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(a * x) + m * hgcrt_mod_left_lcc_class_pair = (a * r) + m * hgcrt_mod_right_lcc_class_pair) -> (exists hgcrt_mod_left_lcc_class_reduced hgcrt_mod_right_lcc_class_reduced. x + M * hgcrt_mod_left_lcc_class_reduced = r + M * hgcrt_mod_right_lcc_class_reduced)) /\\ (((exists hgcrt_mod_left_lcc_class_reduced hgcrt_mod_right_lcc_class_reduced. x + M * hgcrt_mod_left_lcc_class_reduced = r + M * hgcrt_mod_right_lcc_class_reduced) -> (exists hgcrt_mod_left_lcc_class_pair hgcrt_mod_right_lcc_class_pair. (a * x) + m * hgcrt_mod_left_lcc_class_pair = (a * r) + m * hgcrt_mod_right_lcc_class_pair))))", "specialize mod_eq_cancel_gcd_cofactor (a)", "specialize mod_eq_cancel_gcd_cofactor (m)", "specialize mod_eq_cancel_gcd_cofactor (g)", "specialize mod_eq_cancel_gcd_cofactor (A)", "specialize mod_eq_cancel_gcd_cofactor (M)", "specialize mod_eq_cancel_gcd_cofactor (x)", "specialize mod_eq_cancel_gcd_cofactor (r)", "apply mod_eq_cancel_gcd_cofactor", "exact hm0", "exact hg", "exact ha", "exact hm", "cases hc", "split", "intro hx", "apply hc_left", "specialize mod_eq_trans (m)", "specialize mod_eq_trans (a*x)", "specialize mod_eq_trans (b)", "specialize mod_eq_trans (a*r)", "apply mod_eq_trans", "exact hx", "specialize mod_eq_symm (m)", "specialize mod_eq_symm (a*r)", "specialize mod_eq_symm (b)", "apply mod_eq_symm", "exact hr", "intro hx", "specialize mod_eq_trans (m)", "specialize mod_eq_trans (a*x)", "specialize mod_eq_trans (a*r)", "specialize mod_eq_trans (b)", "apply mod_eq_trans", "apply hc_right", "exact hx", "exact hr" ], "script_sha256": "c29c65bd96690c6f7807d5236d7937d0fb36dfe14f230130c49dcf92129b7168", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall a m g A M b r x. (~(m=0)) -> ((((exists hag_left_factor_lcc_class. a = g * hag_left_factor_lcc_class) /\\ (exists hag_right_factor_lcc_class. m = g * hag_right_factor_lcc_class)) /\\ forall hag_divisor_lcc_class. (exists hag_common_left_lcc_class. a = hag_divisor_lcc_class * hag_common_left_lcc_class) -> (exists hag_common_right_lcc_class. m = hag_divisor_lcc_class * hag_common_right_lcc_class) -> exists hag_greatest_factor_lcc_class. g = hag_divisor_lcc_class * hag_greatest_factor_lcc_class)) -> (a=g*A) -> (m=g*M) -> (exists hgcrt_mod_left_lcc_class_reference hgcrt_mod_right_lcc_class_reference. (a * r) + m * hgcrt_mod_left_lcc_class_reference = b + m * hgcrt_mod_right_lcc_class_reference) -> ((((exists hgcrt_mod_left_lcc_class_solution hgcrt_mod_right_lcc_class_solution. 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hm", "rewrite hz", "apply mul_zero_left", "have hM0 : ~(M=0)", "intro hz", "apply hm0", "trans g*M", "exact hm", "rewrite hz", "simp", "have hs : exists x. exists hgcrt_mod_left_lcc_representative_original hgcrt_mod_right_lcc_representative_original. (a * x) + m * hgcrt_mod_left_lcc_representative_original = b + m * hgcrt_mod_right_lcc_representative_original", "specialize linear_congruence_gcd_divisibility_constructs_solution (a)", "specialize linear_congruence_gcd_divisibility_constructs_solution (m)", "specialize linear_congruence_gcd_divisibility_constructs_solution (b)", "specialize linear_congruence_gcd_divisibility_constructs_solution (g)", "apply linear_congruence_gcd_divisibility_constructs_solution", "exact hg", "exact hb", "cases hs", "have hd : exists q r. ((x=M*q+r) /\\ ((exists lcc_gap_representative_division. lcc_gap_representative_division+S (r)=(M))))", "specialize division_remainder_exists (M)", "specialize division_remainder_exists (x)", "apply division_remainder_exists", "exact hM0", "cases hd", "cases hd_witness", "cases hd_witness_witness", "exists x2", "split", "exact hd_witness_witness_right", "have he : exists hgcrt_mod_left_lcc_representative_equal hgcrt_mod_right_lcc_representative_equal. x2 + M * hgcrt_mod_left_lcc_representative_equal = x + M * hgcrt_mod_right_lcc_representative_equal", "specialize mod_eq_symm (M)", "specialize mod_eq_symm (x)", "specialize mod_eq_symm (x2)", "apply mod_eq_symm", "specialize remainder_decomposition_to_mod_eq (M)", "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", "trans M*x1+x2", "exact hd_witness_witness_left", "congr", "apply mul_comm", "refl", "have hc : (((exists hgcrt_mod_left_lcc_representative_target hgcrt_mod_right_lcc_representative_target. (a * x2) + m * hgcrt_mod_left_lcc_representative_target = b + m * hgcrt_mod_right_lcc_representative_target) -> (exists hgcrt_mod_left_lcc_representative_class hgcrt_mod_right_lcc_representative_class. x2 + M * hgcrt_mod_left_lcc_representative_class = x + M * hgcrt_mod_right_lcc_representative_class)) /\\ (((exists hgcrt_mod_left_lcc_representative_class hgcrt_mod_right_lcc_representative_class. x2 + M * hgcrt_mod_left_lcc_representative_class = x + M * hgcrt_mod_right_lcc_representative_class) -> (exists hgcrt_mod_left_lcc_representative_target hgcrt_mod_right_lcc_representative_target. (a * x2) + m * hgcrt_mod_left_lcc_representative_target = b + m * hgcrt_mod_right_lcc_representative_target))))", "specialize linear_congruence_solution_class_iff_reduced_modulus (a)", "specialize linear_congruence_solution_class_iff_reduced_modulus (m)", "specialize linear_congruence_solution_class_iff_reduced_modulus (g)", "specialize linear_congruence_solution_class_iff_reduced_modulus (A)", "specialize linear_congruence_solution_class_iff_reduced_modulus (M)", "specialize linear_congruence_solution_class_iff_reduced_modulus (b)", "specialize linear_congruence_solution_class_iff_reduced_modulus (x)", "specialize linear_congruence_solution_class_iff_reduced_modulus (x2)", "apply linear_congruence_solution_class_iff_reduced_modulus", "exact hm0", "exact hg", "exact ha", "exact hm", "exact hs_witness", "cases hc", "apply hc_right", "exact he" ], "script_sha256": "cc4f5cd23ff9b62e31c0c8b5cd7878127e49fe01c724b091f11871de07d46e18", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall a m g A M b. (~(m=0)) -> ((((exists hag_left_factor_lcc_representative. a = g * hag_left_factor_lcc_representative) /\\ (exists hag_right_factor_lcc_representative. m = g * hag_right_factor_lcc_representative)) /\\ forall hag_divisor_lcc_representative. (exists hag_common_left_lcc_representative. a = hag_divisor_lcc_representative * hag_common_left_lcc_representative) -> (exists hag_common_right_lcc_representative. m = hag_divisor_lcc_representative * hag_common_right_lcc_representative) -> exists hag_greatest_factor_lcc_representative. g = hag_divisor_lcc_representative * hag_greatest_factor_lcc_representative)) -> (a=g*A) -> (m=g*M) -> (exists lcc_bfactor. b=g*lcc_bfactor) -> (exists r. ((exists lcc_gap_representative_bound. lcc_gap_representative_bound+S (r)=(M)) /\\ ((exists hgcrt_mod_left_lcc_representative_solution hgcrt_mod_right_lcc_representative_solution. (a * r) + m * hgcrt_mod_left_lcc_representative_solution = b + m * hgcrt_mod_right_lcc_representative_solution))))", "statement_sha256": "0e81f0f848767b2d7737ccc71efa78143e8519eade906bcaf434b346cd476a9c", "summary": "Construct a genuine solution strictly below m/g, not merely below m, from the actual gcd divisibility witness.", "summary_sha256": "11cd9dfc1329a46549f72aff33a815dcb8fdbe3a5730d3cbf764f1c7dcb52e4f" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "891be115ade6ccb27f2a7a2fc198758e59c6bda66374bb82a775c20da7bf80fa", "bundle_campaign": "congruence-arithmetic", "bundle_node_id": 205, "bundle_sha256": "983051afddc637a4e033546b8f3ddb8dc0ac22aa996b4e28b3822be8895576ad", "campaign": "congruence-arithmetic", "first_enrolled_version": "v34", "parent_catalog_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "rfc_sha256": "f92ea2c941de4b9e99afb992463072f12a70fa1c6fd32cd11ba80ced6d7f8c4b", "source_sha256": 