hmimatchequal)) * jt_e_hmimatch)) /\\ exists fs_q_jt_hmimatchequalright. jt_d_hmimatch = fs_q_jt_hmimatchequalright * S ((S (jt_index_hmimatchequal)) * jt_e_hmimatch) + (jt_right_hmimatchequal))) -> jt_left_hmimatchequal=jt_right_hmimatchequal)))", "specialize jordan_enumeration_index_map_entry (k)", "specialize jordan_enumeration_index_map_entry (A)", "specialize jordan_enumeration_index_map_entry (B)", "specialize jordan_enumeration_index_map_entry (C)", "specialize jordan_enumeration_index_map_entry (D)", "specialize jordan_enumeration_index_map_entry (E)", "specialize jordan_enumeration_index_map_entry (F)", "specialize jordan_enumeration_index_map_entry (G)", "specialize jordan_enumeration_index_map_entry (H)", "specialize jordan_enumeration_index_map_entry (Z)", "specialize jordan_enumeration_index_map_entry (W)", "specialize jordan_enumeration_index_map_entry (u)", "specialize jordan_enumeration_index_map_entry (v)", "specialize jordan_enumeration_index_map_entry (i)", "specialize jordan_enumeration_index_map_entry (r)", "apply jordan_enumeration_index_map_entry", "exact hm", "exact hi", "exact hat", "have hmj : ((exists jt_gap_hmjbound. jt_gap_hmjbound+S (r)=(v)) /\\ (forall jt_b_hmjmatch jt_c_hmjmatch jt_d_hmjmatch jt_e_hmjmatch. (((((exists fs_h_jt_hmjmatchleftcode. fs_h_jt_hmjmatchleftcode + S (jt_b_hmjmatch) = S ((S (j)) * B)) /\\ exists fs_q_jt_hmjmatchleftcode. A = fs_q_jt_hmjmatchleftcode * S ((S (j)) * B) + (jt_b_hmjmatch))) /\\ (((exists fs_h_jt_hmjmatchleftscale. fs_h_jt_hmjmatchleftscale + S (jt_c_hmjmatch) = S ((S (j)) * D)) /\\ exists fs_q_jt_hmjmatchleftscale. C = fs_q_jt_hmjmatchleftscale * S ((S (j)) * D) + (jt_c_hmjmatch))))) -> (((((exists fs_h_jt_hmjmatchrightcode. fs_h_jt_hmjmatchrightcode + S (jt_d_hmjmatch) = S ((S (r)) * F)) /\\ exists fs_q_jt_hmjmatchrightcode. E = fs_q_jt_hmjmatchrightcode * S ((S (r)) * F) + (jt_d_hmjmatch))) /\\ (((exists fs_h_jt_hmjmatchrightscale. fs_h_jt_hmjmatchrightscale + S (jt_e_hmjmatch) = S ((S (r)) * H)) /\\ exists fs_q_jt_hmjmatchrightscale. G = fs_q_jt_hmjmatchrightscale * S ((S (r)) * H) + (jt_e_hmjmatch))))) -> (forall jt_index_hmjmatchequal jt_left_hmjmatchequal jt_right_hmjmatchequal. (exists jt_gap_hmjmatchequalindex. jt_gap_hmjmatchequalindex+S (jt_index_hmjmatchequal)=(k)) -> (((exists fs_h_jt_hmjmatchequalleft. fs_h_jt_hmjmatchequalleft + S (jt_left_hmjmatchequal) = S ((S (jt_index_hmjmatchequal)) * jt_c_hmjmatch)) /\\ exists fs_q_jt_hmjmatchequalleft. jt_b_hmjmatch = fs_q_jt_hmjmatchequalleft * S ((S (jt_index_hmjmatchequal)) * jt_c_hmjmatch) + (jt_left_hmjmatchequal))) -> (((exists fs_h_jt_hmjmatchequalright. fs_h_jt_hmjmatchequalright + S (jt_right_hmjmatchequal) = S ((S (jt_index_hmjmatchequal)) * jt_e_hmjmatch)) /\\ exists fs_q_jt_hmjmatchequalright. jt_d_hmjmatch = fs_q_jt_hmjmatchequalright * S ((S (jt_index_hmjmatchequal)) * jt_e_hmjmatch) + (jt_right_hmjmatchequal))) -> jt_left_hmjmatchequal=jt_right_hmjmatchequal)))", "specialize jordan_enumeration_index_map_entry (k)", "specialize jordan_enumeration_index_map_entry (A)", "specialize jordan_enumeration_index_map_entry (B)", "specialize jordan_enumeration_index_map_entry (C)", "specialize jordan_enumeration_index_map_entry (D)", "specialize jordan_enumeration_index_map_entry (E)", "specialize jordan_enumeration_index_map_entry (F)", "specialize jordan_enumeration_index_map_entry (G)", "specialize jordan_enumeration_index_map_entry (H)", "specialize jordan_enumeration_index_map_entry (Z)", "specialize jordan_enumeration_index_map_entry (W)", "specialize jordan_enumeration_index_map_entry (u)", "specialize jordan_enumeration_index_map_entry (v)", "specialize jordan_enumeration_index_map_entry (j)", "specialize jordan_enumeration_index_map_entry (r)", "apply jordan_enumeration_index_map_entry", "exact hm", "exact hj", "exact hbt", "cases hmi", "cases hmj", "cases hl", "cases hr", "have ht : exists b c. ((((((exists fs_h_jt_target_tupleentrycode. fs_h_jt_target_tupleentrycode + S (b) = S ((S (r)) * F)) /\\ exists fs_q_jt_target_tupleentrycode. E = fs_q_jt_target_tupleentrycode * S ((S (r)) * F) + (b))) /\\ (((exists fs_h_jt_target_tupleentryscale. fs_h_jt_target_tupleentryscale + S (c) = S ((S (r)) * H)) /\\ exists fs_q_jt_target_tupleentryscale. G = fs_q_jt_target_tupleentryscale * S ((S (r)) * H) + (c))))) /\\ (((forall jt_index_target_tuplebound. (exists jt_gap_target_tupleboundindex. jt_gap_target_tupleboundindex+S (jt_index_target_tuplebound)=(k)) -> exists jt_value_target_tuplebound. ((((exists fs_h_jt_target_tupleboundat. fs_h_jt_target_tupleboundat + S (jt_value_target_tuplebound) = S ((S (jt_index_target_tuplebound)) * c)) /\\ exists fs_q_jt_target_tupleboundat. b = fs_q_jt_target_tupleboundat * S ((S (jt_index_target_tuplebound)) * c) + (jt_value_target_tuplebound))) /\\ (exists jt_gap_target_tupleboundvalue. jt_gap_target_tupleboundvalue+S (jt_value_target_tuplebound)=(n)))) /\\ (forall jt_divisor_target_tupleprimitive. (exists jt_factor_target_tupleprimitivemodulus. (n)=(jt_divisor_target_tupleprimitive)*jt_factor_target_tupleprimitivemodulus) -> (forall jt_index_target_tupleprimitivecoordinates jt_value_target_tupleprimitivecoordinates. (exists jt_gap_target_tupleprimitivecoordinatesindex. jt_gap_target_tupleprimitivecoordinatesindex+S (jt_index_target_tupleprimitivecoordinates)=(k)) -> (((exists fs_h_jt_target_tupleprimitivecoordinatesat. fs_h_jt_target_tupleprimitivecoordinatesat + S (jt_value_target_tupleprimitivecoordinates) = S ((S (jt_index_target_tupleprimitivecoordinates)) * c)) /\\ exists fs_q_jt_target_tupleprimitivecoordinatesat. b = fs_q_jt_target_tupleprimitivecoordinatesat * S ((S (jt_index_target_tupleprimitivecoordinates)) * c) + (jt_value_target_tupleprimitivecoordinates))) -> (exists jt_factor_target_tupleprimitivecoordinatesdivides. (jt_value_target_tupleprimitivecoordinates)=(jt_divisor_target_tupleprimitive)*jt_factor_target_tupleprimitivecoordinatesdivides)) -> jt_divisor_target_tupleprimitive=1))))", "specialize hr_left (r)", "apply hr_left", "exact hmi_left", "cases ht", "cases ht_witness", "cases ht_witness_witness", "cases ht_witness_witness_right", "have ha : exists b c. ((((((exists fs_h_jt_first_tupleentrycode. fs_h_jt_first_tupleentrycode + S (b) = S ((S (i)) * B)) /\\ exists fs_q_jt_first_tupleentrycode. A = fs_q_jt_first_tupleentrycode * S ((S (i)) * B) + (b))) /\\ (((exists fs_h_jt_first_tupleentryscale. fs_h_jt_first_tupleentryscale + S (c) = S ((S (i)) * D)) /\\ exists fs_q_jt_first_tupleentryscale. C = fs_q_jt_first_tupleentryscale * S ((S (i)) * D) + (c))))) /\\ (((forall jt_index_first_tuplebound. (exists jt_gap_first_tupleboundindex. jt_gap_first_tupleboundindex+S (jt_index_first_tuplebound)=(k)) -> exists jt_value_first_tuplebound. ((((exists fs_h_jt_first_tupleboundat. fs_h_jt_first_tupleboundat + S (jt_value_first_tuplebound) = S ((S (jt_index_first_tuplebound)) * c)) /\\ exists fs_q_jt_first_tupleboundat. b = fs_q_jt_first_tupleboundat * S ((S (jt_index_first_tuplebound)) * c) + (jt_value_first_tuplebound))) /\\ (exists jt_gap_first_tupleboundvalue. jt_gap_first_tupleboundvalue+S (jt_value_first_tuplebound)=(n)))) /\\ (forall jt_divisor_first_tupleprimitive. (exists jt_factor_first_tupleprimitivemodulus. (n)=(jt_divisor_first_tupleprimitive)*jt_factor_first_tupleprimitivemodulus) -> (forall jt_index_first_tupleprimitivecoordinates jt_value_first_tupleprimitivecoordinates. (exists jt_gap_first_tupleprimitivecoordinatesindex. jt_gap_first_tupleprimitivecoordinatesindex+S (jt_index_first_tupleprimitivecoordinates)=(k)) -> (((exists fs_h_jt_first_tupleprimitivecoordinatesat. fs_h_jt_first_tupleprimitivecoordinatesat + S (jt_value_first_tupleprimitivecoordinates) = S ((S (jt_index_first_tupleprimitivecoordinates)) * c)) /\\ exists fs_q_jt_first_tupleprimitivecoordinatesat. b = fs_q_jt_first_tupleprimitivecoordinatesat * S ((S (jt_index_first_tupleprimitivecoordinates)) * c) + (jt_value_first_tupleprimitivecoordinates))) -> (exists jt_factor_first_tupleprimitivecoordinatesdivides. (jt_value_first_tupleprimitivecoordinates)=(jt_divisor_first_tupleprimitive)*jt_factor_first_tupleprimitivecoordinatesdivides)) -> jt_divisor_first_tupleprimitive=1))))", "specialize hl_left (i)", "apply hl_left", "exact hi", "cases ha", "cases ha_witness", "cases ha_witness_witness", "cases ha_witness_witness_right", "have hb : exists b c. ((((((exists fs_h_jt_second_tupleentrycode. fs_h_jt_second_tupleentrycode + S (b) = S ((S (j)) * B)) /\\ exists fs_q_jt_second_tupleentrycode. A = fs_q_jt_second_tupleentrycode * S ((S (j)) * B) + (b))) /\\ (((exists fs_h_jt_second_tupleentryscale. fs_h_jt_second_tupleentryscale + S (c) = S ((S (j)) * D)) /\\ exists fs_q_jt_second_tupleentryscale. C = fs_q_jt_second_tupleentryscale * S ((S (j)) * D) + (c))))) /\\ (((forall jt_index_second_tuplebound. (exists jt_gap_second_tupleboundindex. jt_gap_second_tupleboundindex+S (jt_index_second_tuplebound)=(k)) -> exists jt_value_second_tuplebound. ((((exists fs_h_jt_second_tupleboundat. fs_h_jt_second_tupleboundat + S (jt_value_second_tuplebound) = S ((S (jt_index_second_tuplebound)) * c)) /\\ exists fs_q_jt_second_tupleboundat. b = fs_q_jt_second_tupleboundat * S ((S (jt_index_second_tuplebound)) * c) + (jt_value_second_tuplebound))) /\\ (exists jt_gap_second_tupleboundvalue. jt_gap_second_tupleboundvalue+S (jt_value_second_tuplebound)=(n)))) /\\ (forall jt_divisor_second_tupleprimitive. (exists jt_factor_second_tupleprimitivemodulus. (n)=(jt_divisor_second_tupleprimitive)*jt_factor_second_tupleprimitivemodulus) -> (forall jt_index_second_tupleprimitivecoordinates jt_value_second_tupleprimitivecoordinates. (exists jt_gap_second_tupleprimitivecoordinatesindex. jt_gap_second_tupleprimitivecoordinatesindex+S (jt_index_second_tupleprimitivecoordinates)=(k)) -> (((exists fs_h_jt_second_tupleprimitivecoordinatesat. fs_h_jt_second_tupleprimitivecoordinatesat + S (jt_value_second_tupleprimitivecoordinates) = S ((S (jt_index_second_tupleprimitivecoordinates)) * c)) /\\ exists fs_q_jt_second_tupleprimitivecoordinatesat. b = fs_q_jt_second_tupleprimitivecoordinatesat * S ((S (jt_index_second_tupleprimitivecoordinates)) * c) + (jt_value_second_tupleprimitivecoordinates))) -> (exists jt_factor_second_tupleprimitivecoordinatesdivides. (jt_value_second_tupleprimitivecoordinates)=(jt_divisor_second_tupleprimitive)*jt_factor_second_tupleprimitivecoordinatesdivides)) -> jt_divisor_second_tupleprimitive=1))))", "specialize hl_left (j)", "apply hl_left", "exact hj", "cases hb", "cases hb_witness", "cases hb_witness_witness", "cases hb_witness_witness_right", "have hea : forall jt_index_first_equal_target jt_left_first_equal_target jt_right_first_equal_target. (exists jt_gap_first_equal_targetindex. jt_gap_first_equal_targetindex+S (jt_index_first_equal_target)=(k)) -> (((exists fs_h_jt_first_equal_targetleft. fs_h_jt_first_equal_targetleft + S (jt_left_first_equal_target) = S ((S (jt_index_first_equal_target)) * x3)) /\\ exists fs_q_jt_first_equal_targetleft. x2 = fs_q_jt_first_equal_targetleft * S ((S (jt_index_first_equal_target)) * x3) + (jt_left_first_equal_target))) -> (((exists fs_h_jt_first_equal_targetright. fs_h_jt_first_equal_targetright + S (jt_right_first_equal_target) = S ((S (jt_index_first_equal_target)) * x1)) /\\ exists fs_q_jt_first_equal_targetright. x = fs_q_jt_first_equal_targetright * S ((S (jt_index_first_equal_target)) * x1) + (jt_right_first_equal_target))) -> jt_left_first_equal_target=jt_right_first_equal_target", "specialize hmi_right (x2)", "specialize hmi_right (x3)", "specialize hmi_right (x)", "specialize hmi_right (x1)", "apply hmi_right", "exact ha_witness_witness_left", "exact ht_witness_witness_left", "have heb : forall jt_index_second_equal_target jt_left_second_equal_target jt_right_second_equal_target. (exists jt_gap_second_equal_targetindex. jt_gap_second_equal_targetindex+S (jt_index_second_equal_target)=(k)) -> (((exists fs_h_jt_second_equal_targetleft. fs_h_jt_second_equal_targetleft + S (jt_left_second_equal_target) = S ((S (jt_index_second_equal_target)) * x5)) /\\ exists fs_q_jt_second_equal_targetleft. x4 = fs_q_jt_second_equal_targetleft * S ((S (jt_index_second_equal_target)) * x5) + (jt_left_second_equal_target))) -> (((exists fs_h_jt_second_equal_targetright. fs_h_jt_second_equal_targetright + S (jt_right_second_equal_target) = S ((S (jt_index_second_equal_target)) * x1)) /\\ exists fs_q_jt_second_equal_targetright. x = fs_q_jt_second_equal_targetright * S ((S (jt_index_second_equal_target)) * x1) + (jt_right_second_equal_target))) -> jt_left_second_equal_target=jt_right_second_equal_target", "specialize hmj_right (x4)", "specialize hmj_right (x5)", "specialize hmj_right (x)", "specialize hmj_right (x1)", "apply hmj_right", "exact hb_witness_witness_left", "exact ht_witness_witness_left", "have hrev : forall jt_index_target_equal_second jt_left_target_equal_second jt_right_target_equal_second. (exists jt_gap_target_equal_secondindex. jt_gap_target_equal_secondindex+S (jt_index_target_equal_second)=(k)) -> (((exists fs_h_jt_target_equal_secondleft. fs_h_jt_target_equal_secondleft + S (jt_left_target_equal_second) = S ((S (jt_index_target_equal_second)) * x1)) /\\ exists fs_q_jt_target_equal_secondleft. x = fs_q_jt_target_equal_secondleft * S ((S (jt_index_target_equal_second)) * x1) + (jt_left_target_equal_second))) -> (((exists fs_h_jt_target_equal_secondright. fs_h_jt_target_equal_secondright + S (jt_right_target_equal_second) = S ((S (jt_index_target_equal_second)) * x5)) /\\ exists fs_q_jt_target_equal_secondright. x4 = fs_q_jt_target_equal_secondright * S ((S (jt_index_target_equal_second)) * x5) + (jt_right_target_equal_second))) -> jt_left_target_equal_second=jt_right_target_equal_second", "specialize jordan_tuple_equal_symm (x4)", "specialize jordan_tuple_equal_symm (x5)", "specialize jordan_tuple_equal_symm (x)", "specialize jordan_tuple_equal_symm (x1)", "specialize jordan_tuple_equal_symm (k)", "apply jordan_tuple_equal_symm", "exact heb", "have heq : forall jt_index_source_equal_source jt_left_source_equal_source jt_right_source_equal_source. (exists jt_gap_source_equal_sourceindex. jt_gap_source_equal_sourceindex+S (jt_index_source_equal_source)=(k)) -> (((exists fs_h_jt_source_equal_sourceleft. fs_h_jt_source_equal_sourceleft + S (jt_left_source_equal_source) = S ((S (jt_index_source_equal_source)) * x3)) /\\ exists fs_q_jt_source_equal_sourceleft. x2 = fs_q_jt_source_equal_sourceleft * S ((S (jt_index_source_equal_source)) * x3) + (jt_left_source_equal_source))) -> (((exists fs_h_jt_source_equal_sourceright. fs_h_jt_source_equal_sourceright + S (jt_right_source_equal_source) = S ((S (jt_index_source_equal_source)) * x5)) /\\ exists fs_q_jt_source_equal_sourceright. x4 = fs_q_jt_source_equal_sourceright * S ((S (jt_index_source_equal_source)) * x5) + (jt_right_source_equal_source))) -> jt_left_source_equal_source=jt_right_source_equal_source", "specialize jordan_tuple_equal_trans (x2)", "specialize jordan_tuple_equal_trans (x3)", "specialize jordan_tuple_equal_trans (x)", "specialize jordan_tuple_equal_trans (x1)", "specialize jordan_tuple_equal_trans (x4)", "specialize jordan_tuple_equal_trans (x5)", "specialize jordan_tuple_equal_trans (k)", "apply jordan_tuple_equal_trans", "exact hea", "exact hrev", "specialize jordan_enumeration_distinct (k)", "specialize jordan_enumeration_distinct (n)", "specialize jordan_enumeration_distinct (A)", "specialize jordan_enumeration_distinct (B)", "specialize jordan_enumeration_distinct (C)", "specialize jordan_enumeration_distinct (D)", "specialize jordan_enumeration_distinct (u)", "specialize jordan_enumeration_distinct (i)", "specialize jordan_enumeration_distinct (j)", "specialize jordan_enumeration_distinct (x2)", "specialize jordan_enumeration_distinct (x3)", "specialize jordan_enumeration_distinct (x4)", "specialize jordan_enumeration_distinct (x5)", "apply jordan_enumeration_distinct", "exact hl", "exact hi", "exact hj", "exact ha_witness_witness_left", "exact hb_witness_witness_left", "exact heq" ], "script_sha256": "140ea6185a4a529ec8fb8a24880e94cc719555a19f7af56c5431b58d9753db98", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "140ea6185a4a529ec8fb8a24880e94cc719555a19f7af56c5431b58d9753db98", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "ee21ed551d2ea2469e34b393ffd50c729e41d07c3a92f11f10d8f7e66540050c" } ], "stable_member": false, "statement": "forall k n A B C D u E F G H v Z W. (((forall jt_i_injective_source. (exists jt_gap_injective_sourcesoundindex. jt_gap_injective_sourcesoundindex+S (jt_i_injective_source)=(u)) -> exists jt_b_injective_source jt_c_injective_source. ((((((exists fs_h_jt_injective_sourcesoundcode. fs_h_jt_injective_sourcesoundcode + S (jt_b_injective_source) = S ((S (jt_i_injective_source)) * B)) /\\ exists fs_q_jt_injective_sourcesoundcode. A = fs_q_jt_injective_sourcesoundcode * S ((S (jt_i_injective_source)) * B) + (jt_b_injective_source))) /\\ (((exists fs_h_jt_injective_sourcesoundscale. fs_h_jt_injective_sourcesoundscale + S (jt_c_injective_source) = S ((S (jt_i_injective_source)) * D)) /\\ exists fs_q_jt_injective_sourcesoundscale. C = fs_q_jt_injective_sourcesoundscale * S ((S (jt_i_injective_source)) * D) + (jt_c_injective_source))))) /\\ (((forall jt_index_injective_sourcebound. (exists jt_gap_injective_sourceboundindex. jt_gap_injective_sourceboundindex+S (jt_index_injective_sourcebound)=(k)) -> exists jt_value_injective_sourcebound. ((((exists fs_h_jt_injective_sourceboundat. fs_h_jt_injective_sourceboundat + S (jt_value_injective_sourcebound) = S ((S (jt_index_injective_sourcebound)) * jt_c_injective_source)) /\\ exists fs_q_jt_injective_sourceboundat. jt_b_injective_source = fs_q_jt_injective_sourceboundat * S ((S (jt_index_injective_sourcebound)) * jt_c_injective_source) + (jt_value_injective_sourcebound))) /\\ (exists jt_gap_injective_sourceboundvalue. jt_gap_injective_sourceboundvalue+S (jt_value_injective_sourcebound)=(n)))) /\\ (forall jt_divisor_injective_sourceprimitive. (exists jt_factor_injective_sourceprimitivemodulus. (n)=(jt_divisor_injective_sourceprimitive)*jt_factor_injective_sourceprimitivemodulus) -> (forall jt_index_injective_sourceprimitivecoordinates jt_value_injective_sourceprimitivecoordinates. (exists jt_gap_injective_sourceprimitivecoordinatesindex. jt_gap_injective_sourceprimitivecoordinatesindex+S (jt_index_injective_sourceprimitivecoordinates)=(k)) -> (((exists fs_h_jt_injective_sourceprimitivecoordinatesat. fs_h_jt_injective_sourceprimitivecoordinatesat + S (jt_value_injective_sourceprimitivecoordinates) = S ((S (jt_index_injective_sourceprimitivecoordinates)) * jt_c_injective_source)) /\\ exists fs_q_jt_injective_sourceprimitivecoordinatesat. jt_b_injective_source = fs_q_jt_injective_sourceprimitivecoordinatesat * S ((S (jt_index_injective_sourceprimitivecoordinates)) * jt_c_injective_source) + (jt_value_injective_sourceprimitivecoordinates))) -> (exists jt_factor_injective_sourceprimitivecoordinatesdivides. (jt_value_injective_sourceprimitivecoordinates)=(jt_divisor_injective_sourceprimitive)*jt_factor_injective_sourceprimitivecoordinatesdivides)) -> jt_divisor_injective_sourceprimitive=1))))) /\\ (((forall jt_b_injective_source jt_c_injective_source. (forall jt_index_injective_sourceinputbound. (exists jt_gap_injective_sourceinputboundindex. jt_gap_injective_sourceinputboundindex+S (jt_index_injective_sourceinputbound)=(k)) -> exists jt_value_injective_sourceinputbound. ((((exists fs_h_jt_injective_sourceinputboundat. fs_h_jt_injective_sourceinputboundat + S (jt_value_injective_sourceinputbound) = S ((S (jt_index_injective_sourceinputbound)) * jt_c_injective_source)) /\\ exists fs_q_jt_injective_sourceinputboundat. jt_b_injective_source = fs_q_jt_injective_sourceinputboundat * S ((S (jt_index_injective_sourceinputbound)) * jt_c_injective_source) + (jt_value_injective_sourceinputbound))) /\\ (exists jt_gap_injective_sourceinputboundvalue. jt_gap_injective_sourceinputboundvalue+S (jt_value_injective_sourceinputbound)=(n)))) -> (forall jt_divisor_injective_sourceinputprimitive. (exists jt_factor_injective_sourceinputprimitivemodulus. (n)=(jt_divisor_injective_sourceinputprimitive)*jt_factor_injective_sourceinputprimitivemodulus) -> (forall jt_index_injective_sourceinputprimitivecoordinates jt_value_injective_sourceinputprimitivecoordinates. (exists jt_gap_injective_sourceinputprimitivecoordinatesindex. jt_gap_injective_sourceinputprimitivecoordinatesindex+S (jt_index_injective_sourceinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_injective_sourceinputprimitivecoordinatesat. fs_h_jt_injective_sourceinputprimitivecoordinatesat + S (jt_value_injective_sourceinputprimitivecoordinates) = S ((S (jt_index_injective_sourceinputprimitivecoordinates)) * jt_c_injective_source)) /\\ exists fs_q_jt_injective_sourceinputprimitivecoordinatesat. jt_b_injective_source = fs_q_jt_injective_sourceinputprimitivecoordinatesat * S ((S (jt_index_injective_sourceinputprimitivecoordinates)) * jt_c_injective_source) + (jt_value_injective_sourceinputprimitivecoordinates))) -> (exists jt_factor_injective_sourceinputprimitivecoordinatesdivides. (jt_value_injective_sourceinputprimitivecoordinates)=(jt_divisor_injective_sourceinputprimitive)*jt_factor_injective_sourceinputprimitivecoordinatesdivides)) -> jt_divisor_injective_sourceinputprimitive=1) -> exists jt_i_injective_source jt_d_injective_source jt_e_injective_source. ((exists jt_gap_injective_sourcecompleteindex. jt_gap_injective_sourcecompleteindex+S (jt_i_injective_source)=(u)) /\\ (((((((exists fs_h_jt_injective_sourcecompletecode. fs_h_jt_injective_sourcecompletecode + S (jt_d_injective_source) = S ((S (jt_i_injective_source)) * B)) /\\ exists fs_q_jt_injective_sourcecompletecode. A = fs_q_jt_injective_sourcecompletecode * S ((S (jt_i_injective_source)) * B) + (jt_d_injective_source))) /\\ (((exists fs_h_jt_injective_sourcecompletescale. fs_h_jt_injective_sourcecompletescale + S (jt_e_injective_source) = S ((S (jt_i_injective_source)) * D)) /\\ exists fs_q_jt_injective_sourcecompletescale. C = fs_q_jt_injective_sourcecompletescale * S ((S (jt_i_injective_source)) * D) + (jt_e_injective_source))))) /\\ (forall jt_index_injective_sourcerepresented jt_left_injective_sourcerepresented jt_right_injective_sourcerepresented. (exists jt_gap_injective_sourcerepresentedindex. jt_gap_injective_sourcerepresentedindex+S (jt_index_injective_sourcerepresented)=(k)) -> (((exists fs_h_jt_injective_sourcerepresentedleft. fs_h_jt_injective_sourcerepresentedleft + S (jt_left_injective_sourcerepresented) = S ((S (jt_index_injective_sourcerepresented)) * jt_c_injective_source)) /\\ exists fs_q_jt_injective_sourcerepresentedleft. jt_b_injective_source = fs_q_jt_injective_sourcerepresentedleft * S ((S (jt_index_injective_sourcerepresented)) * jt_c_injective_source) + (jt_left_injective_sourcerepresented))) -> (((exists fs_h_jt_injective_sourcerepresentedright. fs_h_jt_injective_sourcerepresentedright + S (jt_right_injective_sourcerepresented) = S ((S (jt_index_injective_sourcerepresented)) * jt_e_injective_source)) /\\ exists fs_q_jt_injective_sourcerepresentedright. jt_d_injective_source = fs_q_jt_injective_sourcerepresentedright * S ((S (jt_index_injective_sourcerepresented)) * jt_e_injective_source) + (jt_right_injective_sourcerepresented))) -> jt_left_injective_sourcerepresented=jt_right_injective_sourcerepresented))))) /\\ (forall jt_i_injective_source jt_h_injective_source jt_b_injective_source jt_c_injective_source jt_d_injective_source jt_e_injective_source. (exists jt_gap_injective_sourcefirstindex. jt_gap_injective_sourcefirstindex+S (jt_i_injective_source)=(u)) -> (exists jt_gap_injective_sourcesecondindex. jt_gap_injective_sourcesecondindex+S (jt_h_injective_source)=(u)) -> (((((exists fs_h_jt_injective_sourcefirstcode. fs_h_jt_injective_sourcefirstcode + S (jt_b_injective_source) = S ((S (jt_i_injective_source)) * B)) /\\ exists fs_q_jt_injective_sourcefirstcode. A = fs_q_jt_injective_sourcefirstcode * S ((S (jt_i_injective_source)) * B) + (jt_b_injective_source))) /\\ (((exists fs_h_jt_injective_sourcefirstscale. fs_h_jt_injective_sourcefirstscale + S (jt_c_injective_source) = S ((S (jt_i_injective_source)) * D)) /\\ exists fs_q_jt_injective_sourcefirstscale. C = fs_q_jt_injective_sourcefirstscale * S ((S (jt_i_injective_source)) * D) + (jt_c_injective_source))))) -> (((((exists fs_h_jt_injective_sourcesecondcode. fs_h_jt_injective_sourcesecondcode + S (jt_d_injective_source) = S ((S (jt_h_injective_source)) * B)) /\\ exists fs_q_jt_injective_sourcesecondcode. A = fs_q_jt_injective_sourcesecondcode * S ((S (jt_h_injective_source)) * B) + (jt_d_injective_source))) /\\ (((exists fs_h_jt_injective_sourcesecondscale. fs_h_jt_injective_sourcesecondscale + S (jt_e_injective_source) = S ((S (jt_h_injective_source)) * D)) /\\ exists fs_q_jt_injective_sourcesecondscale. C = fs_q_jt_injective_sourcesecondscale * S ((S (jt_h_injective_source)) * D) + (jt_e_injective_source))))) -> (forall jt_index_injective_sourcesame jt_left_injective_sourcesame jt_right_injective_sourcesame. (exists jt_gap_injective_sourcesameindex. jt_gap_injective_sourcesameindex+S (jt_index_injective_sourcesame)=(k)) -> (((exists fs_h_jt_injective_sourcesameleft. fs_h_jt_injective_sourcesameleft + S (jt_left_injective_sourcesame) = S ((S (jt_index_injective_sourcesame)) * jt_c_injective_source)) /\\ exists fs_q_jt_injective_sourcesameleft. jt_b_injective_source = fs_q_jt_injective_sourcesameleft * S ((S (jt_index_injective_sourcesame)) * jt_c_injective_source) + (jt_left_injective_sourcesame))) -> (((exists fs_h_jt_injective_sourcesameright. fs_h_jt_injective_sourcesameright + S (jt_right_injective_sourcesame) = S ((S (jt_index_injective_sourcesame)) * jt_e_injective_source)) /\\ exists fs_q_jt_injective_sourcesameright. jt_d_injective_source = fs_q_jt_injective_sourcesameright * S ((S (jt_index_injective_sourcesame)) * jt_e_injective_source) + (jt_right_injective_sourcesame))) -> jt_left_injective_sourcesame=jt_right_injective_sourcesame) -> jt_i_injective_source=jt_h_injective_source))))) -> (((forall jt_i_injective_target. (exists jt_gap_injective_targetsoundindex. jt_gap_injective_targetsoundindex+S (jt_i_injective_target)=(v)) -> exists jt_b_injective_target jt_c_injective_target. ((((((exists fs_h_jt_injective_targetsoundcode. fs_h_jt_injective_targetsoundcode + S (jt_b_injective_target) = S ((S (jt_i_injective_target)) * F)) /\\ exists fs_q_jt_injective_targetsoundcode. E = fs_q_jt_injective_targetsoundcode * S ((S (jt_i_injective_target)) * F) + (jt_b_injective_target))) /\\ (((exists fs_h_jt_injective_targetsoundscale. fs_h_jt_injective_targetsoundscale + S (jt_c_injective_target) = S ((S (jt_i_injective_target)) * H)) /\\ exists fs_q_jt_injective_targetsoundscale. G = fs_q_jt_injective_targetsoundscale * S ((S (jt_i_injective_target)) * H) + (jt_c_injective_target))))) /\\ (((forall jt_index_injective_targetbound. (exists jt_gap_injective_targetboundindex. jt_gap_injective_targetboundindex+S (jt_index_injective_targetbound)=(k)) -> exists jt_value_injective_targetbound. ((((exists fs_h_jt_injective_targetboundat. fs_h_jt_injective_targetboundat + S (jt_value_injective_targetbound) = S ((S (jt_index_injective_targetbound)) * jt_c_injective_target)) /\\ exists fs_q_jt_injective_targetboundat. jt_b_injective_target = fs_q_jt_injective_targetboundat * S ((S (jt_index_injective_targetbound)) * jt_c_injective_target) + (jt_value_injective_targetbound))) /\\ (exists jt_gap_injective_targetboundvalue. jt_gap_injective_targetboundvalue+S (jt_value_injective_targetbound)=(n)))) /\\ (forall jt_divisor_injective_targetprimitive. (exists jt_factor_injective_targetprimitivemodulus. (n)=(jt_divisor_injective_targetprimitive)*jt_factor_injective_targetprimitivemodulus) -> (forall jt_index_injective_targetprimitivecoordinates jt_value_injective_targetprimitivecoordinates. (exists jt_gap_injective_targetprimitivecoordinatesindex. jt_gap_injective_targetprimitivecoordinatesindex+S (jt_index_injective_targetprimitivecoordinates)=(k)) -> (((exists fs_h_jt_injective_targetprimitivecoordinatesat. fs_h_jt_injective_targetprimitivecoordinatesat + S (jt_value_injective_targetprimitivecoordinates) = S ((S (jt_index_injective_targetprimitivecoordinates)) * jt_c_injective_target)) /\\ exists fs_q_jt_injective_targetprimitivecoordinatesat. jt_b_injective_target = fs_q_jt_injective_targetprimitivecoordinatesat * S ((S (jt_index_injective_targetprimitivecoordinates)) * jt_c_injective_target) + (jt_value_injective_targetprimitivecoordinates))) -> (exists jt_factor_injective_targetprimitivecoordinatesdivides. (jt_value_injective_targetprimitivecoordinates)=(jt_divisor_injective_targetprimitive)*jt_factor_injective_targetprimitivecoordinatesdivides)) -> jt_divisor_injective_targetprimitive=1))))) /\\ (((forall jt_b_injective_target jt_c_injective_target. (forall jt_index_injective_targetinputbound. (exists jt_gap_injective_targetinputboundindex. jt_gap_injective_targetinputboundindex+S (jt_index_injective_targetinputbound)=(k)) -> exists jt_value_injective_targetinputbound. ((((exists fs_h_jt_injective_targetinputboundat. fs_h_jt_injective_targetinputboundat + S (jt_value_injective_targetinputbound) = S ((S (jt_index_injective_targetinputbound)) * jt_c_injective_target)) /\\ exists fs_q_jt_injective_targetinputboundat. jt_b_injective_target = fs_q_jt_injective_targetinputboundat * S ((S (jt_index_injective_targetinputbound)) * jt_c_injective_target) + (jt_value_injective_targetinputbound))) /\\ (exists jt_gap_injective_targetinputboundvalue. jt_gap_injective_targetinputboundvalue+S (jt_value_injective_targetinputbound)=(n)))) -> (forall jt_divisor_injective_targetinputprimitive. (exists jt_factor_injective_targetinputprimitivemodulus. (n)=(jt_divisor_injective_targetinputprimitive)*jt_factor_injective_targetinputprimitivemodulus) -> (forall jt_index_injective_targetinputprimitivecoordinates jt_value_injective_targetinputprimitivecoordinates. (exists jt_gap_injective_targetinputprimitivecoordinatesindex. jt_gap_injective_targetinputprimitivecoordinatesindex+S (jt_index_injective_targetinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_injective_targetinputprimitivecoordinatesat. fs_h_jt_injective_targetinputprimitivecoordinatesat + S (jt_value_injective_targetinputprimitivecoordinates) = S ((S (jt_index_injective_targetinputprimitivecoordinates)) * jt_c_injective_target)) /\\ exists fs_q_jt_injective_targetinputprimitivecoordinatesat. jt_b_injective_target = fs_q_jt_injective_targetinputprimitivecoordinatesat * S ((S (jt_index_injective_targetinputprimitivecoordinates)) * jt_c_injective_target) + (jt_value_injective_targetinputprimitivecoordinates))) -> (exists jt_factor_injective_targetinputprimitivecoordinatesdivides. (jt_value_injective_targetinputprimitivecoordinates)=(jt_divisor_injective_targetinputprimitive)*jt_factor_injective_targetinputprimitivecoordinatesdivides)) -> jt_divisor_injective_targetinputprimitive=1) -> exists jt_i_injective_target jt_d_injective_target jt_e_injective_target. ((exists jt_gap_injective_targetcompleteindex. jt_gap_injective_targetcompleteindex+S (jt_i_injective_target)=(v)) /\\ (((((((exists fs_h_jt_injective_targetcompletecode. fs_h_jt_injective_targetcompletecode + S (jt_d_injective_target) = S ((S (jt_i_injective_target)) * F)) /\\ exists fs_q_jt_injective_targetcompletecode. E = fs_q_jt_injective_targetcompletecode * S ((S (jt_i_injective_target)) * F) + (jt_d_injective_target))) /\\ (((exists fs_h_jt_injective_targetcompletescale. fs_h_jt_injective_targetcompletescale + S (jt_e_injective_target) = S ((S (jt_i_injective_target)) * H)) /\\ exists fs_q_jt_injective_targetcompletescale. G = fs_q_jt_injective_targetcompletescale * S ((S (jt_i_injective_target)) * H) + (jt_e_injective_target))))) /\\ (forall jt_index_injective_targetrepresented jt_left_injective_targetrepresented jt_right_injective_targetrepresented. (exists jt_gap_injective_targetrepresentedindex. jt_gap_injective_targetrepresentedindex+S (jt_index_injective_targetrepresented)=(k)) -> (((exists fs_h_jt_injective_targetrepresentedleft. fs_h_jt_injective_targetrepresentedleft + S (jt_left_injective_targetrepresented) = S ((S (jt_index_injective_targetrepresented)) * jt_c_injective_target)) /\\ exists fs_q_jt_injective_targetrepresentedleft. jt_b_injective_target = fs_q_jt_injective_targetrepresentedleft * S ((S (jt_index_injective_targetrepresented)) * jt_c_injective_target) + (jt_left_injective_targetrepresented))) -> (((exists fs_h_jt_injective_targetrepresentedright. fs_h_jt_injective_targetrepresentedright + S (jt_right_injective_targetrepresented) = S ((S (jt_index_injective_targetrepresented)) * jt_e_injective_target)) /\\ exists fs_q_jt_injective_targetrepresentedright. jt_d_injective_target = fs_q_jt_injective_targetrepresentedright * S ((S (jt_index_injective_targetrepresented)) * jt_e_injective_target) + (jt_right_injective_targetrepresented))) -> jt_left_injective_targetrepresented=jt_right_injective_targetrepresented))))) /\\ (forall jt_i_injective_target jt_h_injective_target jt_b_injective_target jt_c_injective_target jt_d_injective_target jt_e_injective_target. (exists jt_gap_injective_targetfirstindex. jt_gap_injective_targetfirstindex+S (jt_i_injective_target)=(v)) -> (exists jt_gap_injective_targetsecondindex. jt_gap_injective_targetsecondindex+S (jt_h_injective_target)=(v)) -> (((((exists fs_h_jt_injective_targetfirstcode. fs_h_jt_injective_targetfirstcode + S (jt_b_injective_target) = S ((S (jt_i_injective_target)) * F)) /\\ exists fs_q_jt_injective_targetfirstcode. E = fs_q_jt_injective_targetfirstcode * S ((S (jt_i_injective_target)) * F) + (jt_b_injective_target))) /\\ (((exists fs_h_jt_injective_targetfirstscale. fs_h_jt_injective_targetfirstscale + S (jt_c_injective_target) = S ((S (jt_i_injective_target)) * H)) /\\ exists fs_q_jt_injective_targetfirstscale. G = fs_q_jt_injective_targetfirstscale * S ((S (jt_i_injective_target)) * H) + (jt_c_injective_target))))) -> (((((exists fs_h_jt_injective_targetsecondcode. fs_h_jt_injective_targetsecondcode + S (jt_d_injective_target) = S ((S (jt_h_injective_target)) * F)) /\\ exists fs_q_jt_injective_targetsecondcode. E = fs_q_jt_injective_targetsecondcode * S ((S (jt_h_injective_target)) * F) + (jt_d_injective_target))) /\\ (((exists fs_h_jt_injective_targetsecondscale. fs_h_jt_injective_targetsecondscale + S (jt_e_injective_target) = S ((S (jt_h_injective_target)) * H)) /\\ exists fs_q_jt_injective_targetsecondscale. G = fs_q_jt_injective_targetsecondscale * S ((S (jt_h_injective_target)) * H) + (jt_e_injective_target))))) -> (forall jt_index_injective_targetsame jt_left_injective_targetsame jt_right_injective_targetsame. (exists jt_gap_injective_targetsameindex. jt_gap_injective_targetsameindex+S (jt_index_injective_targetsame)=(k)) -> (((exists fs_h_jt_injective_targetsameleft. fs_h_jt_injective_targetsameleft + S (jt_left_injective_targetsame) = S ((S (jt_index_injective_targetsame)) * jt_c_injective_target)) /\\ exists fs_q_jt_injective_targetsameleft. jt_b_injective_target = fs_q_jt_injective_targetsameleft * S ((S (jt_index_injective_targetsame)) * jt_c_injective_target) + (jt_left_injective_targetsame))) -> (((exists fs_h_jt_injective_targetsameright. fs_h_jt_injective_targetsameright + S (jt_right_injective_targetsame) = S ((S (jt_index_injective_targetsame)) * jt_e_injective_target)) /\\ exists fs_q_jt_injective_targetsameright. jt_d_injective_target = fs_q_jt_injective_targetsameright * S ((S (jt_index_injective_targetsame)) * jt_e_injective_target) + (jt_right_injective_targetsame))) -> jt_left_injective_targetsame=jt_right_injective_targetsame) -> jt_i_injective_target=jt_h_injective_target))))) -> (forall jt_index_injective_map. (exists jt_gap_injective_mapindex. jt_gap_injective_mapindex+S (jt_index_injective_map)=(u)) -> exists jt_image_injective_map. ((((exists fs_h_jt_injective_mapat. fs_h_jt_injective_mapat + S (jt_image_injective_map) = S ((S (jt_index_injective_map)) * W)) /\\ exists fs_q_jt_injective_mapat. Z = fs_q_jt_injective_mapat * S ((S (jt_index_injective_map)) * W) + (jt_image_injective_map))) /\\ (((exists jt_gap_injective_mapbound. jt_gap_injective_mapbound+S (jt_image_injective_map)=(v)) /\\ (forall jt_b_injective_mapmatch jt_c_injective_mapmatch jt_d_injective_mapmatch jt_e_injective_mapmatch. (((((exists fs_h_jt_injective_mapmatchleftcode. fs_h_jt_injective_mapmatchleftcode + S (jt_b_injective_mapmatch) = S ((S (jt_index_injective_map)) * B)) /\\ exists