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"research/arithmetic-library/artifacts/alpha-v34-research-receipt-v1.json", "role": "fresh_release_proof_verification", "selector": "families[slug=congruence-arithmetic]" }, { "document_sha256": "6be052da195a295edce02f4b1955cd9e3dd71d7acefb9ac5794277eda7ef40cc", "kind": "sealed_alpha_v33_parent", "path": "artifacts/peano-library/alpha/catalog-v33.json", "role": "exact_immutable_parent_catalog_bytes", "selector": "catalog" } ], "evidence_status": "alpha_closed", "frontier_campaign": "congruence-arithmetic", "logical_spec_sha256": "acd6def3ecf81954070b4bdd5f018ac5f8a6d795d06fec03168591a9e38d05b2", "membership": "alpha_only", "name": "linear_congruence_progression_bound_iff", "proof_tag": null, "provenance": [ "ha" ], "script": [ "intro M", "intro g", "intro r", "intro t", "intro hr", "split", "intro hb", "have ho : (exists lcc_gap_progression_order. lcc_gap_progression_order+(g)=(t)) \\/ (exists lcc_gap_progression_index. lcc_gap_progression_index+S (t)=(g))", "specialize le_or_lt (g)", "specialize le_or_lt (t)", "apply le_or_lt", "cases ho", "exfalso", "specialize lt_not_le (r+M*t)", "specialize lt_not_le (g*M)", "apply lt_not_le", "exact hb", "specialize le_trans (g*M)", "specialize le_trans (t*M)", "specialize le_trans (r+M*t)", "apply le_trans", "specialize mul_le_mul_right (g)", "specialize mul_le_mul_right (t)", "specialize mul_le_mul_right (M)", "apply mul_le_mul_right", "exact ho_left", "have he : t*M=M*t", "apply mul_comm", "rewrite he", "specialize le_add_left (M*t)", "specialize le_add_left (r)", "apply le_add_left", "exact ho_right", "intro ht", "have hs : exists lcc_gap_progression_next. lcc_gap_progression_next+S (r+M*t)=(M+M*t)", "specialize finite_add_lt_of_lt_of_le (r)", "specialize finite_add_lt_of_lt_of_le (M)", "specialize finite_add_lt_of_lt_of_le (M*t)", "specialize finite_add_lt_of_lt_of_le (M*t)", "apply finite_add_lt_of_lt_of_le", "exact hr", "apply le_refl", "have he : M+M*t=S t*M", "trans M*t+M", "apply add_comm", "trans t*M+M", "congr", "apply mul_comm", "refl", "symm", "apply mul_succ_left", "specialize lt_of_lt_of_le (r+M*t)", "specialize lt_of_lt_of_le (M+M*t)", "specialize lt_of_lt_of_le (g*M)", "apply lt_of_lt_of_le", "exact hs", "rewrite he", "specialize mul_le_mul_right (S t)", "specialize mul_le_mul_right (g)", "specialize mul_le_mul_right (M)", "apply mul_le_mul_right", "exact ht" ], "script_sha256": "7a57f2f1ed63e3edcd26aae0924c12438b7d74ae3fbffe602dda67dd2c7b7898", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall M g r t. (exists lcc_gap_progression_remainder. lcc_gap_progression_remainder+S (r)=(M)) -> ((((exists lcc_gap_progression_small. lcc_gap_progression_small+S (r+M*t)=(g*M)) -> (exists lcc_gap_progression_index. lcc_gap_progression_index+S (t)=(g))) /\\ (((exists lcc_gap_progression_index. lcc_gap_progression_index+S (t)=(g)) -> (exists lcc_gap_progression_small. lcc_gap_progression_small+S (r+M*t)=(g*M))))))", "statement_sha256": "422d69913d182c125f07d898c85cc520cf3a2ebacc8234d26cf808845dd3625c", "summary": "With an actual remainder r (exists lcc_gap_residue_forward_index. lcc_gap_residue_forward_index+S (x1)=(g))) /\\ (((exists lcc_gap_residue_forward_index. lcc_gap_residue_forward_index+S (x1)=(g)) -> (exists lcc_gap_residue_forward. lcc_gap_residue_forward+S (r+M*x1)=(g*M)))))", "specialize linear_congruence_progression_bound_iff (M)", "specialize linear_congruence_progression_bound_iff (g)", "specialize linear_congruence_progression_bound_iff (r)", "specialize linear_congruence_progression_bound_iff (x1)", "apply linear_congruence_progression_bound_iff", "exact hr", "cases hc", "apply hc_left", "rewrite <- he", "exact h_left", "exact he", "intro h", "cases h", "cases h_witness", "split", "rewrite h_witness_right", "have hc : (((exists lcc_gap_residue_backward. lcc_gap_residue_backward+S (r+M*x1)=(g*M)) -> (exists lcc_gap_residue_backward_index. lcc_gap_residue_backward_index+S (x1)=(g))) /\\ (((exists lcc_gap_residue_backward_index. lcc_gap_residue_backward_index+S (x1)=(g)) -> (exists lcc_gap_residue_backward. lcc_gap_residue_backward+S (r+M*x1)=(g*M)))))", "specialize linear_congruence_progression_bound_iff (M)", "specialize linear_congruence_progression_bound_iff (g)", "specialize linear_congruence_progression_bound_iff (r)", "specialize linear_congruence_progression_bound_iff (x1)", "apply linear_congruence_progression_bound_iff", "exact hr", "cases hc", "apply hc_right", "exact h_witness_left", "specialize remainder_decomposition_to_mod_eq (M)", "specialize remainder_decomposition_to_mod_eq (x)", "specialize remainder_decomposition_to_mod_eq (x1)", "specialize remainder_decomposition_to_mod_eq (r)", "apply remainder_decomposition_to_mod_eq", "trans r+M*x1", "exact h_witness_right", "trans M*x1+r", "apply add_comm", "congr", "apply mul_comm", "refl" ], "script_sha256": "0b89c2012d33f48fbc6091a26b443178e1ab28f86e422e8be7ee84936ae5e75f", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall M g r x. (exists lcc_gap_residue_reference. lcc_gap_residue_reference+S (r)=(M)) -> ((((((exists lcc_gap_residue_bound. lcc_gap_residue_bound+S (x)=(g*M)) /\\ ((exists hgcrt_mod_left_lcc_residue_mod hgcrt_mod_right_lcc_residue_mod. x + M * hgcrt_mod_left_lcc_residue_mod = r + M * hgcrt_mod_right_lcc_residue_mod)))) -> (exists t. ((exists lcc_gap_residue_parameter_bound. lcc_gap_residue_parameter_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t))))) /\\ (((exists t. ((exists lcc_gap_residue_parameter_bound. lcc_gap_residue_parameter_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t)))) -> (((exists lcc_gap_residue_bound. lcc_gap_residue_bound+S (x)=(g*M)) /\\ ((exists hgcrt_mod_left_lcc_residue_mod hgcrt_mod_right_lcc_residue_mod. x + M * hgcrt_mod_left_lcc_residue_mod = r + M * hgcrt_mod_right_lcc_residue_mod))))))))", "statement_sha256": "02013eddf26f8d31d5a0df8d602b13c65c13c574a19a056ea4598da744f4849a", "summary": "Construct the exact interval parameter for every bounded member of a residue class, and conversely.", "summary_sha256": "0b5147bd7db2b69097a6c2ccb98e202172998e85dcffd5fd2193003968d103db" }, { "alpha_v34_frontier_enrollment": { "body_receipt_sha256": "c34ef766b1eabc480b1e8b3d913dfb939ed5eef79872e515cc848f4f5211d37c", "bundle_campaign": "congruence-arithmetic", "bundle_node_id": 207, "bundle_sha256": "983051afddc637a4e033546b8f3ddb8dc0ac22aa996b4e28b3822be8895576ad", "campaign": "congruence-arithmetic", "first_enrolled_version": "v34", 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(a * x) + m * hgcrt_mod_left_lcc_bounded_sol = b + m * hgcrt_mod_right_lcc_bounded_sol) -> (exists hgcrt_mod_left_lcc_bounded_red hgcrt_mod_right_lcc_bounded_red. x + M * hgcrt_mod_left_lcc_bounded_red = r + M * hgcrt_mod_right_lcc_bounded_red)) /\\ (((exists hgcrt_mod_left_lcc_bounded_red hgcrt_mod_right_lcc_bounded_red. x + M * hgcrt_mod_left_lcc_bounded_red = r + M * hgcrt_mod_right_lcc_bounded_red) -> (exists hgcrt_mod_left_lcc_bounded_sol hgcrt_mod_right_lcc_bounded_sol. (a * x) + m * hgcrt_mod_left_lcc_bounded_sol = b + m * hgcrt_mod_right_lcc_bounded_sol))))", "specialize linear_congruence_solution_class_iff_reduced_modulus (a)", "specialize linear_congruence_solution_class_iff_reduced_modulus (m)", "specialize linear_congruence_solution_class_iff_reduced_modulus (g)", "specialize linear_congruence_solution_class_iff_reduced_modulus (A)", "specialize linear_congruence_solution_class_iff_reduced_modulus (M)", "specialize linear_congruence_solution_class_iff_reduced_modulus (b)", "specialize linear_congruence_solution_class_iff_reduced_modulus (r)", "specialize linear_congruence_solution_class_iff_reduced_modulus (x)", "apply linear_congruence_solution_class_iff_reduced_modulus", "exact hm0", "exact hg", "exact ha", "exact hm", "exact hs", "cases hc", "have hp : (((((exists lcc_gap_bounded_p. lcc_gap_bounded_p+S (x)=(g*M)) /\\ ((exists hgcrt_mod_left_lcc_bounded_red hgcrt_mod_right_lcc_bounded_red. x + M * hgcrt_mod_left_lcc_bounded_red = r + M * hgcrt_mod_right_lcc_bounded_red)))) -> (exists t. ((exists