fs_q_jt_injective_mapmatchleftcode. A = fs_q_jt_injective_mapmatchleftcode * S ((S (jt_index_injective_map)) * B) + (jt_b_injective_mapmatch))) /\\ (((exists fs_h_jt_injective_mapmatchleftscale. fs_h_jt_injective_mapmatchleftscale + S (jt_c_injective_mapmatch) = S ((S (jt_index_injective_map)) * D)) /\\ exists fs_q_jt_injective_mapmatchleftscale. C = fs_q_jt_injective_mapmatchleftscale * S ((S (jt_index_injective_map)) * D) + (jt_c_injective_mapmatch))))) -> (((((exists fs_h_jt_injective_mapmatchrightcode. fs_h_jt_injective_mapmatchrightcode + S (jt_d_injective_mapmatch) = S ((S (jt_image_injective_map)) * F)) /\\ exists fs_q_jt_injective_mapmatchrightcode. E = fs_q_jt_injective_mapmatchrightcode * S ((S (jt_image_injective_map)) * F) + (jt_d_injective_mapmatch))) /\\ (((exists fs_h_jt_injective_mapmatchrightscale. fs_h_jt_injective_mapmatchrightscale + S (jt_e_injective_mapmatch) = S ((S (jt_image_injective_map)) * H)) /\\ exists fs_q_jt_injective_mapmatchrightscale. G = fs_q_jt_injective_mapmatchrightscale * S ((S (jt_image_injective_map)) * H) + (jt_e_injective_mapmatch))))) -> (forall jt_index_injective_mapmatchequal jt_left_injective_mapmatchequal jt_right_injective_mapmatchequal. (exists jt_gap_injective_mapmatchequalindex. jt_gap_injective_mapmatchequalindex+S (jt_index_injective_mapmatchequal)=(k)) -> (((exists fs_h_jt_injective_mapmatchequalleft. fs_h_jt_injective_mapmatchequalleft + S (jt_left_injective_mapmatchequal) = S ((S (jt_index_injective_mapmatchequal)) * jt_c_injective_mapmatch)) /\\ exists fs_q_jt_injective_mapmatchequalleft. jt_b_injective_mapmatch = fs_q_jt_injective_mapmatchequalleft * S ((S (jt_index_injective_mapmatchequal)) * jt_c_injective_mapmatch) + (jt_left_injective_mapmatchequal))) -> (((exists fs_h_jt_injective_mapmatchequalright. fs_h_jt_injective_mapmatchequalright + S (jt_right_injective_mapmatchequal) = S ((S (jt_index_injective_mapmatchequal)) * jt_e_injective_mapmatch)) /\\ exists fs_q_jt_injective_mapmatchequalright. jt_d_injective_mapmatch = fs_q_jt_injective_mapmatchequalright * S ((S (jt_index_injective_mapmatchequal)) * jt_e_injective_mapmatch) + (jt_right_injective_mapmatchequal))) -> jt_left_injective_mapmatchequal=jt_right_injective_mapmatchequal)))))) -> (((forall jt_index_injective_bound. (exists jt_gap_injective_boundindex. jt_gap_injective_boundindex+S (jt_index_injective_bound)=(u)) -> exists jt_value_injective_bound. ((((exists fs_h_jt_injective_boundat. fs_h_jt_injective_boundat + S (jt_value_injective_bound) = S ((S (jt_index_injective_bound)) * W)) /\\ exists fs_q_jt_injective_boundat. Z = fs_q_jt_injective_boundat * S ((S (jt_index_injective_bound)) * W) + (jt_value_injective_bound))) /\\ (exists jt_gap_injective_boundvalue. jt_gap_injective_boundvalue+S (jt_value_injective_bound)=(v)))) /\\ (forall fp_i_jordan_injective fp_j_jordan_injective fp_value_jordan_injective. (exists fp_gap_jordan_injective_i. fp_gap_jordan_injective_i + S fp_i_jordan_injective = u) -> (exists fp_gap_jordan_injective_j. fp_gap_jordan_injective_j + S fp_j_jordan_injective = u) -> (((exists ff_h_jordan_injective_left. ff_h_jordan_injective_left + S (fp_value_jordan_injective) = S ((S (fp_i_jordan_injective)) * W)) /\\ exists ff_q_jordan_injective_left. Z = ff_q_jordan_injective_left * S ((S (fp_i_jordan_injective)) * W) + (fp_value_jordan_injective))) -> (((exists ff_h_jordan_injective_right. ff_h_jordan_injective_right + S (fp_value_jordan_injective) = S ((S (fp_j_jordan_injective)) * W)) /\\ exists ff_q_jordan_injective_right. Z = ff_q_jordan_injective_right * S ((S (fp_j_jordan_injective)) * W) + (fp_value_jordan_injective))) -> fp_i_jordan_injective = fp_j_jordan_injective)))", "statement_sha256": "ee21ed551d2ea2469e34b393ffd50c729e41d07c3a92f11f10d8f7e66540050c", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Equal decoded map indices force coordinate-equal source tuples and hence equal source positions, not equal raw tuple codes." }, { "admission_dependencies": [ "jordan_enumeration_index_map_exists", "le_refl", "jordan_enumeration_index_map_bounded_injective", "le_or_lt", "finite_bounded_into_oversized_not_injective" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 45, "body_proof_nodes": 102, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "have hm : ∃ Z. ∃ W. ∀ jt_index_cardinality_map. Lt(jt_index_cardinality_map,u) → ∃ x. BetaAt(Z,W,jt_index_cardinality_map,x) ∧ (Lt(x,v) ∧ (∀ y. ∀ z. ∀ n. ∀ m. BetaAt(A,B,jt_index_cardinality_map,y) ∧ BetaAt(C,D,jt_index_cardinality_map,z) → BetaAt(E,F,x,n) ∧ BetaAt(G,H,x,m) → IntegerVectorZero(y,z,n,m,k)))", "specialize jordan_enumeration_index_map_exists (u)", "specialize jordan_enumeration_index_map_exists (k)", "specialize jordan_enumeration_index_map_exists (n)", "specialize jordan_enumeration_index_map_exists (A)", "specialize jordan_enumeration_index_map_exists (B)", "specialize jordan_enumeration_index_map_exists (C)", "specialize jordan_enumeration_index_map_exists (D)", "specialize jordan_enumeration_index_map_exists (u)", "specialize jordan_enumeration_index_map_exists (E)", "specialize jordan_enumeration_index_map_exists (F)", "specialize jordan_enumeration_index_map_exists (G)", "specialize jordan_enumeration_index_map_exists (H)", "specialize jordan_enumeration_index_map_exists (v)", "apply jordan_enumeration_index_map_exists", "exact hl", "exact hr", "specialize le_refl (u)", "apply le_refl", "cases hm", "cases hm_witness", "have hinj : FiniteMatrixSelector(x,x1,u,v)", "specialize jordan_enumeration_index_map_bounded_injective (k)", "specialize jordan_enumeration_index_map_bounded_injective (n)", "specialize jordan_enumeration_index_map_bounded_injective (A)", "specialize jordan_enumeration_index_map_bounded_injective (B)", "specialize jordan_enumeration_index_map_bounded_injective (C)", "specialize jordan_enumeration_index_map_bounded_injective (D)", "specialize jordan_enumeration_index_map_bounded_injective (u)", "specialize jordan_enumeration_index_map_bounded_injective (E)", "specialize jordan_enumeration_index_map_bounded_injective (F)", "specialize jordan_enumeration_index_map_bounded_injective (G)", "specialize jordan_enumeration_index_map_bounded_injective (H)", "specialize jordan_enumeration_index_map_bounded_injective (v)", "specialize jordan_enumeration_index_map_bounded_injective (x)", "specialize jordan_enumeration_index_map_bounded_injective (x1)", "apply jordan_enumeration_index_map_bounded_injective", "exact hl", "exact hr", "exact hm_witness_witness", "cases hinj", "have hc : Le(u,v) ∨ Lt(v,u)", "specialize le_or_lt (u)", "specialize le_or_lt (v)", "apply le_or_lt", "cases hc", "exact hc_left", "exfalso", "specialize finite_bounded_into_oversized_not_injective (x)", "specialize finite_bounded_into_oversized_not_injective (x1)", "specialize finite_bounded_into_oversized_not_injective (u)", "specialize finite_bounded_into_oversized_not_injective (v)", "apply finite_bounded_into_oversized_not_injective", "exact hinj_left", "exact hc_right", "exact hinj_right" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. JordanTupleEnumeration(k,n,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → Le(u,v)", "defined_statement_sha256": "4059856d9432208ced915ca69104277b18c4e1ce8e034534d849717b70686f4c", "definition_uses": { "ND0113": 1, "ND0121": 1, "ND0374": 2, "PD0001": 2, "PD0002": 3, "PD0013": 5 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "8f97e4b79bc26d0febb9dc2d6da7466014cc6d30accbd5a0ad3c6976004d3021", "free_names": [], "script_definition_uses": { "ND0113": 1, "ND0121": 1, "PD0001": 1, "PD0002": 3, "PD0013": 5 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "have hm : " }, { "kind": "text", "text": "∃ Z. ∃ W. ∀ jt_index_cardinality_map. " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(jt_index_cardinality_map,u)" }, { "kind": "text", "text": " → ∃ x. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(Z,W,jt_index_cardinality_map,x)" }, { "kind": "text", "text": " ∧ (" }, { "definition": "PD0002", "kind": "definition", "text": "Lt(x,v)" }, { "kind": "text", "text": " ∧ (∀ y. ∀ z. ∀ n. ∀ m. " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(A,B,jt_index_cardinality_map,y)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(C,D,jt_index_cardinality_map,z)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(E,F,x,n)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(G,H,x,m)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(y,z,n,m,k)" }, { "kind": "text", "text": "))" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_exists (v)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_index_map_exists" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "specialize le_refl (u)" } ], [ { "kind": "text", "text": "apply le_refl" } ], [ { "kind": "text", "text": "cases hm" } ], [ { "kind": "text", "text": "cases hm_witness" } ], [ { "kind": "text", "text": "have hinj : " }, { "definition": "ND0113", "kind": "definition", "text": "FiniteMatrixSelector(x,x1,u,v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_index_map_bounded_injective (x1)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_index_map_bounded_injective" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hm_witness_witness" } ], [ { "kind": "text", "text": "cases hinj" } ], [ { "kind": "text", "text": "have hc : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(u,v)" }, { "kind": "text", "text": " ∨ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(v,u)" } ], [ { "kind": "text", "text": "specialize le_or_lt (u)" } ], [ { "kind": "text", "text": "specialize le_or_lt (v)" } ], [ { "kind": "text", "text": "apply le_or_lt" } ], [ { "kind": "text", "text": "cases hc" } ], [ { "kind": "text", "text": "exact hc_left" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "specialize finite_bounded_into_oversized_not_injective (x)" } ], [ { "kind": "text", "text": "specialize finite_bounded_into_oversized_not_injective (x1)" } ], [ { "kind": "text", "text": "specialize finite_bounded_into_oversized_not_injective (u)" } ], [ { "kind": "text", "text": "specialize finite_bounded_into_oversized_not_injective (v)" } ], [ { "kind": "text", "text": "apply finite_bounded_into_oversized_not_injective" } ], [ { "kind": "text", "text": "exact hinj_left" } ], [ { "kind": "text", "text": "exact hc_right" } ], [ { "kind": "text", "text": "exact hinj_right" } ] ], "statement_definition_uses": { "ND0374": 2, "PD0001": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → " }, { "definition": "PD0001", "kind": "definition", "text": "Le(u,v)" } ] }, "dependencies": [ "jordan_enumeration_index_map_exists", "le_refl", "jordan_enumeration_index_map_bounded_injective", "le_or_lt", "finite_bounded_into_oversized_not_injective" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT0053", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_cardinality_le", "original_ha_bundle_verified": true, "proof_bundle_node_id": 344, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "have hm : exists Z W. forall jt_index_cardinality_map. (exists jt_gap_cardinality_mapindex. jt_gap_cardinality_mapindex+S (jt_index_cardinality_map)=(u)) -> exists jt_image_cardinality_map. ((((exists fs_h_jt_cardinality_mapat. fs_h_jt_cardinality_mapat + S (jt_image_cardinality_map) = S ((S (jt_index_cardinality_map)) * W)) /\\ exists fs_q_jt_cardinality_mapat. Z = fs_q_jt_cardinality_mapat * S ((S (jt_index_cardinality_map)) * W) + (jt_image_cardinality_map))) /\\ (((exists jt_gap_cardinality_mapbound. jt_gap_cardinality_mapbound+S (jt_image_cardinality_map)=(v)) /\\ (forall jt_b_cardinality_mapmatch jt_c_cardinality_mapmatch jt_d_cardinality_mapmatch jt_e_cardinality_mapmatch. (((((exists fs_h_jt_cardinality_mapmatchleftcode. fs_h_jt_cardinality_mapmatchleftcode + S (jt_b_cardinality_mapmatch) = S ((S (jt_index_cardinality_map)) * B)) /\\ exists fs_q_jt_cardinality_mapmatchleftcode. A = fs_q_jt_cardinality_mapmatchleftcode * S ((S (jt_index_cardinality_map)) * B) + (jt_b_cardinality_mapmatch))) /\\ (((exists fs_h_jt_cardinality_mapmatchleftscale. fs_h_jt_cardinality_mapmatchleftscale + S (jt_c_cardinality_mapmatch) = S ((S (jt_index_cardinality_map)) * D)) /\\ exists fs_q_jt_cardinality_mapmatchleftscale. C = fs_q_jt_cardinality_mapmatchleftscale * S ((S (jt_index_cardinality_map)) * D) + (jt_c_cardinality_mapmatch))))) -> (((((exists fs_h_jt_cardinality_mapmatchrightcode. fs_h_jt_cardinality_mapmatchrightcode + S (jt_d_cardinality_mapmatch) = S ((S (jt_image_cardinality_map)) * F)) /\\ exists fs_q_jt_cardinality_mapmatchrightcode. E = fs_q_jt_cardinality_mapmatchrightcode * S ((S (jt_image_cardinality_map)) * F) + (jt_d_cardinality_mapmatch))) /\\ (((exists fs_h_jt_cardinality_mapmatchrightscale. fs_h_jt_cardinality_mapmatchrightscale + S (jt_e_cardinality_mapmatch) = S ((S (jt_image_cardinality_map)) * H)) /\\ exists fs_q_jt_cardinality_mapmatchrightscale. G = fs_q_jt_cardinality_mapmatchrightscale * S ((S (jt_image_cardinality_map)) * H) + (jt_e_cardinality_mapmatch))))) -> (forall jt_index_cardinality_mapmatchequal jt_left_cardinality_mapmatchequal jt_right_cardinality_mapmatchequal. (exists jt_gap_cardinality_mapmatchequalindex. jt_gap_cardinality_mapmatchequalindex+S (jt_index_cardinality_mapmatchequal)=(k)) -> (((exists fs_h_jt_cardinality_mapmatchequalleft. fs_h_jt_cardinality_mapmatchequalleft + S (jt_left_cardinality_mapmatchequal) = S ((S (jt_index_cardinality_mapmatchequal)) * jt_c_cardinality_mapmatch)) /\\ exists fs_q_jt_cardinality_mapmatchequalleft. jt_b_cardinality_mapmatch = fs_q_jt_cardinality_mapmatchequalleft * S ((S (jt_index_cardinality_mapmatchequal)) * jt_c_cardinality_mapmatch) + (jt_left_cardinality_mapmatchequal))) -> (((exists fs_h_jt_cardinality_mapmatchequalright. fs_h_jt_cardinality_mapmatchequalright + S (jt_right_cardinality_mapmatchequal) = S ((S (jt_index_cardinality_mapmatchequal)) * jt_e_cardinality_mapmatch)) /\\ exists fs_q_jt_cardinality_mapmatchequalright. jt_d_cardinality_mapmatch = fs_q_jt_cardinality_mapmatchequalright * S ((S (jt_index_cardinality_mapmatchequal)) * jt_e_cardinality_mapmatch) + (jt_right_cardinality_mapmatchequal))) -> jt_left_cardinality_mapmatchequal=jt_right_cardinality_mapmatchequal)))))", "specialize jordan_enumeration_index_map_exists (u)", "specialize jordan_enumeration_index_map_exists (k)", "specialize jordan_enumeration_index_map_exists (n)", "specialize jordan_enumeration_index_map_exists (A)", "specialize jordan_enumeration_index_map_exists (B)", "specialize jordan_enumeration_index_map_exists (C)", "specialize jordan_enumeration_index_map_exists (D)", "specialize jordan_enumeration_index_map_exists (u)", "specialize jordan_enumeration_index_map_exists (E)", "specialize jordan_enumeration_index_map_exists (F)", "specialize jordan_enumeration_index_map_exists (G)", "specialize jordan_enumeration_index_map_exists (H)", "specialize jordan_enumeration_index_map_exists (v)", "apply jordan_enumeration_index_map_exists", "exact hl", "exact hr", "specialize le_refl (u)", "apply le_refl", "cases hm", "cases hm_witness", "have hinj : ((forall jt_index_cardinality_bound. (exists jt_gap_cardinality_boundindex. jt_gap_cardinality_boundindex+S (jt_index_cardinality_bound)=(u)) -> exists jt_value_cardinality_bound. ((((exists fs_h_jt_cardinality_boundat. fs_h_jt_cardinality_boundat + S (jt_value_cardinality_bound) = S ((S (jt_index_cardinality_bound)) * x1)) /\\ exists fs_q_jt_cardinality_boundat. x = fs_q_jt_cardinality_boundat * S ((S (jt_index_cardinality_bound)) * x1) + (jt_value_cardinality_bound))) /\\ (exists jt_gap_cardinality_boundvalue. jt_gap_cardinality_boundvalue+S (jt_value_cardinality_bound)=(v)))) /\\ (forall fp_i_cardinality_inj fp_j_cardinality_inj fp_value_cardinality_inj. (exists fp_gap_cardinality_inj_i. fp_gap_cardinality_inj_i + S fp_i_cardinality_inj = u) -> (exists fp_gap_cardinality_inj_j. fp_gap_cardinality_inj_j + S fp_j_cardinality_inj = u) -> (((exists ff_h_cardinality_inj_left. ff_h_cardinality_inj_left + S (fp_value_cardinality_inj) = S ((S (fp_i_cardinality_inj)) * x1)) /\\ exists ff_q_cardinality_inj_left. x = ff_q_cardinality_inj_left * S ((S (fp_i_cardinality_inj)) * x1) + (fp_value_cardinality_inj))) -> (((exists ff_h_cardinality_inj_right. ff_h_cardinality_inj_right + S (fp_value_cardinality_inj) = S ((S (fp_j_cardinality_inj)) * x1)) /\\ exists ff_q_cardinality_inj_right. x = ff_q_cardinality_inj_right * S ((S (fp_j_cardinality_inj)) * x1) + (fp_value_cardinality_inj))) -> fp_i_cardinality_inj = fp_j_cardinality_inj))", "specialize jordan_enumeration_index_map_bounded_injective (k)", "specialize jordan_enumeration_index_map_bounded_injective (n)", "specialize jordan_enumeration_index_map_bounded_injective (A)", "specialize jordan_enumeration_index_map_bounded_injective (B)", "specialize jordan_enumeration_index_map_bounded_injective (C)", "specialize jordan_enumeration_index_map_bounded_injective (D)", "specialize jordan_enumeration_index_map_bounded_injective (u)", "specialize jordan_enumeration_index_map_bounded_injective (E)", "specialize jordan_enumeration_index_map_bounded_injective (F)", "specialize jordan_enumeration_index_map_bounded_injective (G)", "specialize jordan_enumeration_index_map_bounded_injective (H)", "specialize jordan_enumeration_index_map_bounded_injective (v)", "specialize jordan_enumeration_index_map_bounded_injective (x)", "specialize jordan_enumeration_index_map_bounded_injective (x1)", "apply jordan_enumeration_index_map_bounded_injective", "exact hl", "exact hr", "exact hm_witness_witness", "cases hinj", "have hc : (exists jt_gap_cardinality_le. jt_gap_cardinality_le+(u)=(v)) \\/ (exists jt_gap_cardinality_overflow. jt_gap_cardinality_overflow+S (v)=(u))", "specialize le_or_lt (u)", "specialize le_or_lt (v)", "apply le_or_lt", "cases hc", "exact hc_left", "exfalso", "specialize finite_bounded_into_oversized_not_injective (x)", "specialize finite_bounded_into_oversized_not_injective (x1)", "specialize finite_bounded_into_oversized_not_injective (u)", "specialize finite_bounded_into_oversized_not_injective (v)", "apply finite_bounded_into_oversized_not_injective", "exact hinj_left", "exact hc_right", "exact hinj_right" ], "script_sha256": "b758098320e64c8404b0e3cad218bac6f5ae970756f21b161a9bd63c8e70154b", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "b758098320e64c8404b0e3cad218bac6f5ae970756f21b161a9bd63c8e70154b", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "8f97e4b79bc26d0febb9dc2d6da7466014cc6d30accbd5a0ad3c6976004d3021" } ], "stable_member": false, "statement": "forall k n A B C D u E F G H v. (((forall jt_i_cardinality_left. (exists jt_gap_cardinality_leftsoundindex. jt_gap_cardinality_leftsoundindex+S (jt_i_cardinality_left)=(u)) -> exists jt_b_cardinality_left jt_c_cardinality_left. ((((((exists fs_h_jt_cardinality_leftsoundcode. fs_h_jt_cardinality_leftsoundcode + S (jt_b_cardinality_left) = S ((S (jt_i_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftsoundcode. A = fs_q_jt_cardinality_leftsoundcode * S ((S (jt_i_cardinality_left)) * B) + (jt_b_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftsoundscale. fs_h_jt_cardinality_leftsoundscale + S (jt_c_cardinality_left) = S ((S (jt_i_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftsoundscale. C = fs_q_jt_cardinality_leftsoundscale * S ((S (jt_i_cardinality_left)) * D) + (jt_c_cardinality_left))))) /\\ (((forall jt_index_cardinality_leftbound. (exists jt_gap_cardinality_leftboundindex. jt_gap_cardinality_leftboundindex+S (jt_index_cardinality_leftbound)=(k)) -> exists jt_value_cardinality_leftbound. ((((exists fs_h_jt_cardinality_leftboundat. fs_h_jt_cardinality_leftboundat + S (jt_value_cardinality_leftbound) = S ((S (jt_index_cardinality_leftbound)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftboundat. jt_b_cardinality_left = fs_q_jt_cardinality_leftboundat * S ((S (jt_index_cardinality_leftbound)) * jt_c_cardinality_left) + (jt_value_cardinality_leftbound))) /\\ (exists jt_gap_cardinality_leftboundvalue. jt_gap_cardinality_leftboundvalue+S (jt_value_cardinality_leftbound)=(n)))) /\\ (forall jt_divisor_cardinality_leftprimitive. (exists jt_factor_cardinality_leftprimitivemodulus. (n)=(jt_divisor_cardinality_leftprimitive)*jt_factor_cardinality_leftprimitivemodulus) -> (forall jt_index_cardinality_leftprimitivecoordinates jt_value_cardinality_leftprimitivecoordinates. (exists jt_gap_cardinality_leftprimitivecoordinatesindex. jt_gap_cardinality_leftprimitivecoordinatesindex+S (jt_index_cardinality_leftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_leftprimitivecoordinatesat. fs_h_jt_cardinality_leftprimitivecoordinatesat + S (jt_value_cardinality_leftprimitivecoordinates) = S ((S (jt_index_cardinality_leftprimitivecoordinates)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftprimitivecoordinatesat. jt_b_cardinality_left = fs_q_jt_cardinality_leftprimitivecoordinatesat * S ((S (jt_index_cardinality_leftprimitivecoordinates)) * jt_c_cardinality_left) + (jt_value_cardinality_leftprimitivecoordinates))) -> (exists jt_factor_cardinality_leftprimitivecoordinatesdivides. (jt_value_cardinality_leftprimitivecoordinates)=(jt_divisor_cardinality_leftprimitive)*jt_factor_cardinality_leftprimitivecoordinatesdivides)) -> jt_divisor_cardinality_leftprimitive=1))))) /\\ (((forall jt_b_cardinality_left jt_c_cardinality_left. (forall jt_index_cardinality_leftinputbound. (exists jt_gap_cardinality_leftinputboundindex. jt_gap_cardinality_leftinputboundindex+S (jt_index_cardinality_leftinputbound)=(k)) -> exists jt_value_cardinality_leftinputbound. ((((exists fs_h_jt_cardinality_leftinputboundat. fs_h_jt_cardinality_leftinputboundat + S (jt_value_cardinality_leftinputbound) = S ((S (jt_index_cardinality_leftinputbound)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftinputboundat. jt_b_cardinality_left = fs_q_jt_cardinality_leftinputboundat * S ((S (jt_index_cardinality_leftinputbound)) * jt_c_cardinality_left) + (jt_value_cardinality_leftinputbound))) /\\ (exists jt_gap_cardinality_leftinputboundvalue. jt_gap_cardinality_leftinputboundvalue+S (jt_value_cardinality_leftinputbound)=(n)))) -> (forall jt_divisor_cardinality_leftinputprimitive. (exists jt_factor_cardinality_leftinputprimitivemodulus. (n)=(jt_divisor_cardinality_leftinputprimitive)*jt_factor_cardinality_leftinputprimitivemodulus) -> (forall jt_index_cardinality_leftinputprimitivecoordinates jt_value_cardinality_leftinputprimitivecoordinates. (exists jt_gap_cardinality_leftinputprimitivecoordinatesindex. jt_gap_cardinality_leftinputprimitivecoordinatesindex+S (jt_index_cardinality_leftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_leftinputprimitivecoordinatesat. fs_h_jt_cardinality_leftinputprimitivecoordinatesat + S (jt_value_cardinality_leftinputprimitivecoordinates) = S ((S (jt_index_cardinality_leftinputprimitivecoordinates)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftinputprimitivecoordinatesat. jt_b_cardinality_left = fs_q_jt_cardinality_leftinputprimitivecoordinatesat * S ((S (jt_index_cardinality_leftinputprimitivecoordinates)) * jt_c_cardinality_left) + (jt_value_cardinality_leftinputprimitivecoordinates))) -> (exists jt_factor_cardinality_leftinputprimitivecoordinatesdivides. (jt_value_cardinality_leftinputprimitivecoordinates)=(jt_divisor_cardinality_leftinputprimitive)*jt_factor_cardinality_leftinputprimitivecoordinatesdivides)) -> jt_divisor_cardinality_leftinputprimitive=1) -> exists jt_i_cardinality_left jt_d_cardinality_left jt_e_cardinality_left. ((exists jt_gap_cardinality_leftcompleteindex. jt_gap_cardinality_leftcompleteindex+S (jt_i_cardinality_left)=(u)) /\\ (((((((exists fs_h_jt_cardinality_leftcompletecode. fs_h_jt_cardinality_leftcompletecode + S (jt_d_cardinality_left) = S ((S (jt_i_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftcompletecode. A = fs_q_jt_cardinality_leftcompletecode * S ((S (jt_i_cardinality_left)) * B) + (jt_d_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftcompletescale. fs_h_jt_cardinality_leftcompletescale + S (jt_e_cardinality_left) = S ((S (jt_i_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftcompletescale. C = fs_q_jt_cardinality_leftcompletescale * S ((S (jt_i_cardinality_left)) * D) + (jt_e_cardinality_left))))) /\\ (forall jt_index_cardinality_leftrepresented jt_left_cardinality_leftrepresented jt_right_cardinality_leftrepresented. (exists jt_gap_cardinality_leftrepresentedindex. jt_gap_cardinality_leftrepresentedindex+S (jt_index_cardinality_leftrepresented)=(k)) -> (((exists fs_h_jt_cardinality_leftrepresentedleft. fs_h_jt_cardinality_leftrepresentedleft + S (jt_left_cardinality_leftrepresented) = S ((S (jt_index_cardinality_leftrepresented)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftrepresentedleft. jt_b_cardinality_left = fs_q_jt_cardinality_leftrepresentedleft * S ((S (jt_index_cardinality_leftrepresented)) * jt_c_cardinality_left) + (jt_left_cardinality_leftrepresented))) -> (((exists fs_h_jt_cardinality_leftrepresentedright. fs_h_jt_cardinality_leftrepresentedright + S (jt_right_cardinality_leftrepresented) = S ((S (jt_index_cardinality_leftrepresented)) * jt_e_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftrepresentedright. jt_d_cardinality_left = fs_q_jt_cardinality_leftrepresentedright * S ((S (jt_index_cardinality_leftrepresented)) * jt_e_cardinality_left) + (jt_right_cardinality_leftrepresented))) -> jt_left_cardinality_leftrepresented=jt_right_cardinality_leftrepresented))))) /\\ (forall jt_i_cardinality_left jt_h_cardinality_left jt_b_cardinality_left jt_c_cardinality_left jt_d_cardinality_left jt_e_cardinality_left. (exists jt_gap_cardinality_leftfirstindex. jt_gap_cardinality_leftfirstindex+S (jt_i_cardinality_left)=(u)) -> (exists jt_gap_cardinality_leftsecondindex. jt_gap_cardinality_leftsecondindex+S (jt_h_cardinality_left)=(u)) -> (((((exists fs_h_jt_cardinality_leftfirstcode. fs_h_jt_cardinality_leftfirstcode + S (jt_b_cardinality_left) = S ((S (jt_i_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftfirstcode. A = fs_q_jt_cardinality_leftfirstcode * S ((S (jt_i_cardinality_left)) * B) + (jt_b_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftfirstscale. fs_h_jt_cardinality_leftfirstscale + S (jt_c_cardinality_left) = S ((S (jt_i_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftfirstscale. C = fs_q_jt_cardinality_leftfirstscale * S ((S (jt_i_cardinality_left)) * D) + (jt_c_cardinality_left))))) -> (((((exists fs_h_jt_cardinality_leftsecondcode. fs_h_jt_cardinality_leftsecondcode + S (jt_d_cardinality_left) = S ((S (jt_h_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftsecondcode. A = fs_q_jt_cardinality_leftsecondcode * S ((S (jt_h_cardinality_left)) * B) + (jt_d_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftsecondscale. fs_h_jt_cardinality_leftsecondscale + S (jt_e_cardinality_left) = S ((S (jt_h_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftsecondscale. C = fs_q_jt_cardinality_leftsecondscale * S ((S (jt_h_cardinality_left)) * D) + (jt_e_cardinality_left))))) -> (forall jt_index_cardinality_leftsame jt_left_cardinality_leftsame jt_right_cardinality_leftsame. (exists jt_gap_cardinality_leftsameindex. jt_gap_cardinality_leftsameindex+S (jt_index_cardinality_leftsame)=(k)) -> (((exists fs_h_jt_cardinality_leftsameleft. fs_h_jt_cardinality_leftsameleft + S (jt_left_cardinality_leftsame) = S ((S (jt_index_cardinality_leftsame)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftsameleft. jt_b_cardinality_left = fs_q_jt_cardinality_leftsameleft * S ((S (jt_index_cardinality_leftsame)) * jt_c_cardinality_left) + (jt_left_cardinality_leftsame))) -> (((exists fs_h_jt_cardinality_leftsameright. fs_h_jt_cardinality_leftsameright + S (jt_right_cardinality_leftsame) = S ((S (jt_index_cardinality_leftsame)) * jt_e_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftsameright. jt_d_cardinality_left = fs_q_jt_cardinality_leftsameright * S ((S (jt_index_cardinality_leftsame)) * jt_e_cardinality_left) + (jt_right_cardinality_leftsame))) -> jt_left_cardinality_leftsame=jt_right_cardinality_leftsame) -> jt_i_cardinality_left=jt_h_cardinality_left))))) -> (((forall jt_i_cardinality_right. (exists jt_gap_cardinality_rightsoundindex. jt_gap_cardinality_rightsoundindex+S (jt_i_cardinality_right)=(v)) -> exists jt_b_cardinality_right jt_c_cardinality_right. ((((((exists fs_h_jt_cardinality_rightsoundcode. fs_h_jt_cardinality_rightsoundcode + S (jt_b_cardinality_right) = S ((S (jt_i_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightsoundcode. E = fs_q_jt_cardinality_rightsoundcode * S ((S (jt_i_cardinality_right)) * F) + (jt_b_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightsoundscale. fs_h_jt_cardinality_rightsoundscale + S (jt_c_cardinality_right) = S ((S (jt_i_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightsoundscale. G = fs_q_jt_cardinality_rightsoundscale * S ((S (jt_i_cardinality_right)) * H) + (jt_c_cardinality_right))))) /\\ (((forall jt_index_cardinality_rightbound. (exists jt_gap_cardinality_rightboundindex. jt_gap_cardinality_rightboundindex+S (jt_index_cardinality_rightbound)=(k)) -> exists jt_value_cardinality_rightbound. ((((exists fs_h_jt_cardinality_rightboundat. fs_h_jt_cardinality_rightboundat + S (jt_value_cardinality_rightbound) = S ((S (jt_index_cardinality_rightbound)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightboundat. jt_b_cardinality_right = fs_q_jt_cardinality_rightboundat * S ((S (jt_index_cardinality_rightbound)) * jt_c_cardinality_right) + (jt_value_cardinality_rightbound))) /\\ (exists jt_gap_cardinality_rightboundvalue. jt_gap_cardinality_rightboundvalue+S (jt_value_cardinality_rightbound)=(n)))) /\\ (forall jt_divisor_cardinality_rightprimitive. (exists jt_factor_cardinality_rightprimitivemodulus. (n)=(jt_divisor_cardinality_rightprimitive)*jt_factor_cardinality_rightprimitivemodulus) -> (forall jt_index_cardinality_rightprimitivecoordinates jt_value_cardinality_rightprimitivecoordinates. (exists jt_gap_cardinality_rightprimitivecoordinatesindex. jt_gap_cardinality_rightprimitivecoordinatesindex+S (jt_index_cardinality_rightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_rightprimitivecoordinatesat. fs_h_jt_cardinality_rightprimitivecoordinatesat + S (jt_value_cardinality_rightprimitivecoordinates) = S ((S (jt_index_cardinality_rightprimitivecoordinates)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightprimitivecoordinatesat. jt_b_cardinality_right = fs_q_jt_cardinality_rightprimitivecoordinatesat * S ((S (jt_index_cardinality_rightprimitivecoordinates)) * jt_c_cardinality_right) + (jt_value_cardinality_rightprimitivecoordinates))) -> (exists jt_factor_cardinality_rightprimitivecoordinatesdivides. (jt_value_cardinality_rightprimitivecoordinates)=(jt_divisor_cardinality_rightprimitive)*jt_factor_cardinality_rightprimitivecoordinatesdivides)) -> jt_divisor_cardinality_rightprimitive=1))))) /\\ (((forall jt_b_cardinality_right jt_c_cardinality_right. (forall jt_index_cardinality_rightinputbound. (exists jt_gap_cardinality_rightinputboundindex. jt_gap_cardinality_rightinputboundindex+S (jt_index_cardinality_rightinputbound)=(k)) -> exists jt_value_cardinality_rightinputbound. ((((exists fs_h_jt_cardinality_rightinputboundat. fs_h_jt_cardinality_rightinputboundat + S (jt_value_cardinality_rightinputbound) = S ((S (jt_index_cardinality_rightinputbound)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightinputboundat. jt_b_cardinality_right = fs_q_jt_cardinality_rightinputboundat * S ((S (jt_index_cardinality_rightinputbound)) * jt_c_cardinality_right) + (jt_value_cardinality_rightinputbound))) /\\ (exists jt_gap_cardinality_rightinputboundvalue. jt_gap_cardinality_rightinputboundvalue+S (jt_value_cardinality_rightinputbound)=(n)))) -> (forall jt_divisor_cardinality_rightinputprimitive. (exists jt_factor_cardinality_rightinputprimitivemodulus. (n)=(jt_divisor_cardinality_rightinputprimitive)*jt_factor_cardinality_rightinputprimitivemodulus) -> (forall jt_index_cardinality_rightinputprimitivecoordinates jt_value_cardinality_rightinputprimitivecoordinates. (exists jt_gap_cardinality_rightinputprimitivecoordinatesindex. jt_gap_cardinality_rightinputprimitivecoordinatesindex+S (jt_index_cardinality_rightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_rightinputprimitivecoordinatesat. fs_h_jt_cardinality_rightinputprimitivecoordinatesat + S (jt_value_cardinality_rightinputprimitivecoordinates) = S ((S (jt_index_cardinality_rightinputprimitivecoordinates)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightinputprimitivecoordinatesat. jt_b_cardinality_right = fs_q_jt_cardinality_rightinputprimitivecoordinatesat * S ((S (jt_index_cardinality_rightinputprimitivecoordinates)) * jt_c_cardinality_right) + (jt_value_cardinality_rightinputprimitivecoordinates))) -> (exists jt_factor_cardinality_rightinputprimitivecoordinatesdivides. (jt_value_cardinality_rightinputprimitivecoordinates)=(jt_divisor_cardinality_rightinputprimitive)*jt_factor_cardinality_rightinputprimitivecoordinatesdivides)) -> jt_divisor_cardinality_rightinputprimitive=1) -> exists jt_i_cardinality_right jt_d_cardinality_right jt_e_cardinality_right. ((exists jt_gap_cardinality_rightcompleteindex. jt_gap_cardinality_rightcompleteindex+S (jt_i_cardinality_right)=(v)) /\\ (((((((exists fs_h_jt_cardinality_rightcompletecode. fs_h_jt_cardinality_rightcompletecode + S (jt_d_cardinality_right) = S ((S (jt_i_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightcompletecode. E = fs_q_jt_cardinality_rightcompletecode * S ((S (jt_i_cardinality_right)) * F) + (jt_d_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightcompletescale. fs_h_jt_cardinality_rightcompletescale + S (jt_e_cardinality_right) = S ((S (jt_i_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightcompletescale. G = fs_q_jt_cardinality_rightcompletescale * S ((S (jt_i_cardinality_right)) * H) + (jt_e_cardinality_right))))) /\\ (forall jt_index_cardinality_rightrepresented jt_left_cardinality_rightrepresented jt_right_cardinality_rightrepresented. (exists jt_gap_cardinality_rightrepresentedindex. jt_gap_cardinality_rightrepresentedindex+S (jt_index_cardinality_rightrepresented)=(k)) -> (((exists fs_h_jt_cardinality_rightrepresentedleft. fs_h_jt_cardinality_rightrepresentedleft + S (jt_left_cardinality_rightrepresented) = S ((S (jt_index_cardinality_rightrepresented)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightrepresentedleft. jt_b_cardinality_right = fs_q_jt_cardinality_rightrepresentedleft * S ((S (jt_index_cardinality_rightrepresented)) * jt_c_cardinality_right) + (jt_left_cardinality_rightrepresented))) -> (((exists fs_h_jt_cardinality_rightrepresentedright. fs_h_jt_cardinality_rightrepresentedright + S (jt_right_cardinality_rightrepresented) = S ((S (jt_index_cardinality_rightrepresented)) * jt_e_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightrepresentedright. jt_d_cardinality_right = fs_q_jt_cardinality_rightrepresentedright * S ((S (jt_index_cardinality_rightrepresented)) * jt_e_cardinality_right) + (jt_right_cardinality_rightrepresented))) -> jt_left_cardinality_rightrepresented=jt_right_cardinality_rightrepresented))))) /\\ (forall jt_i_cardinality_right jt_h_cardinality_right jt_b_cardinality_right jt_c_cardinality_right jt_d_cardinality_right jt_e_cardinality_right. (exists jt_gap_cardinality_rightfirstindex. jt_gap_cardinality_rightfirstindex+S (jt_i_cardinality_right)=(v)) -> (exists jt_gap_cardinality_rightsecondindex. jt_gap_cardinality_rightsecondindex+S (jt_h_cardinality_right)=(v)) -> (((((exists fs_h_jt_cardinality_rightfirstcode. fs_h_jt_cardinality_rightfirstcode + S (jt_b_cardinality_right) = S ((S (jt_i_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightfirstcode. E = fs_q_jt_cardinality_rightfirstcode * S ((S (jt_i_cardinality_right)) * F) + (jt_b_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightfirstscale. fs_h_jt_cardinality_rightfirstscale + S (jt_c_cardinality_right) = S ((S (jt_i_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightfirstscale. G = fs_q_jt_cardinality_rightfirstscale * S ((S (jt_i_cardinality_right)) * H) + (jt_c_cardinality_right))))) -> (((((exists fs_h_jt_cardinality_rightsecondcode. fs_h_jt_cardinality_rightsecondcode + S (jt_d_cardinality_right) = S ((S (jt_h_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightsecondcode. E = fs_q_jt_cardinality_rightsecondcode * S ((S (jt_h_cardinality_right)) * F) + (jt_d_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightsecondscale. fs_h_jt_cardinality_rightsecondscale + S (jt_e_cardinality_right) = S ((S (jt_h_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightsecondscale. G = fs_q_jt_cardinality_rightsecondscale * S ((S (jt_h_cardinality_right)) * H) + (jt_e_cardinality_right))))) -> (forall jt_index_cardinality_rightsame jt_left_cardinality_rightsame jt_right_cardinality_rightsame. (exists jt_gap_cardinality_rightsameindex. jt_gap_cardinality_rightsameindex+S (jt_index_cardinality_rightsame)=(k)) -> (((exists fs_h_jt_cardinality_rightsameleft. fs_h_jt_cardinality_rightsameleft + S (jt_left_cardinality_rightsame) = S ((S (jt_index_cardinality_rightsame)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightsameleft. jt_b_cardinality_right = fs_q_jt_cardinality_rightsameleft * S ((S (jt_index_cardinality_rightsame)) * jt_c_cardinality_right) + (jt_left_cardinality_rightsame))) -> (((exists fs_h_jt_cardinality_rightsameright. fs_h_jt_cardinality_rightsameright + S (jt_right_cardinality_rightsame) = S ((S (jt_index_cardinality_rightsame)) * jt_e_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightsameright. jt_d_cardinality_right = fs_q_jt_cardinality_rightsameright * S ((S (jt_index_cardinality_rightsame)) * jt_e_cardinality_right) + (jt_right_cardinality_rightsame))) -> jt_left_cardinality_rightsame=jt_right_cardinality_rightsame) -> jt_i_cardinality_right=jt_h_cardinality_right))))) -> (exists jt_gap_cardinality_result. jt_gap_cardinality_result+(u)=(v))", "statement_sha256": "8f97e4b79bc26d0febb9dc2d6da7466014cc6d30accbd5a0ad3c6976004d3021", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A genuinely constructed bounded injection implies the source count is at most the target count, including empty lists." }, { "admission_dependencies": [ "jordan_enumeration_cardinality_le", "le_antisymm" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 33, "body_proof_nodes": 55, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "have hle : Le(u,v)", "specialize jordan_enumeration_cardinality_le (k)", "specialize jordan_enumeration_cardinality_le (n)", "specialize jordan_enumeration_cardinality_le (A)", "specialize jordan_enumeration_cardinality_le (B)", "specialize jordan_enumeration_cardinality_le (C)", "specialize jordan_enumeration_cardinality_le (D)", "specialize jordan_enumeration_cardinality_le (u)", "specialize jordan_enumeration_cardinality_le (E)", "specialize jordan_enumeration_cardinality_le (F)", "specialize jordan_enumeration_cardinality_le (G)", "specialize jordan_enumeration_cardinality_le (H)", "specialize jordan_enumeration_cardinality_le (v)", "apply jordan_enumeration_cardinality_le", "exact hl", "exact hr", "have hge : Le(v,u)", "specialize jordan_enumeration_cardinality_le (k)", "specialize jordan_enumeration_cardinality_le (n)", "specialize jordan_enumeration_cardinality_le (E)", "specialize jordan_enumeration_cardinality_le (F)", "specialize jordan_enumeration_cardinality_le (G)", "specialize jordan_enumeration_cardinality_le (H)", "specialize jordan_enumeration_cardinality_le (v)", "specialize jordan_enumeration_cardinality_le (A)", "specialize jordan_enumeration_cardinality_le (B)", "specialize jordan_enumeration_cardinality_le (C)", "specialize jordan_enumeration_cardinality_le (D)", "specialize jordan_enumeration_cardinality_le (u)", "apply jordan_enumeration_cardinality_le", "exact hr", "exact hl", "specialize le_antisymm (u)", "specialize le_antisymm (v)", "apply le_antisymm", "exact hle", "exact hge" ], "defined_statement": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. JordanTupleEnumeration(k,n,A,B,C,D,u) → JordanTupleEnumeration(k,n,E,F,G,H,v) → u = v", "defined_statement_sha256": "c85a992f425bf5680303fddc146349cd2f04eaa95b204be7a9377334e94b3127", "definition_uses": { "ND0374": 2, "PD0001": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "5689dad2dd80c191ff88d95fe2acba2a87dc3857279baa4d710a45a44721579c", "free_names": [], "script_definition_uses": { "PD0001": 2 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro A" } ], [ { "kind": "text", "text": "intro B" } ], [ { "kind": "text", "text": "intro C" } ], [ { "kind": "text", "text": "intro D" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro E" } ], [ { "kind": "text", "text": "intro F" } ], [ { "kind": "text", "text": "intro G" } ], [ { "kind": "text", "text": "intro H" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "have hle : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(u,v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (v)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_cardinality_le" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "have hge : " }, { "definition": "PD0001", "kind": "definition", "text": "Le(v,u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (E)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (F)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (G)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (H)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (v)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (A)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (B)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (C)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (D)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_le (u)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_cardinality_le" } ], [ { "kind": "text", "text": "exact hr" } ], [ { "kind": "text", "text": "exact hl" } ], [ { "kind": "text", "text": "specialize le_antisymm (u)" } ], [ { "kind": "text", "text": "specialize le_antisymm (v)" } ], [ { "kind": "text", "text": "apply le_antisymm" } ], [ { "kind": "text", "text": "exact hle" } ], [ { "kind": "text", "text": "exact hge" } ] ], "statement_definition_uses": { "ND0374": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ A. ∀ B. ∀ C. ∀ D. ∀ u. ∀ E. ∀ F. ∀ G. ∀ H. ∀ v. " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,A,B,C,D,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0374", "kind": "definition", "text": "JordanTupleEnumeration(k,n,E,F,G,H,v)" }, { "kind": "text", "text": " → u = v" } ] }, "dependencies": [ "jordan_enumeration_cardinality_le", "le_antisymm" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT0054", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_enumeration_cardinality_unique", "original_ha_bundle_verified": true, "proof_bundle_node_id": 345, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro A", "intro B", "intro C", "intro D", "intro u", "intro E", "intro F", "intro G", "intro H", "intro v", "intro hl", "intro hr", "have hle : exists jt_gap_unique_forward. jt_gap_unique_forward+(u)=(v)", "specialize jordan_enumeration_cardinality_le (k)", "specialize jordan_enumeration_cardinality_le (n)", "specialize jordan_enumeration_cardinality_le (A)", "specialize jordan_enumeration_cardinality_le (B)", "specialize jordan_enumeration_cardinality_le (C)", "specialize jordan_enumeration_cardinality_le (D)", "specialize jordan_enumeration_cardinality_le (u)", "specialize jordan_enumeration_cardinality_le (E)", "specialize jordan_enumeration_cardinality_le (F)", "specialize jordan_enumeration_cardinality_le (G)", "specialize jordan_enumeration_cardinality_le (H)", "specialize jordan_enumeration_cardinality_le (v)", "apply jordan_enumeration_cardinality_le", "exact hl", "exact hr", "have hge : exists jt_gap_unique_backward. jt_gap_unique_backward+(v)=(u)", "specialize jordan_enumeration_cardinality_le (k)", "specialize jordan_enumeration_cardinality_le (n)", "specialize jordan_enumeration_cardinality_le (E)", "specialize jordan_enumeration_cardinality_le (F)", "specialize jordan_enumeration_cardinality_le (G)", "specialize jordan_enumeration_cardinality_le (H)", "specialize jordan_enumeration_cardinality_le (v)", "specialize jordan_enumeration_cardinality_le (A)", "specialize jordan_enumeration_cardinality_le (B)", "specialize jordan_enumeration_cardinality_le (C)", "specialize jordan_enumeration_cardinality_le (D)", "specialize jordan_enumeration_cardinality_le (u)", "apply jordan_enumeration_cardinality_le", "exact hr", "exact hl", "specialize le_antisymm (u)", "specialize le_antisymm (v)", "apply le_antisymm", "exact hle", "exact hge" ], "script_sha256": "390f3a48351a8cc7bf62a16d91d17a1c84e1621cce1095cb99db8da331623db2", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "390f3a48351a8cc7bf62a16d91d17a1c84e1621cce1095cb99db8da331623db2", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "5689dad2dd80c191ff88d95fe2acba2a87dc3857279baa4d710a45a44721579c" } ], "stable_member": false, "statement": "forall k n A B C D u E F G H v. (((forall jt_i_cardinality_left. (exists jt_gap_cardinality_leftsoundindex. jt_gap_cardinality_leftsoundindex+S (jt_i_cardinality_left)=(u)) -> exists jt_b_cardinality_left jt_c_cardinality_left. ((((((exists fs_h_jt_cardinality_leftsoundcode. fs_h_jt_cardinality_leftsoundcode + S (jt_b_cardinality_left) = S ((S (jt_i_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftsoundcode. A = fs_q_jt_cardinality_leftsoundcode * S ((S (jt_i_cardinality_left)) * B) + (jt_b_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftsoundscale. fs_h_jt_cardinality_leftsoundscale + S (jt_c_cardinality_left) = S ((S (jt_i_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftsoundscale. C = fs_q_jt_cardinality_leftsoundscale * S ((S (jt_i_cardinality_left)) * D) + (jt_c_cardinality_left))))) /\\ (((forall jt_index_cardinality_leftbound. (exists jt_gap_cardinality_leftboundindex. jt_gap_cardinality_leftboundindex+S (jt_index_cardinality_leftbound)=(k)) -> exists jt_value_cardinality_leftbound. ((((exists fs_h_jt_cardinality_leftboundat. fs_h_jt_cardinality_leftboundat + S (jt_value_cardinality_leftbound) = S ((S (jt_index_cardinality_leftbound)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftboundat. jt_b_cardinality_left = fs_q_jt_cardinality_leftboundat * S ((S (jt_index_cardinality_leftbound)) * jt_c_cardinality_left) + (jt_value_cardinality_leftbound))) /\\ (exists jt_gap_cardinality_leftboundvalue. jt_gap_cardinality_leftboundvalue+S (jt_value_cardinality_leftbound)=(n)))) /\\ (forall jt_divisor_cardinality_leftprimitive. (exists jt_factor_cardinality_leftprimitivemodulus. (n)=(jt_divisor_cardinality_leftprimitive)*jt_factor_cardinality_leftprimitivemodulus) -> (forall jt_index_cardinality_leftprimitivecoordinates jt_value_cardinality_leftprimitivecoordinates. (exists jt_gap_cardinality_leftprimitivecoordinatesindex. jt_gap_cardinality_leftprimitivecoordinatesindex+S (jt_index_cardinality_leftprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_leftprimitivecoordinatesat. fs_h_jt_cardinality_leftprimitivecoordinatesat + S (jt_value_cardinality_leftprimitivecoordinates) = S ((S (jt_index_cardinality_leftprimitivecoordinates)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftprimitivecoordinatesat. jt_b_cardinality_left = fs_q_jt_cardinality_leftprimitivecoordinatesat * S ((S (jt_index_cardinality_leftprimitivecoordinates)) * jt_c_cardinality_left) + (jt_value_cardinality_leftprimitivecoordinates))) -> (exists jt_factor_cardinality_leftprimitivecoordinatesdivides. (jt_value_cardinality_leftprimitivecoordinates)=(jt_divisor_cardinality_leftprimitive)*jt_factor_cardinality_leftprimitivecoordinatesdivides)) -> jt_divisor_cardinality_leftprimitive=1))))) /\\ (((forall jt_b_cardinality_left jt_c_cardinality_left. (forall jt_index_cardinality_leftinputbound. (exists jt_gap_cardinality_leftinputboundindex. jt_gap_cardinality_leftinputboundindex+S (jt_index_cardinality_leftinputbound)=(k)) -> exists jt_value_cardinality_leftinputbound. ((((exists fs_h_jt_cardinality_leftinputboundat. fs_h_jt_cardinality_leftinputboundat + S (jt_value_cardinality_leftinputbound) = S ((S (jt_index_cardinality_leftinputbound)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftinputboundat. jt_b_cardinality_left = fs_q_jt_cardinality_leftinputboundat * S ((S (jt_index_cardinality_leftinputbound)) * jt_c_cardinality_left) + (jt_value_cardinality_leftinputbound))) /\\ (exists jt_gap_cardinality_leftinputboundvalue. jt_gap_cardinality_leftinputboundvalue+S (jt_value_cardinality_leftinputbound)=(n)))) -> (forall jt_divisor_cardinality_leftinputprimitive. (exists jt_factor_cardinality_leftinputprimitivemodulus. (n)=(jt_divisor_cardinality_leftinputprimitive)*jt_factor_cardinality_leftinputprimitivemodulus) -> (forall jt_index_cardinality_leftinputprimitivecoordinates jt_value_cardinality_leftinputprimitivecoordinates. (exists jt_gap_cardinality_leftinputprimitivecoordinatesindex. jt_gap_cardinality_leftinputprimitivecoordinatesindex+S (jt_index_cardinality_leftinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_leftinputprimitivecoordinatesat. fs_h_jt_cardinality_leftinputprimitivecoordinatesat + S (jt_value_cardinality_leftinputprimitivecoordinates) = S ((S (jt_index_cardinality_leftinputprimitivecoordinates)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftinputprimitivecoordinatesat. jt_b_cardinality_left = fs_q_jt_cardinality_leftinputprimitivecoordinatesat * S ((S (jt_index_cardinality_leftinputprimitivecoordinates)) * jt_c_cardinality_left) + (jt_value_cardinality_leftinputprimitivecoordinates))) -> (exists jt_factor_cardinality_leftinputprimitivecoordinatesdivides. (jt_value_cardinality_leftinputprimitivecoordinates)=(jt_divisor_cardinality_leftinputprimitive)*jt_factor_cardinality_leftinputprimitivecoordinatesdivides)) -> jt_divisor_cardinality_leftinputprimitive=1) -> exists jt_i_cardinality_left jt_d_cardinality_left jt_e_cardinality_left. ((exists jt_gap_cardinality_leftcompleteindex. jt_gap_cardinality_leftcompleteindex+S (jt_i_cardinality_left)=(u)) /\\ (((((((exists fs_h_jt_cardinality_leftcompletecode. fs_h_jt_cardinality_leftcompletecode + S (jt_d_cardinality_left) = S ((S (jt_i_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftcompletecode. A = fs_q_jt_cardinality_leftcompletecode * S ((S (jt_i_cardinality_left)) * B) + (jt_d_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftcompletescale. fs_h_jt_cardinality_leftcompletescale + S (jt_e_cardinality_left) = S ((S (jt_i_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftcompletescale. C = fs_q_jt_cardinality_leftcompletescale * S ((S (jt_i_cardinality_left)) * D) + (jt_e_cardinality_left))))) /\\ (forall jt_index_cardinality_leftrepresented jt_left_cardinality_leftrepresented jt_right_cardinality_leftrepresented. (exists jt_gap_cardinality_leftrepresentedindex. jt_gap_cardinality_leftrepresentedindex+S (jt_index_cardinality_leftrepresented)=(k)) -> (((exists fs_h_jt_cardinality_leftrepresentedleft. fs_h_jt_cardinality_leftrepresentedleft + S (jt_left_cardinality_leftrepresented) = S ((S (jt_index_cardinality_leftrepresented)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftrepresentedleft. jt_b_cardinality_left = fs_q_jt_cardinality_leftrepresentedleft * S ((S (jt_index_cardinality_leftrepresented)) * jt_c_cardinality_left) + (jt_left_cardinality_leftrepresented))) -> (((exists fs_h_jt_cardinality_leftrepresentedright. fs_h_jt_cardinality_leftrepresentedright + S (jt_right_cardinality_leftrepresented) = S ((S (jt_index_cardinality_leftrepresented)) * jt_e_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftrepresentedright. jt_d_cardinality_left = fs_q_jt_cardinality_leftrepresentedright * S ((S (jt_index_cardinality_leftrepresented)) * jt_e_cardinality_left) + (jt_right_cardinality_leftrepresented))) -> jt_left_cardinality_leftrepresented=jt_right_cardinality_leftrepresented))))) /\\ (forall jt_i_cardinality_left jt_h_cardinality_left jt_b_cardinality_left jt_c_cardinality_left jt_d_cardinality_left jt_e_cardinality_left. (exists jt_gap_cardinality_leftfirstindex. jt_gap_cardinality_leftfirstindex+S (jt_i_cardinality_left)=(u)) -> (exists jt_gap_cardinality_leftsecondindex. jt_gap_cardinality_leftsecondindex+S (jt_h_cardinality_left)=(u)) -> (((((exists fs_h_jt_cardinality_leftfirstcode. fs_h_jt_cardinality_leftfirstcode + S (jt_b_cardinality_left) = S ((S (jt_i_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftfirstcode. A = fs_q_jt_cardinality_leftfirstcode * S ((S (jt_i_cardinality_left)) * B) + (jt_b_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftfirstscale. fs_h_jt_cardinality_leftfirstscale + S (jt_c_cardinality_left) = S ((S (jt_i_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftfirstscale. C = fs_q_jt_cardinality_leftfirstscale * S ((S (jt_i_cardinality_left)) * D) + (jt_c_cardinality_left))))) -> (((((exists fs_h_jt_cardinality_leftsecondcode. fs_h_jt_cardinality_leftsecondcode + S (jt_d_cardinality_left) = S ((S (jt_h_cardinality_left)) * B)) /\\ exists fs_q_jt_cardinality_leftsecondcode. A = fs_q_jt_cardinality_leftsecondcode * S ((S (jt_h_cardinality_left)) * B) + (jt_d_cardinality_left))) /\\ (((exists fs_h_jt_cardinality_leftsecondscale. fs_h_jt_cardinality_leftsecondscale + S (jt_e_cardinality_left) = S ((S (jt_h_cardinality_left)) * D)) /\\ exists fs_q_jt_cardinality_leftsecondscale. C = fs_q_jt_cardinality_leftsecondscale * S ((S (jt_h_cardinality_left)) * D) + (jt_e_cardinality_left))))) -> (forall jt_index_cardinality_leftsame jt_left_cardinality_leftsame jt_right_cardinality_leftsame. (exists jt_gap_cardinality_leftsameindex. jt_gap_cardinality_leftsameindex+S (jt_index_cardinality_leftsame)=(k)) -> (((exists fs_h_jt_cardinality_leftsameleft. fs_h_jt_cardinality_leftsameleft + S (jt_left_cardinality_leftsame) = S ((S (jt_index_cardinality_leftsame)) * jt_c_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftsameleft. jt_b_cardinality_left = fs_q_jt_cardinality_leftsameleft * S ((S (jt_index_cardinality_leftsame)) * jt_c_cardinality_left) + (jt_left_cardinality_leftsame))) -> (((exists fs_h_jt_cardinality_leftsameright. fs_h_jt_cardinality_leftsameright + S (jt_right_cardinality_leftsame) = S ((S (jt_index_cardinality_leftsame)) * jt_e_cardinality_left)) /\\ exists fs_q_jt_cardinality_leftsameright. jt_d_cardinality_left = fs_q_jt_cardinality_leftsameright * S ((S (jt_index_cardinality_leftsame)) * jt_e_cardinality_left) + (jt_right_cardinality_leftsame))) -> jt_left_cardinality_leftsame=jt_right_cardinality_leftsame) -> jt_i_cardinality_left=jt_h_cardinality_left))))) -> (((forall jt_i_cardinality_right. (exists jt_gap_cardinality_rightsoundindex. jt_gap_cardinality_rightsoundindex+S (jt_i_cardinality_right)=(v)) -> exists jt_b_cardinality_right jt_c_cardinality_right. ((((((exists fs_h_jt_cardinality_rightsoundcode. fs_h_jt_cardinality_rightsoundcode + S (jt_b_cardinality_right) = S ((S (jt_i_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightsoundcode. E = fs_q_jt_cardinality_rightsoundcode * S ((S (jt_i_cardinality_right)) * F) + (jt_b_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightsoundscale. fs_h_jt_cardinality_rightsoundscale + S (jt_c_cardinality_right) = S ((S (jt_i_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightsoundscale. G = fs_q_jt_cardinality_rightsoundscale * S ((S (jt_i_cardinality_right)) * H) + (jt_c_cardinality_right))))) /\\ (((forall jt_index_cardinality_rightbound. (exists jt_gap_cardinality_rightboundindex. jt_gap_cardinality_rightboundindex+S (jt_index_cardinality_rightbound)=(k)) -> exists jt_value_cardinality_rightbound. ((((exists fs_h_jt_cardinality_rightboundat. fs_h_jt_cardinality_rightboundat + S (jt_value_cardinality_rightbound) = S ((S (jt_index_cardinality_rightbound)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightboundat. jt_b_cardinality_right = fs_q_jt_cardinality_rightboundat * S ((S (jt_index_cardinality_rightbound)) * jt_c_cardinality_right) + (jt_value_cardinality_rightbound))) /\\ (exists jt_gap_cardinality_rightboundvalue. jt_gap_cardinality_rightboundvalue+S (jt_value_cardinality_rightbound)=(n)))) /\\ (forall jt_divisor_cardinality_rightprimitive. (exists jt_factor_cardinality_rightprimitivemodulus. (n)=(jt_divisor_cardinality_rightprimitive)*jt_factor_cardinality_rightprimitivemodulus) -> (forall jt_index_cardinality_rightprimitivecoordinates jt_value_cardinality_rightprimitivecoordinates. (exists jt_gap_cardinality_rightprimitivecoordinatesindex. jt_gap_cardinality_rightprimitivecoordinatesindex+S (jt_index_cardinality_rightprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_rightprimitivecoordinatesat. fs_h_jt_cardinality_rightprimitivecoordinatesat + S (jt_value_cardinality_rightprimitivecoordinates) = S ((S (jt_index_cardinality_rightprimitivecoordinates)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightprimitivecoordinatesat. jt_b_cardinality_right = fs_q_jt_cardinality_rightprimitivecoordinatesat * S ((S (jt_index_cardinality_rightprimitivecoordinates)) * jt_c_cardinality_right) + (jt_value_cardinality_rightprimitivecoordinates))) -> (exists jt_factor_cardinality_rightprimitivecoordinatesdivides. (jt_value_cardinality_rightprimitivecoordinates)=(jt_divisor_cardinality_rightprimitive)*jt_factor_cardinality_rightprimitivecoordinatesdivides)) -> jt_divisor_cardinality_rightprimitive=1))))) /\\ (((forall jt_b_cardinality_right jt_c_cardinality_right. (forall jt_index_cardinality_rightinputbound. (exists jt_gap_cardinality_rightinputboundindex. jt_gap_cardinality_rightinputboundindex+S (jt_index_cardinality_rightinputbound)=(k)) -> exists jt_value_cardinality_rightinputbound. ((((exists fs_h_jt_cardinality_rightinputboundat. fs_h_jt_cardinality_rightinputboundat + S (jt_value_cardinality_rightinputbound) = S ((S (jt_index_cardinality_rightinputbound)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightinputboundat. jt_b_cardinality_right = fs_q_jt_cardinality_rightinputboundat * S ((S (jt_index_cardinality_rightinputbound)) * jt_c_cardinality_right) + (jt_value_cardinality_rightinputbound))) /\\ (exists jt_gap_cardinality_rightinputboundvalue. jt_gap_cardinality_rightinputboundvalue+S (jt_value_cardinality_rightinputbound)=(n)))) -> (forall jt_divisor_cardinality_rightinputprimitive. (exists jt_factor_cardinality_rightinputprimitivemodulus. (n)=(jt_divisor_cardinality_rightinputprimitive)*jt_factor_cardinality_rightinputprimitivemodulus) -> (forall jt_index_cardinality_rightinputprimitivecoordinates jt_value_cardinality_rightinputprimitivecoordinates. (exists jt_gap_cardinality_rightinputprimitivecoordinatesindex. jt_gap_cardinality_rightinputprimitivecoordinatesindex+S (jt_index_cardinality_rightinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_cardinality_rightinputprimitivecoordinatesat. fs_h_jt_cardinality_rightinputprimitivecoordinatesat + S (jt_value_cardinality_rightinputprimitivecoordinates) = S ((S (jt_index_cardinality_rightinputprimitivecoordinates)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightinputprimitivecoordinatesat. jt_b_cardinality_right = fs_q_jt_cardinality_rightinputprimitivecoordinatesat * S ((S (jt_index_cardinality_rightinputprimitivecoordinates)) * jt_c_cardinality_right) + (jt_value_cardinality_rightinputprimitivecoordinates))) -> (exists jt_factor_cardinality_rightinputprimitivecoordinatesdivides. (jt_value_cardinality_rightinputprimitivecoordinates)=(jt_divisor_cardinality_rightinputprimitive)*jt_factor_cardinality_rightinputprimitivecoordinatesdivides)) -> jt_divisor_cardinality_rightinputprimitive=1) -> exists jt_i_cardinality_right jt_d_cardinality_right jt_e_cardinality_right. ((exists jt_gap_cardinality_rightcompleteindex. jt_gap_cardinality_rightcompleteindex+S (jt_i_cardinality_right)=(v)) /\\ (((((((exists fs_h_jt_cardinality_rightcompletecode. fs_h_jt_cardinality_rightcompletecode + S (jt_d_cardinality_right) = S ((S (jt_i_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightcompletecode. E = fs_q_jt_cardinality_rightcompletecode * S ((S (jt_i_cardinality_right)) * F) + (jt_d_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightcompletescale. fs_h_jt_cardinality_rightcompletescale + S (jt_e_cardinality_right) = S ((S (jt_i_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightcompletescale. G = fs_q_jt_cardinality_rightcompletescale * S ((S (jt_i_cardinality_right)) * H) + (jt_e_cardinality_right))))) /\\ (forall jt_index_cardinality_rightrepresented jt_left_cardinality_rightrepresented jt_right_cardinality_rightrepresented. (exists jt_gap_cardinality_rightrepresentedindex. jt_gap_cardinality_rightrepresentedindex+S (jt_index_cardinality_rightrepresented)=(k)) -> (((exists fs_h_jt_cardinality_rightrepresentedleft. fs_h_jt_cardinality_rightrepresentedleft + S (jt_left_cardinality_rightrepresented) = S ((S (jt_index_cardinality_rightrepresented)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightrepresentedleft. jt_b_cardinality_right = fs_q_jt_cardinality_rightrepresentedleft * S ((S (jt_index_cardinality_rightrepresented)) * jt_c_cardinality_right) + (jt_left_cardinality_rightrepresented))) -> (((exists fs_h_jt_cardinality_rightrepresentedright. fs_h_jt_cardinality_rightrepresentedright + S (jt_right_cardinality_rightrepresented) = S ((S (jt_index_cardinality_rightrepresented)) * jt_e_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightrepresentedright. jt_d_cardinality_right = fs_q_jt_cardinality_rightrepresentedright * S ((S (jt_index_cardinality_rightrepresented)) * jt_e_cardinality_right) + (jt_right_cardinality_rightrepresented))) -> jt_left_cardinality_rightrepresented=jt_right_cardinality_rightrepresented))))) /\\ (forall jt_i_cardinality_right jt_h_cardinality_right jt_b_cardinality_right jt_c_cardinality_right jt_d_cardinality_right jt_e_cardinality_right. (exists jt_gap_cardinality_rightfirstindex. jt_gap_cardinality_rightfirstindex+S (jt_i_cardinality_right)=(v)) -> (exists jt_gap_cardinality_rightsecondindex. jt_gap_cardinality_rightsecondindex+S (jt_h_cardinality_right)=(v)) -> (((((exists fs_h_jt_cardinality_rightfirstcode. fs_h_jt_cardinality_rightfirstcode + S (jt_b_cardinality_right) = S ((S (jt_i_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightfirstcode. E = fs_q_jt_cardinality_rightfirstcode * S ((S (jt_i_cardinality_right)) * F) + (jt_b_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightfirstscale. fs_h_jt_cardinality_rightfirstscale + S (jt_c_cardinality_right) = S ((S (jt_i_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightfirstscale. G = fs_q_jt_cardinality_rightfirstscale * S ((S (jt_i_cardinality_right)) * H) + (jt_c_cardinality_right))))) -> (((((exists fs_h_jt_cardinality_rightsecondcode. fs_h_jt_cardinality_rightsecondcode + S (jt_d_cardinality_right) = S ((S (jt_h_cardinality_right)) * F)) /\\ exists fs_q_jt_cardinality_rightsecondcode. E = fs_q_jt_cardinality_rightsecondcode * S ((S (jt_h_cardinality_right)) * F) + (jt_d_cardinality_right))) /\\ (((exists fs_h_jt_cardinality_rightsecondscale. fs_h_jt_cardinality_rightsecondscale + S (jt_e_cardinality_right) = S ((S (jt_h_cardinality_right)) * H)) /\\ exists fs_q_jt_cardinality_rightsecondscale. G = fs_q_jt_cardinality_rightsecondscale * S ((S (jt_h_cardinality_right)) * H) + (jt_e_cardinality_right))))) -> (forall jt_index_cardinality_rightsame jt_left_cardinality_rightsame jt_right_cardinality_rightsame. (exists jt_gap_cardinality_rightsameindex. jt_gap_cardinality_rightsameindex+S (jt_index_cardinality_rightsame)=(k)) -> (((exists fs_h_jt_cardinality_rightsameleft. fs_h_jt_cardinality_rightsameleft + S (jt_left_cardinality_rightsame) = S ((S (jt_index_cardinality_rightsame)) * jt_c_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightsameleft. jt_b_cardinality_right = fs_q_jt_cardinality_rightsameleft * S ((S (jt_index_cardinality_rightsame)) * jt_c_cardinality_right) + (jt_left_cardinality_rightsame))) -> (((exists fs_h_jt_cardinality_rightsameright. fs_h_jt_cardinality_rightsameright + S (jt_right_cardinality_rightsame) = S ((S (jt_index_cardinality_rightsame)) * jt_e_cardinality_right)) /\\ exists fs_q_jt_cardinality_rightsameright. jt_d_cardinality_right = fs_q_jt_cardinality_rightsameright * S ((S (jt_index_cardinality_rightsame)) * jt_e_cardinality_right) + (jt_right_cardinality_rightsame))) -> jt_left_cardinality_rightsame=jt_right_cardinality_rightsame) -> jt_i_cardinality_right=jt_h_cardinality_right))))) -> (u=v)", "statement_sha256": "5689dad2dd80c191ff88d95fe2acba2a87dc3857279baa4d710a45a44721579c", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Two complete duplicate-free enumerations of the same primitive coordinate tuples have equal lengths." }, { "admission_dependencies": [ "jordan_enumeration_cardinality_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 58, "body_proof_nodes": 108, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro n", "intro u", "intro v", "intro hl", "intro hr", "cases hl", "cases hl_right", "cases hr", "cases hr_right", "cases hl_right_right", "cases hl_right_right_witness", "cases hl_right_right_witness_witness", "cases hl_right_right_witness_witness_witness", "cases hr_right_right", "cases hr_right_right_witness", "cases hr_right_right_witness_witness", "cases hr_right_right_witness_witness_witness", "specialize jordan_enumeration_cardinality_unique (k)", "specialize jordan_enumeration_cardinality_unique (n)", "specialize jordan_enumeration_cardinality_unique (x)", "specialize jordan_enumeration_cardinality_unique (x1)", "specialize jordan_enumeration_cardinality_unique (x2)", "specialize jordan_enumeration_cardinality_unique (x3)", "specialize jordan_enumeration_cardinality_unique (u)", "specialize jordan_enumeration_cardinality_unique (x4)", "specialize jordan_enumeration_cardinality_unique (x5)", "specialize jordan_enumeration_cardinality_unique (x6)", "specialize jordan_enumeration_cardinality_unique (x7)", "specialize jordan_enumeration_cardinality_unique (v)", "apply jordan_enumeration_cardinality_unique", "exact hl_right_right_witness_witness_witness_witness", "exact hr_right_right_witness_witness_witness_witness" ], "defined_statement": "∀ k. ∀ n. ∀ u. ∀ v. JordanTotient(k,n,u) → JordanTotient(k,n,v) → u = v", "defined_statement_sha256": "71dfe7363efba78e293169704989db8d966143ab12af87785bd2084ef37decb2", "definition_uses": { "ND0375": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "e8ac43d89f3ac491062245faf7472b67b63a2b800c8d64daa127153e67041a08", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro hl" } ], [ { "kind": "text", "text": "intro hr" } ], [ { "kind": "text", "text": "cases hl" } ], [ { "kind": "text", "text": "cases hl_right" } ], [ { "kind": "text", "text": "cases hr" } ], [ { "kind": "text", "text": "cases hr_right" } ], [ { "kind": "text", "text": "cases hl_right_right" } ], [ { "kind": "text", "text": "cases hl_right_right_witness" } ], [ { "kind": "text", "text": "cases hl_right_right_witness_witness" } ], [ { "kind": "text", "text": "cases hl_right_right_witness_witness_witness" } ], [ { "kind": "text", "text": "cases hr_right_right" } ], [ { "kind": "text", "text": "cases hr_right_right_witness" } ], [ { "kind": "text", "text": "cases hr_right_right_witness_witness" } ], [ { "kind": "text", "text": "cases hr_right_right_witness_witness_witness" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (k)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (n)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x1)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x2)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x3)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (u)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x4)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x5)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x6)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (x7)" } ], [ { "kind": "text", "text": "specialize jordan_enumeration_cardinality_unique (v)" } ], [ { "kind": "text", "text": "apply jordan_enumeration_cardinality_unique" } ], [ { "kind": "text", "text": "exact hl_right_right_witness_witness_witness_witness" } ], [ { "kind": "text", "text": "exact hr_right_right_witness_witness_witness_witness" } ] ], "statement_definition_uses": { "ND0375": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ n. ∀ u. ∀ v. " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,n,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,n,v)" }, { "kind": "text", "text": " → u = v" } ] }, "dependencies": [ "jordan_enumeration_cardinality_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_count_uniqueness_candidate_theorems", "first_admitted_version": "v35", "id": "JT0055", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_count_unique", "original_ha_bundle_verified": true, "proof_bundle_node_id": 346, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro n", "intro u", "intro v", "intro hl", "intro hr", "cases hl", "cases hl_right", "cases hr", "cases hr_right", "cases hl_right_right", "cases hl_right_right_witness", "cases hl_right_right_witness_witness", "cases hl_right_right_witness_witness_witness", "cases hr_right_right", "cases hr_right_right_witness", "cases hr_right_right_witness_witness", "cases hr_right_right_witness_witness_witness", "specialize jordan_enumeration_cardinality_unique (k)", "specialize jordan_enumeration_cardinality_unique (n)", "specialize jordan_enumeration_cardinality_unique (x)", "specialize jordan_enumeration_cardinality_unique (x1)", "specialize jordan_enumeration_cardinality_unique (x2)", "specialize jordan_enumeration_cardinality_unique (x3)", "specialize jordan_enumeration_cardinality_unique (u)", "specialize jordan_enumeration_cardinality_unique (x4)", "specialize jordan_enumeration_cardinality_unique (x5)", "specialize jordan_enumeration_cardinality_unique (x6)", "specialize jordan_enumeration_cardinality_unique (x7)", "specialize jordan_enumeration_cardinality_unique (v)", "apply jordan_enumeration_cardinality_unique", "exact hl_right_right_witness_witness_witness_witness", "exact hr_right_right_witness_witness_witness_witness" ], "script_sha256": "25fbe4941160599a4e99649e8638f45ba942144673156db782d1cebedd36cd6b", "source_filename": "jordan_count_uniqueness_candidate.py", "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "sources": [ { "factory": "make_jordan_count_uniqueness_candidate_theorems", "script_sha256": "25fbe4941160599a4e99649e8638f45ba942144673156db782d1cebedd36cd6b", "selected": true, "source_module": "peano_lab.library.jordan_count_uniqueness_candidate", "source_sha256": "06e609a6f14b837eeb8d913d92e6090d4703dfd8b0d25aa4e348fcbd50b57074", "statement_sha256": "e8ac43d89f3ac491062245faf7472b67b63a2b800c8d64daa127153e67041a08" } ], "stable_member": false, "statement": "forall k n u v. (((~((k)=0)) /\\ (((~((n)=0)) /\\ (exists jt_codes_unique_jordan_left jt_code_scale_unique_jordan_left jt_scales_unique_jordan_left jt_scale_scale_unique_jordan_left. ((forall jt_i_unique_jordan_leftenum. (exists jt_gap_unique_jordan_leftenumsoundindex. jt_gap_unique_jordan_leftenumsoundindex+S (jt_i_unique_jordan_leftenum)=(u)) -> exists jt_b_unique_jordan_leftenum jt_c_unique_jordan_leftenum. ((((((exists fs_h_jt_unique_jordan_leftenumsoundcode. fs_h_jt_unique_jordan_leftenumsoundcode + S (jt_b_unique_jordan_leftenum) = S ((S (jt_i_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumsoundcode. jt_codes_unique_jordan_left = fs_q_jt_unique_jordan_leftenumsoundcode * S ((S (jt_i_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left) + (jt_b_unique_jordan_leftenum))) /\\ (((exists fs_h_jt_unique_jordan_leftenumsoundscale. fs_h_jt_unique_jordan_leftenumsoundscale + S (jt_c_unique_jordan_leftenum) = S ((S (jt_i_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumsoundscale. jt_scales_unique_jordan_left = fs_q_jt_unique_jordan_leftenumsoundscale * S ((S (jt_i_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left) + (jt_c_unique_jordan_leftenum))))) /\\ (((forall jt_index_unique_jordan_leftenumbound. (exists jt_gap_unique_jordan_leftenumboundindex. jt_gap_unique_jordan_leftenumboundindex+S (jt_index_unique_jordan_leftenumbound)=(k)) -> exists jt_value_unique_jordan_leftenumbound. ((((exists fs_h_jt_unique_jordan_leftenumboundat. fs_h_jt_unique_jordan_leftenumboundat + S (jt_value_unique_jordan_leftenumbound) = S ((S (jt_index_unique_jordan_leftenumbound)) * jt_c_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenumboundat. jt_b_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenumboundat * S ((S (jt_index_unique_jordan_leftenumbound)) * jt_c_unique_jordan_leftenum) + (jt_value_unique_jordan_leftenumbound))) /\\ (exists jt_gap_unique_jordan_leftenumboundvalue. jt_gap_unique_jordan_leftenumboundvalue+S (jt_value_unique_jordan_leftenumbound)=(n)))) /\\ (forall jt_divisor_unique_jordan_leftenumprimitive. (exists jt_factor_unique_jordan_leftenumprimitivemodulus. (n)=(jt_divisor_unique_jordan_leftenumprimitive)*jt_factor_unique_jordan_leftenumprimitivemodulus) -> (forall jt_index_unique_jordan_leftenumprimitivecoordinates