lcc_gap_bounded_parameter_bound. lcc_gap_bounded_parameter_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t))))) /\\ (((exists t. ((exists lcc_gap_bounded_parameter_bound. lcc_gap_bounded_parameter_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t)))) -> (((exists lcc_gap_bounded_p. lcc_gap_bounded_p+S (x)=(g*M)) /\\ ((exists hgcrt_mod_left_lcc_bounded_red hgcrt_mod_right_lcc_bounded_red. x + M * hgcrt_mod_left_lcc_bounded_red = r + M * hgcrt_mod_right_lcc_bounded_red)))))))", "specialize linear_congruence_bounded_residue_parametrized (M)", "specialize linear_congruence_bounded_residue_parametrized (g)", "specialize linear_congruence_bounded_residue_parametrized (r)", "specialize linear_congruence_bounded_residue_parametrized (x)", "apply linear_congruence_bounded_residue_parametrized", "exact hr", "cases hp", "split", "intro h", "cases h", "apply hp_left", "split", "rewrite <- hm", "exact h_left", "apply hc_left", "exact h_right", "intro h", "have he : ((exists lcc_gap_bounded_back. lcc_gap_bounded_back+S (x)=(g*M)) /\\ ((exists hgcrt_mod_left_lcc_bounded_red hgcrt_mod_right_lcc_bounded_red. x + M * hgcrt_mod_left_lcc_bounded_red = r + M * hgcrt_mod_right_lcc_bounded_red)))", "apply hp_right", "exact h", "cases he", "split", "rewrite hm", "exact he_left", "apply hc_right", "exact he_right" ], "script_sha256": "22d3d8132384e31d8bf327bdba63c72067b2ec298063fda3887e8c354b92ac67", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall a m g A M b r x. (~(m=0)) -> ((((exists hag_left_factor_lcc_bounded. a = g * hag_left_factor_lcc_bounded) /\\ (exists hag_right_factor_lcc_bounded. m = g * hag_right_factor_lcc_bounded)) /\\ forall hag_divisor_lcc_bounded. (exists hag_common_left_lcc_bounded. a = hag_divisor_lcc_bounded * hag_common_left_lcc_bounded) -> (exists hag_common_right_lcc_bounded. m = hag_divisor_lcc_bounded * hag_common_right_lcc_bounded) -> exists hag_greatest_factor_lcc_bounded. g = hag_divisor_lcc_bounded * hag_greatest_factor_lcc_bounded)) -> (a=g*A) -> (m=g*M) -> (exists lcc_gap_bounded_r. lcc_gap_bounded_r+S (r)=(M)) -> (exists hgcrt_mod_left_lcc_bounded_reference hgcrt_mod_right_lcc_bounded_reference. (a * r) + m * hgcrt_mod_left_lcc_bounded_reference = b + m * hgcrt_mod_right_lcc_bounded_reference) -> ((((((exists lcc_gap_bounded_x. lcc_gap_bounded_x+S (x)=(m)) /\\ ((exists hgcrt_mod_left_lcc_bounded_sol hgcrt_mod_right_lcc_bounded_sol. (a * x) + m * hgcrt_mod_left_lcc_bounded_sol = b + m * hgcrt_mod_right_lcc_bounded_sol)))) -> (exists t. ((exists lcc_gap_bounded_parameter_bound. lcc_gap_bounded_parameter_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t))))) /\\ (((exists t. ((exists lcc_gap_bounded_parameter_bound. lcc_gap_bounded_parameter_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t)))) -> (((exists lcc_gap_bounded_x. lcc_gap_bounded_x+S (x)=(m)) /\\ ((exists hgcrt_mod_left_lcc_bounded_sol hgcrt_mod_right_lcc_bounded_sol. (a * x) + m * hgcrt_mod_left_lcc_bounded_sol = b + m * hgcrt_mod_right_lcc_bounded_sol))))))))", "statement_sha256": "b52a0bf8b5ef028594cb4ef8e9ab30e7fbe92c3baf91d4356370d601a24925f9", "summary": "All solutions below the original nonzero modulus are exactly r+M*t for t ((((exists hag_left_factor_lcc_enumeration. a = g * hag_left_factor_lcc_enumeration) /\\ (exists hag_right_factor_lcc_enumeration. m = g * hag_right_factor_lcc_enumeration)) /\\ forall hag_divisor_lcc_enumeration. (exists hag_common_left_lcc_enumeration. a = hag_divisor_lcc_enumeration * hag_common_left_lcc_enumeration) -> (exists hag_common_right_lcc_enumeration. m = hag_divisor_lcc_enumeration * hag_common_right_lcc_enumeration) -> exists hag_greatest_factor_lcc_enumeration. g = hag_divisor_lcc_enumeration * hag_greatest_factor_lcc_enumeration)) -> (a=g*A) -> (m=g*M) -> (exists lcc_enum_bfactor. b=g*lcc_enum_bfactor) -> (exists r. ((exists lcc_gap_enumeration_r. lcc_gap_enumeration_r+S (r)=(M)) /\\ (((exists hgcrt_mod_left_lcc_enumeration_reference hgcrt_mod_right_lcc_enumeration_reference. (a * r) + m * hgcrt_mod_left_lcc_enumeration_reference = b + m * hgcrt_mod_right_lcc_enumeration_reference) /\\ (((forall x. (((((exists lcc_gap_enumeration_x. lcc_gap_enumeration_x+S (x)=(m)) /\\ ((exists hgcrt_mod_left_lcc_enumeration_solution hgcrt_mod_right_lcc_enumeration_solution. (a * x) + m * hgcrt_mod_left_lcc_enumeration_solution = b + m * hgcrt_mod_right_lcc_enumeration_solution)))) -> (exists t. ((exists lcc_gap_enumeration_param_bound. lcc_gap_enumeration_param_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t))))) /\\ (((exists t. ((exists lcc_gap_enumeration_param_bound. lcc_gap_enumeration_param_bound+S (t)=(g)) /\\ (((x)=(r)+(M)*t)))) -> (((exists lcc_gap_enumeration_x. lcc_gap_enumeration_x+S (x)=(m)) /\\ ((exists hgcrt_mod_left_lcc_enumeration_solution hgcrt_mod_right_lcc_enumeration_solution. (a * x) + m * hgcrt_mod_left_lcc_enumeration_solution = b + m * hgcrt_mod_right_lcc_enumeration_solution)))))))) /\\ ((forall t u. (exists lcc_gap_enumeration_t. lcc_gap_enumeration_t+S (t)=(g)) -> (exists lcc_gap_enumeration_u. lcc_gap_enumeration_u+S (u)=(g)) -> ((r)+M*t=(r)+M*u) -> t=u))))))))", "statement_sha256": "489b9733a5124b9e9e82074322f4aa82b37cb54e89cc0dfa508658546c84a5c4", "summary": "Construct r and an actual bijection from t (a*x=b)) /\\ (((a*x=b) -> (exists hgcrt_mod_left_lcc_zero_x hgcrt_mod_right_lcc_zero_x. (a * x) + 0 * hgcrt_mod_left_lcc_zero_x = b + 0 * hgcrt_mod_right_lcc_zero_x))))", "specialize mod_eq_zero_iff_eq (a*x)", "specialize mod_eq_zero_iff_eq (b)", "apply mod_eq_zero_iff_eq", "cases he_x", "have hxeq : a*x=b", "apply he_x_left", "exact hx", "have he_y : (((exists hgcrt_mod_left_lcc_zero_y hgcrt_mod_right_lcc_zero_y. (a * y) + 0 * hgcrt_mod_left_lcc_zero_y = b + 0 * hgcrt_mod_right_lcc_zero_y) -> (a*y=b)) /\\ (((a*y=b) -> (exists hgcrt_mod_left_lcc_zero_y hgcrt_mod_right_lcc_zero_y. (a * y) + 0 * hgcrt_mod_left_lcc_zero_y = b + 0 * hgcrt_mod_right_lcc_zero_y))))", "specialize mod_eq_zero_iff_eq (a*y)", "specialize mod_eq_zero_iff_eq (b)", "apply mod_eq_zero_iff_eq", "cases he_y", "have hyeq : a*y=b", "apply he_y_left", "exact hy", "specialize mul_left_cancel_nonzero (a)", "specialize mul_left_cancel_nonzero (x)", "specialize mul_left_cancel_nonzero (y)", "apply mul_left_cancel_nonzero", "exact ha", "trans b", "exact hxeq", "symm", "exact hyeq" ], "script_sha256": "85b3c7e493cdd1078acb9b154384a630483d9d082600ad879fa4bf970db4e24a", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall a b x y. (~(a=0)) -> (exists hgcrt_mod_left_lcc_zero_unique_x hgcrt_mod_right_lcc_zero_unique_x. (a * x) + 0 * hgcrt_mod_left_lcc_zero_unique_x = b + 0 * hgcrt_mod_right_lcc_zero_unique_x) -> (exists hgcrt_mod_left_lcc_zero_unique_y hgcrt_mod_right_lcc_zero_unique_y. 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(exists ff_b_all_predecessor_power ff_c_all_predecessor_power. ((forall ff_i_all_predecessor_power_repeat. (exists ff_lt_all_predecessor_power_repeat_bound. ff_lt_all_predecessor_power_repeat_bound + S ff_i_all_predecessor_power_repeat = x) -> (((exists ff_h_all_predecessor_power_repeat_decoded. ff_h_all_predecessor_power_repeat_decoded + S (a) = S ((S (ff_i_all_predecessor_power_repeat)) * ff_c_all_predecessor_power)) /\\ exists ff_q_all_predecessor_power_repeat_decoded. ff_b_all_predecessor_power = ff_q_all_predecessor_power_repeat_decoded * S ((S (ff_i_all_predecessor_power_repeat)) * ff_c_all_predecessor_power) + (a)))) /\\ (exists ff_u_all_predecessor_power_product ff_v_all_predecessor_power_product. ((((exists ff_h_all_predecessor_power_product_start. ff_h_all_predecessor_power_product_start + S (1) = S ((S (0)) * ff_v_all_predecessor_power_product)) /\\ exists ff_q_all_predecessor_power_product_start. ff_u_all_predecessor_power_product = ff_q_all_predecessor_power_product_start * S ((S (0)) * ff_v_all_predecessor_power_product) + (1))) /\\ ((((exists ff_h_all_predecessor_power_product_terminal. ff_h_all_predecessor_power_product_terminal + S (r) = S ((S (x)) * ff_v_all_predecessor_power_product)) /\\ exists ff_q_all_predecessor_power_product_terminal. ff_u_all_predecessor_power_product = ff_q_all_predecessor_power_product_terminal * S ((S (x)) * ff_v_all_predecessor_power_product) + (r))) /\\ forall ff_i_all_predecessor_power_product. (exists ff_lt_all_predecessor_power_product_bound. ff_lt_all_predecessor_power_product_bound + S ff_i_all_predecessor_power_product = x) -> exists ff_p_all_predecessor_power_product ff_r_all_predecessor_power_product ff_s_all_predecessor_power_product. ((((exists ff_h_all_predecessor_power_product_factor. ff_h_all_predecessor_power_product_factor + S (ff_p_all_predecessor_power_product) = S ((S (ff_i_all_predecessor_power_product)) * ff_c_all_predecessor_power)) /\\ exists ff_q_all_predecessor_power_product_factor. ff_b_all_predecessor_power = ff_q_all_predecessor_power_product_factor * S ((S (ff_i_all_predecessor_power_product)) * ff_c_all_predecessor_power) + (ff_p_all_predecessor_power_product))) /\\ ((((exists ff_h_all_predecessor_power_product_partial. ff_h_all_predecessor_power_product_partial + S (ff_r_all_predecessor_power_product) = S ((S (ff_i_all_predecessor_power_product)) * ff_v_all_predecessor_power_product)) /\\ exists ff_q_all_predecessor_power_product_partial. ff_u_all_predecessor_power_product = ff_q_all_predecessor_power_product_partial * S ((S (ff_i_all_predecessor_power_product)) * ff_v_all_predecessor_power_product) + (ff_r_all_predecessor_power_product))) /\\ ((((exists ff_h_all_predecessor_power_product_successor. ff_h_all_predecessor_power_product_successor + S (ff_s_all_predecessor_power_product) = S ((S (S ff_i_all_predecessor_power_product)) * ff_v_all_predecessor_power_product)) /\\ exists ff_q_all_predecessor_power_product_successor. ff_u_all_predecessor_power_product = ff_q_all_predecessor_power_product_successor * S ((S (S ff_i_all_predecessor_power_product)) * ff_v_all_predecessor_power_product) + (ff_s_all_predecessor_power_product))) /\\ ff_s_all_predecessor_power_product = ff_r_all_predecessor_power_product * ff_p_all_predecessor_power_product)))))))) /\\ A = r * a", "specialize pow_successor_decompose a", "specialize pow_successor_decompose x", "specialize pow_successor_decompose p", "specialize pow_successor_decompose A", "apply pow_successor_decompose", "exact hps_witness", "exact hpow", "cases hdecomp", "cases hdecomp_witness", "have hsplit : (forall frp_divisor_all_coprime. (exists frp_left_factor_all_coprime. p = frp_divisor_all_coprime * frp_left_factor_all_coprime) -> (exists frp_right_factor_all_coprime. a = frp_divisor_all_coprime * frp_right_factor_all_coprime) -> frp_divisor_all_coprime = 1) \\/ (exists fep_factor_all_divides. a = p * fep_factor_all_divides)", "specialize prime_coprime_or_divides p", "specialize prime_coprime_or_divides a", "apply prime_coprime_or_divides", "exact hp", "cases hsplit", "have hnotdiv : ~(exists k. a = p * k)", "intro hdiv", "have hpdiv : exists q. p = p * q", "specialize multiple_refl p", "exact multiple_refl", "have hpone : p = 1", "specialize hsplit_left p", "apply hsplit_left", "exact hpdiv", "exact hdiv", "cases hp", "apply hp_left", "exact hpone", "have hprev : exists fep_mod_left_all_predecessor_result fep_mod_right_all_predecessor_result. x1 + p * fep_mod_left_all_predecessor_result = 1 + p * fep_mod_right_all_predecessor_result", "specialize fermat_predecessor_exponent_mod_one p", "specialize fermat_predecessor_exponent_mod_one x", "specialize fermat_predecessor_exponent_mod_one a", "specialize fermat_predecessor_exponent_mod_one x1", "apply fermat_predecessor_exponent_mod_one", "exact hps_witness", "exact hp", "exact hnotdiv", "exact hdecomp_witness_left", "have hscaled : exists fep_product_mod_left_all_scaled fep_product_mod_right_all_scaled. (x1 * a) + p * fep_product_mod_left_all_scaled = (1 * a) + p * fep_product_mod_right_all_scaled", "specialize mod_eq_mul_right p", "specialize mod_eq_mul_right x1", "specialize mod_eq_mul_right 1", "specialize mod_eq_mul_right a", "apply mod_eq_mul_right", "exact hprev", "rewrite hdecomp_witness_right", "specialize one_mul a", "rewrite one_mul at hscaled", "exact hscaled", "cases hsplit_right", "have hAdiv : exists s. A = p * s", "rewrite hdecomp_witness_right", "specialize multiple_mul_left p", "specialize multiple_mul_left a", "specialize multiple_mul_left x1", "apply multiple_mul_left", "exists x2", "exact hsplit_right_witness", "cases hAdiv", "exists x2", "exists x3", "rewrite hAdiv_witness", "rewrite hsplit_right_witness", "specialize add_comm (p * x3)", "specialize add_comm (p * x2)", "exact add_comm" ], "script_sha256": "8eb0acf82d2d34d4fda5867d9b142af0cf9b4ad01554f8da57f15a7038c9d9f7", "source": { "kind": "candidate_module", "path": "peano-lab/py/peano_lab/library/linear_congruence_classification_candidate.py", "sha256": "12b1a98ce830704485f1ea78475fba8b10e39031ffbef00b1b5dfc8ffdef7f47" }, "statement": "forall p a A. ((~(p = 1) /\\ forall frm_prime_left_all_prime frm_prime_right_all_prime. p = frm_prime_left_all_prime * frm_prime_right_all_prime -> frm_prime_left_all_prime = 1 \\/ frm_prime_right_all_prime = 1)) -> (exists ff_b_all_power ff_c_all_power. ((forall ff_i_all_power_repeat. (exists ff_lt_all_power_repeat_bound. ff_lt_all_power_repeat_bound + S ff_i_all_power_repeat = p) -> (((exists ff_h_all_power_repeat_decoded. ff_h_all_power_repeat_decoded + S (a) = S ((S (ff_i_all_power_repeat)) * ff_c_all_power)) /\\ exists ff_q_all_power_repeat_decoded. ff_b_all_power = ff_q_all_power_repeat_decoded * S ((S (ff_i_all_power_repeat)) * ff_c_all_power) + (a)))) /\\ (exists ff_u_all_power_product ff_v_all_power_product. ((((exists ff_h_all_power_product_start. ff_h_all_power_product_start + S (1) = S ((S (0)) * ff_v_all_power_product)) /\\ exists ff_q_all_power_product_start. ff_u_all_power_product = ff_q_all_power_product_start * S ((S (0)) * ff_v_all_power_product) + (1))) /\\ ((((exists ff_h_all_power_product_terminal. ff_h_all_power_product_terminal + S (A) = S ((S (p)) * ff_v_all_power_product)) /\\ exists ff_q_all_power_product_terminal. ff_u_all_power_product = ff_q_all_power_product_terminal * S ((S (p)) * ff_v_all_power_product) + (A))) /\\ forall ff_i_all_power_product. (exists ff_lt_all_power_product_bound. ff_lt_all_power_product_bound + S ff_i_all_power_product = p) -> exists ff_p_all_power_product ff_r_all_power_product ff_s_all_power_product. ((((exists ff_h_all_power_product_factor. ff_h_all_power_product_factor + S (ff_p_all_power_product) = S ((S (ff_i_all_power_product)) * ff_c_all_power)) /\\ exists ff_q_all_power_product_factor. ff_b_all_power = ff_q_all_power_product_factor * S ((S (ff_i_all_power_product)) * ff_c_all_power) + (ff_p_all_power_product))) /\\ ((((exists ff_h_all_power_product_partial. ff_h_all_power_product_partial + S (ff_r_all_power_product) = S ((S (ff_i_all_power_product)) * ff_v_all_power_product)) /\\ exists ff_q_all_power_product_partial. ff_u_all_power_product = ff_q_all_power_product_partial * S ((S (ff_i_all_power_product)) * ff_v_all_power_product) + (ff_r_all_power_product))) /\\ ((((exists ff_h_all_power_product_successor. ff_h_all_power_product_successor + S (ff_s_all_power_product) = S ((S (S ff_i_all_power_product)) * ff_v_all_power_product)) /\\ exists ff_q_all_power_product_successor. ff_u_all_power_product = ff_q_all_power_product_successor * S ((S (S ff_i_all_power_product)) * ff_v_all_power_product) + (ff_s_all_power_product))) /\\ ff_s_all_power_product = ff_r_all_power_product * ff_p_all_power_product)))))))) -> (exists fep_mod_left_all_result fep_mod_right_all_result. A + p * fep_mod_left_all_result = a + p * fep_mod_right_all_result)", "statement_sha256": "6a1162d7a8f6279242317f8ac7b9e93ca4f53d4dcf5563ca4a048d8dec75bb23", "summary": "Fermat's little theorem for every natural base in relational-power form.", "summary_sha256": "74bcf651ee910b3e169c8d4cbd6f884dc9074f5311deeb0b407ee8e100938f3b" } ]