jt_value_unique_jordan_leftenumprimitivecoordinates. (exists jt_gap_unique_jordan_leftenumprimitivecoordinatesindex. jt_gap_unique_jordan_leftenumprimitivecoordinatesindex+S (jt_index_unique_jordan_leftenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unique_jordan_leftenumprimitivecoordinatesat. fs_h_jt_unique_jordan_leftenumprimitivecoordinatesat + S (jt_value_unique_jordan_leftenumprimitivecoordinates) = S ((S (jt_index_unique_jordan_leftenumprimitivecoordinates)) * jt_c_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenumprimitivecoordinatesat. jt_b_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenumprimitivecoordinatesat * S ((S (jt_index_unique_jordan_leftenumprimitivecoordinates)) * jt_c_unique_jordan_leftenum) + (jt_value_unique_jordan_leftenumprimitivecoordinates))) -> (exists jt_factor_unique_jordan_leftenumprimitivecoordinatesdivides. (jt_value_unique_jordan_leftenumprimitivecoordinates)=(jt_divisor_unique_jordan_leftenumprimitive)*jt_factor_unique_jordan_leftenumprimitivecoordinatesdivides)) -> jt_divisor_unique_jordan_leftenumprimitive=1))))) /\\ (((forall jt_b_unique_jordan_leftenum jt_c_unique_jordan_leftenum. (forall jt_index_unique_jordan_leftenuminputbound. (exists jt_gap_unique_jordan_leftenuminputboundindex. jt_gap_unique_jordan_leftenuminputboundindex+S (jt_index_unique_jordan_leftenuminputbound)=(k)) -> exists jt_value_unique_jordan_leftenuminputbound. ((((exists fs_h_jt_unique_jordan_leftenuminputboundat. fs_h_jt_unique_jordan_leftenuminputboundat + S (jt_value_unique_jordan_leftenuminputbound) = S ((S (jt_index_unique_jordan_leftenuminputbound)) * jt_c_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenuminputboundat. jt_b_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenuminputboundat * S ((S (jt_index_unique_jordan_leftenuminputbound)) * jt_c_unique_jordan_leftenum) + (jt_value_unique_jordan_leftenuminputbound))) /\\ (exists jt_gap_unique_jordan_leftenuminputboundvalue. jt_gap_unique_jordan_leftenuminputboundvalue+S (jt_value_unique_jordan_leftenuminputbound)=(n)))) -> (forall jt_divisor_unique_jordan_leftenuminputprimitive. (exists jt_factor_unique_jordan_leftenuminputprimitivemodulus. (n)=(jt_divisor_unique_jordan_leftenuminputprimitive)*jt_factor_unique_jordan_leftenuminputprimitivemodulus) -> (forall jt_index_unique_jordan_leftenuminputprimitivecoordinates jt_value_unique_jordan_leftenuminputprimitivecoordinates. (exists jt_gap_unique_jordan_leftenuminputprimitivecoordinatesindex. jt_gap_unique_jordan_leftenuminputprimitivecoordinatesindex+S (jt_index_unique_jordan_leftenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unique_jordan_leftenuminputprimitivecoordinatesat. fs_h_jt_unique_jordan_leftenuminputprimitivecoordinatesat + S (jt_value_unique_jordan_leftenuminputprimitivecoordinates) = S ((S (jt_index_unique_jordan_leftenuminputprimitivecoordinates)) * jt_c_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenuminputprimitivecoordinatesat. jt_b_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenuminputprimitivecoordinatesat * S ((S (jt_index_unique_jordan_leftenuminputprimitivecoordinates)) * jt_c_unique_jordan_leftenum) + (jt_value_unique_jordan_leftenuminputprimitivecoordinates))) -> (exists jt_factor_unique_jordan_leftenuminputprimitivecoordinatesdivides. (jt_value_unique_jordan_leftenuminputprimitivecoordinates)=(jt_divisor_unique_jordan_leftenuminputprimitive)*jt_factor_unique_jordan_leftenuminputprimitivecoordinatesdivides)) -> jt_divisor_unique_jordan_leftenuminputprimitive=1) -> exists jt_i_unique_jordan_leftenum jt_d_unique_jordan_leftenum jt_e_unique_jordan_leftenum. ((exists jt_gap_unique_jordan_leftenumcompleteindex. jt_gap_unique_jordan_leftenumcompleteindex+S (jt_i_unique_jordan_leftenum)=(u)) /\\ (((((((exists fs_h_jt_unique_jordan_leftenumcompletecode. fs_h_jt_unique_jordan_leftenumcompletecode + S (jt_d_unique_jordan_leftenum) = S ((S (jt_i_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumcompletecode. jt_codes_unique_jordan_left = fs_q_jt_unique_jordan_leftenumcompletecode * S ((S (jt_i_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left) + (jt_d_unique_jordan_leftenum))) /\\ (((exists fs_h_jt_unique_jordan_leftenumcompletescale. fs_h_jt_unique_jordan_leftenumcompletescale + S (jt_e_unique_jordan_leftenum) = S ((S (jt_i_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumcompletescale. jt_scales_unique_jordan_left = fs_q_jt_unique_jordan_leftenumcompletescale * S ((S (jt_i_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left) + (jt_e_unique_jordan_leftenum))))) /\\ (forall jt_index_unique_jordan_leftenumrepresented jt_left_unique_jordan_leftenumrepresented jt_right_unique_jordan_leftenumrepresented. (exists jt_gap_unique_jordan_leftenumrepresentedindex. jt_gap_unique_jordan_leftenumrepresentedindex+S (jt_index_unique_jordan_leftenumrepresented)=(k)) -> (((exists fs_h_jt_unique_jordan_leftenumrepresentedleft. fs_h_jt_unique_jordan_leftenumrepresentedleft + S (jt_left_unique_jordan_leftenumrepresented) = S ((S (jt_index_unique_jordan_leftenumrepresented)) * jt_c_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenumrepresentedleft. jt_b_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenumrepresentedleft * S ((S (jt_index_unique_jordan_leftenumrepresented)) * jt_c_unique_jordan_leftenum) + (jt_left_unique_jordan_leftenumrepresented))) -> (((exists fs_h_jt_unique_jordan_leftenumrepresentedright. fs_h_jt_unique_jordan_leftenumrepresentedright + S (jt_right_unique_jordan_leftenumrepresented) = S ((S (jt_index_unique_jordan_leftenumrepresented)) * jt_e_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenumrepresentedright. jt_d_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenumrepresentedright * S ((S (jt_index_unique_jordan_leftenumrepresented)) * jt_e_unique_jordan_leftenum) + (jt_right_unique_jordan_leftenumrepresented))) -> jt_left_unique_jordan_leftenumrepresented=jt_right_unique_jordan_leftenumrepresented))))) /\\ (forall jt_i_unique_jordan_leftenum jt_h_unique_jordan_leftenum jt_b_unique_jordan_leftenum jt_c_unique_jordan_leftenum jt_d_unique_jordan_leftenum jt_e_unique_jordan_leftenum. (exists jt_gap_unique_jordan_leftenumfirstindex. jt_gap_unique_jordan_leftenumfirstindex+S (jt_i_unique_jordan_leftenum)=(u)) -> (exists jt_gap_unique_jordan_leftenumsecondindex. jt_gap_unique_jordan_leftenumsecondindex+S (jt_h_unique_jordan_leftenum)=(u)) -> (((((exists fs_h_jt_unique_jordan_leftenumfirstcode. fs_h_jt_unique_jordan_leftenumfirstcode + S (jt_b_unique_jordan_leftenum) = S ((S (jt_i_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumfirstcode. jt_codes_unique_jordan_left = fs_q_jt_unique_jordan_leftenumfirstcode * S ((S (jt_i_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left) + (jt_b_unique_jordan_leftenum))) /\\ (((exists fs_h_jt_unique_jordan_leftenumfirstscale. fs_h_jt_unique_jordan_leftenumfirstscale + S (jt_c_unique_jordan_leftenum) = S ((S (jt_i_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumfirstscale. jt_scales_unique_jordan_left = fs_q_jt_unique_jordan_leftenumfirstscale * S ((S (jt_i_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left) + (jt_c_unique_jordan_leftenum))))) -> (((((exists fs_h_jt_unique_jordan_leftenumsecondcode. fs_h_jt_unique_jordan_leftenumsecondcode + S (jt_d_unique_jordan_leftenum) = S ((S (jt_h_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumsecondcode. jt_codes_unique_jordan_left = fs_q_jt_unique_jordan_leftenumsecondcode * S ((S (jt_h_unique_jordan_leftenum)) * jt_code_scale_unique_jordan_left) + (jt_d_unique_jordan_leftenum))) /\\ (((exists fs_h_jt_unique_jordan_leftenumsecondscale. fs_h_jt_unique_jordan_leftenumsecondscale + S (jt_e_unique_jordan_leftenum) = S ((S (jt_h_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left)) /\\ exists fs_q_jt_unique_jordan_leftenumsecondscale. jt_scales_unique_jordan_left = fs_q_jt_unique_jordan_leftenumsecondscale * S ((S (jt_h_unique_jordan_leftenum)) * jt_scale_scale_unique_jordan_left) + (jt_e_unique_jordan_leftenum))))) -> (forall jt_index_unique_jordan_leftenumsame jt_left_unique_jordan_leftenumsame jt_right_unique_jordan_leftenumsame. (exists jt_gap_unique_jordan_leftenumsameindex. jt_gap_unique_jordan_leftenumsameindex+S (jt_index_unique_jordan_leftenumsame)=(k)) -> (((exists fs_h_jt_unique_jordan_leftenumsameleft. fs_h_jt_unique_jordan_leftenumsameleft + S (jt_left_unique_jordan_leftenumsame) = S ((S (jt_index_unique_jordan_leftenumsame)) * jt_c_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenumsameleft. jt_b_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenumsameleft * S ((S (jt_index_unique_jordan_leftenumsame)) * jt_c_unique_jordan_leftenum) + (jt_left_unique_jordan_leftenumsame))) -> (((exists fs_h_jt_unique_jordan_leftenumsameright. fs_h_jt_unique_jordan_leftenumsameright + S (jt_right_unique_jordan_leftenumsame) = S ((S (jt_index_unique_jordan_leftenumsame)) * jt_e_unique_jordan_leftenum)) /\\ exists fs_q_jt_unique_jordan_leftenumsameright. jt_d_unique_jordan_leftenum = fs_q_jt_unique_jordan_leftenumsameright * S ((S (jt_index_unique_jordan_leftenumsame)) * jt_e_unique_jordan_leftenum) + (jt_right_unique_jordan_leftenumsame))) -> jt_left_unique_jordan_leftenumsame=jt_right_unique_jordan_leftenumsame) -> jt_i_unique_jordan_leftenum=jt_h_unique_jordan_leftenum))))))))) -> (((~((k)=0)) /\\ (((~((n)=0)) /\\ (exists jt_codes_unique_jordan_right jt_code_scale_unique_jordan_right jt_scales_unique_jordan_right jt_scale_scale_unique_jordan_right. ((forall jt_i_unique_jordan_rightenum. (exists jt_gap_unique_jordan_rightenumsoundindex. jt_gap_unique_jordan_rightenumsoundindex+S (jt_i_unique_jordan_rightenum)=(v)) -> exists jt_b_unique_jordan_rightenum jt_c_unique_jordan_rightenum. ((((((exists fs_h_jt_unique_jordan_rightenumsoundcode. fs_h_jt_unique_jordan_rightenumsoundcode + S (jt_b_unique_jordan_rightenum) = S ((S (jt_i_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumsoundcode. jt_codes_unique_jordan_right = fs_q_jt_unique_jordan_rightenumsoundcode * S ((S (jt_i_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right) + (jt_b_unique_jordan_rightenum))) /\\ (((exists fs_h_jt_unique_jordan_rightenumsoundscale. fs_h_jt_unique_jordan_rightenumsoundscale + S (jt_c_unique_jordan_rightenum) = S ((S (jt_i_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumsoundscale. jt_scales_unique_jordan_right = fs_q_jt_unique_jordan_rightenumsoundscale * S ((S (jt_i_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right) + (jt_c_unique_jordan_rightenum))))) /\\ (((forall jt_index_unique_jordan_rightenumbound. (exists jt_gap_unique_jordan_rightenumboundindex. jt_gap_unique_jordan_rightenumboundindex+S (jt_index_unique_jordan_rightenumbound)=(k)) -> exists jt_value_unique_jordan_rightenumbound. ((((exists fs_h_jt_unique_jordan_rightenumboundat. fs_h_jt_unique_jordan_rightenumboundat + S (jt_value_unique_jordan_rightenumbound) = S ((S (jt_index_unique_jordan_rightenumbound)) * jt_c_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenumboundat. jt_b_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenumboundat * S ((S (jt_index_unique_jordan_rightenumbound)) * jt_c_unique_jordan_rightenum) + (jt_value_unique_jordan_rightenumbound))) /\\ (exists jt_gap_unique_jordan_rightenumboundvalue. jt_gap_unique_jordan_rightenumboundvalue+S (jt_value_unique_jordan_rightenumbound)=(n)))) /\\ (forall jt_divisor_unique_jordan_rightenumprimitive. (exists jt_factor_unique_jordan_rightenumprimitivemodulus. (n)=(jt_divisor_unique_jordan_rightenumprimitive)*jt_factor_unique_jordan_rightenumprimitivemodulus) -> (forall jt_index_unique_jordan_rightenumprimitivecoordinates jt_value_unique_jordan_rightenumprimitivecoordinates. (exists jt_gap_unique_jordan_rightenumprimitivecoordinatesindex. jt_gap_unique_jordan_rightenumprimitivecoordinatesindex+S (jt_index_unique_jordan_rightenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unique_jordan_rightenumprimitivecoordinatesat. fs_h_jt_unique_jordan_rightenumprimitivecoordinatesat + S (jt_value_unique_jordan_rightenumprimitivecoordinates) = S ((S (jt_index_unique_jordan_rightenumprimitivecoordinates)) * jt_c_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenumprimitivecoordinatesat. jt_b_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenumprimitivecoordinatesat * S ((S (jt_index_unique_jordan_rightenumprimitivecoordinates)) * jt_c_unique_jordan_rightenum) + (jt_value_unique_jordan_rightenumprimitivecoordinates))) -> (exists jt_factor_unique_jordan_rightenumprimitivecoordinatesdivides. (jt_value_unique_jordan_rightenumprimitivecoordinates)=(jt_divisor_unique_jordan_rightenumprimitive)*jt_factor_unique_jordan_rightenumprimitivecoordinatesdivides)) -> jt_divisor_unique_jordan_rightenumprimitive=1))))) /\\ (((forall jt_b_unique_jordan_rightenum jt_c_unique_jordan_rightenum. (forall jt_index_unique_jordan_rightenuminputbound. (exists jt_gap_unique_jordan_rightenuminputboundindex. jt_gap_unique_jordan_rightenuminputboundindex+S (jt_index_unique_jordan_rightenuminputbound)=(k)) -> exists jt_value_unique_jordan_rightenuminputbound. ((((exists fs_h_jt_unique_jordan_rightenuminputboundat. fs_h_jt_unique_jordan_rightenuminputboundat + S (jt_value_unique_jordan_rightenuminputbound) = S ((S (jt_index_unique_jordan_rightenuminputbound)) * jt_c_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenuminputboundat. jt_b_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenuminputboundat * S ((S (jt_index_unique_jordan_rightenuminputbound)) * jt_c_unique_jordan_rightenum) + (jt_value_unique_jordan_rightenuminputbound))) /\\ (exists jt_gap_unique_jordan_rightenuminputboundvalue. jt_gap_unique_jordan_rightenuminputboundvalue+S (jt_value_unique_jordan_rightenuminputbound)=(n)))) -> (forall jt_divisor_unique_jordan_rightenuminputprimitive. (exists jt_factor_unique_jordan_rightenuminputprimitivemodulus. (n)=(jt_divisor_unique_jordan_rightenuminputprimitive)*jt_factor_unique_jordan_rightenuminputprimitivemodulus) -> (forall jt_index_unique_jordan_rightenuminputprimitivecoordinates jt_value_unique_jordan_rightenuminputprimitivecoordinates. (exists jt_gap_unique_jordan_rightenuminputprimitivecoordinatesindex. jt_gap_unique_jordan_rightenuminputprimitivecoordinatesindex+S (jt_index_unique_jordan_rightenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unique_jordan_rightenuminputprimitivecoordinatesat. fs_h_jt_unique_jordan_rightenuminputprimitivecoordinatesat + S (jt_value_unique_jordan_rightenuminputprimitivecoordinates) = S ((S (jt_index_unique_jordan_rightenuminputprimitivecoordinates)) * jt_c_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenuminputprimitivecoordinatesat. jt_b_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenuminputprimitivecoordinatesat * S ((S (jt_index_unique_jordan_rightenuminputprimitivecoordinates)) * jt_c_unique_jordan_rightenum) + (jt_value_unique_jordan_rightenuminputprimitivecoordinates))) -> (exists jt_factor_unique_jordan_rightenuminputprimitivecoordinatesdivides. (jt_value_unique_jordan_rightenuminputprimitivecoordinates)=(jt_divisor_unique_jordan_rightenuminputprimitive)*jt_factor_unique_jordan_rightenuminputprimitivecoordinatesdivides)) -> jt_divisor_unique_jordan_rightenuminputprimitive=1) -> exists jt_i_unique_jordan_rightenum jt_d_unique_jordan_rightenum jt_e_unique_jordan_rightenum. ((exists jt_gap_unique_jordan_rightenumcompleteindex. jt_gap_unique_jordan_rightenumcompleteindex+S (jt_i_unique_jordan_rightenum)=(v)) /\\ (((((((exists fs_h_jt_unique_jordan_rightenumcompletecode. fs_h_jt_unique_jordan_rightenumcompletecode + S (jt_d_unique_jordan_rightenum) = S ((S (jt_i_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumcompletecode. jt_codes_unique_jordan_right = fs_q_jt_unique_jordan_rightenumcompletecode * S ((S (jt_i_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right) + (jt_d_unique_jordan_rightenum))) /\\ (((exists fs_h_jt_unique_jordan_rightenumcompletescale. fs_h_jt_unique_jordan_rightenumcompletescale + S (jt_e_unique_jordan_rightenum) = S ((S (jt_i_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumcompletescale. jt_scales_unique_jordan_right = fs_q_jt_unique_jordan_rightenumcompletescale * S ((S (jt_i_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right) + (jt_e_unique_jordan_rightenum))))) /\\ (forall jt_index_unique_jordan_rightenumrepresented jt_left_unique_jordan_rightenumrepresented jt_right_unique_jordan_rightenumrepresented. (exists jt_gap_unique_jordan_rightenumrepresentedindex. jt_gap_unique_jordan_rightenumrepresentedindex+S (jt_index_unique_jordan_rightenumrepresented)=(k)) -> (((exists fs_h_jt_unique_jordan_rightenumrepresentedleft. fs_h_jt_unique_jordan_rightenumrepresentedleft + S (jt_left_unique_jordan_rightenumrepresented) = S ((S (jt_index_unique_jordan_rightenumrepresented)) * jt_c_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenumrepresentedleft. jt_b_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenumrepresentedleft * S ((S (jt_index_unique_jordan_rightenumrepresented)) * jt_c_unique_jordan_rightenum) + (jt_left_unique_jordan_rightenumrepresented))) -> (((exists fs_h_jt_unique_jordan_rightenumrepresentedright. fs_h_jt_unique_jordan_rightenumrepresentedright + S (jt_right_unique_jordan_rightenumrepresented) = S ((S (jt_index_unique_jordan_rightenumrepresented)) * jt_e_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenumrepresentedright. jt_d_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenumrepresentedright * S ((S (jt_index_unique_jordan_rightenumrepresented)) * jt_e_unique_jordan_rightenum) + (jt_right_unique_jordan_rightenumrepresented))) -> jt_left_unique_jordan_rightenumrepresented=jt_right_unique_jordan_rightenumrepresented))))) /\\ (forall jt_i_unique_jordan_rightenum jt_h_unique_jordan_rightenum jt_b_unique_jordan_rightenum jt_c_unique_jordan_rightenum jt_d_unique_jordan_rightenum jt_e_unique_jordan_rightenum. (exists jt_gap_unique_jordan_rightenumfirstindex. jt_gap_unique_jordan_rightenumfirstindex+S (jt_i_unique_jordan_rightenum)=(v)) -> (exists jt_gap_unique_jordan_rightenumsecondindex. jt_gap_unique_jordan_rightenumsecondindex+S (jt_h_unique_jordan_rightenum)=(v)) -> (((((exists fs_h_jt_unique_jordan_rightenumfirstcode. fs_h_jt_unique_jordan_rightenumfirstcode + S (jt_b_unique_jordan_rightenum) = S ((S (jt_i_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumfirstcode. jt_codes_unique_jordan_right = fs_q_jt_unique_jordan_rightenumfirstcode * S ((S (jt_i_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right) + (jt_b_unique_jordan_rightenum))) /\\ (((exists fs_h_jt_unique_jordan_rightenumfirstscale. fs_h_jt_unique_jordan_rightenumfirstscale + S (jt_c_unique_jordan_rightenum) = S ((S (jt_i_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumfirstscale. jt_scales_unique_jordan_right = fs_q_jt_unique_jordan_rightenumfirstscale * S ((S (jt_i_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right) + (jt_c_unique_jordan_rightenum))))) -> (((((exists fs_h_jt_unique_jordan_rightenumsecondcode. fs_h_jt_unique_jordan_rightenumsecondcode + S (jt_d_unique_jordan_rightenum) = S ((S (jt_h_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumsecondcode. jt_codes_unique_jordan_right = fs_q_jt_unique_jordan_rightenumsecondcode * S ((S (jt_h_unique_jordan_rightenum)) * jt_code_scale_unique_jordan_right) + (jt_d_unique_jordan_rightenum))) /\\ (((exists fs_h_jt_unique_jordan_rightenumsecondscale. fs_h_jt_unique_jordan_rightenumsecondscale + S (jt_e_unique_jordan_rightenum) = S ((S (jt_h_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right)) /\\ exists fs_q_jt_unique_jordan_rightenumsecondscale. jt_scales_unique_jordan_right = fs_q_jt_unique_jordan_rightenumsecondscale * S ((S (jt_h_unique_jordan_rightenum)) * jt_scale_scale_unique_jordan_right) + (jt_e_unique_jordan_rightenum))))) -> (forall jt_index_unique_jordan_rightenumsame jt_left_unique_jordan_rightenumsame jt_right_unique_jordan_rightenumsame. (exists jt_gap_unique_jordan_rightenumsameindex. jt_gap_unique_jordan_rightenumsameindex+S (jt_index_unique_jordan_rightenumsame)=(k)) -> (((exists fs_h_jt_unique_jordan_rightenumsameleft. fs_h_jt_unique_jordan_rightenumsameleft + S (jt_left_unique_jordan_rightenumsame) = S ((S (jt_index_unique_jordan_rightenumsame)) * jt_c_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenumsameleft. jt_b_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenumsameleft * S ((S (jt_index_unique_jordan_rightenumsame)) * jt_c_unique_jordan_rightenum) + (jt_left_unique_jordan_rightenumsame))) -> (((exists fs_h_jt_unique_jordan_rightenumsameright. fs_h_jt_unique_jordan_rightenumsameright + S (jt_right_unique_jordan_rightenumsame) = S ((S (jt_index_unique_jordan_rightenumsame)) * jt_e_unique_jordan_rightenum)) /\\ exists fs_q_jt_unique_jordan_rightenumsameright. jt_d_unique_jordan_rightenum = fs_q_jt_unique_jordan_rightenumsameright * S ((S (jt_index_unique_jordan_rightenumsame)) * jt_e_unique_jordan_rightenum) + (jt_right_unique_jordan_rightenumsame))) -> jt_left_unique_jordan_rightenumsame=jt_right_unique_jordan_rightenumsame) -> jt_i_unique_jordan_rightenum=jt_h_unique_jordan_rightenum))))))))) -> (u=v)", "statement_sha256": "e8ac43d89f3ac491062245faf7472b67b63a2b800c8d64daa127153e67041a08", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The independently defined Jordan relation has a unique count, regardless of all chosen beta encodings." }, { "admission_dependencies": [ "jordan_totient_coprime_product", "jordan_totient_count_unique" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 22, "body_proof_nodes": 32, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro a", "intro b", "intro u", "intro v", "intro w", "intro hcop", "intro ha", "intro hb", "intro hw", "have hp : JordanTotient(k,a · b,u · v)", "specialize jordan_totient_coprime_product (k)", "specialize jordan_totient_coprime_product (a)", "specialize jordan_totient_coprime_product (b)", "specialize jordan_totient_coprime_product (u)", "specialize jordan_totient_coprime_product (v)", "apply jordan_totient_coprime_product", "exact hcop", "exact ha", "exact hb", "specialize jordan_totient_count_unique (k)", "specialize jordan_totient_count_unique (a*b)", "specialize jordan_totient_count_unique (w)", "specialize jordan_totient_count_unique (u*v)", "apply jordan_totient_count_unique", "exact hw", "exact hp" ], "defined_statement": "∀ k. ∀ a. ∀ b. ∀ u. ∀ v. ∀ w. Coprime(a,b) → JordanTotient(k,a,u) → JordanTotient(k,b,v) → JordanTotient(k,a · b,w) → w = u · v", "defined_statement_sha256": "b4e89aab7aca3c45a23e7fbda421bf51db82fbe91af85cc86d25e8f0d831e334", "definition_uses": { "ND0375": 4, "PD0005": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "f906f76472bff7fa58b3907e0e99149a6345cfcb15d7a19e72326169e26a4e13", "free_names": [], "script_definition_uses": { "ND0375": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro a" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro u" } ], [ { "kind": "text", "text": "intro v" } ], [ { "kind": "text", "text": "intro w" } ], [ { "kind": "text", "text": "intro hcop" } ], [ { "kind": "text", "text": "intro ha" } ], [ { "kind": "text", "text": "intro hb" } ], [ { "kind": "text", "text": "intro hw" } ], [ { "kind": "text", "text": "have hp : " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a · b,u · v)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (a)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (b)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (u)" } ], [ { "kind": "text", "text": "specialize jordan_totient_coprime_product (v)" } ], [ { "kind": "text", "text": "apply jordan_totient_coprime_product" } ], [ { "kind": "text", "text": "exact hcop" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hb" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (a*b)" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (w)" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (u*v)" } ], [ { "kind": "text", "text": "apply jordan_totient_count_unique" } ], [ { "kind": "text", "text": "exact hw" } ], [ { "kind": "text", "text": "exact hp" } ] ], "statement_definition_uses": { "ND0375": 3, "PD0005": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ a. ∀ b. ∀ u. ∀ v. ∀ w. " }, { "definition": "PD0005", "kind": "definition", "text": "Coprime(a,b)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a,u)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,b,v)" }, { "kind": "text", "text": " → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,a · b,w)" }, { "kind": "text", "text": " → w = u · v" } ] }, "dependencies": [ "jordan_totient_coprime_product", "jordan_totient_count_unique" ], "enrolled_in_alpha": true, "factory": "make_jordan_multiplicativity_unique_candidate_theorems", "first_admitted_version": "v35", "id": "JT0056", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_multiplicativity_unique_counts", "original_ha_bundle_verified": true, "proof_bundle_node_id": 347, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro a", "intro b", "intro u", "intro v", "intro w", "intro hcop", "intro ha", "intro hb", "intro hw", "have hp : ((~((k)=0)) /\\ (((~((a*b)=0)) /\\ (exists jt_codes_constructed_product jt_code_scale_constructed_product jt_scales_constructed_product jt_scale_scale_constructed_product. ((forall jt_i_constructed_productenum. (exists jt_gap_constructed_productenumsoundindex. jt_gap_constructed_productenumsoundindex+S (jt_i_constructed_productenum)=(u*v)) -> exists jt_b_constructed_productenum jt_c_constructed_productenum. ((((((exists fs_h_jt_constructed_productenumsoundcode. fs_h_jt_constructed_productenumsoundcode + S (jt_b_constructed_productenum) = S ((S (jt_i_constructed_productenum)) * jt_code_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumsoundcode. jt_codes_constructed_product = fs_q_jt_constructed_productenumsoundcode * S ((S (jt_i_constructed_productenum)) * jt_code_scale_constructed_product) + (jt_b_constructed_productenum))) /\\ (((exists fs_h_jt_constructed_productenumsoundscale. fs_h_jt_constructed_productenumsoundscale + S (jt_c_constructed_productenum) = S ((S (jt_i_constructed_productenum)) * jt_scale_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumsoundscale. jt_scales_constructed_product = fs_q_jt_constructed_productenumsoundscale * S ((S (jt_i_constructed_productenum)) * jt_scale_scale_constructed_product) + (jt_c_constructed_productenum))))) /\\ (((forall jt_index_constructed_productenumbound. (exists jt_gap_constructed_productenumboundindex. jt_gap_constructed_productenumboundindex+S (jt_index_constructed_productenumbound)=(k)) -> exists jt_value_constructed_productenumbound. ((((exists fs_h_jt_constructed_productenumboundat. fs_h_jt_constructed_productenumboundat + S (jt_value_constructed_productenumbound) = S ((S (jt_index_constructed_productenumbound)) * jt_c_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenumboundat. jt_b_constructed_productenum = fs_q_jt_constructed_productenumboundat * S ((S (jt_index_constructed_productenumbound)) * jt_c_constructed_productenum) + (jt_value_constructed_productenumbound))) /\\ (exists jt_gap_constructed_productenumboundvalue. jt_gap_constructed_productenumboundvalue+S (jt_value_constructed_productenumbound)=(a*b)))) /\\ (forall jt_divisor_constructed_productenumprimitive. (exists jt_factor_constructed_productenumprimitivemodulus. (a*b)=(jt_divisor_constructed_productenumprimitive)*jt_factor_constructed_productenumprimitivemodulus) -> (forall jt_index_constructed_productenumprimitivecoordinates jt_value_constructed_productenumprimitivecoordinates. (exists jt_gap_constructed_productenumprimitivecoordinatesindex. jt_gap_constructed_productenumprimitivecoordinatesindex+S (jt_index_constructed_productenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_constructed_productenumprimitivecoordinatesat. fs_h_jt_constructed_productenumprimitivecoordinatesat + S (jt_value_constructed_productenumprimitivecoordinates) = S ((S (jt_index_constructed_productenumprimitivecoordinates)) * jt_c_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenumprimitivecoordinatesat. jt_b_constructed_productenum = fs_q_jt_constructed_productenumprimitivecoordinatesat * S ((S (jt_index_constructed_productenumprimitivecoordinates)) * jt_c_constructed_productenum) + (jt_value_constructed_productenumprimitivecoordinates))) -> (exists jt_factor_constructed_productenumprimitivecoordinatesdivides. (jt_value_constructed_productenumprimitivecoordinates)=(jt_divisor_constructed_productenumprimitive)*jt_factor_constructed_productenumprimitivecoordinatesdivides)) -> jt_divisor_constructed_productenumprimitive=1))))) /\\ (((forall jt_b_constructed_productenum jt_c_constructed_productenum. (forall jt_index_constructed_productenuminputbound. (exists jt_gap_constructed_productenuminputboundindex. jt_gap_constructed_productenuminputboundindex+S (jt_index_constructed_productenuminputbound)=(k)) -> exists jt_value_constructed_productenuminputbound. ((((exists fs_h_jt_constructed_productenuminputboundat. fs_h_jt_constructed_productenuminputboundat + S (jt_value_constructed_productenuminputbound) = S ((S (jt_index_constructed_productenuminputbound)) * jt_c_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenuminputboundat. jt_b_constructed_productenum = fs_q_jt_constructed_productenuminputboundat * S ((S (jt_index_constructed_productenuminputbound)) * jt_c_constructed_productenum) + (jt_value_constructed_productenuminputbound))) /\\ (exists jt_gap_constructed_productenuminputboundvalue. jt_gap_constructed_productenuminputboundvalue+S (jt_value_constructed_productenuminputbound)=(a*b)))) -> (forall jt_divisor_constructed_productenuminputprimitive. (exists jt_factor_constructed_productenuminputprimitivemodulus. (a*b)=(jt_divisor_constructed_productenuminputprimitive)*jt_factor_constructed_productenuminputprimitivemodulus) -> (forall jt_index_constructed_productenuminputprimitivecoordinates jt_value_constructed_productenuminputprimitivecoordinates. (exists jt_gap_constructed_productenuminputprimitivecoordinatesindex. jt_gap_constructed_productenuminputprimitivecoordinatesindex+S (jt_index_constructed_productenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_constructed_productenuminputprimitivecoordinatesat. fs_h_jt_constructed_productenuminputprimitivecoordinatesat + S (jt_value_constructed_productenuminputprimitivecoordinates) = S ((S (jt_index_constructed_productenuminputprimitivecoordinates)) * jt_c_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenuminputprimitivecoordinatesat. jt_b_constructed_productenum = fs_q_jt_constructed_productenuminputprimitivecoordinatesat * S ((S (jt_index_constructed_productenuminputprimitivecoordinates)) * jt_c_constructed_productenum) + (jt_value_constructed_productenuminputprimitivecoordinates))) -> (exists jt_factor_constructed_productenuminputprimitivecoordinatesdivides. (jt_value_constructed_productenuminputprimitivecoordinates)=(jt_divisor_constructed_productenuminputprimitive)*jt_factor_constructed_productenuminputprimitivecoordinatesdivides)) -> jt_divisor_constructed_productenuminputprimitive=1) -> exists jt_i_constructed_productenum jt_d_constructed_productenum jt_e_constructed_productenum. ((exists jt_gap_constructed_productenumcompleteindex. jt_gap_constructed_productenumcompleteindex+S (jt_i_constructed_productenum)=(u*v)) /\\ (((((((exists fs_h_jt_constructed_productenumcompletecode. fs_h_jt_constructed_productenumcompletecode + S (jt_d_constructed_productenum) = S ((S (jt_i_constructed_productenum)) * jt_code_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumcompletecode. jt_codes_constructed_product = fs_q_jt_constructed_productenumcompletecode * S ((S (jt_i_constructed_productenum)) * jt_code_scale_constructed_product) + (jt_d_constructed_productenum))) /\\ (((exists fs_h_jt_constructed_productenumcompletescale. fs_h_jt_constructed_productenumcompletescale + S (jt_e_constructed_productenum) = S ((S (jt_i_constructed_productenum)) * jt_scale_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumcompletescale. jt_scales_constructed_product = fs_q_jt_constructed_productenumcompletescale * S ((S (jt_i_constructed_productenum)) * jt_scale_scale_constructed_product) + (jt_e_constructed_productenum))))) /\\ (forall jt_index_constructed_productenumrepresented jt_left_constructed_productenumrepresented jt_right_constructed_productenumrepresented. (exists jt_gap_constructed_productenumrepresentedindex. jt_gap_constructed_productenumrepresentedindex+S (jt_index_constructed_productenumrepresented)=(k)) -> (((exists fs_h_jt_constructed_productenumrepresentedleft. fs_h_jt_constructed_productenumrepresentedleft + S (jt_left_constructed_productenumrepresented) = S ((S (jt_index_constructed_productenumrepresented)) * jt_c_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenumrepresentedleft. jt_b_constructed_productenum = fs_q_jt_constructed_productenumrepresentedleft * S ((S (jt_index_constructed_productenumrepresented)) * jt_c_constructed_productenum) + (jt_left_constructed_productenumrepresented))) -> (((exists fs_h_jt_constructed_productenumrepresentedright. fs_h_jt_constructed_productenumrepresentedright + S (jt_right_constructed_productenumrepresented) = S ((S (jt_index_constructed_productenumrepresented)) * jt_e_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenumrepresentedright. jt_d_constructed_productenum = fs_q_jt_constructed_productenumrepresentedright * S ((S (jt_index_constructed_productenumrepresented)) * jt_e_constructed_productenum) + (jt_right_constructed_productenumrepresented))) -> jt_left_constructed_productenumrepresented=jt_right_constructed_productenumrepresented))))) /\\ (forall jt_i_constructed_productenum jt_h_constructed_productenum jt_b_constructed_productenum jt_c_constructed_productenum jt_d_constructed_productenum jt_e_constructed_productenum. (exists jt_gap_constructed_productenumfirstindex. jt_gap_constructed_productenumfirstindex+S (jt_i_constructed_productenum)=(u*v)) -> (exists jt_gap_constructed_productenumsecondindex. jt_gap_constructed_productenumsecondindex+S (jt_h_constructed_productenum)=(u*v)) -> (((((exists fs_h_jt_constructed_productenumfirstcode. fs_h_jt_constructed_productenumfirstcode + S (jt_b_constructed_productenum) = S ((S (jt_i_constructed_productenum)) * jt_code_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumfirstcode. jt_codes_constructed_product = fs_q_jt_constructed_productenumfirstcode * S ((S (jt_i_constructed_productenum)) * jt_code_scale_constructed_product) + (jt_b_constructed_productenum))) /\\ (((exists fs_h_jt_constructed_productenumfirstscale. fs_h_jt_constructed_productenumfirstscale + S (jt_c_constructed_productenum) = S ((S (jt_i_constructed_productenum)) * jt_scale_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumfirstscale. jt_scales_constructed_product = fs_q_jt_constructed_productenumfirstscale * S ((S (jt_i_constructed_productenum)) * jt_scale_scale_constructed_product) + (jt_c_constructed_productenum))))) -> (((((exists fs_h_jt_constructed_productenumsecondcode. fs_h_jt_constructed_productenumsecondcode + S (jt_d_constructed_productenum) = S ((S (jt_h_constructed_productenum)) * jt_code_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumsecondcode. jt_codes_constructed_product = fs_q_jt_constructed_productenumsecondcode * S ((S (jt_h_constructed_productenum)) * jt_code_scale_constructed_product) + (jt_d_constructed_productenum))) /\\ (((exists fs_h_jt_constructed_productenumsecondscale. fs_h_jt_constructed_productenumsecondscale + S (jt_e_constructed_productenum) = S ((S (jt_h_constructed_productenum)) * jt_scale_scale_constructed_product)) /\\ exists fs_q_jt_constructed_productenumsecondscale. jt_scales_constructed_product = fs_q_jt_constructed_productenumsecondscale * S ((S (jt_h_constructed_productenum)) * jt_scale_scale_constructed_product) + (jt_e_constructed_productenum))))) -> (forall jt_index_constructed_productenumsame jt_left_constructed_productenumsame jt_right_constructed_productenumsame. (exists jt_gap_constructed_productenumsameindex. jt_gap_constructed_productenumsameindex+S (jt_index_constructed_productenumsame)=(k)) -> (((exists fs_h_jt_constructed_productenumsameleft. fs_h_jt_constructed_productenumsameleft + S (jt_left_constructed_productenumsame) = S ((S (jt_index_constructed_productenumsame)) * jt_c_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenumsameleft. jt_b_constructed_productenum = fs_q_jt_constructed_productenumsameleft * S ((S (jt_index_constructed_productenumsame)) * jt_c_constructed_productenum) + (jt_left_constructed_productenumsame))) -> (((exists fs_h_jt_constructed_productenumsameright. fs_h_jt_constructed_productenumsameright + S (jt_right_constructed_productenumsame) = S ((S (jt_index_constructed_productenumsame)) * jt_e_constructed_productenum)) /\\ exists fs_q_jt_constructed_productenumsameright. jt_d_constructed_productenum = fs_q_jt_constructed_productenumsameright * S ((S (jt_index_constructed_productenumsame)) * jt_e_constructed_productenum) + (jt_right_constructed_productenumsame))) -> jt_left_constructed_productenumsame=jt_right_constructed_productenumsame) -> jt_i_constructed_productenum=jt_h_constructed_productenum))))))))", "specialize jordan_totient_coprime_product (k)", "specialize jordan_totient_coprime_product (a)", "specialize jordan_totient_coprime_product (b)", "specialize jordan_totient_coprime_product (u)", "specialize jordan_totient_coprime_product (v)", "apply jordan_totient_coprime_product", "exact hcop", "exact ha", "exact hb", "specialize jordan_totient_count_unique (k)", "specialize jordan_totient_count_unique (a*b)", "specialize jordan_totient_count_unique (w)", "specialize jordan_totient_count_unique (u*v)", "apply jordan_totient_count_unique", "exact hw", "exact hp" ], "script_sha256": "f2315b11e7f273169bb1e31e0f027727458249a85d542ec4450fbb0d9be68739", "source_filename": "jordan_multiplicativity_unique_candidate.py", "source_module": "peano_lab.library.jordan_multiplicativity_unique_candidate", "sources": [ { "factory": "make_jordan_multiplicativity_unique_candidate_theorems", "script_sha256": "f2315b11e7f273169bb1e31e0f027727458249a85d542ec4450fbb0d9be68739", "selected": true, "source_module": "peano_lab.library.jordan_multiplicativity_unique_candidate", "source_sha256": "c2e3f94ee6777659c30b449472b471697c0e6039f28f9052c4c1ebf2cb8fe99f", "statement_sha256": "f906f76472bff7fa58b3907e0e99149a6345cfcb15d7a19e72326169e26a4e13" } ], "stable_member": false, "statement": "forall k a b u v w. (forall jt_divisor_arbitrary_coprime. (exists jt_factor_arbitrary_coprimea. (a)=(jt_divisor_arbitrary_coprime)*jt_factor_arbitrary_coprimea) -> (exists jt_factor_arbitrary_coprimeb. (b)=(jt_divisor_arbitrary_coprime)*jt_factor_arbitrary_coprimeb) -> jt_divisor_arbitrary_coprime=1) -> (((~((k)=0)) /\\ (((~((a)=0)) /\\ (exists jt_codes_arbitrary_left jt_code_scale_arbitrary_left jt_scales_arbitrary_left jt_scale_scale_arbitrary_left. ((forall jt_i_arbitrary_leftenum. (exists jt_gap_arbitrary_leftenumsoundindex. jt_gap_arbitrary_leftenumsoundindex+S (jt_i_arbitrary_leftenum)=(u)) -> exists jt_b_arbitrary_leftenum jt_c_arbitrary_leftenum. ((((((exists fs_h_jt_arbitrary_leftenumsoundcode. fs_h_jt_arbitrary_leftenumsoundcode + S (jt_b_arbitrary_leftenum) = S ((S (jt_i_arbitrary_leftenum)) * jt_code_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumsoundcode. jt_codes_arbitrary_left = fs_q_jt_arbitrary_leftenumsoundcode * S ((S (jt_i_arbitrary_leftenum)) * jt_code_scale_arbitrary_left) + (jt_b_arbitrary_leftenum))) /\\ (((exists fs_h_jt_arbitrary_leftenumsoundscale. fs_h_jt_arbitrary_leftenumsoundscale + S (jt_c_arbitrary_leftenum) = S ((S (jt_i_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumsoundscale. jt_scales_arbitrary_left = fs_q_jt_arbitrary_leftenumsoundscale * S ((S (jt_i_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left) + (jt_c_arbitrary_leftenum))))) /\\ (((forall jt_index_arbitrary_leftenumbound. (exists jt_gap_arbitrary_leftenumboundindex. jt_gap_arbitrary_leftenumboundindex+S (jt_index_arbitrary_leftenumbound)=(k)) -> exists jt_value_arbitrary_leftenumbound. ((((exists fs_h_jt_arbitrary_leftenumboundat. fs_h_jt_arbitrary_leftenumboundat + S (jt_value_arbitrary_leftenumbound) = S ((S (jt_index_arbitrary_leftenumbound)) * jt_c_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenumboundat. jt_b_arbitrary_leftenum = fs_q_jt_arbitrary_leftenumboundat * S ((S (jt_index_arbitrary_leftenumbound)) * jt_c_arbitrary_leftenum) + (jt_value_arbitrary_leftenumbound))) /\\ (exists jt_gap_arbitrary_leftenumboundvalue. jt_gap_arbitrary_leftenumboundvalue+S (jt_value_arbitrary_leftenumbound)=(a)))) /\\ (forall jt_divisor_arbitrary_leftenumprimitive. (exists jt_factor_arbitrary_leftenumprimitivemodulus. (a)=(jt_divisor_arbitrary_leftenumprimitive)*jt_factor_arbitrary_leftenumprimitivemodulus) -> (forall jt_index_arbitrary_leftenumprimitivecoordinates jt_value_arbitrary_leftenumprimitivecoordinates. (exists jt_gap_arbitrary_leftenumprimitivecoordinatesindex. jt_gap_arbitrary_leftenumprimitivecoordinatesindex+S (jt_index_arbitrary_leftenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_arbitrary_leftenumprimitivecoordinatesat. fs_h_jt_arbitrary_leftenumprimitivecoordinatesat + S (jt_value_arbitrary_leftenumprimitivecoordinates) = S ((S (jt_index_arbitrary_leftenumprimitivecoordinates)) * jt_c_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenumprimitivecoordinatesat. jt_b_arbitrary_leftenum = fs_q_jt_arbitrary_leftenumprimitivecoordinatesat * S ((S (jt_index_arbitrary_leftenumprimitivecoordinates)) * jt_c_arbitrary_leftenum) + (jt_value_arbitrary_leftenumprimitivecoordinates))) -> (exists jt_factor_arbitrary_leftenumprimitivecoordinatesdivides. (jt_value_arbitrary_leftenumprimitivecoordinates)=(jt_divisor_arbitrary_leftenumprimitive)*jt_factor_arbitrary_leftenumprimitivecoordinatesdivides)) -> jt_divisor_arbitrary_leftenumprimitive=1))))) /\\ (((forall jt_b_arbitrary_leftenum jt_c_arbitrary_leftenum. (forall jt_index_arbitrary_leftenuminputbound. (exists jt_gap_arbitrary_leftenuminputboundindex. jt_gap_arbitrary_leftenuminputboundindex+S (jt_index_arbitrary_leftenuminputbound)=(k)) -> exists jt_value_arbitrary_leftenuminputbound. ((((exists fs_h_jt_arbitrary_leftenuminputboundat. fs_h_jt_arbitrary_leftenuminputboundat + S (jt_value_arbitrary_leftenuminputbound) = S ((S (jt_index_arbitrary_leftenuminputbound)) * jt_c_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenuminputboundat. jt_b_arbitrary_leftenum = fs_q_jt_arbitrary_leftenuminputboundat * S ((S (jt_index_arbitrary_leftenuminputbound)) * jt_c_arbitrary_leftenum) + (jt_value_arbitrary_leftenuminputbound))) /\\ (exists jt_gap_arbitrary_leftenuminputboundvalue. jt_gap_arbitrary_leftenuminputboundvalue+S (jt_value_arbitrary_leftenuminputbound)=(a)))) -> (forall jt_divisor_arbitrary_leftenuminputprimitive. (exists jt_factor_arbitrary_leftenuminputprimitivemodulus. (a)=(jt_divisor_arbitrary_leftenuminputprimitive)*jt_factor_arbitrary_leftenuminputprimitivemodulus) -> (forall jt_index_arbitrary_leftenuminputprimitivecoordinates jt_value_arbitrary_leftenuminputprimitivecoordinates. (exists jt_gap_arbitrary_leftenuminputprimitivecoordinatesindex. jt_gap_arbitrary_leftenuminputprimitivecoordinatesindex+S (jt_index_arbitrary_leftenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_arbitrary_leftenuminputprimitivecoordinatesat. fs_h_jt_arbitrary_leftenuminputprimitivecoordinatesat + S (jt_value_arbitrary_leftenuminputprimitivecoordinates) = S ((S (jt_index_arbitrary_leftenuminputprimitivecoordinates)) * jt_c_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenuminputprimitivecoordinatesat. jt_b_arbitrary_leftenum = fs_q_jt_arbitrary_leftenuminputprimitivecoordinatesat * S ((S (jt_index_arbitrary_leftenuminputprimitivecoordinates)) * jt_c_arbitrary_leftenum) + (jt_value_arbitrary_leftenuminputprimitivecoordinates))) -> (exists jt_factor_arbitrary_leftenuminputprimitivecoordinatesdivides. (jt_value_arbitrary_leftenuminputprimitivecoordinates)=(jt_divisor_arbitrary_leftenuminputprimitive)*jt_factor_arbitrary_leftenuminputprimitivecoordinatesdivides)) -> jt_divisor_arbitrary_leftenuminputprimitive=1) -> exists jt_i_arbitrary_leftenum jt_d_arbitrary_leftenum jt_e_arbitrary_leftenum. ((exists jt_gap_arbitrary_leftenumcompleteindex. jt_gap_arbitrary_leftenumcompleteindex+S (jt_i_arbitrary_leftenum)=(u)) /\\ (((((((exists fs_h_jt_arbitrary_leftenumcompletecode. fs_h_jt_arbitrary_leftenumcompletecode + S (jt_d_arbitrary_leftenum) = S ((S (jt_i_arbitrary_leftenum)) * jt_code_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumcompletecode. jt_codes_arbitrary_left = fs_q_jt_arbitrary_leftenumcompletecode * S ((S (jt_i_arbitrary_leftenum)) * jt_code_scale_arbitrary_left) + (jt_d_arbitrary_leftenum))) /\\ (((exists fs_h_jt_arbitrary_leftenumcompletescale. fs_h_jt_arbitrary_leftenumcompletescale + S (jt_e_arbitrary_leftenum) = S ((S (jt_i_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumcompletescale. jt_scales_arbitrary_left = fs_q_jt_arbitrary_leftenumcompletescale * S ((S (jt_i_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left) + (jt_e_arbitrary_leftenum))))) /\\ (forall jt_index_arbitrary_leftenumrepresented jt_left_arbitrary_leftenumrepresented jt_right_arbitrary_leftenumrepresented. (exists jt_gap_arbitrary_leftenumrepresentedindex. jt_gap_arbitrary_leftenumrepresentedindex+S (jt_index_arbitrary_leftenumrepresented)=(k)) -> (((exists fs_h_jt_arbitrary_leftenumrepresentedleft. fs_h_jt_arbitrary_leftenumrepresentedleft + S (jt_left_arbitrary_leftenumrepresented) = S ((S (jt_index_arbitrary_leftenumrepresented)) * jt_c_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenumrepresentedleft. jt_b_arbitrary_leftenum = fs_q_jt_arbitrary_leftenumrepresentedleft * S ((S (jt_index_arbitrary_leftenumrepresented)) * jt_c_arbitrary_leftenum) + (jt_left_arbitrary_leftenumrepresented))) -> (((exists fs_h_jt_arbitrary_leftenumrepresentedright. fs_h_jt_arbitrary_leftenumrepresentedright + S (jt_right_arbitrary_leftenumrepresented) = S ((S (jt_index_arbitrary_leftenumrepresented)) * jt_e_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenumrepresentedright. jt_d_arbitrary_leftenum = fs_q_jt_arbitrary_leftenumrepresentedright * S ((S (jt_index_arbitrary_leftenumrepresented)) * jt_e_arbitrary_leftenum) + (jt_right_arbitrary_leftenumrepresented))) -> jt_left_arbitrary_leftenumrepresented=jt_right_arbitrary_leftenumrepresented))))) /\\ (forall jt_i_arbitrary_leftenum jt_h_arbitrary_leftenum jt_b_arbitrary_leftenum jt_c_arbitrary_leftenum jt_d_arbitrary_leftenum jt_e_arbitrary_leftenum. (exists jt_gap_arbitrary_leftenumfirstindex. jt_gap_arbitrary_leftenumfirstindex+S (jt_i_arbitrary_leftenum)=(u)) -> (exists jt_gap_arbitrary_leftenumsecondindex. jt_gap_arbitrary_leftenumsecondindex+S (jt_h_arbitrary_leftenum)=(u)) -> (((((exists fs_h_jt_arbitrary_leftenumfirstcode. fs_h_jt_arbitrary_leftenumfirstcode + S (jt_b_arbitrary_leftenum) = S ((S (jt_i_arbitrary_leftenum)) * jt_code_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumfirstcode. jt_codes_arbitrary_left = fs_q_jt_arbitrary_leftenumfirstcode * S ((S (jt_i_arbitrary_leftenum)) * jt_code_scale_arbitrary_left) + (jt_b_arbitrary_leftenum))) /\\ (((exists fs_h_jt_arbitrary_leftenumfirstscale. fs_h_jt_arbitrary_leftenumfirstscale + S (jt_c_arbitrary_leftenum) = S ((S (jt_i_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumfirstscale. jt_scales_arbitrary_left = fs_q_jt_arbitrary_leftenumfirstscale * S ((S (jt_i_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left) + (jt_c_arbitrary_leftenum))))) -> (((((exists fs_h_jt_arbitrary_leftenumsecondcode. fs_h_jt_arbitrary_leftenumsecondcode + S (jt_d_arbitrary_leftenum) = S ((S (jt_h_arbitrary_leftenum)) * jt_code_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumsecondcode. jt_codes_arbitrary_left = fs_q_jt_arbitrary_leftenumsecondcode * S ((S (jt_h_arbitrary_leftenum)) * jt_code_scale_arbitrary_left) + (jt_d_arbitrary_leftenum))) /\\ (((exists fs_h_jt_arbitrary_leftenumsecondscale. fs_h_jt_arbitrary_leftenumsecondscale + S (jt_e_arbitrary_leftenum) = S ((S (jt_h_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left)) /\\ exists fs_q_jt_arbitrary_leftenumsecondscale. jt_scales_arbitrary_left = fs_q_jt_arbitrary_leftenumsecondscale * S ((S (jt_h_arbitrary_leftenum)) * jt_scale_scale_arbitrary_left) + (jt_e_arbitrary_leftenum))))) -> (forall jt_index_arbitrary_leftenumsame jt_left_arbitrary_leftenumsame jt_right_arbitrary_leftenumsame. (exists jt_gap_arbitrary_leftenumsameindex. jt_gap_arbitrary_leftenumsameindex+S (jt_index_arbitrary_leftenumsame)=(k)) -> (((exists fs_h_jt_arbitrary_leftenumsameleft. fs_h_jt_arbitrary_leftenumsameleft + S (jt_left_arbitrary_leftenumsame) = S ((S (jt_index_arbitrary_leftenumsame)) * jt_c_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenumsameleft. jt_b_arbitrary_leftenum = fs_q_jt_arbitrary_leftenumsameleft * S ((S (jt_index_arbitrary_leftenumsame)) * jt_c_arbitrary_leftenum) + (jt_left_arbitrary_leftenumsame))) -> (((exists fs_h_jt_arbitrary_leftenumsameright. fs_h_jt_arbitrary_leftenumsameright + S (jt_right_arbitrary_leftenumsame) = S ((S (jt_index_arbitrary_leftenumsame)) * jt_e_arbitrary_leftenum)) /\\ exists fs_q_jt_arbitrary_leftenumsameright. jt_d_arbitrary_leftenum = fs_q_jt_arbitrary_leftenumsameright * S ((S (jt_index_arbitrary_leftenumsame)) * jt_e_arbitrary_leftenum) + (jt_right_arbitrary_leftenumsame))) -> jt_left_arbitrary_leftenumsame=jt_right_arbitrary_leftenumsame) -> jt_i_arbitrary_leftenum=jt_h_arbitrary_leftenum))))))))) -> (((~((k)=0)) /\\ (((~((b)=0)) /\\ (exists jt_codes_arbitrary_right jt_code_scale_arbitrary_right jt_scales_arbitrary_right jt_scale_scale_arbitrary_right. ((forall jt_i_arbitrary_rightenum. (exists jt_gap_arbitrary_rightenumsoundindex. jt_gap_arbitrary_rightenumsoundindex+S (jt_i_arbitrary_rightenum)=(v)) -> exists jt_b_arbitrary_rightenum jt_c_arbitrary_rightenum. ((((((exists fs_h_jt_arbitrary_rightenumsoundcode. fs_h_jt_arbitrary_rightenumsoundcode + S (jt_b_arbitrary_rightenum) = S ((S (jt_i_arbitrary_rightenum)) * jt_code_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumsoundcode. jt_codes_arbitrary_right = fs_q_jt_arbitrary_rightenumsoundcode * S ((S (jt_i_arbitrary_rightenum)) * jt_code_scale_arbitrary_right) + (jt_b_arbitrary_rightenum))) /\\ (((exists fs_h_jt_arbitrary_rightenumsoundscale. fs_h_jt_arbitrary_rightenumsoundscale + S (jt_c_arbitrary_rightenum) = S ((S (jt_i_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumsoundscale. jt_scales_arbitrary_right = fs_q_jt_arbitrary_rightenumsoundscale * S ((S (jt_i_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right) + (jt_c_arbitrary_rightenum))))) /\\ (((forall jt_index_arbitrary_rightenumbound. (exists jt_gap_arbitrary_rightenumboundindex. jt_gap_arbitrary_rightenumboundindex+S (jt_index_arbitrary_rightenumbound)=(k)) -> exists jt_value_arbitrary_rightenumbound. ((((exists fs_h_jt_arbitrary_rightenumboundat. fs_h_jt_arbitrary_rightenumboundat + S (jt_value_arbitrary_rightenumbound) = S ((S (jt_index_arbitrary_rightenumbound)) * jt_c_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenumboundat. jt_b_arbitrary_rightenum = fs_q_jt_arbitrary_rightenumboundat * S ((S (jt_index_arbitrary_rightenumbound)) * jt_c_arbitrary_rightenum) + (jt_value_arbitrary_rightenumbound))) /\\ (exists jt_gap_arbitrary_rightenumboundvalue. jt_gap_arbitrary_rightenumboundvalue+S (jt_value_arbitrary_rightenumbound)=(b)))) /\\ (forall jt_divisor_arbitrary_rightenumprimitive. (exists jt_factor_arbitrary_rightenumprimitivemodulus. (b)=(jt_divisor_arbitrary_rightenumprimitive)*jt_factor_arbitrary_rightenumprimitivemodulus) -> (forall jt_index_arbitrary_rightenumprimitivecoordinates jt_value_arbitrary_rightenumprimitivecoordinates. (exists jt_gap_arbitrary_rightenumprimitivecoordinatesindex. jt_gap_arbitrary_rightenumprimitivecoordinatesindex+S (jt_index_arbitrary_rightenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_arbitrary_rightenumprimitivecoordinatesat. fs_h_jt_arbitrary_rightenumprimitivecoordinatesat + S (jt_value_arbitrary_rightenumprimitivecoordinates) = S ((S (jt_index_arbitrary_rightenumprimitivecoordinates)) * jt_c_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenumprimitivecoordinatesat. jt_b_arbitrary_rightenum = fs_q_jt_arbitrary_rightenumprimitivecoordinatesat * S ((S (jt_index_arbitrary_rightenumprimitivecoordinates)) * jt_c_arbitrary_rightenum) + (jt_value_arbitrary_rightenumprimitivecoordinates))) -> (exists jt_factor_arbitrary_rightenumprimitivecoordinatesdivides. (jt_value_arbitrary_rightenumprimitivecoordinates)=(jt_divisor_arbitrary_rightenumprimitive)*jt_factor_arbitrary_rightenumprimitivecoordinatesdivides)) -> jt_divisor_arbitrary_rightenumprimitive=1))))) /\\ (((forall jt_b_arbitrary_rightenum jt_c_arbitrary_rightenum. (forall jt_index_arbitrary_rightenuminputbound. (exists jt_gap_arbitrary_rightenuminputboundindex. jt_gap_arbitrary_rightenuminputboundindex+S (jt_index_arbitrary_rightenuminputbound)=(k)) -> exists jt_value_arbitrary_rightenuminputbound. ((((exists fs_h_jt_arbitrary_rightenuminputboundat. fs_h_jt_arbitrary_rightenuminputboundat + S (jt_value_arbitrary_rightenuminputbound) = S ((S (jt_index_arbitrary_rightenuminputbound)) * jt_c_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenuminputboundat. jt_b_arbitrary_rightenum = fs_q_jt_arbitrary_rightenuminputboundat * S ((S (jt_index_arbitrary_rightenuminputbound)) * jt_c_arbitrary_rightenum) + (jt_value_arbitrary_rightenuminputbound))) /\\ (exists jt_gap_arbitrary_rightenuminputboundvalue. jt_gap_arbitrary_rightenuminputboundvalue+S (jt_value_arbitrary_rightenuminputbound)=(b)))) -> (forall jt_divisor_arbitrary_rightenuminputprimitive. (exists jt_factor_arbitrary_rightenuminputprimitivemodulus. (b)=(jt_divisor_arbitrary_rightenuminputprimitive)*jt_factor_arbitrary_rightenuminputprimitivemodulus) -> (forall jt_index_arbitrary_rightenuminputprimitivecoordinates jt_value_arbitrary_rightenuminputprimitivecoordinates. (exists jt_gap_arbitrary_rightenuminputprimitivecoordinatesindex. jt_gap_arbitrary_rightenuminputprimitivecoordinatesindex+S (jt_index_arbitrary_rightenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_arbitrary_rightenuminputprimitivecoordinatesat. fs_h_jt_arbitrary_rightenuminputprimitivecoordinatesat + S (jt_value_arbitrary_rightenuminputprimitivecoordinates) = S ((S (jt_index_arbitrary_rightenuminputprimitivecoordinates)) * jt_c_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenuminputprimitivecoordinatesat. jt_b_arbitrary_rightenum = fs_q_jt_arbitrary_rightenuminputprimitivecoordinatesat * S ((S (jt_index_arbitrary_rightenuminputprimitivecoordinates)) * jt_c_arbitrary_rightenum) + (jt_value_arbitrary_rightenuminputprimitivecoordinates))) -> (exists jt_factor_arbitrary_rightenuminputprimitivecoordinatesdivides. (jt_value_arbitrary_rightenuminputprimitivecoordinates)=(jt_divisor_arbitrary_rightenuminputprimitive)*jt_factor_arbitrary_rightenuminputprimitivecoordinatesdivides)) -> jt_divisor_arbitrary_rightenuminputprimitive=1) -> exists jt_i_arbitrary_rightenum jt_d_arbitrary_rightenum jt_e_arbitrary_rightenum. ((exists jt_gap_arbitrary_rightenumcompleteindex. jt_gap_arbitrary_rightenumcompleteindex+S (jt_i_arbitrary_rightenum)=(v)) /\\ (((((((exists fs_h_jt_arbitrary_rightenumcompletecode. fs_h_jt_arbitrary_rightenumcompletecode + S (jt_d_arbitrary_rightenum) = S ((S (jt_i_arbitrary_rightenum)) * jt_code_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumcompletecode. jt_codes_arbitrary_right = fs_q_jt_arbitrary_rightenumcompletecode * S ((S (jt_i_arbitrary_rightenum)) * jt_code_scale_arbitrary_right) + (jt_d_arbitrary_rightenum))) /\\ (((exists fs_h_jt_arbitrary_rightenumcompletescale. fs_h_jt_arbitrary_rightenumcompletescale + S (jt_e_arbitrary_rightenum) = S ((S (jt_i_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumcompletescale. jt_scales_arbitrary_right = fs_q_jt_arbitrary_rightenumcompletescale * S ((S (jt_i_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right) + (jt_e_arbitrary_rightenum))))) /\\ (forall jt_index_arbitrary_rightenumrepresented jt_left_arbitrary_rightenumrepresented jt_right_arbitrary_rightenumrepresented. (exists jt_gap_arbitrary_rightenumrepresentedindex. jt_gap_arbitrary_rightenumrepresentedindex+S (jt_index_arbitrary_rightenumrepresented)=(k)) -> (((exists fs_h_jt_arbitrary_rightenumrepresentedleft. fs_h_jt_arbitrary_rightenumrepresentedleft + S (jt_left_arbitrary_rightenumrepresented) = S ((S (jt_index_arbitrary_rightenumrepresented)) * jt_c_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenumrepresentedleft. jt_b_arbitrary_rightenum = fs_q_jt_arbitrary_rightenumrepresentedleft * S ((S (jt_index_arbitrary_rightenumrepresented)) * jt_c_arbitrary_rightenum) + (jt_left_arbitrary_rightenumrepresented))) -> (((exists fs_h_jt_arbitrary_rightenumrepresentedright. fs_h_jt_arbitrary_rightenumrepresentedright + S (jt_right_arbitrary_rightenumrepresented) = S ((S (jt_index_arbitrary_rightenumrepresented)) * jt_e_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenumrepresentedright. jt_d_arbitrary_rightenum = fs_q_jt_arbitrary_rightenumrepresentedright * S ((S (jt_index_arbitrary_rightenumrepresented)) * jt_e_arbitrary_rightenum) + (jt_right_arbitrary_rightenumrepresented))) -> jt_left_arbitrary_rightenumrepresented=jt_right_arbitrary_rightenumrepresented))))) /\\ (forall jt_i_arbitrary_rightenum jt_h_arbitrary_rightenum jt_b_arbitrary_rightenum jt_c_arbitrary_rightenum jt_d_arbitrary_rightenum jt_e_arbitrary_rightenum. (exists jt_gap_arbitrary_rightenumfirstindex. jt_gap_arbitrary_rightenumfirstindex+S (jt_i_arbitrary_rightenum)=(v)) -> (exists jt_gap_arbitrary_rightenumsecondindex. jt_gap_arbitrary_rightenumsecondindex+S (jt_h_arbitrary_rightenum)=(v)) -> (((((exists fs_h_jt_arbitrary_rightenumfirstcode. fs_h_jt_arbitrary_rightenumfirstcode + S (jt_b_arbitrary_rightenum) = S ((S (jt_i_arbitrary_rightenum)) * jt_code_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumfirstcode. jt_codes_arbitrary_right = fs_q_jt_arbitrary_rightenumfirstcode * S ((S (jt_i_arbitrary_rightenum)) * jt_code_scale_arbitrary_right) + (jt_b_arbitrary_rightenum))) /\\ (((exists fs_h_jt_arbitrary_rightenumfirstscale. fs_h_jt_arbitrary_rightenumfirstscale + S (jt_c_arbitrary_rightenum) = S ((S (jt_i_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumfirstscale. jt_scales_arbitrary_right = fs_q_jt_arbitrary_rightenumfirstscale * S ((S (jt_i_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right) + (jt_c_arbitrary_rightenum))))) -> (((((exists fs_h_jt_arbitrary_rightenumsecondcode. fs_h_jt_arbitrary_rightenumsecondcode + S (jt_d_arbitrary_rightenum) = S ((S (jt_h_arbitrary_rightenum)) * jt_code_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumsecondcode. jt_codes_arbitrary_right = fs_q_jt_arbitrary_rightenumsecondcode * S ((S (jt_h_arbitrary_rightenum)) * jt_code_scale_arbitrary_right) + (jt_d_arbitrary_rightenum))) /\\ (((exists fs_h_jt_arbitrary_rightenumsecondscale. fs_h_jt_arbitrary_rightenumsecondscale + S (jt_e_arbitrary_rightenum) = S ((S (jt_h_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right)) /\\ exists fs_q_jt_arbitrary_rightenumsecondscale. jt_scales_arbitrary_right = fs_q_jt_arbitrary_rightenumsecondscale * S ((S (jt_h_arbitrary_rightenum)) * jt_scale_scale_arbitrary_right) + (jt_e_arbitrary_rightenum))))) -> (forall jt_index_arbitrary_rightenumsame jt_left_arbitrary_rightenumsame jt_right_arbitrary_rightenumsame. (exists jt_gap_arbitrary_rightenumsameindex. jt_gap_arbitrary_rightenumsameindex+S (jt_index_arbitrary_rightenumsame)=(k)) -> (((exists fs_h_jt_arbitrary_rightenumsameleft. fs_h_jt_arbitrary_rightenumsameleft + S (jt_left_arbitrary_rightenumsame) = S ((S (jt_index_arbitrary_rightenumsame)) * jt_c_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenumsameleft. jt_b_arbitrary_rightenum = fs_q_jt_arbitrary_rightenumsameleft * S ((S (jt_index_arbitrary_rightenumsame)) * jt_c_arbitrary_rightenum) + (jt_left_arbitrary_rightenumsame))) -> (((exists fs_h_jt_arbitrary_rightenumsameright. fs_h_jt_arbitrary_rightenumsameright + S (jt_right_arbitrary_rightenumsame) = S ((S (jt_index_arbitrary_rightenumsame)) * jt_e_arbitrary_rightenum)) /\\ exists fs_q_jt_arbitrary_rightenumsameright. jt_d_arbitrary_rightenum = fs_q_jt_arbitrary_rightenumsameright * S ((S (jt_index_arbitrary_rightenumsame)) * jt_e_arbitrary_rightenum) + (jt_right_arbitrary_rightenumsame))) -> jt_left_arbitrary_rightenumsame=jt_right_arbitrary_rightenumsame) -> jt_i_arbitrary_rightenum=jt_h_arbitrary_rightenum))))))))) -> (((~((k)=0)) /\\ (((~((a*b)=0)) /\\ (exists jt_codes_arbitrary_product jt_code_scale_arbitrary_product jt_scales_arbitrary_product jt_scale_scale_arbitrary_product. ((forall jt_i_arbitrary_productenum. (exists jt_gap_arbitrary_productenumsoundindex. jt_gap_arbitrary_productenumsoundindex+S (jt_i_arbitrary_productenum)=(w)) -> exists jt_b_arbitrary_productenum jt_c_arbitrary_productenum. ((((((exists fs_h_jt_arbitrary_productenumsoundcode. fs_h_jt_arbitrary_productenumsoundcode + S (jt_b_arbitrary_productenum) = S ((S (jt_i_arbitrary_productenum)) * jt_code_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumsoundcode. jt_codes_arbitrary_product = fs_q_jt_arbitrary_productenumsoundcode * S ((S (jt_i_arbitrary_productenum)) * jt_code_scale_arbitrary_product) + (jt_b_arbitrary_productenum))) /\\ (((exists fs_h_jt_arbitrary_productenumsoundscale. fs_h_jt_arbitrary_productenumsoundscale + S (jt_c_arbitrary_productenum) = S ((S (jt_i_arbitrary_productenum)) * jt_scale_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumsoundscale. jt_scales_arbitrary_product = fs_q_jt_arbitrary_productenumsoundscale * S ((S (jt_i_arbitrary_productenum)) * jt_scale_scale_arbitrary_product) + (jt_c_arbitrary_productenum))))) /\\ (((forall jt_index_arbitrary_productenumbound. (exists jt_gap_arbitrary_productenumboundindex. jt_gap_arbitrary_productenumboundindex+S (jt_index_arbitrary_productenumbound)=(k)) -> exists jt_value_arbitrary_productenumbound. ((((exists fs_h_jt_arbitrary_productenumboundat. fs_h_jt_arbitrary_productenumboundat + S (jt_value_arbitrary_productenumbound) = S ((S (jt_index_arbitrary_productenumbound)) * jt_c_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenumboundat. jt_b_arbitrary_productenum = fs_q_jt_arbitrary_productenumboundat * S ((S (jt_index_arbitrary_productenumbound)) * jt_c_arbitrary_productenum) + (jt_value_arbitrary_productenumbound))) /\\ (exists jt_gap_arbitrary_productenumboundvalue. jt_gap_arbitrary_productenumboundvalue+S (jt_value_arbitrary_productenumbound)=(a*b)))) /\\ (forall jt_divisor_arbitrary_productenumprimitive. (exists jt_factor_arbitrary_productenumprimitivemodulus. (a*b)=(jt_divisor_arbitrary_productenumprimitive)*jt_factor_arbitrary_productenumprimitivemodulus) -> (forall jt_index_arbitrary_productenumprimitivecoordinates jt_value_arbitrary_productenumprimitivecoordinates. (exists jt_gap_arbitrary_productenumprimitivecoordinatesindex. jt_gap_arbitrary_productenumprimitivecoordinatesindex+S (jt_index_arbitrary_productenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_arbitrary_productenumprimitivecoordinatesat. fs_h_jt_arbitrary_productenumprimitivecoordinatesat + S (jt_value_arbitrary_productenumprimitivecoordinates) = S ((S (jt_index_arbitrary_productenumprimitivecoordinates)) * jt_c_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenumprimitivecoordinatesat. jt_b_arbitrary_productenum = fs_q_jt_arbitrary_productenumprimitivecoordinatesat * S ((S (jt_index_arbitrary_productenumprimitivecoordinates)) * jt_c_arbitrary_productenum) + (jt_value_arbitrary_productenumprimitivecoordinates))) -> (exists jt_factor_arbitrary_productenumprimitivecoordinatesdivides. (jt_value_arbitrary_productenumprimitivecoordinates)=(jt_divisor_arbitrary_productenumprimitive)*jt_factor_arbitrary_productenumprimitivecoordinatesdivides)) -> jt_divisor_arbitrary_productenumprimitive=1))))) /\\ (((forall jt_b_arbitrary_productenum jt_c_arbitrary_productenum. (forall jt_index_arbitrary_productenuminputbound. (exists jt_gap_arbitrary_productenuminputboundindex. jt_gap_arbitrary_productenuminputboundindex+S (jt_index_arbitrary_productenuminputbound)=(k)) -> exists jt_value_arbitrary_productenuminputbound. ((((exists fs_h_jt_arbitrary_productenuminputboundat. fs_h_jt_arbitrary_productenuminputboundat + S (jt_value_arbitrary_productenuminputbound) = S ((S (jt_index_arbitrary_productenuminputbound)) * jt_c_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenuminputboundat. jt_b_arbitrary_productenum = fs_q_jt_arbitrary_productenuminputboundat * S ((S (jt_index_arbitrary_productenuminputbound)) * jt_c_arbitrary_productenum) + (jt_value_arbitrary_productenuminputbound))) /\\ (exists jt_gap_arbitrary_productenuminputboundvalue. jt_gap_arbitrary_productenuminputboundvalue+S (jt_value_arbitrary_productenuminputbound)=(a*b)))) -> (forall jt_divisor_arbitrary_productenuminputprimitive. (exists jt_factor_arbitrary_productenuminputprimitivemodulus. (a*b)=(jt_divisor_arbitrary_productenuminputprimitive)*jt_factor_arbitrary_productenuminputprimitivemodulus) -> (forall jt_index_arbitrary_productenuminputprimitivecoordinates jt_value_arbitrary_productenuminputprimitivecoordinates. (exists jt_gap_arbitrary_productenuminputprimitivecoordinatesindex. jt_gap_arbitrary_productenuminputprimitivecoordinatesindex+S (jt_index_arbitrary_productenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_arbitrary_productenuminputprimitivecoordinatesat. fs_h_jt_arbitrary_productenuminputprimitivecoordinatesat + S (jt_value_arbitrary_productenuminputprimitivecoordinates) = S ((S (jt_index_arbitrary_productenuminputprimitivecoordinates)) * jt_c_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenuminputprimitivecoordinatesat. jt_b_arbitrary_productenum = fs_q_jt_arbitrary_productenuminputprimitivecoordinatesat * S ((S (jt_index_arbitrary_productenuminputprimitivecoordinates)) * jt_c_arbitrary_productenum) + (jt_value_arbitrary_productenuminputprimitivecoordinates))) -> (exists jt_factor_arbitrary_productenuminputprimitivecoordinatesdivides. (jt_value_arbitrary_productenuminputprimitivecoordinates)=(jt_divisor_arbitrary_productenuminputprimitive)*jt_factor_arbitrary_productenuminputprimitivecoordinatesdivides)) -> jt_divisor_arbitrary_productenuminputprimitive=1) -> exists jt_i_arbitrary_productenum jt_d_arbitrary_productenum jt_e_arbitrary_productenum. ((exists jt_gap_arbitrary_productenumcompleteindex. jt_gap_arbitrary_productenumcompleteindex+S (jt_i_arbitrary_productenum)=(w)) /\\ (((((((exists fs_h_jt_arbitrary_productenumcompletecode. fs_h_jt_arbitrary_productenumcompletecode + S (jt_d_arbitrary_productenum) = S ((S (jt_i_arbitrary_productenum)) * jt_code_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumcompletecode. jt_codes_arbitrary_product = fs_q_jt_arbitrary_productenumcompletecode * S ((S (jt_i_arbitrary_productenum)) * jt_code_scale_arbitrary_product) + (jt_d_arbitrary_productenum))) /\\ (((exists fs_h_jt_arbitrary_productenumcompletescale. fs_h_jt_arbitrary_productenumcompletescale + S (jt_e_arbitrary_productenum) = S ((S (jt_i_arbitrary_productenum)) * jt_scale_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumcompletescale. jt_scales_arbitrary_product = fs_q_jt_arbitrary_productenumcompletescale * S ((S (jt_i_arbitrary_productenum)) * jt_scale_scale_arbitrary_product) + (jt_e_arbitrary_productenum))))) /\\ (forall jt_index_arbitrary_productenumrepresented jt_left_arbitrary_productenumrepresented jt_right_arbitrary_productenumrepresented. (exists jt_gap_arbitrary_productenumrepresentedindex. jt_gap_arbitrary_productenumrepresentedindex+S (jt_index_arbitrary_productenumrepresented)=(k)) -> (((exists fs_h_jt_arbitrary_productenumrepresentedleft. fs_h_jt_arbitrary_productenumrepresentedleft + S (jt_left_arbitrary_productenumrepresented) = S ((S (jt_index_arbitrary_productenumrepresented)) * jt_c_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenumrepresentedleft. jt_b_arbitrary_productenum = fs_q_jt_arbitrary_productenumrepresentedleft * S ((S (jt_index_arbitrary_productenumrepresented)) * jt_c_arbitrary_productenum) + (jt_left_arbitrary_productenumrepresented))) -> (((exists fs_h_jt_arbitrary_productenumrepresentedright. fs_h_jt_arbitrary_productenumrepresentedright + S (jt_right_arbitrary_productenumrepresented) = S ((S (jt_index_arbitrary_productenumrepresented)) * jt_e_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenumrepresentedright. jt_d_arbitrary_productenum = fs_q_jt_arbitrary_productenumrepresentedright * S ((S (jt_index_arbitrary_productenumrepresented)) * jt_e_arbitrary_productenum) + (jt_right_arbitrary_productenumrepresented))) -> jt_left_arbitrary_productenumrepresented=jt_right_arbitrary_productenumrepresented))))) /\\ (forall jt_i_arbitrary_productenum jt_h_arbitrary_productenum jt_b_arbitrary_productenum jt_c_arbitrary_productenum jt_d_arbitrary_productenum jt_e_arbitrary_productenum. (exists jt_gap_arbitrary_productenumfirstindex. jt_gap_arbitrary_productenumfirstindex+S (jt_i_arbitrary_productenum)=(w)) -> (exists jt_gap_arbitrary_productenumsecondindex. jt_gap_arbitrary_productenumsecondindex+S (jt_h_arbitrary_productenum)=(w)) -> (((((exists fs_h_jt_arbitrary_productenumfirstcode. fs_h_jt_arbitrary_productenumfirstcode + S (jt_b_arbitrary_productenum) = S ((S (jt_i_arbitrary_productenum)) * jt_code_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumfirstcode. jt_codes_arbitrary_product = fs_q_jt_arbitrary_productenumfirstcode * S ((S (jt_i_arbitrary_productenum)) * jt_code_scale_arbitrary_product) + (jt_b_arbitrary_productenum))) /\\ (((exists fs_h_jt_arbitrary_productenumfirstscale. fs_h_jt_arbitrary_productenumfirstscale + S (jt_c_arbitrary_productenum) = S ((S (jt_i_arbitrary_productenum)) * jt_scale_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumfirstscale. jt_scales_arbitrary_product = fs_q_jt_arbitrary_productenumfirstscale * S ((S (jt_i_arbitrary_productenum)) * jt_scale_scale_arbitrary_product) + (jt_c_arbitrary_productenum))))) -> (((((exists fs_h_jt_arbitrary_productenumsecondcode. fs_h_jt_arbitrary_productenumsecondcode + S (jt_d_arbitrary_productenum) = S ((S (jt_h_arbitrary_productenum)) * jt_code_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumsecondcode. jt_codes_arbitrary_product = fs_q_jt_arbitrary_productenumsecondcode * S ((S (jt_h_arbitrary_productenum)) * jt_code_scale_arbitrary_product) + (jt_d_arbitrary_productenum))) /\\ (((exists fs_h_jt_arbitrary_productenumsecondscale. fs_h_jt_arbitrary_productenumsecondscale + S (jt_e_arbitrary_productenum) = S ((S (jt_h_arbitrary_productenum)) * jt_scale_scale_arbitrary_product)) /\\ exists fs_q_jt_arbitrary_productenumsecondscale. jt_scales_arbitrary_product = fs_q_jt_arbitrary_productenumsecondscale * S ((S (jt_h_arbitrary_productenum)) * jt_scale_scale_arbitrary_product) + (jt_e_arbitrary_productenum))))) -> (forall jt_index_arbitrary_productenumsame jt_left_arbitrary_productenumsame jt_right_arbitrary_productenumsame. (exists jt_gap_arbitrary_productenumsameindex. jt_gap_arbitrary_productenumsameindex+S (jt_index_arbitrary_productenumsame)=(k)) -> (((exists fs_h_jt_arbitrary_productenumsameleft. fs_h_jt_arbitrary_productenumsameleft + S (jt_left_arbitrary_productenumsame) = S ((S (jt_index_arbitrary_productenumsame)) * jt_c_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenumsameleft. jt_b_arbitrary_productenum = fs_q_jt_arbitrary_productenumsameleft * S ((S (jt_index_arbitrary_productenumsame)) * jt_c_arbitrary_productenum) + (jt_left_arbitrary_productenumsame))) -> (((exists fs_h_jt_arbitrary_productenumsameright. fs_h_jt_arbitrary_productenumsameright + S (jt_right_arbitrary_productenumsame) = S ((S (jt_index_arbitrary_productenumsame)) * jt_e_arbitrary_productenum)) /\\ exists fs_q_jt_arbitrary_productenumsameright. jt_d_arbitrary_productenum = fs_q_jt_arbitrary_productenumsameright * S ((S (jt_index_arbitrary_productenumsame)) * jt_e_arbitrary_productenum) + (jt_right_arbitrary_productenumsame))) -> jt_left_arbitrary_productenumsame=jt_right_arbitrary_productenumsame) -> jt_i_arbitrary_productenum=jt_h_arbitrary_productenum))))))))) -> (w=u*v)", "statement_sha256": "f906f76472bff7fa58b3907e0e99149a6345cfcb15d7a19e72326169e26a4e13", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Any three genuine Jordan counts at coprime moduli obey multiplication, by the independently constructed product enumeration and count uniqueness." }, { "admission_dependencies": [ "beta_at_unique", "le_zero", "le_of_succ_le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 21, "body_proof_nodes": 37, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro k", "intro i", "intro a", "intro hBound", "intro hi", "intro ha", "have hValue : ∃ z. BetaAt(b,c,i,z) ∧ Lt(z,1)", "specialize hBound (i)", "apply hBound", "exact hi", "cases hValue", "cases hValue_witness", "trans x", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (i)", "specialize beta_at_unique (a)", "specialize beta_at_unique (x)", "apply beta_at_unique", "exact ha", "exact hValue_witness_left", "apply le_zero", "apply le_of_succ_le_succ", "exact hValue_witness_right" ], "defined_statement": "∀ b. ∀ c. ∀ k. ∀ i. ∀ a. 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" }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,z)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(z,1)" } ], [ { "kind": "text", "text": "specialize hBound (i)" } ], [ { "kind": "text", "text": "apply hBound" } ], [ { "kind": "text", "text": "exact hi" } ], [ { "kind": "text", "text": "cases hValue" } ], [ { "kind": "text", "text": "cases hValue_witness" } ], [ { "kind": "text", "text": "trans x" } ], [ { "kind": "text", "text": "specialize beta_at_unique (b)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (c)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (i)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (a)" } ], [ { "kind": "text", "text": "specialize beta_at_unique (x)" } ], [ { "kind": "text", "text": "apply beta_at_unique" } ], [ { "kind": "text", "text": "exact ha" } ], [ { "kind": "text", "text": "exact hValue_witness_left" } ], [ { "kind": "text", "text": "apply le_zero" } ], [ { "kind": "text", "text": "apply le_of_succ_le_succ" } ], [ { "kind": "text", "text": "exact hValue_witness_right" } ] ], "statement_definition_uses": { "ND0262": 1, "PD0002": 1, "PD0013": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ b. ∀ c. ∀ k. ∀ i. ∀ a. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,1)" }, { "kind": "text", "text": " → " }, { "definition": "PD0002", "kind": "definition", "text": "Lt(i,k)" }, { "kind": "text", "text": " → " }, { "definition": "PD0013", "kind": "definition", "text": "BetaAt(b,c,i,a)" }, { "kind": "text", "text": " → a = 0" } ] }, "dependencies": [ "beta_at_unique", "le_zero", "le_of_succ_le_succ" ], "enrolled_in_alpha": true, "factory": "make_jordan_unit_modulus_candidate_theorems", "first_admitted_version": "v35", "id": "JT0057", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuple_bounded_one_entry_zero", "original_ha_bundle_verified": true, "proof_bundle_node_id": 348, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro k", "intro i", "intro a", "intro hBound", "intro hi", "intro ha", "have hValue : exists z. ((((exists fs_h_jt_unit_value. fs_h_jt_unit_value + S (z) = S ((S (i)) * c)) /\\ exists fs_q_jt_unit_value. b = fs_q_jt_unit_value * S ((S (i)) * c) + (z))) /\\ (exists jt_gap_unit_bound. jt_gap_unit_bound+S (z)=(1)))", "specialize hBound (i)", "apply hBound", "exact hi", "cases hValue", "cases hValue_witness", "trans x", "specialize beta_at_unique (b)", "specialize beta_at_unique (c)", "specialize beta_at_unique (i)", "specialize beta_at_unique (a)", "specialize beta_at_unique (x)", "apply beta_at_unique", "exact ha", "exact hValue_witness_left", "apply le_zero", "apply le_of_succ_le_succ", "exact hValue_witness_right" ], "script_sha256": "e4e5d5e3501c21efe4dbd1ff1de9eb769f0c40685c9ccaa5cdeff39ece9d7261", "source_filename": "jordan_unit_modulus_candidate.py", "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "sources": [ { "factory": "make_jordan_unit_modulus_candidate_theorems", "script_sha256": "e4e5d5e3501c21efe4dbd1ff1de9eb769f0c40685c9ccaa5cdeff39ece9d7261", "selected": true, "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "source_sha256": "8fdeb1c10bed3e445b700b10ebbf2c9ff16e29802756ba21001cd5c4fff100a8", "statement_sha256": "6cd41c6ea181a252e9bfca24f3ba30ad367832b69623b78f6a8e41a522dc5908" } ], "stable_member": false, "statement": "forall b c k i a. (forall jt_index_unit_tuple. (exists jt_gap_unit_tupleindex. jt_gap_unit_tupleindex+S (jt_index_unit_tuple)=(k)) -> exists jt_value_unit_tuple. ((((exists fs_h_jt_unit_tupleat. fs_h_jt_unit_tupleat + S (jt_value_unit_tuple) = S ((S (jt_index_unit_tuple)) * c)) /\\ exists fs_q_jt_unit_tupleat. b = fs_q_jt_unit_tupleat * S ((S (jt_index_unit_tuple)) * c) + (jt_value_unit_tuple))) /\\ (exists jt_gap_unit_tuplevalue. jt_gap_unit_tuplevalue+S (jt_value_unit_tuple)=(1)))) -> (exists jt_gap_unit_index. jt_gap_unit_index+S (i)=(k)) -> (((exists fs_h_jt_unit_entry. fs_h_jt_unit_entry + S (a) = S ((S (i)) * c)) /\\ exists fs_q_jt_unit_entry. b = fs_q_jt_unit_entry * S ((S (i)) * c) + (a))) -> (a=0)", "statement_sha256": "6cd41c6ea181a252e9bfca24f3ba30ad367832b69623b78f6a8e41a522dc5908", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every decoded coordinate in a tuple bounded by one equals zero." }, { "admission_dependencies": [ "finite_beta_zero_code" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 12, "body_proof_nodes": 22, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro i", "intro hi", "exists 0", "split", "specialize finite_beta_zero_code (i)", "apply finite_beta_zero_code", "exists 0", "simp" ], "defined_statement": "∀ k. BetaPrefixInto(0,0,k,1)", "defined_statement_sha256": "33b749bc685331c647a47294d853b9571af672848364111c5734f8801a0abf96", "definition_uses": { "ND0262": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "12c001aa4ecff0c256deb90f77ae88e14b9d707ce24839f7ead999772593c1d3", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro i" } ], [ { "kind": "text", "text": "intro hi" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize finite_beta_zero_code (i)" } ], [ { "kind": "text", "text": "apply finite_beta_zero_code" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "simp" } ] ], "statement_definition_uses": { "ND0262": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(0,0,k,1)" } ] }, "dependencies": [ "finite_beta_zero_code" ], "enrolled_in_alpha": true, "factory": "make_jordan_unit_modulus_candidate_theorems", "first_admitted_version": "v35", "id": "JT0058", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_zero_tuple_bounded_one", "original_ha_bundle_verified": true, "proof_bundle_node_id": 349, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro i", "intro hi", "exists 0", "split", "specialize finite_beta_zero_code (i)", "apply finite_beta_zero_code", "exists 0", "simp" ], "script_sha256": "33ec0c65e192b765e090d3fe79ffe79b0f1dffd9cc1831f6fd97e775200189fa", "source_filename": "jordan_unit_modulus_candidate.py", "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "sources": [ { "factory": "make_jordan_unit_modulus_candidate_theorems", "script_sha256": "33ec0c65e192b765e090d3fe79ffe79b0f1dffd9cc1831f6fd97e775200189fa", "selected": true, "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "source_sha256": "8fdeb1c10bed3e445b700b10ebbf2c9ff16e29802756ba21001cd5c4fff100a8", "statement_sha256": "12c001aa4ecff0c256deb90f77ae88e14b9d707ce24839f7ead999772593c1d3" } ], "stable_member": false, "statement": "forall k. (forall jt_index_unit_zero_tuple. (exists jt_gap_unit_zero_tupleindex. jt_gap_unit_zero_tupleindex+S (jt_index_unit_zero_tuple)=(k)) -> exists jt_value_unit_zero_tuple. ((((exists fs_h_jt_unit_zero_tupleat. fs_h_jt_unit_zero_tupleat + S (jt_value_unit_zero_tuple) = S ((S (jt_index_unit_zero_tuple)) * 0)) /\\ exists fs_q_jt_unit_zero_tupleat. 0 = fs_q_jt_unit_zero_tupleat * S ((S (jt_index_unit_zero_tuple)) * 0) + (jt_value_unit_zero_tuple))) /\\ (exists jt_gap_unit_zero_tuplevalue. jt_gap_unit_zero_tuplevalue+S (jt_value_unit_zero_tuple)=(1))))", "statement_sha256": "12c001aa4ecff0c256deb90f77ae88e14b9d707ce24839f7ead999772593c1d3", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "The literal beta tuple (0,0) has every coordinate below one, at every length." }, { "admission_dependencies": [ "jordan_tuple_bounded_one_entry_zero" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 25, "body_proof_nodes": 40, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro hLeft", "intro hRight", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "trans 0", "apply jordan_tuple_bounded_one_entry_zero", "exact hLeft", "exact hi", "exact ha", "symm", "apply jordan_tuple_bounded_one_entry_zero", "exact hRight", "exact hi", "exact hz" ], "defined_statement": "∀ b. ∀ c. ∀ d. ∀ e. ∀ k. 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" }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(b,c,k,1)" }, { "kind": "text", "text": " → " }, { "definition": "ND0262", "kind": "definition", "text": "BetaPrefixInto(d,e,k,1)" }, { "kind": "text", "text": " → " }, { "definition": "ND0121", "kind": "definition", "text": "IntegerVectorZero(b,c,d,e,k)" } ] }, "dependencies": [ "jordan_tuple_bounded_one_entry_zero" ], "enrolled_in_alpha": true, "factory": "make_jordan_unit_modulus_candidate_theorems", "first_admitted_version": "v35", "id": "JT0059", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_tuples_bounded_one_equal", "original_ha_bundle_verified": true, "proof_bundle_node_id": 350, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro b", "intro c", "intro d", "intro e", "intro k", "intro hLeft", "intro hRight", "intro i", "intro a", "intro z", "intro hi", "intro ha", "intro hz", "trans 0", "apply jordan_tuple_bounded_one_entry_zero", "exact hLeft", "exact hi", "exact ha", "symm", "apply jordan_tuple_bounded_one_entry_zero", "exact hRight", "exact hi", "exact hz" ], "script_sha256": "27d248056cbd59e15e8d5a7093787d805036e4e407bf50b62f141317851f4701", "source_filename": "jordan_unit_modulus_candidate.py", "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "sources": [ { "factory": "make_jordan_unit_modulus_candidate_theorems", "script_sha256": "27d248056cbd59e15e8d5a7093787d805036e4e407bf50b62f141317851f4701", "selected": true, "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "source_sha256": "8fdeb1c10bed3e445b700b10ebbf2c9ff16e29802756ba21001cd5c4fff100a8", "statement_sha256": "5f7cdd5236757c7980b87520cfd2d574651915df0a2c67f823f9ab8927342fa3" } ], "stable_member": false, "statement": "forall b c d e k. (forall jt_index_unit_left. (exists jt_gap_unit_leftindex. jt_gap_unit_leftindex+S (jt_index_unit_left)=(k)) -> exists jt_value_unit_left. ((((exists fs_h_jt_unit_leftat. fs_h_jt_unit_leftat + S (jt_value_unit_left) = S ((S (jt_index_unit_left)) * c)) /\\ exists fs_q_jt_unit_leftat. b = fs_q_jt_unit_leftat * S ((S (jt_index_unit_left)) * c) + (jt_value_unit_left))) /\\ (exists jt_gap_unit_leftvalue. jt_gap_unit_leftvalue+S (jt_value_unit_left)=(1)))) -> (forall jt_index_unit_right. (exists jt_gap_unit_rightindex. jt_gap_unit_rightindex+S (jt_index_unit_right)=(k)) -> exists jt_value_unit_right. ((((exists fs_h_jt_unit_rightat. fs_h_jt_unit_rightat + S (jt_value_unit_right) = S ((S (jt_index_unit_right)) * e)) /\\ exists fs_q_jt_unit_rightat. d = fs_q_jt_unit_rightat * S ((S (jt_index_unit_right)) * e) + (jt_value_unit_right))) /\\ (exists jt_gap_unit_rightvalue. jt_gap_unit_rightvalue+S (jt_value_unit_right)=(1)))) -> (forall jt_index_unit_equal jt_left_unit_equal jt_right_unit_equal. (exists jt_gap_unit_equalindex. jt_gap_unit_equalindex+S (jt_index_unit_equal)=(k)) -> (((exists fs_h_jt_unit_equalleft. fs_h_jt_unit_equalleft + S (jt_left_unit_equal) = S ((S (jt_index_unit_equal)) * c)) /\\ exists fs_q_jt_unit_equalleft. b = fs_q_jt_unit_equalleft * S ((S (jt_index_unit_equal)) * c) + (jt_left_unit_equal))) -> (((exists fs_h_jt_unit_equalright. fs_h_jt_unit_equalright + S (jt_right_unit_equal) = S ((S (jt_index_unit_equal)) * e)) /\\ exists fs_q_jt_unit_equalright. d = fs_q_jt_unit_equalright * S ((S (jt_index_unit_equal)) * e) + (jt_right_unit_equal))) -> jt_left_unit_equal=jt_right_unit_equal)", "statement_sha256": "5f7cdd5236757c7980b87520cfd2d574651915df0a2c67f823f9ab8927342fa3", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Any two canonical tuples modulo one represent the same coordinate tuple." }, { "admission_dependencies": [ "finite_beta_zero_code", "jordan_zero_tuple_bounded_one", "jordan_primitive_tuple_modulus_one", "jordan_tuples_bounded_one_equal", "le_zero", "le_of_succ_le_succ" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 32, "body_proof_nodes": 133, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "split", "intro i", "intro hi", "exists 0", "exists 0", "split", "split", "specialize finite_beta_zero_code (i)", "apply finite_beta_zero_code", "specialize finite_beta_zero_code (i)", "apply finite_beta_zero_code", "split", "specialize jordan_zero_tuple_bounded_one (k)", "apply jordan_zero_tuple_bounded_one", "specialize jordan_primitive_tuple_modulus_one (0)", "specialize jordan_primitive_tuple_modulus_one (0)", "specialize jordan_primitive_tuple_modulus_one (k)", "apply jordan_primitive_tuple_modulus_one", "split", "intro b", "intro c", "intro hBound", "intro hPrimitive", "exists 0", "exists 0", "exists 0", "split", "exists 0", "simp", "split", "split", "specialize finite_beta_zero_code (0)", "apply finite_beta_zero_code", "specialize finite_beta_zero_code (0)", "apply finite_beta_zero_code", "specialize jordan_tuples_bounded_one_equal (b)", "specialize jordan_tuples_bounded_one_equal (c)", "specialize jordan_tuples_bounded_one_equal (0)", "specialize jordan_tuples_bounded_one_equal (0)", "specialize jordan_tuples_bounded_one_equal (k)", "apply jordan_tuples_bounded_one_equal", "exact hBound", "specialize jordan_zero_tuple_bounded_one (k)", "apply jordan_zero_tuple_bounded_one", "intro i", "intro h", "intro b", "intro c", "intro d", "intro e", "intro hi", "intro hh", "intro hFirst", "intro hSecond", "intro hEqual", "trans 0", "apply le_zero", "apply le_of_succ_le_succ", "exact hi", "symm", "apply le_zero", "apply le_of_succ_le_succ", "exact hh" ], "defined_statement": "∀ k. 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(((forall jt_i_unit_enumeration. (exists jt_gap_unit_enumerationsoundindex. jt_gap_unit_enumerationsoundindex+S (jt_i_unit_enumeration)=(1)) -> exists jt_b_unit_enumeration jt_c_unit_enumeration. ((((((exists fs_h_jt_unit_enumerationsoundcode. fs_h_jt_unit_enumerationsoundcode + S (jt_b_unit_enumeration) = S ((S (jt_i_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationsoundcode. 0 = fs_q_jt_unit_enumerationsoundcode * S ((S (jt_i_unit_enumeration)) * 0) + (jt_b_unit_enumeration))) /\\ (((exists fs_h_jt_unit_enumerationsoundscale. fs_h_jt_unit_enumerationsoundscale + S (jt_c_unit_enumeration) = S ((S (jt_i_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationsoundscale. 0 = fs_q_jt_unit_enumerationsoundscale * S ((S (jt_i_unit_enumeration)) * 0) + (jt_c_unit_enumeration))))) /\\ (((forall jt_index_unit_enumerationbound. (exists jt_gap_unit_enumerationboundindex. jt_gap_unit_enumerationboundindex+S (jt_index_unit_enumerationbound)=(k)) -> exists jt_value_unit_enumerationbound. ((((exists fs_h_jt_unit_enumerationboundat. fs_h_jt_unit_enumerationboundat + S (jt_value_unit_enumerationbound) = S ((S (jt_index_unit_enumerationbound)) * jt_c_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationboundat. jt_b_unit_enumeration = fs_q_jt_unit_enumerationboundat * S ((S (jt_index_unit_enumerationbound)) * jt_c_unit_enumeration) + (jt_value_unit_enumerationbound))) /\\ (exists jt_gap_unit_enumerationboundvalue. jt_gap_unit_enumerationboundvalue+S (jt_value_unit_enumerationbound)=(1)))) /\\ (forall jt_divisor_unit_enumerationprimitive. (exists jt_factor_unit_enumerationprimitivemodulus. (1)=(jt_divisor_unit_enumerationprimitive)*jt_factor_unit_enumerationprimitivemodulus) -> (forall jt_index_unit_enumerationprimitivecoordinates jt_value_unit_enumerationprimitivecoordinates. (exists jt_gap_unit_enumerationprimitivecoordinatesindex. jt_gap_unit_enumerationprimitivecoordinatesindex+S (jt_index_unit_enumerationprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unit_enumerationprimitivecoordinatesat. fs_h_jt_unit_enumerationprimitivecoordinatesat + S (jt_value_unit_enumerationprimitivecoordinates) = S ((S (jt_index_unit_enumerationprimitivecoordinates)) * jt_c_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationprimitivecoordinatesat. jt_b_unit_enumeration = fs_q_jt_unit_enumerationprimitivecoordinatesat * S ((S (jt_index_unit_enumerationprimitivecoordinates)) * jt_c_unit_enumeration) + (jt_value_unit_enumerationprimitivecoordinates))) -> (exists jt_factor_unit_enumerationprimitivecoordinatesdivides. (jt_value_unit_enumerationprimitivecoordinates)=(jt_divisor_unit_enumerationprimitive)*jt_factor_unit_enumerationprimitivecoordinatesdivides)) -> jt_divisor_unit_enumerationprimitive=1))))) /\\ (((forall jt_b_unit_enumeration jt_c_unit_enumeration. (forall jt_index_unit_enumerationinputbound. (exists jt_gap_unit_enumerationinputboundindex. jt_gap_unit_enumerationinputboundindex+S (jt_index_unit_enumerationinputbound)=(k)) -> exists jt_value_unit_enumerationinputbound. ((((exists fs_h_jt_unit_enumerationinputboundat. fs_h_jt_unit_enumerationinputboundat + S (jt_value_unit_enumerationinputbound) = S ((S (jt_index_unit_enumerationinputbound)) * jt_c_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationinputboundat. jt_b_unit_enumeration = fs_q_jt_unit_enumerationinputboundat * S ((S (jt_index_unit_enumerationinputbound)) * jt_c_unit_enumeration) + (jt_value_unit_enumerationinputbound))) /\\ (exists jt_gap_unit_enumerationinputboundvalue. jt_gap_unit_enumerationinputboundvalue+S (jt_value_unit_enumerationinputbound)=(1)))) -> (forall jt_divisor_unit_enumerationinputprimitive. (exists jt_factor_unit_enumerationinputprimitivemodulus. (1)=(jt_divisor_unit_enumerationinputprimitive)*jt_factor_unit_enumerationinputprimitivemodulus) -> (forall jt_index_unit_enumerationinputprimitivecoordinates jt_value_unit_enumerationinputprimitivecoordinates. (exists jt_gap_unit_enumerationinputprimitivecoordinatesindex. jt_gap_unit_enumerationinputprimitivecoordinatesindex+S (jt_index_unit_enumerationinputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unit_enumerationinputprimitivecoordinatesat. fs_h_jt_unit_enumerationinputprimitivecoordinatesat + S (jt_value_unit_enumerationinputprimitivecoordinates) = S ((S (jt_index_unit_enumerationinputprimitivecoordinates)) * jt_c_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationinputprimitivecoordinatesat. jt_b_unit_enumeration = fs_q_jt_unit_enumerationinputprimitivecoordinatesat * S ((S (jt_index_unit_enumerationinputprimitivecoordinates)) * jt_c_unit_enumeration) + (jt_value_unit_enumerationinputprimitivecoordinates))) -> (exists jt_factor_unit_enumerationinputprimitivecoordinatesdivides. (jt_value_unit_enumerationinputprimitivecoordinates)=(jt_divisor_unit_enumerationinputprimitive)*jt_factor_unit_enumerationinputprimitivecoordinatesdivides)) -> jt_divisor_unit_enumerationinputprimitive=1) -> exists jt_i_unit_enumeration jt_d_unit_enumeration jt_e_unit_enumeration. ((exists jt_gap_unit_enumerationcompleteindex. jt_gap_unit_enumerationcompleteindex+S (jt_i_unit_enumeration)=(1)) /\\ (((((((exists fs_h_jt_unit_enumerationcompletecode. fs_h_jt_unit_enumerationcompletecode + S (jt_d_unit_enumeration) = S ((S (jt_i_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationcompletecode. 0 = fs_q_jt_unit_enumerationcompletecode * S ((S (jt_i_unit_enumeration)) * 0) + (jt_d_unit_enumeration))) /\\ (((exists fs_h_jt_unit_enumerationcompletescale. fs_h_jt_unit_enumerationcompletescale + S (jt_e_unit_enumeration) = S ((S (jt_i_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationcompletescale. 0 = fs_q_jt_unit_enumerationcompletescale * S ((S (jt_i_unit_enumeration)) * 0) + (jt_e_unit_enumeration))))) /\\ (forall jt_index_unit_enumerationrepresented jt_left_unit_enumerationrepresented jt_right_unit_enumerationrepresented. (exists jt_gap_unit_enumerationrepresentedindex. jt_gap_unit_enumerationrepresentedindex+S (jt_index_unit_enumerationrepresented)=(k)) -> (((exists fs_h_jt_unit_enumerationrepresentedleft. fs_h_jt_unit_enumerationrepresentedleft + S (jt_left_unit_enumerationrepresented) = S ((S (jt_index_unit_enumerationrepresented)) * jt_c_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationrepresentedleft. jt_b_unit_enumeration = fs_q_jt_unit_enumerationrepresentedleft * S ((S (jt_index_unit_enumerationrepresented)) * jt_c_unit_enumeration) + (jt_left_unit_enumerationrepresented))) -> (((exists fs_h_jt_unit_enumerationrepresentedright. fs_h_jt_unit_enumerationrepresentedright + S (jt_right_unit_enumerationrepresented) = S ((S (jt_index_unit_enumerationrepresented)) * jt_e_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationrepresentedright. jt_d_unit_enumeration = fs_q_jt_unit_enumerationrepresentedright * S ((S (jt_index_unit_enumerationrepresented)) * jt_e_unit_enumeration) + (jt_right_unit_enumerationrepresented))) -> jt_left_unit_enumerationrepresented=jt_right_unit_enumerationrepresented))))) /\\ (forall jt_i_unit_enumeration jt_h_unit_enumeration jt_b_unit_enumeration jt_c_unit_enumeration jt_d_unit_enumeration jt_e_unit_enumeration. (exists jt_gap_unit_enumerationfirstindex. jt_gap_unit_enumerationfirstindex+S (jt_i_unit_enumeration)=(1)) -> (exists jt_gap_unit_enumerationsecondindex. jt_gap_unit_enumerationsecondindex+S (jt_h_unit_enumeration)=(1)) -> (((((exists fs_h_jt_unit_enumerationfirstcode. fs_h_jt_unit_enumerationfirstcode + S (jt_b_unit_enumeration) = S ((S (jt_i_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationfirstcode. 0 = fs_q_jt_unit_enumerationfirstcode * S ((S (jt_i_unit_enumeration)) * 0) + (jt_b_unit_enumeration))) /\\ (((exists fs_h_jt_unit_enumerationfirstscale. fs_h_jt_unit_enumerationfirstscale + S (jt_c_unit_enumeration) = S ((S (jt_i_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationfirstscale. 0 = fs_q_jt_unit_enumerationfirstscale * S ((S (jt_i_unit_enumeration)) * 0) + (jt_c_unit_enumeration))))) -> (((((exists fs_h_jt_unit_enumerationsecondcode. fs_h_jt_unit_enumerationsecondcode + S (jt_d_unit_enumeration) = S ((S (jt_h_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationsecondcode. 0 = fs_q_jt_unit_enumerationsecondcode * S ((S (jt_h_unit_enumeration)) * 0) + (jt_d_unit_enumeration))) /\\ (((exists fs_h_jt_unit_enumerationsecondscale. fs_h_jt_unit_enumerationsecondscale + S (jt_e_unit_enumeration) = S ((S (jt_h_unit_enumeration)) * 0)) /\\ exists fs_q_jt_unit_enumerationsecondscale. 0 = fs_q_jt_unit_enumerationsecondscale * S ((S (jt_h_unit_enumeration)) * 0) + (jt_e_unit_enumeration))))) -> (forall jt_index_unit_enumerationsame jt_left_unit_enumerationsame jt_right_unit_enumerationsame. (exists jt_gap_unit_enumerationsameindex. jt_gap_unit_enumerationsameindex+S (jt_index_unit_enumerationsame)=(k)) -> (((exists fs_h_jt_unit_enumerationsameleft. fs_h_jt_unit_enumerationsameleft + S (jt_left_unit_enumerationsame) = S ((S (jt_index_unit_enumerationsame)) * jt_c_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationsameleft. jt_b_unit_enumeration = fs_q_jt_unit_enumerationsameleft * S ((S (jt_index_unit_enumerationsame)) * jt_c_unit_enumeration) + (jt_left_unit_enumerationsame))) -> (((exists fs_h_jt_unit_enumerationsameright. fs_h_jt_unit_enumerationsameright + S (jt_right_unit_enumerationsame) = S ((S (jt_index_unit_enumerationsame)) * jt_e_unit_enumeration)) /\\ exists fs_q_jt_unit_enumerationsameright. jt_d_unit_enumeration = fs_q_jt_unit_enumerationsameright * S ((S (jt_index_unit_enumerationsame)) * jt_e_unit_enumeration) + (jt_right_unit_enumerationsame))) -> jt_left_unit_enumerationsame=jt_right_unit_enumerationsame) -> jt_i_unit_enumeration=jt_h_unit_enumeration)))))", "statement_sha256": "99850462c7abd45c720848d9ae82a8ef88bb9f23f26600c2fa9836f3de829edd", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "An actual one-position beta list is sound, complete and duplicate-free for primitive tuples modulo one." }, { "admission_dependencies": [ "succ_ne_zero", "jordan_unit_modulus_singleton_enumeration" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 13, "body_proof_nodes": 21, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro hk", "split", "exact hk", "split", "specialize succ_ne_zero (0)", "apply succ_ne_zero", "exists 0", "exists 0", "exists 0", "exists 0", "specialize jordan_unit_modulus_singleton_enumeration (k)", "apply jordan_unit_modulus_singleton_enumeration" ], "defined_statement": "∀ k. ¬k = 0 → JordanTotient(k,1,1)", "defined_statement_sha256": "bd5eb97ce00df8b8d1f61e0d8394b53d8857fa05dd08e30247628ea5c19b9377", "definition_uses": { "ND0375": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "7ad86a69ae04d3ffb8bd19aa4da5d03fa51db2525cfd47859be7a338eafa9fcb", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hk" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "exact hk" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "specialize succ_ne_zero (0)" } ], [ { "kind": "text", "text": "apply succ_ne_zero" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "exists 0" } ], [ { "kind": "text", "text": "specialize jordan_unit_modulus_singleton_enumeration (k)" } ], [ { "kind": "text", "text": "apply jordan_unit_modulus_singleton_enumeration" } ] ], "statement_definition_uses": { "ND0375": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ¬k = 0 → " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,1,1)" } ] }, "dependencies": [ "succ_ne_zero", "jordan_unit_modulus_singleton_enumeration" ], "enrolled_in_alpha": true, "factory": "make_jordan_unit_modulus_candidate_theorems", "first_admitted_version": "v35", "id": "JT005B", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_at_one", "original_ha_bundle_verified": true, "proof_bundle_node_id": 352, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro hk", "split", "exact hk", "split", "specialize succ_ne_zero (0)", "apply succ_ne_zero", "exists 0", "exists 0", "exists 0", "exists 0", "specialize jordan_unit_modulus_singleton_enumeration (k)", "apply jordan_unit_modulus_singleton_enumeration" ], "script_sha256": "d2d63551225c81813ea697b6564cac592c3b086584505543c8c50b0041bc20a5", "source_filename": "jordan_unit_modulus_candidate.py", "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "sources": [ { "factory": "make_jordan_unit_modulus_candidate_theorems", "script_sha256": "d2d63551225c81813ea697b6564cac592c3b086584505543c8c50b0041bc20a5", "selected": true, "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "source_sha256": "8fdeb1c10bed3e445b700b10ebbf2c9ff16e29802756ba21001cd5c4fff100a8", "statement_sha256": "7ad86a69ae04d3ffb8bd19aa4da5d03fa51db2525cfd47859be7a338eafa9fcb" } ], "stable_member": false, "statement": "forall k. (~(k=0)) -> (((~((k)=0)) /\\ (((~((1)=0)) /\\ (exists jt_codes_unit_count jt_code_scale_unit_count jt_scales_unit_count jt_scale_scale_unit_count. ((forall jt_i_unit_countenum. (exists jt_gap_unit_countenumsoundindex. jt_gap_unit_countenumsoundindex+S (jt_i_unit_countenum)=(1)) -> exists jt_b_unit_countenum jt_c_unit_countenum. ((((((exists fs_h_jt_unit_countenumsoundcode. fs_h_jt_unit_countenumsoundcode + S (jt_b_unit_countenum) = S ((S (jt_i_unit_countenum)) * jt_code_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumsoundcode. jt_codes_unit_count = fs_q_jt_unit_countenumsoundcode * S ((S (jt_i_unit_countenum)) * jt_code_scale_unit_count) + (jt_b_unit_countenum))) /\\ (((exists fs_h_jt_unit_countenumsoundscale. fs_h_jt_unit_countenumsoundscale + S (jt_c_unit_countenum) = S ((S (jt_i_unit_countenum)) * jt_scale_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumsoundscale. jt_scales_unit_count = fs_q_jt_unit_countenumsoundscale * S ((S (jt_i_unit_countenum)) * jt_scale_scale_unit_count) + (jt_c_unit_countenum))))) /\\ (((forall jt_index_unit_countenumbound. (exists jt_gap_unit_countenumboundindex. jt_gap_unit_countenumboundindex+S (jt_index_unit_countenumbound)=(k)) -> exists jt_value_unit_countenumbound. ((((exists fs_h_jt_unit_countenumboundat. fs_h_jt_unit_countenumboundat + S (jt_value_unit_countenumbound) = S ((S (jt_index_unit_countenumbound)) * jt_c_unit_countenum)) /\\ exists fs_q_jt_unit_countenumboundat. jt_b_unit_countenum = fs_q_jt_unit_countenumboundat * S ((S (jt_index_unit_countenumbound)) * jt_c_unit_countenum) + (jt_value_unit_countenumbound))) /\\ (exists jt_gap_unit_countenumboundvalue. jt_gap_unit_countenumboundvalue+S (jt_value_unit_countenumbound)=(1)))) /\\ (forall jt_divisor_unit_countenumprimitive. (exists jt_factor_unit_countenumprimitivemodulus. (1)=(jt_divisor_unit_countenumprimitive)*jt_factor_unit_countenumprimitivemodulus) -> (forall jt_index_unit_countenumprimitivecoordinates jt_value_unit_countenumprimitivecoordinates. (exists jt_gap_unit_countenumprimitivecoordinatesindex. jt_gap_unit_countenumprimitivecoordinatesindex+S (jt_index_unit_countenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unit_countenumprimitivecoordinatesat. fs_h_jt_unit_countenumprimitivecoordinatesat + S (jt_value_unit_countenumprimitivecoordinates) = S ((S (jt_index_unit_countenumprimitivecoordinates)) * jt_c_unit_countenum)) /\\ exists fs_q_jt_unit_countenumprimitivecoordinatesat. jt_b_unit_countenum = fs_q_jt_unit_countenumprimitivecoordinatesat * S ((S (jt_index_unit_countenumprimitivecoordinates)) * jt_c_unit_countenum) + (jt_value_unit_countenumprimitivecoordinates))) -> (exists jt_factor_unit_countenumprimitivecoordinatesdivides. (jt_value_unit_countenumprimitivecoordinates)=(jt_divisor_unit_countenumprimitive)*jt_factor_unit_countenumprimitivecoordinatesdivides)) -> jt_divisor_unit_countenumprimitive=1))))) /\\ (((forall jt_b_unit_countenum jt_c_unit_countenum. (forall jt_index_unit_countenuminputbound. (exists jt_gap_unit_countenuminputboundindex. jt_gap_unit_countenuminputboundindex+S (jt_index_unit_countenuminputbound)=(k)) -> exists jt_value_unit_countenuminputbound. ((((exists fs_h_jt_unit_countenuminputboundat. fs_h_jt_unit_countenuminputboundat + S (jt_value_unit_countenuminputbound) = S ((S (jt_index_unit_countenuminputbound)) * jt_c_unit_countenum)) /\\ exists fs_q_jt_unit_countenuminputboundat. jt_b_unit_countenum = fs_q_jt_unit_countenuminputboundat * S ((S (jt_index_unit_countenuminputbound)) * jt_c_unit_countenum) + (jt_value_unit_countenuminputbound))) /\\ (exists jt_gap_unit_countenuminputboundvalue. jt_gap_unit_countenuminputboundvalue+S (jt_value_unit_countenuminputbound)=(1)))) -> (forall jt_divisor_unit_countenuminputprimitive. (exists jt_factor_unit_countenuminputprimitivemodulus. (1)=(jt_divisor_unit_countenuminputprimitive)*jt_factor_unit_countenuminputprimitivemodulus) -> (forall jt_index_unit_countenuminputprimitivecoordinates jt_value_unit_countenuminputprimitivecoordinates. (exists jt_gap_unit_countenuminputprimitivecoordinatesindex. jt_gap_unit_countenuminputprimitivecoordinatesindex+S (jt_index_unit_countenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unit_countenuminputprimitivecoordinatesat. fs_h_jt_unit_countenuminputprimitivecoordinatesat + S (jt_value_unit_countenuminputprimitivecoordinates) = S ((S (jt_index_unit_countenuminputprimitivecoordinates)) * jt_c_unit_countenum)) /\\ exists fs_q_jt_unit_countenuminputprimitivecoordinatesat. jt_b_unit_countenum = fs_q_jt_unit_countenuminputprimitivecoordinatesat * S ((S (jt_index_unit_countenuminputprimitivecoordinates)) * jt_c_unit_countenum) + (jt_value_unit_countenuminputprimitivecoordinates))) -> (exists jt_factor_unit_countenuminputprimitivecoordinatesdivides. (jt_value_unit_countenuminputprimitivecoordinates)=(jt_divisor_unit_countenuminputprimitive)*jt_factor_unit_countenuminputprimitivecoordinatesdivides)) -> jt_divisor_unit_countenuminputprimitive=1) -> exists jt_i_unit_countenum jt_d_unit_countenum jt_e_unit_countenum. ((exists jt_gap_unit_countenumcompleteindex. jt_gap_unit_countenumcompleteindex+S (jt_i_unit_countenum)=(1)) /\\ (((((((exists fs_h_jt_unit_countenumcompletecode. fs_h_jt_unit_countenumcompletecode + S (jt_d_unit_countenum) = S ((S (jt_i_unit_countenum)) * jt_code_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumcompletecode. jt_codes_unit_count = fs_q_jt_unit_countenumcompletecode * S ((S (jt_i_unit_countenum)) * jt_code_scale_unit_count) + (jt_d_unit_countenum))) /\\ (((exists fs_h_jt_unit_countenumcompletescale. fs_h_jt_unit_countenumcompletescale + S (jt_e_unit_countenum) = S ((S (jt_i_unit_countenum)) * jt_scale_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumcompletescale. jt_scales_unit_count = fs_q_jt_unit_countenumcompletescale * S ((S (jt_i_unit_countenum)) * jt_scale_scale_unit_count) + (jt_e_unit_countenum))))) /\\ (forall jt_index_unit_countenumrepresented jt_left_unit_countenumrepresented jt_right_unit_countenumrepresented. (exists jt_gap_unit_countenumrepresentedindex. jt_gap_unit_countenumrepresentedindex+S (jt_index_unit_countenumrepresented)=(k)) -> (((exists fs_h_jt_unit_countenumrepresentedleft. fs_h_jt_unit_countenumrepresentedleft + S (jt_left_unit_countenumrepresented) = S ((S (jt_index_unit_countenumrepresented)) * jt_c_unit_countenum)) /\\ exists fs_q_jt_unit_countenumrepresentedleft. jt_b_unit_countenum = fs_q_jt_unit_countenumrepresentedleft * S ((S (jt_index_unit_countenumrepresented)) * jt_c_unit_countenum) + (jt_left_unit_countenumrepresented))) -> (((exists fs_h_jt_unit_countenumrepresentedright. fs_h_jt_unit_countenumrepresentedright + S (jt_right_unit_countenumrepresented) = S ((S (jt_index_unit_countenumrepresented)) * jt_e_unit_countenum)) /\\ exists fs_q_jt_unit_countenumrepresentedright. jt_d_unit_countenum = fs_q_jt_unit_countenumrepresentedright * S ((S (jt_index_unit_countenumrepresented)) * jt_e_unit_countenum) + (jt_right_unit_countenumrepresented))) -> jt_left_unit_countenumrepresented=jt_right_unit_countenumrepresented))))) /\\ (forall jt_i_unit_countenum jt_h_unit_countenum jt_b_unit_countenum jt_c_unit_countenum jt_d_unit_countenum jt_e_unit_countenum. (exists jt_gap_unit_countenumfirstindex. jt_gap_unit_countenumfirstindex+S (jt_i_unit_countenum)=(1)) -> (exists jt_gap_unit_countenumsecondindex. jt_gap_unit_countenumsecondindex+S (jt_h_unit_countenum)=(1)) -> (((((exists fs_h_jt_unit_countenumfirstcode. fs_h_jt_unit_countenumfirstcode + S (jt_b_unit_countenum) = S ((S (jt_i_unit_countenum)) * jt_code_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumfirstcode. jt_codes_unit_count = fs_q_jt_unit_countenumfirstcode * S ((S (jt_i_unit_countenum)) * jt_code_scale_unit_count) + (jt_b_unit_countenum))) /\\ (((exists fs_h_jt_unit_countenumfirstscale. fs_h_jt_unit_countenumfirstscale + S (jt_c_unit_countenum) = S ((S (jt_i_unit_countenum)) * jt_scale_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumfirstscale. jt_scales_unit_count = fs_q_jt_unit_countenumfirstscale * S ((S (jt_i_unit_countenum)) * jt_scale_scale_unit_count) + (jt_c_unit_countenum))))) -> (((((exists fs_h_jt_unit_countenumsecondcode. fs_h_jt_unit_countenumsecondcode + S (jt_d_unit_countenum) = S ((S (jt_h_unit_countenum)) * jt_code_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumsecondcode. jt_codes_unit_count = fs_q_jt_unit_countenumsecondcode * S ((S (jt_h_unit_countenum)) * jt_code_scale_unit_count) + (jt_d_unit_countenum))) /\\ (((exists fs_h_jt_unit_countenumsecondscale. fs_h_jt_unit_countenumsecondscale + S (jt_e_unit_countenum) = S ((S (jt_h_unit_countenum)) * jt_scale_scale_unit_count)) /\\ exists fs_q_jt_unit_countenumsecondscale. jt_scales_unit_count = fs_q_jt_unit_countenumsecondscale * S ((S (jt_h_unit_countenum)) * jt_scale_scale_unit_count) + (jt_e_unit_countenum))))) -> (forall jt_index_unit_countenumsame jt_left_unit_countenumsame jt_right_unit_countenumsame. (exists jt_gap_unit_countenumsameindex. jt_gap_unit_countenumsameindex+S (jt_index_unit_countenumsame)=(k)) -> (((exists fs_h_jt_unit_countenumsameleft. fs_h_jt_unit_countenumsameleft + S (jt_left_unit_countenumsame) = S ((S (jt_index_unit_countenumsame)) * jt_c_unit_countenum)) /\\ exists fs_q_jt_unit_countenumsameleft. jt_b_unit_countenum = fs_q_jt_unit_countenumsameleft * S ((S (jt_index_unit_countenumsame)) * jt_c_unit_countenum) + (jt_left_unit_countenumsame))) -> (((exists fs_h_jt_unit_countenumsameright. fs_h_jt_unit_countenumsameright + S (jt_right_unit_countenumsame) = S ((S (jt_index_unit_countenumsame)) * jt_e_unit_countenum)) /\\ exists fs_q_jt_unit_countenumsameright. jt_d_unit_countenum = fs_q_jt_unit_countenumsameright * S ((S (jt_index_unit_countenumsame)) * jt_e_unit_countenum) + (jt_right_unit_countenumsame))) -> jt_left_unit_countenumsame=jt_right_unit_countenumsame) -> jt_i_unit_countenum=jt_h_unit_countenum)))))))))", "statement_sha256": "7ad86a69ae04d3ffb8bd19aa4da5d03fa51db2525cfd47859be7a338eafa9fcb", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "For every positive rank k, J_k(1)=1, proved by the explicit singleton enumeration." }, { "admission_dependencies": [ "jordan_totient_count_unique", "jordan_totient_at_one" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 12, "body_proof_nodes": 18, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro k", "intro j", "intro hCount", "have hPositive : ~(k=0)", "cases hCount", "exact hCount_left", "specialize jordan_totient_count_unique (k)", "specialize jordan_totient_count_unique (1)", "specialize jordan_totient_count_unique (j)", "specialize jordan_totient_count_unique (1)", "apply jordan_totient_count_unique", "exact hCount", "specialize jordan_totient_at_one (k)", "apply jordan_totient_at_one", "exact hPositive" ], "defined_statement": "∀ k. ∀ j. JordanTotient(k,1,j) → j = 1", "defined_statement_sha256": "e0cb50242198b10bd2e9bafe3451228a04d9b11256c06b5dd221bbc9c86ecfeb", "definition_uses": { "ND0375": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "5b09d9d06efe4146b0de5993af4aa823488e535f7667d9497839d0885ae9a3d8", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro hCount" } ], [ { "kind": "text", "text": "have hPositive : ~(k=0)" } ], [ { "kind": "text", "text": "cases hCount" } ], [ { "kind": "text", "text": "exact hCount_left" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (k)" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (1)" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (j)" } ], [ { "kind": "text", "text": "specialize jordan_totient_count_unique (1)" } ], [ { "kind": "text", "text": "apply jordan_totient_count_unique" } ], [ { "kind": "text", "text": "exact hCount" } ], [ { "kind": "text", "text": "specialize jordan_totient_at_one (k)" } ], [ { "kind": "text", "text": "apply jordan_totient_at_one" } ], [ { "kind": "text", "text": "exact hPositive" } ] ], "statement_definition_uses": { "ND0375": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ k. ∀ j. " }, { "definition": "ND0375", "kind": "definition", "text": "JordanTotient(k,1,j)" }, { "kind": "text", "text": " → j = 1" } ] }, "dependencies": [ "jordan_totient_count_unique", "jordan_totient_at_one" ], "enrolled_in_alpha": true, "factory": "make_jordan_unit_modulus_candidate_theorems", "first_admitted_version": "v35", "id": "JT005C", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_totient_at_one_unique", "original_ha_bundle_verified": true, "proof_bundle_node_id": 353, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro k", "intro j", "intro hCount", "have hPositive : ~(k=0)", "cases hCount", "exact hCount_left", "specialize jordan_totient_count_unique (k)", "specialize jordan_totient_count_unique (1)", "specialize jordan_totient_count_unique (j)", "specialize jordan_totient_count_unique (1)", "apply jordan_totient_count_unique", "exact hCount", "specialize jordan_totient_at_one (k)", "apply jordan_totient_at_one", "exact hPositive" ], "script_sha256": "1a73a435694d8825d940ae72de8449a346f758d36085b4f1e862406612670de4", "source_filename": "jordan_unit_modulus_candidate.py", "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "sources": [ { "factory": "make_jordan_unit_modulus_candidate_theorems", "script_sha256": "1a73a435694d8825d940ae72de8449a346f758d36085b4f1e862406612670de4", "selected": true, "source_module": "peano_lab.library.jordan_unit_modulus_candidate", "source_sha256": "8fdeb1c10bed3e445b700b10ebbf2c9ff16e29802756ba21001cd5c4fff100a8", "statement_sha256": "5b09d9d06efe4146b0de5993af4aa823488e535f7667d9497839d0885ae9a3d8" } ], "stable_member": false, "statement": "forall k j. (((~((k)=0)) /\\ (((~((1)=0)) /\\ (exists jt_codes_unit_arbitrary_count jt_code_scale_unit_arbitrary_count jt_scales_unit_arbitrary_count jt_scale_scale_unit_arbitrary_count. ((forall jt_i_unit_arbitrary_countenum. (exists jt_gap_unit_arbitrary_countenumsoundindex. jt_gap_unit_arbitrary_countenumsoundindex+S (jt_i_unit_arbitrary_countenum)=(j)) -> exists jt_b_unit_arbitrary_countenum jt_c_unit_arbitrary_countenum. ((((((exists fs_h_jt_unit_arbitrary_countenumsoundcode. fs_h_jt_unit_arbitrary_countenumsoundcode + S (jt_b_unit_arbitrary_countenum) = S ((S (jt_i_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumsoundcode. jt_codes_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumsoundcode * S ((S (jt_i_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count) + (jt_b_unit_arbitrary_countenum))) /\\ (((exists fs_h_jt_unit_arbitrary_countenumsoundscale. fs_h_jt_unit_arbitrary_countenumsoundscale + S (jt_c_unit_arbitrary_countenum) = S ((S (jt_i_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumsoundscale. jt_scales_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumsoundscale * S ((S (jt_i_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count) + (jt_c_unit_arbitrary_countenum))))) /\\ (((forall jt_index_unit_arbitrary_countenumbound. (exists jt_gap_unit_arbitrary_countenumboundindex. jt_gap_unit_arbitrary_countenumboundindex+S (jt_index_unit_arbitrary_countenumbound)=(k)) -> exists jt_value_unit_arbitrary_countenumbound. ((((exists fs_h_jt_unit_arbitrary_countenumboundat. fs_h_jt_unit_arbitrary_countenumboundat + S (jt_value_unit_arbitrary_countenumbound) = S ((S (jt_index_unit_arbitrary_countenumbound)) * jt_c_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenumboundat. jt_b_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenumboundat * S ((S (jt_index_unit_arbitrary_countenumbound)) * jt_c_unit_arbitrary_countenum) + (jt_value_unit_arbitrary_countenumbound))) /\\ (exists jt_gap_unit_arbitrary_countenumboundvalue. jt_gap_unit_arbitrary_countenumboundvalue+S (jt_value_unit_arbitrary_countenumbound)=(1)))) /\\ (forall jt_divisor_unit_arbitrary_countenumprimitive. (exists jt_factor_unit_arbitrary_countenumprimitivemodulus. (1)=(jt_divisor_unit_arbitrary_countenumprimitive)*jt_factor_unit_arbitrary_countenumprimitivemodulus) -> (forall jt_index_unit_arbitrary_countenumprimitivecoordinates jt_value_unit_arbitrary_countenumprimitivecoordinates. (exists jt_gap_unit_arbitrary_countenumprimitivecoordinatesindex. jt_gap_unit_arbitrary_countenumprimitivecoordinatesindex+S (jt_index_unit_arbitrary_countenumprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unit_arbitrary_countenumprimitivecoordinatesat. fs_h_jt_unit_arbitrary_countenumprimitivecoordinatesat + S (jt_value_unit_arbitrary_countenumprimitivecoordinates) = S ((S (jt_index_unit_arbitrary_countenumprimitivecoordinates)) * jt_c_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenumprimitivecoordinatesat. jt_b_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenumprimitivecoordinatesat * S ((S (jt_index_unit_arbitrary_countenumprimitivecoordinates)) * jt_c_unit_arbitrary_countenum) + (jt_value_unit_arbitrary_countenumprimitivecoordinates))) -> (exists jt_factor_unit_arbitrary_countenumprimitivecoordinatesdivides. (jt_value_unit_arbitrary_countenumprimitivecoordinates)=(jt_divisor_unit_arbitrary_countenumprimitive)*jt_factor_unit_arbitrary_countenumprimitivecoordinatesdivides)) -> jt_divisor_unit_arbitrary_countenumprimitive=1))))) /\\ (((forall jt_b_unit_arbitrary_countenum jt_c_unit_arbitrary_countenum. (forall jt_index_unit_arbitrary_countenuminputbound. (exists jt_gap_unit_arbitrary_countenuminputboundindex. jt_gap_unit_arbitrary_countenuminputboundindex+S (jt_index_unit_arbitrary_countenuminputbound)=(k)) -> exists jt_value_unit_arbitrary_countenuminputbound. ((((exists fs_h_jt_unit_arbitrary_countenuminputboundat. fs_h_jt_unit_arbitrary_countenuminputboundat + S (jt_value_unit_arbitrary_countenuminputbound) = S ((S (jt_index_unit_arbitrary_countenuminputbound)) * jt_c_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenuminputboundat. jt_b_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenuminputboundat * S ((S (jt_index_unit_arbitrary_countenuminputbound)) * jt_c_unit_arbitrary_countenum) + (jt_value_unit_arbitrary_countenuminputbound))) /\\ (exists jt_gap_unit_arbitrary_countenuminputboundvalue. jt_gap_unit_arbitrary_countenuminputboundvalue+S (jt_value_unit_arbitrary_countenuminputbound)=(1)))) -> (forall jt_divisor_unit_arbitrary_countenuminputprimitive. (exists jt_factor_unit_arbitrary_countenuminputprimitivemodulus. (1)=(jt_divisor_unit_arbitrary_countenuminputprimitive)*jt_factor_unit_arbitrary_countenuminputprimitivemodulus) -> (forall jt_index_unit_arbitrary_countenuminputprimitivecoordinates jt_value_unit_arbitrary_countenuminputprimitivecoordinates. (exists jt_gap_unit_arbitrary_countenuminputprimitivecoordinatesindex. jt_gap_unit_arbitrary_countenuminputprimitivecoordinatesindex+S (jt_index_unit_arbitrary_countenuminputprimitivecoordinates)=(k)) -> (((exists fs_h_jt_unit_arbitrary_countenuminputprimitivecoordinatesat. fs_h_jt_unit_arbitrary_countenuminputprimitivecoordinatesat + S (jt_value_unit_arbitrary_countenuminputprimitivecoordinates) = S ((S (jt_index_unit_arbitrary_countenuminputprimitivecoordinates)) * jt_c_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenuminputprimitivecoordinatesat. jt_b_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenuminputprimitivecoordinatesat * S ((S (jt_index_unit_arbitrary_countenuminputprimitivecoordinates)) * jt_c_unit_arbitrary_countenum) + (jt_value_unit_arbitrary_countenuminputprimitivecoordinates))) -> (exists jt_factor_unit_arbitrary_countenuminputprimitivecoordinatesdivides. (jt_value_unit_arbitrary_countenuminputprimitivecoordinates)=(jt_divisor_unit_arbitrary_countenuminputprimitive)*jt_factor_unit_arbitrary_countenuminputprimitivecoordinatesdivides)) -> jt_divisor_unit_arbitrary_countenuminputprimitive=1) -> exists jt_i_unit_arbitrary_countenum jt_d_unit_arbitrary_countenum jt_e_unit_arbitrary_countenum. ((exists jt_gap_unit_arbitrary_countenumcompleteindex. jt_gap_unit_arbitrary_countenumcompleteindex+S (jt_i_unit_arbitrary_countenum)=(j)) /\\ (((((((exists fs_h_jt_unit_arbitrary_countenumcompletecode. fs_h_jt_unit_arbitrary_countenumcompletecode + S (jt_d_unit_arbitrary_countenum) = S ((S (jt_i_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumcompletecode. jt_codes_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumcompletecode * S ((S (jt_i_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count) + (jt_d_unit_arbitrary_countenum))) /\\ (((exists fs_h_jt_unit_arbitrary_countenumcompletescale. fs_h_jt_unit_arbitrary_countenumcompletescale + S (jt_e_unit_arbitrary_countenum) = S ((S (jt_i_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumcompletescale. jt_scales_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumcompletescale * S ((S (jt_i_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count) + (jt_e_unit_arbitrary_countenum))))) /\\ (forall jt_index_unit_arbitrary_countenumrepresented jt_left_unit_arbitrary_countenumrepresented jt_right_unit_arbitrary_countenumrepresented. (exists jt_gap_unit_arbitrary_countenumrepresentedindex. jt_gap_unit_arbitrary_countenumrepresentedindex+S (jt_index_unit_arbitrary_countenumrepresented)=(k)) -> (((exists fs_h_jt_unit_arbitrary_countenumrepresentedleft. fs_h_jt_unit_arbitrary_countenumrepresentedleft + S (jt_left_unit_arbitrary_countenumrepresented) = S ((S (jt_index_unit_arbitrary_countenumrepresented)) * jt_c_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenumrepresentedleft. jt_b_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenumrepresentedleft * S ((S (jt_index_unit_arbitrary_countenumrepresented)) * jt_c_unit_arbitrary_countenum) + (jt_left_unit_arbitrary_countenumrepresented))) -> (((exists fs_h_jt_unit_arbitrary_countenumrepresentedright. fs_h_jt_unit_arbitrary_countenumrepresentedright + S (jt_right_unit_arbitrary_countenumrepresented) = S ((S (jt_index_unit_arbitrary_countenumrepresented)) * jt_e_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenumrepresentedright. jt_d_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenumrepresentedright * S ((S (jt_index_unit_arbitrary_countenumrepresented)) * jt_e_unit_arbitrary_countenum) + (jt_right_unit_arbitrary_countenumrepresented))) -> jt_left_unit_arbitrary_countenumrepresented=jt_right_unit_arbitrary_countenumrepresented))))) /\\ (forall jt_i_unit_arbitrary_countenum jt_h_unit_arbitrary_countenum jt_b_unit_arbitrary_countenum jt_c_unit_arbitrary_countenum jt_d_unit_arbitrary_countenum jt_e_unit_arbitrary_countenum. (exists jt_gap_unit_arbitrary_countenumfirstindex. jt_gap_unit_arbitrary_countenumfirstindex+S (jt_i_unit_arbitrary_countenum)=(j)) -> (exists jt_gap_unit_arbitrary_countenumsecondindex. jt_gap_unit_arbitrary_countenumsecondindex+S (jt_h_unit_arbitrary_countenum)=(j)) -> (((((exists fs_h_jt_unit_arbitrary_countenumfirstcode. fs_h_jt_unit_arbitrary_countenumfirstcode + S (jt_b_unit_arbitrary_countenum) = S ((S (jt_i_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumfirstcode. jt_codes_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumfirstcode * S ((S (jt_i_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count) + (jt_b_unit_arbitrary_countenum))) /\\ (((exists fs_h_jt_unit_arbitrary_countenumfirstscale. fs_h_jt_unit_arbitrary_countenumfirstscale + S (jt_c_unit_arbitrary_countenum) = S ((S (jt_i_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumfirstscale. jt_scales_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumfirstscale * S ((S (jt_i_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count) + (jt_c_unit_arbitrary_countenum))))) -> (((((exists fs_h_jt_unit_arbitrary_countenumsecondcode. fs_h_jt_unit_arbitrary_countenumsecondcode + S (jt_d_unit_arbitrary_countenum) = S ((S (jt_h_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumsecondcode. jt_codes_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumsecondcode * S ((S (jt_h_unit_arbitrary_countenum)) * jt_code_scale_unit_arbitrary_count) + (jt_d_unit_arbitrary_countenum))) /\\ (((exists fs_h_jt_unit_arbitrary_countenumsecondscale. fs_h_jt_unit_arbitrary_countenumsecondscale + S (jt_e_unit_arbitrary_countenum) = S ((S (jt_h_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count)) /\\ exists fs_q_jt_unit_arbitrary_countenumsecondscale. jt_scales_unit_arbitrary_count = fs_q_jt_unit_arbitrary_countenumsecondscale * S ((S (jt_h_unit_arbitrary_countenum)) * jt_scale_scale_unit_arbitrary_count) + (jt_e_unit_arbitrary_countenum))))) -> (forall jt_index_unit_arbitrary_countenumsame jt_left_unit_arbitrary_countenumsame jt_right_unit_arbitrary_countenumsame. (exists jt_gap_unit_arbitrary_countenumsameindex. jt_gap_unit_arbitrary_countenumsameindex+S (jt_index_unit_arbitrary_countenumsame)=(k)) -> (((exists fs_h_jt_unit_arbitrary_countenumsameleft. fs_h_jt_unit_arbitrary_countenumsameleft + S (jt_left_unit_arbitrary_countenumsame) = S ((S (jt_index_unit_arbitrary_countenumsame)) * jt_c_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenumsameleft. jt_b_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenumsameleft * S ((S (jt_index_unit_arbitrary_countenumsame)) * jt_c_unit_arbitrary_countenum) + (jt_left_unit_arbitrary_countenumsame))) -> (((exists fs_h_jt_unit_arbitrary_countenumsameright. fs_h_jt_unit_arbitrary_countenumsameright + S (jt_right_unit_arbitrary_countenumsame) = S ((S (jt_index_unit_arbitrary_countenumsame)) * jt_e_unit_arbitrary_countenum)) /\\ exists fs_q_jt_unit_arbitrary_countenumsameright. jt_d_unit_arbitrary_countenum = fs_q_jt_unit_arbitrary_countenumsameright * S ((S (jt_index_unit_arbitrary_countenumsame)) * jt_e_unit_arbitrary_countenum) + (jt_right_unit_arbitrary_countenumsame))) -> jt_left_unit_arbitrary_countenumsame=jt_right_unit_arbitrary_countenumsame) -> jt_i_unit_arbitrary_countenum=jt_h_unit_arbitrary_countenum))))))))) -> (j=1)", "statement_sha256": "5b09d9d06efe4146b0de5993af4aa823488e535f7667d9497839d0885ae9a3d8", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every genuine Jordan count modulo one equals one, independently of its beta encoding." }, { "admission_dependencies": [], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 19, "body_proof_nodes": 28, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro p", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hpn", "intro hprimitive", "intro hall", "cases hp", "apply hp_left", "specialize hprimitive (p)", "apply hprimitive", "exact hpn", "exact hall" ], "defined_statement": "∀ p. ∀ n. ∀ b. ∀ c. ∀ k. Prime(p) → Dvd(p,n) → JordanPrimitiveTuple(n,b,c,k) → ¬JordanTupleAllDivisible(p,b,c,k)", "defined_statement_sha256": "acd72f7e22a7a58218d24f237a3ff4f3bfff3fb2018e9b334011786208578bf0", "definition_uses": { "ND0371": 1, "ND0372": 1, "PD0003": 1, "PD0004": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "7b9413d92687b12588dedd426bde5998a3506504acad82adb60b6f7a334cbcc3", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hpn" } ], [ { "kind": "text", "text": "intro hprimitive" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "cases hp" } ], [ { "kind": "text", "text": "apply hp_left" } ], [ { "kind": "text", "text": "specialize hprimitive (p)" } ], [ { "kind": "text", "text": "apply hprimitive" } ], [ { "kind": "text", "text": "exact hpn" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0371": 1, "ND0372": 1, "PD0003": 1, "PD0004": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ p. ∀ n. ∀ b. ∀ c. ∀ k. " }, { "definition": "PD0004", "kind": "definition", "text": "Prime(p)" }, { "kind": "text", "text": " → " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(p,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " → ¬" }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(p,b,c,k)" } ] }, "dependencies": [], "enrolled_in_alpha": true, "factory": "make_jordan_prime_power_characterization_candidate_theorems", "first_admitted_version": "v35", "id": "JT005D", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_primitive_tuple_avoids_prime_common_divisor", "original_ha_bundle_verified": true, "proof_bundle_node_id": 354, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro p", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hpn", "intro hprimitive", "intro hall", "cases hp", "apply hp_left", "specialize hprimitive (p)", "apply hprimitive", "exact hpn", "exact hall" ], "script_sha256": "82a35cdda96dacc619dbb7979ad8842f6ea26b2578543bdaf54329e133a07012", "source_filename": "jordan_prime_power_characterization_candidate.py", "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "sources": [ { "factory": "make_jordan_prime_power_characterization_candidate_theorems", "script_sha256": "82a35cdda96dacc619dbb7979ad8842f6ea26b2578543bdaf54329e133a07012", "selected": true, "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "source_sha256": "6f7d65e91dfe8c818b2bf5a3d0e8b1754f75fe631cc943543c5081820a299f11", "statement_sha256": "7b9413d92687b12588dedd426bde5998a3506504acad82adb60b6f7a334cbcc3" } ], "stable_member": false, "statement": "forall p n b c k. (~((p) = 1) /\\ forall pvs_left_jordan_base pvs_right_jordan_base. (p) = pvs_left_jordan_base * pvs_right_jordan_base -> pvs_left_jordan_base = 1 \\/ pvs_right_jordan_base = 1) -> (exists jt_factor_jordan_modulus. (n)=(p)*jt_factor_jordan_modulus) -> (forall jt_divisor_power_primitive. (exists jt_factor_power_primitivemodulus. (n)=(jt_divisor_power_primitive)*jt_factor_power_primitivemodulus) -> (forall jt_index_power_primitivecoordinates jt_value_power_primitivecoordinates. (exists jt_gap_power_primitivecoordinatesindex. jt_gap_power_primitivecoordinatesindex+S (jt_index_power_primitivecoordinates)=(k)) -> (((exists fs_h_jt_power_primitivecoordinatesat. fs_h_jt_power_primitivecoordinatesat + S (jt_value_power_primitivecoordinates) = S ((S (jt_index_power_primitivecoordinates)) * c)) /\\ exists fs_q_jt_power_primitivecoordinatesat. b = fs_q_jt_power_primitivecoordinatesat * S ((S (jt_index_power_primitivecoordinates)) * c) + (jt_value_power_primitivecoordinates))) -> (exists jt_factor_power_primitivecoordinatesdivides. (jt_value_power_primitivecoordinates)=(jt_divisor_power_primitive)*jt_factor_power_primitivecoordinatesdivides)) -> jt_divisor_power_primitive=1) -> ~(forall jt_index_power_all_divisible jt_value_power_all_divisible. (exists jt_gap_power_all_divisibleindex. jt_gap_power_all_divisibleindex+S (jt_index_power_all_divisible)=(k)) -> (((exists fs_h_jt_power_all_divisibleat. fs_h_jt_power_all_divisibleat + S (jt_value_power_all_divisible) = S ((S (jt_index_power_all_divisible)) * c)) /\\ exists fs_q_jt_power_all_divisibleat. b = fs_q_jt_power_all_divisibleat * S ((S (jt_index_power_all_divisible)) * c) + (jt_value_power_all_divisible))) -> (exists jt_factor_power_all_divisibledivides. (jt_value_power_all_divisible)=(p)*jt_factor_power_all_divisibledivides))", "statement_sha256": "7b9413d92687b12588dedd426bde5998a3506504acad82adb60b6f7a334cbcc3", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "A prime divisor of the modulus cannot divide every coordinate of a primitive tuple." }, { "admission_dependencies": [ "pow_nonzero_of_one_le", "one_le_of_ne_zero", "prime_nonzero", "eq_decidable", "mul_zero_left", "prime_divisor_exists", "multiple_trans", "prime_divisor_of_prime_power", "jordan_tuple_divisor_downward" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 37, "body_proof_nodes": 97, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro p", "intro e", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hpow", "intro hnot", "have hn : ~(n=0)", "intro hz", "specialize pow_nonzero_of_one_le (p)", "specialize pow_nonzero_of_one_le (e)", "specialize pow_nonzero_of_one_le (n)", "apply pow_nonzero_of_one_le", "specialize one_le_of_ne_zero (p)", "apply one_le_of_ne_zero", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "exact hpow", "exact hz", "intro d", "intro hd", "intro hall", "specialize eq_decidable d", "specialize eq_decidable 1", "cases eq_decidable", "exact eq_decidable_left", "exfalso", "apply hnot", "have hdnonzero : ~(d=0)", "intro hz", "apply hn", "cases hd", "trans d*x", "exact hd_witness", "rewrite hz", "specialize mul_zero_left (x)", "apply mul_zero_left", "have hprime : ∃ q. Prime(q) ∧ Dvd(q,d)", "specialize prime_divisor_exists (d)", "apply prime_divisor_exists", "exact hdnonzero", "exact eq_decidable_right", "cases hprime", "cases hprime_witness", "have hqn : Dvd(x,n)", "specialize multiple_trans (d)", "specialize multiple_trans (x)", "specialize multiple_trans (n)", "apply multiple_trans", "exact hd", "exact hprime_witness_right", "have hqp : x=p", "specialize prime_divisor_of_prime_power (p)", "specialize prime_divisor_of_prime_power (x)", "specialize prime_divisor_of_prime_power (e)", "specialize prime_divisor_of_prime_power (n)", "apply prime_divisor_of_prime_power", "exact hp", "exact hprime_witness_left", "exact hpow", "exact hqn", "rewrite <- hqp", "specialize jordan_tuple_divisor_downward (x)", "specialize jordan_tuple_divisor_downward (d)", "specialize jordan_tuple_divisor_downward (b)", "specialize jordan_tuple_divisor_downward (c)", "specialize jordan_tuple_divisor_downward (k)", "apply jordan_tuple_divisor_downward", "exact hprime_witness_right", "exact hall" ], "defined_statement": "∀ p. ∀ e. ∀ n. ∀ b. ∀ c. ∀ k. Prime(p) → Pow(p,e,n) → ¬JordanTupleAllDivisible(p,b,c,k) → JordanPrimitiveTuple(n,b,c,k)", "defined_statement_sha256": "6ad5de8218685ba295d656ea4d54498fd861f87e5b9acb63e66b955fbf1e9f80", "definition_uses": { "ND0371": 1, "ND0372": 1, "PD0003": 2, "PD0004": 2, "PD0020": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "b6d7d853520589e5b9d5c3e6050341c6d2868c4db33581078c2a3f75e703a508", "free_names": [], "script_definition_uses": { "PD0003": 2, "PD0004": 1 }, "script_parts": [ [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro e" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hpow" } ], [ { "kind": "text", "text": "intro hnot" } ], [ { "kind": "text", "text": "have hn : ~(n=0)" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "specialize pow_nonzero_of_one_le (p)" } ], [ { "kind": "text", "text": "specialize pow_nonzero_of_one_le (e)" } ], [ { "kind": "text", "text": "specialize pow_nonzero_of_one_le (n)" } ], [ { "kind": "text", "text": "apply pow_nonzero_of_one_le" } ], [ { "kind": "text", "text": "specialize one_le_of_ne_zero (p)" } ], [ { "kind": "text", "text": "apply one_le_of_ne_zero" } ], [ { "kind": "text", "text": "intro hpzero" } ], [ { "kind": "text", "text": "specialize prime_nonzero (p)" } ], [ { "kind": "text", "text": "apply prime_nonzero" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "exact hpzero" } ], [ { "kind": "text", "text": "exact hpow" } ], [ { "kind": "text", "text": "exact hz" } ], [ { "kind": "text", "text": "intro d" } ], [ { "kind": "text", "text": "intro hd" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize eq_decidable d" } ], [ { "kind": "text", "text": "specialize eq_decidable 1" } ], [ { "kind": "text", "text": "cases eq_decidable" } ], [ { "kind": "text", "text": "exact eq_decidable_left" } ], [ { "kind": "text", "text": "exfalso" } ], [ { "kind": "text", "text": "apply hnot" } ], [ { "kind": "text", "text": "have hdnonzero : ~(d=0)" } ], [ { "kind": "text", "text": "intro hz" } ], [ { "kind": "text", "text": "apply hn" } ], [ { "kind": "text", "text": "cases hd" } ], [ { "kind": "text", "text": "trans d*x" } ], [ { "kind": "text", "text": "exact hd_witness" } ], [ { "kind": "text", "text": "rewrite hz" } ], [ { "kind": "text", "text": "specialize mul_zero_left (x)" } ], [ { "kind": "text", "text": "apply mul_zero_left" } ], [ { "kind": "text", "text": "have hprime : " }, { "kind": "text", "text": "∃ q. " }, { "definition": "PD0004", "kind": "definition", "text": "Prime(q)" }, { "kind": "text", "text": " ∧ " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(q,d)" } ], [ { "kind": "text", "text": "specialize prime_divisor_exists (d)" } ], [ { "kind": "text", "text": "apply prime_divisor_exists" } ], [ { "kind": "text", "text": "exact hdnonzero" } ], [ { "kind": "text", "text": "exact eq_decidable_right" } ], [ { "kind": "text", "text": "cases hprime" } ], [ { "kind": "text", "text": "cases hprime_witness" } ], [ { "kind": "text", "text": "have hqn : " }, { "definition": "PD0003", "kind": "definition", "text": "Dvd(x,n)" } ], [ { "kind": "text", "text": "specialize multiple_trans (d)" } ], [ { "kind": "text", "text": "specialize multiple_trans (x)" } ], [ { "kind": "text", "text": "specialize multiple_trans (n)" } ], [ { "kind": "text", "text": "apply multiple_trans" } ], [ { "kind": "text", "text": "exact hd" } ], [ { "kind": "text", "text": "exact hprime_witness_right" } ], [ { "kind": "text", "text": "have hqp : x=p" } ], [ { "kind": "text", "text": "specialize prime_divisor_of_prime_power (p)" } ], [ { "kind": "text", "text": "specialize prime_divisor_of_prime_power (x)" } ], [ { "kind": "text", "text": "specialize prime_divisor_of_prime_power (e)" } ], [ { "kind": "text", "text": "specialize prime_divisor_of_prime_power (n)" } ], [ { "kind": "text", "text": "apply prime_divisor_of_prime_power" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "exact hprime_witness_left" } ], [ { "kind": "text", "text": "exact hpow" } ], [ { "kind": "text", "text": "exact hqn" } ], [ { "kind": "text", "text": "rewrite <- hqp" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (x)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (d)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (b)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (c)" } ], [ { "kind": "text", "text": "specialize jordan_tuple_divisor_downward (k)" } ], [ { "kind": "text", "text": "apply jordan_tuple_divisor_downward" } ], [ { "kind": "text", "text": "exact hprime_witness_right" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0371": 1, "ND0372": 1, "PD0004": 1, "PD0020": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ p. ∀ e. ∀ n. ∀ b. ∀ c. ∀ k. " }, { "definition": "PD0004", "kind": "definition", "text": "Prime(p)" }, { "kind": "text", "text": " → " }, { "definition": "PD0020", "kind": "definition", "text": "Pow(p,e,n)" }, { "kind": "text", "text": " → ¬" }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(p,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" } ] }, "dependencies": [ "pow_nonzero_of_one_le", "one_le_of_ne_zero", "prime_nonzero", "eq_decidable", "mul_zero_left", "prime_divisor_exists", "multiple_trans", "prime_divisor_of_prime_power", "jordan_tuple_divisor_downward" ], "enrolled_in_alpha": true, "factory": "make_jordan_prime_power_characterization_candidate_theorems", "first_admitted_version": "v35", "id": "JT005E", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_prime_power_tuple_primitive_of_not_all_divisible", "original_ha_bundle_verified": true, "proof_bundle_node_id": 355, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro p", "intro e", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hpow", "intro hnot", "have hn : ~(n=0)", "intro hz", "specialize pow_nonzero_of_one_le (p)", "specialize pow_nonzero_of_one_le (e)", "specialize pow_nonzero_of_one_le (n)", "apply pow_nonzero_of_one_le", "specialize one_le_of_ne_zero (p)", "apply one_le_of_ne_zero", "intro hpzero", "specialize prime_nonzero (p)", "apply prime_nonzero", "exact hp", "exact hpzero", "exact hpow", "exact hz", "intro d", "intro hd", "intro hall", "specialize eq_decidable d", "specialize eq_decidable 1", "cases eq_decidable", "exact eq_decidable_left", "exfalso", "apply hnot", "have hdnonzero : ~(d=0)", "intro hz", "apply hn", "cases hd", "trans d*x", "exact hd_witness", "rewrite hz", "specialize mul_zero_left (x)", "apply mul_zero_left", "have hprime : exists q. ((~((q) = 1) /\\ forall pvs_left_jordan_common_prime pvs_right_jordan_common_prime. (q) = pvs_left_jordan_common_prime * pvs_right_jordan_common_prime -> pvs_left_jordan_common_prime = 1 \\/ pvs_right_jordan_common_prime = 1) /\\ (exists jt_factor_jordan_common_divisor. (d)=(q)*jt_factor_jordan_common_divisor))", "specialize prime_divisor_exists (d)", "apply prime_divisor_exists", "exact hdnonzero", "exact eq_decidable_right", "cases hprime", "cases hprime_witness", "have hqn : exists jt_factor_jordan_modulus_prime. (n)=(x)*jt_factor_jordan_modulus_prime", "specialize multiple_trans (d)", "specialize multiple_trans (x)", "specialize multiple_trans (n)", "apply multiple_trans", "exact hd", "exact hprime_witness_right", "have hqp : x=p", "specialize prime_divisor_of_prime_power (p)", "specialize prime_divisor_of_prime_power (x)", "specialize prime_divisor_of_prime_power (e)", "specialize prime_divisor_of_prime_power (n)", "apply prime_divisor_of_prime_power", "exact hp", "exact hprime_witness_left", "exact hpow", "exact hqn", "rewrite <- hqp", "specialize jordan_tuple_divisor_downward (x)", "specialize jordan_tuple_divisor_downward (d)", "specialize jordan_tuple_divisor_downward (b)", "specialize jordan_tuple_divisor_downward (c)", "specialize jordan_tuple_divisor_downward (k)", "apply jordan_tuple_divisor_downward", "exact hprime_witness_right", "exact hall" ], "script_sha256": "6757aedf60518b92cfdd8bf5522fbce66ef03c7a29404e3a8e20bb092614b968", "source_filename": "jordan_prime_power_characterization_candidate.py", "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "sources": [ { "factory": "make_jordan_prime_power_characterization_candidate_theorems", "script_sha256": "6757aedf60518b92cfdd8bf5522fbce66ef03c7a29404e3a8e20bb092614b968", "selected": true, "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "source_sha256": "6f7d65e91dfe8c818b2bf5a3d0e8b1754f75fe631cc943543c5081820a299f11", "statement_sha256": "b6d7d853520589e5b9d5c3e6050341c6d2868c4db33581078c2a3f75e703a508" } ], "stable_member": false, "statement": "forall p e n b c k. (~((p) = 1) /\\ forall pvs_left_jordan_base pvs_right_jordan_base. (p) = pvs_left_jordan_base * pvs_right_jordan_base -> pvs_left_jordan_base = 1 \\/ pvs_right_jordan_base = 1) -> (exists pa_b_pvs_jordan_power pa_c_pvs_jordan_power. ((forall pa_i_pvs_jordan_power_repeat. (exists pa_lt_pvs_jordan_power_repeat_bound. pa_lt_pvs_jordan_power_repeat_bound + S pa_i_pvs_jordan_power_repeat = e) -> (((exists pa_h_pvs_jordan_power_repeat_decoded. pa_h_pvs_jordan_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_jordan_power_repeat)) * pa_c_pvs_jordan_power)) /\\ exists pa_q_pvs_jordan_power_repeat_decoded. pa_b_pvs_jordan_power = pa_q_pvs_jordan_power_repeat_decoded * S ((S (pa_i_pvs_jordan_power_repeat)) * pa_c_pvs_jordan_power) + (p)))) /\\ (exists pa_u_pvs_jordan_power_product pa_v_pvs_jordan_power_product. ((((exists pa_h_pvs_jordan_power_product_start. pa_h_pvs_jordan_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_jordan_power_product)) /\\ exists pa_q_pvs_jordan_power_product_start. pa_u_pvs_jordan_power_product = pa_q_pvs_jordan_power_product_start * S ((S (0)) * pa_v_pvs_jordan_power_product) + (1))) /\\ ((((exists pa_h_pvs_jordan_power_product_terminal. pa_h_pvs_jordan_power_product_terminal + S (n) = S ((S (e)) * pa_v_pvs_jordan_power_product)) /\\ exists pa_q_pvs_jordan_power_product_terminal. pa_u_pvs_jordan_power_product = pa_q_pvs_jordan_power_product_terminal * S ((S (e)) * pa_v_pvs_jordan_power_product) + (n))) /\\ forall pa_i_pvs_jordan_power_product. (exists pa_lt_pvs_jordan_power_product_bound. pa_lt_pvs_jordan_power_product_bound + S pa_i_pvs_jordan_power_product = e) -> exists pa_p_pvs_jordan_power_product pa_r_pvs_jordan_power_product pa_s_pvs_jordan_power_product. ((((exists pa_h_pvs_jordan_power_product_factor. pa_h_pvs_jordan_power_product_factor + S (pa_p_pvs_jordan_power_product) = S ((S (pa_i_pvs_jordan_power_product)) * pa_c_pvs_jordan_power)) /\\ exists pa_q_pvs_jordan_power_product_factor. pa_b_pvs_jordan_power = pa_q_pvs_jordan_power_product_factor * S ((S (pa_i_pvs_jordan_power_product)) * pa_c_pvs_jordan_power) + (pa_p_pvs_jordan_power_product))) /\\ ((((exists pa_h_pvs_jordan_power_product_partial. pa_h_pvs_jordan_power_product_partial + S (pa_r_pvs_jordan_power_product) = S ((S (pa_i_pvs_jordan_power_product)) * pa_v_pvs_jordan_power_product)) /\\ exists pa_q_pvs_jordan_power_product_partial. pa_u_pvs_jordan_power_product = pa_q_pvs_jordan_power_product_partial * S ((S (pa_i_pvs_jordan_power_product)) * pa_v_pvs_jordan_power_product) + (pa_r_pvs_jordan_power_product))) /\\ ((((exists pa_h_pvs_jordan_power_product_successor. pa_h_pvs_jordan_power_product_successor + S (pa_s_pvs_jordan_power_product) = S ((S (S pa_i_pvs_jordan_power_product)) * pa_v_pvs_jordan_power_product)) /\\ exists pa_q_pvs_jordan_power_product_successor. pa_u_pvs_jordan_power_product = pa_q_pvs_jordan_power_product_successor * S ((S (S pa_i_pvs_jordan_power_product)) * pa_v_pvs_jordan_power_product) + (pa_s_pvs_jordan_power_product))) /\\ pa_s_pvs_jordan_power_product = pa_r_pvs_jordan_power_product * pa_p_pvs_jordan_power_product)))))))) -> (~(forall jt_index_power_all_divisible jt_value_power_all_divisible. (exists jt_gap_power_all_divisibleindex. jt_gap_power_all_divisibleindex+S (jt_index_power_all_divisible)=(k)) -> (((exists fs_h_jt_power_all_divisibleat. fs_h_jt_power_all_divisibleat + S (jt_value_power_all_divisible) = S ((S (jt_index_power_all_divisible)) * c)) /\\ exists fs_q_jt_power_all_divisibleat. b = fs_q_jt_power_all_divisibleat * S ((S (jt_index_power_all_divisible)) * c) + (jt_value_power_all_divisible))) -> (exists jt_factor_power_all_divisibledivides. (jt_value_power_all_divisible)=(p)*jt_factor_power_all_divisibledivides))) -> forall jt_divisor_power_primitive. (exists jt_factor_power_primitivemodulus. (n)=(jt_divisor_power_primitive)*jt_factor_power_primitivemodulus) -> (forall jt_index_power_primitivecoordinates jt_value_power_primitivecoordinates. (exists jt_gap_power_primitivecoordinatesindex. jt_gap_power_primitivecoordinatesindex+S (jt_index_power_primitivecoordinates)=(k)) -> (((exists fs_h_jt_power_primitivecoordinatesat. fs_h_jt_power_primitivecoordinatesat + S (jt_value_power_primitivecoordinates) = S ((S (jt_index_power_primitivecoordinates)) * c)) /\\ exists fs_q_jt_power_primitivecoordinatesat. b = fs_q_jt_power_primitivecoordinatesat * S ((S (jt_index_power_primitivecoordinates)) * c) + (jt_value_power_primitivecoordinates))) -> (exists jt_factor_power_primitivecoordinatesdivides. (jt_value_power_primitivecoordinates)=(jt_divisor_power_primitive)*jt_factor_power_primitivecoordinatesdivides)) -> jt_divisor_power_primitive=1", "statement_sha256": "b6d7d853520589e5b9d5c3e6050341c6d2868c4db33581078c2a3f75e703a508", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Every nonunit common divisor has an actual prime divisor; prime-power support then forces a forbidden common factor p." }, { "admission_dependencies": [ "jordan_primitive_tuple_avoids_prime_common_divisor", "pow_positive_exponent_base_divides", "succ_ne_zero", "jordan_prime_power_tuple_primitive_of_not_all_divisible" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 41, "body_proof_nodes": 99, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro p", "intro h", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hpow", "split", "intro hprimitive", "intro hall", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (p)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (n)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (b)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (c)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (k)", "apply jordan_primitive_tuple_avoids_prime_common_divisor", "exact hp", "specialize pow_positive_exponent_base_divides (p)", "specialize pow_positive_exponent_base_divides (S h)", "specialize pow_positive_exponent_base_divides (n)", "apply pow_positive_exponent_base_divides", "intro hszero", "specialize succ_ne_zero (h)", "apply succ_ne_zero", "exact hszero", "exact hpow", "exact hprimitive", "exact hall", "intro hnot", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (p)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (S h)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (n)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (b)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (c)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (k)", "apply jordan_prime_power_tuple_primitive_of_not_all_divisible", "exact hp", "exact hpow", "exact hnot" ], "defined_statement": "∀ p. ∀ h. ∀ n. ∀ b. ∀ c. ∀ k. Prime(p) → Pow(p,S h,n) → (JordanPrimitiveTuple(n,b,c,k) → ¬JordanTupleAllDivisible(p,b,c,k)) ∧ (¬JordanTupleAllDivisible(p,b,c,k) → JordanPrimitiveTuple(n,b,c,k))", "defined_statement_sha256": "1934072ca4bfb7802ec869125d409ab65e0c5a8f4e8d096e68be2c2e03bdaf46", "definition_uses": { "ND0371": 2, "ND0372": 2, "PD0004": 1, "PD0020": 1 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "84696e7e875bc4b25237a038ac8f985066642f8e2e9313d672136e66f19d617b", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hpow" } ], [ { "kind": "text", "text": "split" } ], [ { "kind": "text", "text": "intro hprimitive" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (p)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (n)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_avoids_prime_common_divisor" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "specialize pow_positive_exponent_base_divides (p)" } ], [ { "kind": "text", "text": "specialize pow_positive_exponent_base_divides (S h)" } ], [ { "kind": "text", "text": "specialize pow_positive_exponent_base_divides (n)" } ], [ { "kind": "text", "text": "apply pow_positive_exponent_base_divides" } ], [ { "kind": "text", "text": "intro hszero" } ], [ { "kind": "text", "text": "specialize succ_ne_zero (h)" } ], [ { "kind": "text", "text": "apply succ_ne_zero" } ], [ { "kind": "text", "text": "exact hszero" } ], [ { "kind": "text", "text": "exact hpow" } ], [ { "kind": "text", "text": "exact hprimitive" } ], [ { "kind": "text", "text": "exact hall" } ], [ { "kind": "text", "text": "intro hnot" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (p)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (S h)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (n)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (b)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (c)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (k)" } ], [ { "kind": "text", "text": "apply jordan_prime_power_tuple_primitive_of_not_all_divisible" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "exact hpow" } ], [ { "kind": "text", "text": "exact hnot" } ] ], "statement_definition_uses": { "ND0371": 2, "ND0372": 2, "PD0004": 1, "PD0020": 1 }, "statement_parts": [ { "kind": "text", "text": "∀ p. ∀ h. ∀ n. ∀ b. ∀ c. ∀ k. " }, { "definition": "PD0004", "kind": "definition", "text": "Prime(p)" }, { "kind": "text", "text": " → " }, { "definition": "PD0020", "kind": "definition", "text": "Pow(p,S h,n)" }, { "kind": "text", "text": " → (" }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": " → ¬" }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(p,b,c,k)" }, { "kind": "text", "text": ") ∧ (¬" }, { "definition": "ND0371", "kind": "definition", "text": "JordanTupleAllDivisible(p,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" }, { "kind": "text", "text": ")" } ] }, "dependencies": [ "jordan_primitive_tuple_avoids_prime_common_divisor", "pow_positive_exponent_base_divides", "succ_ne_zero", "jordan_prime_power_tuple_primitive_of_not_all_divisible" ], "enrolled_in_alpha": true, "factory": "make_jordan_prime_power_characterization_candidate_theorems", "first_admitted_version": "v35", "id": "JT005F", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_prime_power_tuple_primitive_characterization", "original_ha_bundle_verified": true, "proof_bundle_node_id": 356, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro p", "intro h", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hpow", "split", "intro hprimitive", "intro hall", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (p)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (n)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (b)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (c)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (k)", "apply jordan_primitive_tuple_avoids_prime_common_divisor", "exact hp", "specialize pow_positive_exponent_base_divides (p)", "specialize pow_positive_exponent_base_divides (S h)", "specialize pow_positive_exponent_base_divides (n)", "apply pow_positive_exponent_base_divides", "intro hszero", "specialize succ_ne_zero (h)", "apply succ_ne_zero", "exact hszero", "exact hpow", "exact hprimitive", "exact hall", "intro hnot", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (p)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (S h)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (n)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (b)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (c)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (k)", "apply jordan_prime_power_tuple_primitive_of_not_all_divisible", "exact hp", "exact hpow", "exact hnot" ], "script_sha256": "6fbddd7800c1d195adb7a1cea670abdaf8e3a54dbc460bb57e7fbfee597d6d92", "source_filename": "jordan_prime_power_characterization_candidate.py", "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "sources": [ { "factory": "make_jordan_prime_power_characterization_candidate_theorems", "script_sha256": "6fbddd7800c1d195adb7a1cea670abdaf8e3a54dbc460bb57e7fbfee597d6d92", "selected": true, "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "source_sha256": "6f7d65e91dfe8c818b2bf5a3d0e8b1754f75fe631cc943543c5081820a299f11", "statement_sha256": "84696e7e875bc4b25237a038ac8f985066642f8e2e9313d672136e66f19d617b" } ], "stable_member": false, "statement": "forall p h n b c k. (~((p) = 1) /\\ forall pvs_left_jordan_base pvs_right_jordan_base. (p) = pvs_left_jordan_base * pvs_right_jordan_base -> pvs_left_jordan_base = 1 \\/ pvs_right_jordan_base = 1) -> (exists pa_b_pvs_jordan_positive_power pa_c_pvs_jordan_positive_power. ((forall pa_i_pvs_jordan_positive_power_repeat. (exists pa_lt_pvs_jordan_positive_power_repeat_bound. pa_lt_pvs_jordan_positive_power_repeat_bound + S pa_i_pvs_jordan_positive_power_repeat = S h) -> (((exists pa_h_pvs_jordan_positive_power_repeat_decoded. pa_h_pvs_jordan_positive_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_jordan_positive_power_repeat)) * pa_c_pvs_jordan_positive_power)) /\\ exists pa_q_pvs_jordan_positive_power_repeat_decoded. pa_b_pvs_jordan_positive_power = pa_q_pvs_jordan_positive_power_repeat_decoded * S ((S (pa_i_pvs_jordan_positive_power_repeat)) * pa_c_pvs_jordan_positive_power) + (p)))) /\\ (exists pa_u_pvs_jordan_positive_power_product pa_v_pvs_jordan_positive_power_product. ((((exists pa_h_pvs_jordan_positive_power_product_start. pa_h_pvs_jordan_positive_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_jordan_positive_power_product)) /\\ exists pa_q_pvs_jordan_positive_power_product_start. pa_u_pvs_jordan_positive_power_product = pa_q_pvs_jordan_positive_power_product_start * S ((S (0)) * pa_v_pvs_jordan_positive_power_product) + (1))) /\\ ((((exists pa_h_pvs_jordan_positive_power_product_terminal. pa_h_pvs_jordan_positive_power_product_terminal + S (n) = S ((S (S h)) * pa_v_pvs_jordan_positive_power_product)) /\\ exists pa_q_pvs_jordan_positive_power_product_terminal. pa_u_pvs_jordan_positive_power_product = pa_q_pvs_jordan_positive_power_product_terminal * S ((S (S h)) * pa_v_pvs_jordan_positive_power_product) + (n))) /\\ forall pa_i_pvs_jordan_positive_power_product. (exists pa_lt_pvs_jordan_positive_power_product_bound. pa_lt_pvs_jordan_positive_power_product_bound + S pa_i_pvs_jordan_positive_power_product = S h) -> exists pa_p_pvs_jordan_positive_power_product pa_r_pvs_jordan_positive_power_product pa_s_pvs_jordan_positive_power_product. ((((exists pa_h_pvs_jordan_positive_power_product_factor. pa_h_pvs_jordan_positive_power_product_factor + S (pa_p_pvs_jordan_positive_power_product) = S ((S (pa_i_pvs_jordan_positive_power_product)) * pa_c_pvs_jordan_positive_power)) /\\ exists pa_q_pvs_jordan_positive_power_product_factor. pa_b_pvs_jordan_positive_power = pa_q_pvs_jordan_positive_power_product_factor * S ((S (pa_i_pvs_jordan_positive_power_product)) * pa_c_pvs_jordan_positive_power) + (pa_p_pvs_jordan_positive_power_product))) /\\ ((((exists pa_h_pvs_jordan_positive_power_product_partial. pa_h_pvs_jordan_positive_power_product_partial + S (pa_r_pvs_jordan_positive_power_product) = S ((S (pa_i_pvs_jordan_positive_power_product)) * pa_v_pvs_jordan_positive_power_product)) /\\ exists pa_q_pvs_jordan_positive_power_product_partial. pa_u_pvs_jordan_positive_power_product = pa_q_pvs_jordan_positive_power_product_partial * S ((S (pa_i_pvs_jordan_positive_power_product)) * pa_v_pvs_jordan_positive_power_product) + (pa_r_pvs_jordan_positive_power_product))) /\\ ((((exists pa_h_pvs_jordan_positive_power_product_successor. pa_h_pvs_jordan_positive_power_product_successor + S (pa_s_pvs_jordan_positive_power_product) = S ((S (S pa_i_pvs_jordan_positive_power_product)) * pa_v_pvs_jordan_positive_power_product)) /\\ exists pa_q_pvs_jordan_positive_power_product_successor. pa_u_pvs_jordan_positive_power_product = pa_q_pvs_jordan_positive_power_product_successor * S ((S (S pa_i_pvs_jordan_positive_power_product)) * pa_v_pvs_jordan_positive_power_product) + (pa_s_pvs_jordan_positive_power_product))) /\\ pa_s_pvs_jordan_positive_power_product = pa_r_pvs_jordan_positive_power_product * pa_p_pvs_jordan_positive_power_product)))))))) -> (((forall jt_divisor_power_primitive. (exists jt_factor_power_primitivemodulus. (n)=(jt_divisor_power_primitive)*jt_factor_power_primitivemodulus) -> (forall jt_index_power_primitivecoordinates jt_value_power_primitivecoordinates. (exists jt_gap_power_primitivecoordinatesindex. jt_gap_power_primitivecoordinatesindex+S (jt_index_power_primitivecoordinates)=(k)) -> (((exists fs_h_jt_power_primitivecoordinatesat. fs_h_jt_power_primitivecoordinatesat + S (jt_value_power_primitivecoordinates) = S ((S (jt_index_power_primitivecoordinates)) * c)) /\\ exists fs_q_jt_power_primitivecoordinatesat. b = fs_q_jt_power_primitivecoordinatesat * S ((S (jt_index_power_primitivecoordinates)) * c) + (jt_value_power_primitivecoordinates))) -> (exists jt_factor_power_primitivecoordinatesdivides. (jt_value_power_primitivecoordinates)=(jt_divisor_power_primitive)*jt_factor_power_primitivecoordinatesdivides)) -> jt_divisor_power_primitive=1) -> (~(forall jt_index_power_all_divisible jt_value_power_all_divisible. (exists jt_gap_power_all_divisibleindex. jt_gap_power_all_divisibleindex+S (jt_index_power_all_divisible)=(k)) -> (((exists fs_h_jt_power_all_divisibleat. fs_h_jt_power_all_divisibleat + S (jt_value_power_all_divisible) = S ((S (jt_index_power_all_divisible)) * c)) /\\ exists fs_q_jt_power_all_divisibleat. b = fs_q_jt_power_all_divisibleat * S ((S (jt_index_power_all_divisible)) * c) + (jt_value_power_all_divisible))) -> (exists jt_factor_power_all_divisibledivides. (jt_value_power_all_divisible)=(p)*jt_factor_power_all_divisibledivides)))) /\\ ((~(forall jt_index_power_all_divisible jt_value_power_all_divisible. (exists jt_gap_power_all_divisibleindex. jt_gap_power_all_divisibleindex+S (jt_index_power_all_divisible)=(k)) -> (((exists fs_h_jt_power_all_divisibleat. fs_h_jt_power_all_divisibleat + S (jt_value_power_all_divisible) = S ((S (jt_index_power_all_divisible)) * c)) /\\ exists fs_q_jt_power_all_divisibleat. b = fs_q_jt_power_all_divisibleat * S ((S (jt_index_power_all_divisible)) * c) + (jt_value_power_all_divisible))) -> (exists jt_factor_power_all_divisibledivides. (jt_value_power_all_divisible)=(p)*jt_factor_power_all_divisibledivides))) -> (forall jt_divisor_power_primitive. (exists jt_factor_power_primitivemodulus. (n)=(jt_divisor_power_primitive)*jt_factor_power_primitivemodulus) -> (forall jt_index_power_primitivecoordinates jt_value_power_primitivecoordinates. (exists jt_gap_power_primitivecoordinatesindex. jt_gap_power_primitivecoordinatesindex+S (jt_index_power_primitivecoordinates)=(k)) -> (((exists fs_h_jt_power_primitivecoordinatesat. fs_h_jt_power_primitivecoordinatesat + S (jt_value_power_primitivecoordinates) = S ((S (jt_index_power_primitivecoordinates)) * c)) /\\ exists fs_q_jt_power_primitivecoordinatesat. b = fs_q_jt_power_primitivecoordinatesat * S ((S (jt_index_power_primitivecoordinates)) * c) + (jt_value_power_primitivecoordinates))) -> (exists jt_factor_power_primitivecoordinatesdivides. (jt_value_power_primitivecoordinates)=(jt_divisor_power_primitive)*jt_factor_power_primitivecoordinatesdivides)) -> jt_divisor_power_primitive=1)))", "statement_sha256": "84696e7e875bc4b25237a038ac8f985066642f8e2e9313d672136e66f19d617b", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Over every positive power of a prime, a tuple is primitive exactly when p does not divide all its coordinates." }, { "admission_dependencies": [ "jordan_prime_power_tuple_primitive_of_not_all_divisible", "jordan_primitive_tuple_avoids_prime_common_divisor", "pow_positive_exponent_base_divides", "succ_ne_zero" ], "admitted_to_alpha": true, "admitted_to_stable": false, "alpha_checked_use": true, "alpha_edition_version": "v35", "alpha_evidence": "alpha_closed", "alpha_first_enrolled_version": "v35", "body_proof_depth": 56, "body_proof_nodes": 99, "campaign_milestone": "G008", "checked_use": true, "defined": { "defined_script": [ "intro p", "intro h", "intro j", "intro m", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hm", "intro hn", "intro hprimitive", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (p)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (S j)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (n)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (b)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (c)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (k)", "apply jordan_prime_power_tuple_primitive_of_not_all_divisible", "exact hp", "exact hn", "intro hall", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (p)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (m)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (b)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (c)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (k)", "apply jordan_primitive_tuple_avoids_prime_common_divisor", "exact hp", "specialize pow_positive_exponent_base_divides (p)", "specialize pow_positive_exponent_base_divides (S h)", "specialize pow_positive_exponent_base_divides (m)", "apply pow_positive_exponent_base_divides", "intro hszero", "specialize succ_ne_zero (h)", "apply succ_ne_zero", "exact hszero", "exact hm", "exact hprimitive", "exact hall" ], "defined_statement": "∀ p. ∀ h. ∀ j. ∀ m. ∀ n. ∀ b. ∀ c. ∀ k. Prime(p) → Pow(p,S h,m) → Pow(p,S j,n) → JordanPrimitiveTuple(m,b,c,k) → JordanPrimitiveTuple(n,b,c,k)", "defined_statement_sha256": "2a476f30db292c35df899bc3765995460f8580c3d11a8760bd1c6913a6718ed7", "definition_uses": { "ND0372": 2, "PD0004": 1, "PD0020": 2 }, "exact_ast_equivalence": true, "expanded_statement_sha256": "612a3716b3aed0ebb86fd8b210b974a2a6d99739f166b23db7cbd8c30a68f8a7", "free_names": [], "script_definition_uses": {}, "script_parts": [ [ { "kind": "text", "text": "intro p" } ], [ { "kind": "text", "text": "intro h" } ], [ { "kind": "text", "text": "intro j" } ], [ { "kind": "text", "text": "intro m" } ], [ { "kind": "text", "text": "intro n" } ], [ { "kind": "text", "text": "intro b" } ], [ { "kind": "text", "text": "intro c" } ], [ { "kind": "text", "text": "intro k" } ], [ { "kind": "text", "text": "intro hp" } ], [ { "kind": "text", "text": "intro hm" } ], [ { "kind": "text", "text": "intro hn" } ], [ { "kind": "text", "text": "intro hprimitive" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (p)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (S j)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (n)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (b)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (c)" } ], [ { "kind": "text", "text": "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (k)" } ], [ { "kind": "text", "text": "apply jordan_prime_power_tuple_primitive_of_not_all_divisible" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "exact hn" } ], [ { "kind": "text", "text": "intro hall" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (p)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (m)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (b)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (c)" } ], [ { "kind": "text", "text": "specialize jordan_primitive_tuple_avoids_prime_common_divisor (k)" } ], [ { "kind": "text", "text": "apply jordan_primitive_tuple_avoids_prime_common_divisor" } ], [ { "kind": "text", "text": "exact hp" } ], [ { "kind": "text", "text": "specialize pow_positive_exponent_base_divides (p)" } ], [ { "kind": "text", "text": "specialize pow_positive_exponent_base_divides (S h)" } ], [ { "kind": "text", "text": "specialize pow_positive_exponent_base_divides (m)" } ], [ { "kind": "text", "text": "apply pow_positive_exponent_base_divides" } ], [ { "kind": "text", "text": "intro hszero" } ], [ { "kind": "text", "text": "specialize succ_ne_zero (h)" } ], [ { "kind": "text", "text": "apply succ_ne_zero" } ], [ { "kind": "text", "text": "exact hszero" } ], [ { "kind": "text", "text": "exact hm" } ], [ { "kind": "text", "text": "exact hprimitive" } ], [ { "kind": "text", "text": "exact hall" } ] ], "statement_definition_uses": { "ND0372": 2, "PD0004": 1, "PD0020": 2 }, "statement_parts": [ { "kind": "text", "text": "∀ p. ∀ h. ∀ j. ∀ m. ∀ n. ∀ b. ∀ c. ∀ k. " }, { "definition": "PD0004", "kind": "definition", "text": "Prime(p)" }, { "kind": "text", "text": " → " }, { "definition": "PD0020", "kind": "definition", "text": "Pow(p,S h,m)" }, { "kind": "text", "text": " → " }, { "definition": "PD0020", "kind": "definition", "text": "Pow(p,S j,n)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(m,b,c,k)" }, { "kind": "text", "text": " → " }, { "definition": "ND0372", "kind": "definition", "text": "JordanPrimitiveTuple(n,b,c,k)" } ] }, "dependencies": [ "jordan_prime_power_tuple_primitive_of_not_all_divisible", "jordan_primitive_tuple_avoids_prime_common_divisor", "pow_positive_exponent_base_divides", "succ_ne_zero" ], "enrolled_in_alpha": true, "factory": "make_jordan_prime_power_characterization_candidate_theorems", "first_admitted_version": "v35", "id": "JT0060", "independent_lean_bundle_verified": true, "inventory_role": "first_admitted_alpha_v35", "name": "jordan_prime_power_tuple_primitivity_invariant", "original_ha_bundle_verified": true, "proof_bundle_node_id": 357, "proof_bundle_sha256": "9164d35758d1fa15d18ec792a429cbb33fd4c511df5651b9f15d37bececf5ea7", "script": [ "intro p", "intro h", "intro j", "intro m", "intro n", "intro b", "intro c", "intro k", "intro hp", "intro hm", "intro hn", "intro hprimitive", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (p)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (S j)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (n)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (b)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (c)", "specialize jordan_prime_power_tuple_primitive_of_not_all_divisible (k)", "apply jordan_prime_power_tuple_primitive_of_not_all_divisible", "exact hp", "exact hn", "intro hall", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (p)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (m)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (b)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (c)", "specialize jordan_primitive_tuple_avoids_prime_common_divisor (k)", "apply jordan_primitive_tuple_avoids_prime_common_divisor", "exact hp", "specialize pow_positive_exponent_base_divides (p)", "specialize pow_positive_exponent_base_divides (S h)", "specialize pow_positive_exponent_base_divides (m)", "apply pow_positive_exponent_base_divides", "intro hszero", "specialize succ_ne_zero (h)", "apply succ_ne_zero", "exact hszero", "exact hm", "exact hprimitive", "exact hall" ], "script_sha256": "3c7ffc81617b373d3da14b9e5b0ffff9c1afb91fbf9ff99e051ce12722b047c4", "source_filename": "jordan_prime_power_characterization_candidate.py", "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "sources": [ { "factory": "make_jordan_prime_power_characterization_candidate_theorems", "script_sha256": "3c7ffc81617b373d3da14b9e5b0ffff9c1afb91fbf9ff99e051ce12722b047c4", "selected": true, "source_module": "peano_lab.library.jordan_prime_power_characterization_candidate", "source_sha256": "6f7d65e91dfe8c818b2bf5a3d0e8b1754f75fe631cc943543c5081820a299f11", "statement_sha256": "612a3716b3aed0ebb86fd8b210b974a2a6d99739f166b23db7cbd8c30a68f8a7" } ], "stable_member": false, "statement": "forall p h j m n b c k. (~((p) = 1) /\\ forall pvs_left_jordan_base pvs_right_jordan_base. (p) = pvs_left_jordan_base * pvs_right_jordan_base -> pvs_left_jordan_base = 1 \\/ pvs_right_jordan_base = 1) -> (exists pa_b_pvs_jordan_first_power pa_c_pvs_jordan_first_power. ((forall pa_i_pvs_jordan_first_power_repeat. (exists pa_lt_pvs_jordan_first_power_repeat_bound. pa_lt_pvs_jordan_first_power_repeat_bound + S pa_i_pvs_jordan_first_power_repeat = S h) -> (((exists pa_h_pvs_jordan_first_power_repeat_decoded. pa_h_pvs_jordan_first_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_jordan_first_power_repeat)) * pa_c_pvs_jordan_first_power)) /\\ exists pa_q_pvs_jordan_first_power_repeat_decoded. pa_b_pvs_jordan_first_power = pa_q_pvs_jordan_first_power_repeat_decoded * S ((S (pa_i_pvs_jordan_first_power_repeat)) * pa_c_pvs_jordan_first_power) + (p)))) /\\ (exists pa_u_pvs_jordan_first_power_product pa_v_pvs_jordan_first_power_product. ((((exists pa_h_pvs_jordan_first_power_product_start. pa_h_pvs_jordan_first_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_jordan_first_power_product)) /\\ exists pa_q_pvs_jordan_first_power_product_start. pa_u_pvs_jordan_first_power_product = pa_q_pvs_jordan_first_power_product_start * S ((S (0)) * pa_v_pvs_jordan_first_power_product) + (1))) /\\ ((((exists pa_h_pvs_jordan_first_power_product_terminal. pa_h_pvs_jordan_first_power_product_terminal + S (m) = S ((S (S h)) * pa_v_pvs_jordan_first_power_product)) /\\ exists pa_q_pvs_jordan_first_power_product_terminal. pa_u_pvs_jordan_first_power_product = pa_q_pvs_jordan_first_power_product_terminal * S ((S (S h)) * pa_v_pvs_jordan_first_power_product) + (m))) /\\ forall pa_i_pvs_jordan_first_power_product. (exists pa_lt_pvs_jordan_first_power_product_bound. pa_lt_pvs_jordan_first_power_product_bound + S pa_i_pvs_jordan_first_power_product = S h) -> exists pa_p_pvs_jordan_first_power_product pa_r_pvs_jordan_first_power_product pa_s_pvs_jordan_first_power_product. ((((exists pa_h_pvs_jordan_first_power_product_factor. pa_h_pvs_jordan_first_power_product_factor + S (pa_p_pvs_jordan_first_power_product) = S ((S (pa_i_pvs_jordan_first_power_product)) * pa_c_pvs_jordan_first_power)) /\\ exists pa_q_pvs_jordan_first_power_product_factor. pa_b_pvs_jordan_first_power = pa_q_pvs_jordan_first_power_product_factor * S ((S (pa_i_pvs_jordan_first_power_product)) * pa_c_pvs_jordan_first_power) + (pa_p_pvs_jordan_first_power_product))) /\\ ((((exists pa_h_pvs_jordan_first_power_product_partial. pa_h_pvs_jordan_first_power_product_partial + S (pa_r_pvs_jordan_first_power_product) = S ((S (pa_i_pvs_jordan_first_power_product)) * pa_v_pvs_jordan_first_power_product)) /\\ exists pa_q_pvs_jordan_first_power_product_partial. pa_u_pvs_jordan_first_power_product = pa_q_pvs_jordan_first_power_product_partial * S ((S (pa_i_pvs_jordan_first_power_product)) * pa_v_pvs_jordan_first_power_product) + (pa_r_pvs_jordan_first_power_product))) /\\ ((((exists pa_h_pvs_jordan_first_power_product_successor. pa_h_pvs_jordan_first_power_product_successor + S (pa_s_pvs_jordan_first_power_product) = S ((S (S pa_i_pvs_jordan_first_power_product)) * pa_v_pvs_jordan_first_power_product)) /\\ exists pa_q_pvs_jordan_first_power_product_successor. pa_u_pvs_jordan_first_power_product = pa_q_pvs_jordan_first_power_product_successor * S ((S (S pa_i_pvs_jordan_first_power_product)) * pa_v_pvs_jordan_first_power_product) + (pa_s_pvs_jordan_first_power_product))) /\\ pa_s_pvs_jordan_first_power_product = pa_r_pvs_jordan_first_power_product * pa_p_pvs_jordan_first_power_product)))))))) -> (exists pa_b_pvs_jordan_second_power pa_c_pvs_jordan_second_power. ((forall pa_i_pvs_jordan_second_power_repeat. (exists pa_lt_pvs_jordan_second_power_repeat_bound. pa_lt_pvs_jordan_second_power_repeat_bound + S pa_i_pvs_jordan_second_power_repeat = S j) -> (((exists pa_h_pvs_jordan_second_power_repeat_decoded. pa_h_pvs_jordan_second_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_jordan_second_power_repeat)) * pa_c_pvs_jordan_second_power)) /\\ exists pa_q_pvs_jordan_second_power_repeat_decoded. pa_b_pvs_jordan_second_power = pa_q_pvs_jordan_second_power_repeat_decoded * S ((S (pa_i_pvs_jordan_second_power_repeat)) * pa_c_pvs_jordan_second_power) + (p)))) /\\ (exists pa_u_pvs_jordan_second_power_product pa_v_pvs_jordan_second_power_product. ((((exists pa_h_pvs_jordan_second_power_product_start. pa_h_pvs_jordan_second_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_jordan_second_power_product)) /\\ exists pa_q_pvs_jordan_second_power_product_start. pa_u_pvs_jordan_second_power_product = pa_q_pvs_jordan_second_power_product_start * S ((S (0)) * pa_v_pvs_jordan_second_power_product) + (1))) /\\ ((((exists pa_h_pvs_jordan_second_power_product_terminal. pa_h_pvs_jordan_second_power_product_terminal + S (n) = S ((S (S j)) * pa_v_pvs_jordan_second_power_product)) /\\ exists pa_q_pvs_jordan_second_power_product_terminal. pa_u_pvs_jordan_second_power_product = pa_q_pvs_jordan_second_power_product_terminal * S ((S (S j)) * pa_v_pvs_jordan_second_power_product) + (n))) /\\ forall pa_i_pvs_jordan_second_power_product. (exists pa_lt_pvs_jordan_second_power_product_bound. pa_lt_pvs_jordan_second_power_product_bound + S pa_i_pvs_jordan_second_power_product = S j) -> exists pa_p_pvs_jordan_second_power_product pa_r_pvs_jordan_second_power_product pa_s_pvs_jordan_second_power_product. ((((exists pa_h_pvs_jordan_second_power_product_factor. pa_h_pvs_jordan_second_power_product_factor + S (pa_p_pvs_jordan_second_power_product) = S ((S (pa_i_pvs_jordan_second_power_product)) * pa_c_pvs_jordan_second_power)) /\\ exists pa_q_pvs_jordan_second_power_product_factor. pa_b_pvs_jordan_second_power = pa_q_pvs_jordan_second_power_product_factor * S ((S (pa_i_pvs_jordan_second_power_product)) * pa_c_pvs_jordan_second_power) + (pa_p_pvs_jordan_second_power_product))) /\\ ((((exists pa_h_pvs_jordan_second_power_product_partial. pa_h_pvs_jordan_second_power_product_partial + S (pa_r_pvs_jordan_second_power_product) = S ((S (pa_i_pvs_jordan_second_power_product)) * pa_v_pvs_jordan_second_power_product)) /\\ exists pa_q_pvs_jordan_second_power_product_partial. pa_u_pvs_jordan_second_power_product = pa_q_pvs_jordan_second_power_product_partial * S ((S (pa_i_pvs_jordan_second_power_product)) * pa_v_pvs_jordan_second_power_product) + (pa_r_pvs_jordan_second_power_product))) /\\ ((((exists pa_h_pvs_jordan_second_power_product_successor. pa_h_pvs_jordan_second_power_product_successor + S (pa_s_pvs_jordan_second_power_product) = S ((S (S pa_i_pvs_jordan_second_power_product)) * pa_v_pvs_jordan_second_power_product)) /\\ exists pa_q_pvs_jordan_second_power_product_successor. pa_u_pvs_jordan_second_power_product = pa_q_pvs_jordan_second_power_product_successor * S ((S (S pa_i_pvs_jordan_second_power_product)) * pa_v_pvs_jordan_second_power_product) + (pa_s_pvs_jordan_second_power_product))) /\\ pa_s_pvs_jordan_second_power_product = pa_r_pvs_jordan_second_power_product * pa_p_pvs_jordan_second_power_product)))))))) -> (forall jt_divisor_jordan_first_primitive. (exists jt_factor_jordan_first_primitivemodulus. (m)=(jt_divisor_jordan_first_primitive)*jt_factor_jordan_first_primitivemodulus) -> (forall jt_index_jordan_first_primitivecoordinates jt_value_jordan_first_primitivecoordinates. (exists jt_gap_jordan_first_primitivecoordinatesindex. jt_gap_jordan_first_primitivecoordinatesindex+S (jt_index_jordan_first_primitivecoordinates)=(k)) -> (((exists fs_h_jt_jordan_first_primitivecoordinatesat. fs_h_jt_jordan_first_primitivecoordinatesat + S (jt_value_jordan_first_primitivecoordinates) = S ((S (jt_index_jordan_first_primitivecoordinates)) * c)) /\\ exists fs_q_jt_jordan_first_primitivecoordinatesat. b = fs_q_jt_jordan_first_primitivecoordinatesat * S ((S (jt_index_jordan_first_primitivecoordinates)) * c) + (jt_value_jordan_first_primitivecoordinates))) -> (exists jt_factor_jordan_first_primitivecoordinatesdivides. (jt_value_jordan_first_primitivecoordinates)=(jt_divisor_jordan_first_primitive)*jt_factor_jordan_first_primitivecoordinatesdivides)) -> jt_divisor_jordan_first_primitive=1) -> forall jt_divisor_power_primitive. (exists jt_factor_power_primitivemodulus. (n)=(jt_divisor_power_primitive)*jt_factor_power_primitivemodulus) -> (forall jt_index_power_primitivecoordinates jt_value_power_primitivecoordinates. (exists jt_gap_power_primitivecoordinatesindex. jt_gap_power_primitivecoordinatesindex+S (jt_index_power_primitivecoordinates)=(k)) -> (((exists fs_h_jt_power_primitivecoordinatesat. fs_h_jt_power_primitivecoordinatesat + S (jt_value_power_primitivecoordinates) = S ((S (jt_index_power_primitivecoordinates)) * c)) /\\ exists fs_q_jt_power_primitivecoordinatesat. b = fs_q_jt_power_primitivecoordinatesat * S ((S (jt_index_power_primitivecoordinates)) * c) + (jt_value_power_primitivecoordinates))) -> (exists jt_factor_power_primitivecoordinatesdivides. (jt_value_power_primitivecoordinates)=(jt_divisor_power_primitive)*jt_factor_power_primitivecoordinatesdivides)) -> jt_divisor_power_primitive=1", "statement_sha256": "612a3716b3aed0ebb86fd8b210b974a2a6d99739f166b23db7cbd8c30a68f8a7", "status": "Alpha v35 checked-use · first admitted v35 · independently kernel and Lean verified; not Stable", "summary": "Primitivity of an actual beta tuple is invariant under changing the positive exponent of its prime-power modulus." } ], "original_ha_bundle_verified": true, "parent_alpha_checked_use_count": 4223, "parent_alpha_edition_version": "v34", "path_policy": "proof_dependency_edges_only", "proof_adjacency": { "jordan_canonical_crt_tuple_exists": { "critical_root_path": [ "JT002A", "JT002C", "JT0032" ], "dependencies": [ "jordan_crt_tuple_exists", "jordan_tuple_normalize_exists", "jordan_tuple_congruence_divisor", "jordan_tuple_congruence_trans", "jordan_tuple_congruence_symm", "jordan_crt_tuple_left", "jordan_crt_tuple_right" ], "dependents": [ 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"JT0007", "JT005E", "JT0060" ] }, "publication_scope": "alpha_checked_use_publication", "raw_inherited_support_count": 262, "release_source_binding_sha256": "d3cce2f96380c8bccb59a5cdbfdb20a24b6dff396eefd34609392d3424f8264b", "reserved_tag_slots": { "JT0001": { "alias": "jordan_tuple_equal_refl", "canonical_admission_name": "integer_vector_equal_components_zero", "counted_as_new": false } }, "root_names": [ "jordan_order_zero_excluded", "jordan_modulus_zero_excluded", "jordan_totient_multiplicativity_unique_counts", "jordan_totient_at_one_unique", "jordan_prime_power_tuple_primitive_characterization", "jordan_prime_power_tuple_primitivity_invariant", "jordan_totient_multiplicativity_exists" ], "schema": "peano-lab-alpha-v35-jordan-explorer-v1", "source_aliases": { "jordan_tuple_equal_refl": "integer_vector_equal_components_zero" }, "source_owned_theorem_count": 96, "stable_admitted_node_count": 0, "stable_edition_count": 432, "stable_member": false, "tags": { "jordan_canonical_crt_tuple_exists": "JT0032", "jordan_canonical_crt_tuple_unique": "JT003C", "jordan_crt_component_recovery": "JT003B", "jordan_crt_tuple_empty": "JT002A", "jordan_crt_tuple_exists": "JT002C", "jordan_crt_tuple_extend": "JT002B", "jordan_crt_tuple_left": "JT002D", "jordan_crt_tuple_right": "JT002E", "jordan_divisibility_congruence_transport": "JT000D", "jordan_enumeration_actual_value": "JT003D", "jordan_enumeration_cardinality_le": "JT0053", "jordan_enumeration_cardinality_unique": "JT0054", "jordan_enumeration_complete": "JT003E", "jordan_enumeration_distinct": "JT003F", "jordan_enumeration_index_map_append": "JT004F", "jordan_enumeration_index_map_bounded_injective": "JT0052", "jordan_enumeration_index_map_empty": "JT004E", "jordan_enumeration_index_map_entry": "JT0051", "jordan_enumeration_index_map_exists": "JT0050", "jordan_enumeration_position_match_exists": "JT004D", "jordan_enumeration_position_match_from_entries": "JT004C", "jordan_enumeration_reduce_primitive": "JT0045", "jordan_modulus_zero_excluded": "JT000A", "jordan_order_zero_excluded": "JT0009", "jordan_prime_power_tuple_primitive_characterization": "JT005F", "jordan_prime_power_tuple_primitive_of_not_all_divisible": "JT005E", "jordan_prime_power_tuple_primitivity_invariant": "JT0060", "jordan_primitive_crt_tuple_exists": "JT0033", "jordan_primitive_tuple_avoids_prime_common_divisor": "JT005D", "jordan_primitive_tuple_congruence_transport": "JT000F", "jordan_primitive_tuple_coprime_product": "JT000B", "jordan_primitive_tuple_decidable": "JT0015", "jordan_primitive_tuple_divisor_modulus": "JT0006", "jordan_primitive_tuple_modulus_one": "JT0008", "jordan_primitive_tuple_product_components": "JT000C", "jordan_primitive_tuple_transport": "JT0005", "jordan_product_enumeration_exists": "JT0049", "jordan_rectangle_crt_actual_entry": "JT0043", "jordan_rectangle_crt_append": "JT0040", "jordan_rectangle_crt_covers": "JT0047", "jordan_rectangle_crt_distinct": "JT0046", "jordan_rectangle_crt_enumeration": "JT0048", "jordan_rectangle_crt_exists": "JT0042", "jordan_rectangle_crt_pair_value": "JT0044", "jordan_rectangle_crt_successor": "JT0041", "jordan_rectangle_flat_bound": "JT0036", "jordan_rectangle_pair_unique": "JT0037", "jordan_rectangle_quotient_bound": "JT0035", "jordan_rectangle_width_nonzero": "JT0034", "jordan_totient_at_one": "JT005B", "jordan_totient_at_one_unique": "JT005C", "jordan_totient_coprime_product": "JT004A", "jordan_totient_count_unique": "JT0055", "jordan_totient_exists": "JT0029", "jordan_totient_from_complete_scan": "JT0025", "jordan_totient_multiplicativity_exists": "JT004B", "jordan_totient_multiplicativity_unique_counts": "JT0056", "jordan_tuple_all_divisible_decidable": "JT0012", "jordan_tuple_all_divisible_empty": "JT0010", "jordan_tuple_all_divisible_extend": "JT0011", "jordan_tuple_bounded_congruence_equal": "JT0039", "jordan_tuple_bounded_one_entry_zero": "JT0057", "jordan_tuple_bounded_transport": "JT0020", "jordan_tuple_common_divisor_transport": "JT0004", "jordan_tuple_congruence_coprime_product": "JT003A", "jordan_tuple_congruence_divisor": "JT0031", "jordan_tuple_congruence_symm": "JT000E", "jordan_tuple_congruence_trans": "JT0030", "jordan_tuple_divisor_downward": "JT0007", "jordan_tuple_divisor_test_decidable": "JT0013", "jordan_tuple_equal_congruence": "JT0038", "jordan_tuple_equal_decidable": "JT0019", "jordan_tuple_equal_drop_last": "JT0017", "jordan_tuple_equal_empty": "JT0016", "jordan_tuple_equal_entry": "JT001F", "jordan_tuple_equal_extend": "JT0018", "jordan_tuple_equal_symm": "JT0002", "jordan_tuple_equal_trans": "JT0003", "jordan_tuple_listed_decidable": "JT001C", "jordan_tuple_listed_empty": "JT001A", "jordan_tuple_listed_equal_transport": "JT0022", "jordan_tuple_listed_lift": "JT001B", "jordan_tuple_normalize_exists": "JT002F", "jordan_tuple_outer_append_exists": "JT0021", "jordan_tuple_prefix_equal": "JT001E", "jordan_tuple_primitive_bounded_decidable": "JT0014", "jordan_tuple_representatives_exists": "JT0028", "jordan_tuple_scan_append": "JT0026", "jordan_tuple_scan_complete": "JT0024", "jordan_tuple_scan_empty": "JT001D", "jordan_tuple_scan_exists": "JT0027", "jordan_tuple_scan_skip": "JT0023", "jordan_tuples_bounded_one_equal": "JT0059", "jordan_unit_modulus_singleton_enumeration": "JT005A", "jordan_zero_tuple_bounded_one": "JT0058" } }