{ "IR_parents_closed": 0, "authority": "historical_evidence_not_admission", "definitions": [ { "ast": [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 2 ] ] ], "dependencies": [], "id": "PD0002", "name": "Lt", "parameters": [ "a", "b" ], "source": "exists h. h + S a = b" }, { "ast": [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 2 ] ] ] ], [ "var", 4 ] ] ] ] ], "dependencies": [], "id": "PD0013", "name": "BetaAt", "parameters": [ "b", "c", "i", "x" ], "source": "((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))" }, { "ast": [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "PD0014", "name": "Product", "parameters": [ "b", "c", "l", "z" ], "source": "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)))))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 5 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ], [ "var", 4 ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "PD0019", "name": "Repeat", "parameters": [ "b", "c", "a", "l" ], "source": "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)))" }, { "ast": [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 5 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "var", 4 ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 1 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0014", "PD0019" ], "id": "PD0020", "name": "Pow", "parameters": [ "a", "e", "z" ], "source": "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)))))))" }, { "ast": [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 1 ] ], [ "var", 2 ] ] ], "dependencies": [], "id": "PD0001", "name": "Le", "parameters": [ "a", "b" ], "source": "exists h. h + a = b" }, { "ast": [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ], [ "var", 1 ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "mul", [ "succ", [ "var", 2 ] ], [ "succ", [ "var", 2 ] ] ] ] ] ], "dependencies": [ "PD0001", "PD0002" ], "id": "PD0051", "name": "FloorSqrt", "parameters": [ "n", "s" ], "source": "((exists bcs_sqrt_lower_gap_bertrand_defined_floor_sqrt. bcs_sqrt_lower_gap_bertrand_defined_floor_sqrt + (s) * (s) = (n)) /\\ exists bcs_sqrt_upper_gap_bertrand_defined_floor_sqrt. bcs_sqrt_upper_gap_bertrand_defined_floor_sqrt + S (n) = S (s) * S (s))" }, { "ast": [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "var", 3 ], [ "var", 3 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ] ] ], [ "mul", [ "succ", [ "var", 3 ] ], [ "succ", [ "var", 3 ] ] ] ] ] ] ], "dependencies": [ "PD0020", "PD0051" ], "id": "ND0446", "name": "DyadicSqrt2", "parameters": [ "k", "d", "s" ], "source": "(∃ x. ∃ y. (∀ z. S z ≤ k → 3 ≤ S (S z · y) ∧ (∃ n. x = n · S (S z · y) + 2)) ∧ (∃ z. ∃ n. 2 ≤ S (1 · n) ∧ (∃ m. z = m · S (1 · n) + 1) ∧ (S d ≤ S (S k · n) ∧ (∃ m. z = m · S (S k · n) + d) ∧ (∀ m. S m ≤ k → ∃ i. ∃ j. ∃ u. S i ≤ S (S m · y) ∧ (∃ v. x = v · S (S m · y) + i) ∧ (S j ≤ S (S m · n) ∧ (∃ v. z = v · S (S m · n) + j) ∧ (S u ≤ S (S S m · n) ∧ (∃ v. z = v · S (S S m · n) + u) ∧ u = j · i)))))) ∧ (s · s ≤ 2 · (d · d) ∧ S (2 · (d · d)) ≤ S s · S s)" }, { "ast": [ "or", [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ], [ "eq", [ "var", 3 ], [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ] ], [ "eq", [ "mul", [ "var", 2 ], [ "var", 3 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 1 ] ] ] ] ] ], "dependencies": [], "id": "ND0424", "name": "NatAtanhCoefficient", "parameters": [ "d", "i", "a" ], "source": "(∃ x. i = 2 · x ∧ a = 0) ∨ (∃ x. i = S (2 · x) ∧ i · a = 2 · d)" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 6 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ] ], [ "eq", [ "mul", [ "var", 2 ], [ "var", 1 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0424" ], "id": "ND0425", "name": "NatAtanhPrefix", "parameters": [ "d", "ab", "ac", "N" ], "source": "∀ x. ∀ y. S x ≤ N → S y ≤ S (S x · ac) ∧ (∃ z. ab = z · S (S x · ac) + y) → (∃ z. x = 2 · z ∧ y = 0) ∨ (∃ z. x = S (2 · z) ∧ x · y = 2 · d)" }, { "ast": [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "var", 5 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 8 ] ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 8 ] ] ], [ "var", 7 ] ] ] ], [ "add", [ "mul", [ "var", 5 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0013" ], "id": "ND0426", "name": "NatScaledHornerStep", "parameters": [ "ab", "ac", "t", "hb", "hc", "k" ], "source": "∃ x. ∃ y. S x ≤ S (S k · hc) ∧ (∃ z. hb = z · S (S k · hc) + x) ∧ (S y ≤ S (S k · ac) ∧ (∃ z. ab = z · S (S k · ac) + y) ∧ (S (t · x + y) ≤ S (S S k · hc) ∧ (∃ z. hb = z · S (S S k · hc) + (t · x + y))))" }, { "ast": [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 6 ] ] ] ], [ "zero" ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 5 ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 9 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "var", 6 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 9 ] ] ] ], [ "add", [ "mul", [ "var", 6 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0013", "PD0002", "ND0426" ], "id": "ND0427", "name": "NatScaledHornerTrace", "parameters": [ "ab", "ac", "t", "N", "hb", "hc" ], "source": "1 ≤ S (1 · hc) ∧ (∃ x. hb = x · S (1 · hc) + 0) ∧ (∀ x. S x ≤ N → ∃ y. ∃ z. S y ≤ S (S x · hc) ∧ (∃ n. hb = n · S (S x · hc) + y) ∧ (S z ≤ S (S x · ac) ∧ (∃ n. ab = n · S (S x · ac) + z) ∧ (S (t · y + z) ≤ S (S S x · hc) ∧ (∃ n. hb = n · S (S S x · hc) + (t · y + z)))))" }, { "ast": [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "var", 8 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ], [ "add", [ "mul", [ "var", 8 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 1 ] ] ] ], [ "var", 7 ] ] ] ] ] ] ] ], "dependencies": [ "ND0427", "PD0013" ], "id": "ND0437", "name": "NatHornerValue", "parameters": [ "ab", "ac", "t", "H", "y" ], "source": "∃ x. ∃ z. 1 ≤ S (1 · z) ∧ (∃ n. x = n · S (1 · z) + 0) ∧ (∀ n. S n ≤ H → ∃ m. ∃ k. S m ≤ S (S n · z) ∧ (∃ i. x = i · S (S n · z) + m) ∧ (S k ≤ S (S n · ac) ∧ (∃ i. ab = i · S (S n · ac) + k) ∧ (S (t · m + k) ≤ S (S S n · z) ∧ (∃ i. x = i · S (S S n · z) + (t · m + k))))) ∧ (S y ≤ S (S H · z) ∧ (∃ n. x = n · S (S H · z) + y))" }, { "ast": [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 3 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 2 ] ] ] ], [ "var", 5 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 3 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 4 ], [ "zero" ] ] ] ], "dependencies": [ "PD0002", "PD0001", "PD0013" ], "id": "ND0291", "name": "BetaZeroExtend", "parameters": [ "b", "c", "L", "i", "a" ], "source": "(((exists pfa_gap_lowercontinuationinside. pfa_gap_lowercontinuationinside + S ((i)) = ((L))) /\\ ((((exists ff_h_pfp_lowercontinuationentry. ff_h_pfp_lowercontinuationentry + S ((a)) = S ((S ((i))) * (c))) /\\ exists ff_q_pfp_lowercontinuationentry. (b) = ff_q_pfp_lowercontinuationentry * S ((S ((i))) * (c)) + ((a))))))) \\/ (((exists pfc_gap_lowercontinuationoutside. pfc_gap_lowercontinuationoutside+((L))=((i))) /\\ ((((a))=0))))" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 6 ] ] ], [ "imp", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 5 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 5 ] ], [ "var", 2 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ], "dependencies": [ "PD0002", "ND0291" ], "id": "ND0408", "name": "NatZeroBefore", "parameters": [ "b", "c", "L", "k" ], "source": "∀ x. ∀ y. S x ≤ k → S x ≤ L ∧ (S y ≤ S (S x · c) ∧ (∃ z. b = z · S (S x · c) + y)) ∨ L ≤ x ∧ y = 0 → y = 0" }, { "ast": [ "and", [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "var", 3 ], [ "var", 3 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ] ] ], [ "mul", [ "succ", [ "var", 3 ] ], [ "succ", [ "var", 3 ] ] ] ] ] ] ], [ "and", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ] ], [ "eq", [ "mul", [ "var", 2 ], [ "var", 1 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 6 ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ], [ "add", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0446", "ND0425", "ND0437", "PD0020" ], "id": "ND0447", "name": "NatSqrt2LogArgument", "parameters": [ "k", "d", "s", "D", "ab", "ac", "x", "p" ], "source": "(∃ y. ∃ z. (∀ n. S n ≤ k → 3 ≤ S (S n · z) ∧ (∃ m. y = m · S (S n · z) + 2)) ∧ (∃ n. ∃ m. 2 ≤ S (1 · m) ∧ (∃ i. n = i · S (1 · m) + 1) ∧ (S d ≤ S (S k · m) ∧ (∃ i. n = i · S (S k · m) + d) ∧ (∀ i. S i ≤ k → ∃ j. ∃ u. ∃ v. S j ≤ S (S i · z) ∧ (∃ w. y = w · S (S i · z) + j) ∧ (S u ≤ S (S i · m) ∧ (∃ w. n = w · S (S i · m) + u) ∧ (S v ≤ S (S S i · m) ∧ (∃ w. n = w · S (S S i · m) + v) ∧ v = u · j)))))) ∧ (s · s ≤ 2 · (d · d) ∧ S (2 · (d · d)) ≤ S s · S s) ∧ ((∀ y. ∀ z. S y ≤ S (2 · k) → S z ≤ S (S y · ac) ∧ (∃ n. ab = n · S (S y · ac) + z) → (∃ n. y = 2 · n ∧ z = 0) ∨ (∃ n. y = S (2 · n) ∧ y · z = 2 · D)) ∧ ((∃ y. ∃ z. 1 ≤ S (1 · z) ∧ (∃ n. y = n · S (1 · z) + 0) ∧ (∀ n. S n ≤ S (2 · k) → ∃ m. ∃ i. S m ≤ S (S n · z) ∧ (∃ j. y = j · S (S n · z) + m) ∧ (S i ≤ S (S n · ac) ∧ (∃ j. ab = j · S (S n · ac) + i) ∧ (S (3 · m + i) ≤ S (S S n · z) ∧ (∃ j. y = j · S (S S n · z) + (3 · m + i))))) ∧ (S x ≤ S (S S (2 · k) · z) ∧ (∃ n. y = n · S (S S (2 · k) · z) + x))) ∧ (∃ y. ∃ z. (∀ n. S n ≤ S (2 · k) → 4 ≤ S (S n · z) ∧ (∃ m. y = m · S (S n · z) + 3)) ∧ (∃ n. ∃ m. 2 ≤ S (1 · m) ∧ (∃ i. n = i · S (1 · m) + 1) ∧ (S p ≤ S (S S (2 · k) · m) ∧ (∃ i. n = i · S (S S (2 · k) · m) + p) ∧ (∀ i. S i ≤ S (2 · k) → ∃ j. ∃ u. ∃ v. S j ≤ S (S i · z) ∧ (∃ w. y = w · S (S i · z) + j) ∧ (S u ≤ S (S i · m) ∧ (∃ w. n = w · S (S i · m) + u) ∧ (S v ≤ S (S S i · m) ∧ (∃ w. n = w · S (S S i · m) + v) ∧ v = u · j))))))))" }, { "ast": [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 5 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 7 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 1 ], [ "mul", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ], [ "var", 0 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0423", "name": "NatExpWeightTable", "parameters": [ "d", "N", "wb", "wc" ], "source": "2 ≤ S (S N · wc) ∧ (∃ x. wb = x · S (S N · wc) + 1) ∧ (∀ x. ∀ y. ∀ z. S x ≤ N → S y ≤ S (S x · wc) ∧ (∃ n. wb = n · S (S x · wc) + y) → S z ≤ S (S S x · wc) ∧ (∃ n. wb = n · S (S S x · wc) + z) → y = S x · d · z)" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 10 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 8 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "var", 8 ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 9 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "PD0020" ], "id": "ND0442", "name": "NatPowerWeightedPrefix", "parameters": [ "x", "wb", "wc", "tb", "tc", "K" ], "source": "∀ y. ∀ z. ∀ n. ∀ m. S y ≤ K → S z ≤ S (S y · wc) ∧ (∃ k. wb = k · S (S y · wc) + z) → (∃ k. ∃ i. (∀ j. S j ≤ y → S x ≤ S (S j · i) ∧ (∃ u. k = u · S (S j · i) + x)) ∧ (∃ j. ∃ u. 2 ≤ S (1 · u) ∧ (∃ v. j = v · S (1 · u) + 1) ∧ (S n ≤ S (S y · u) ∧ (∃ v. j = v · S (S y · u) + n) ∧ (∀ v. S v ≤ y → ∃ w. ∃ x0. ∃ x1. S w ≤ S (S v · i) ∧ (∃ x2. k = x2 · S (S v · i) + w) ∧ (S x0 ≤ S (S v · u) ∧ (∃ x2. j = x2 · S (S v · u) + x0) ∧ (S x1 ≤ S (S S v · u) ∧ (∃ x2. j = x2 · S (S S v · u) + x1) ∧ x1 = x0 · w)))))) → S m ≤ S (S y · tc) ∧ (∃ k. tb = k · S (S y · tc) + m) → m = z · n" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 5 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "var", 4 ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ], [ "add", [ "var", 4 ], [ "var", 1 ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "PD0018", "name": "Range", "parameters": [ "b", "c", "a", "l" ], "source": "forall ff_i_defined_range. (exists ff_lt_defined_range_bound. ff_lt_defined_range_bound + S ff_i_defined_range = l) -> (((exists ff_h_defined_range_decoded. ff_h_defined_range_decoded + S (a + ff_i_defined_range) = S ((S (ff_i_defined_range)) * c)) /\\ exists ff_q_defined_range_decoded. b = ff_q_defined_range_decoded * S ((S (ff_i_defined_range)) * c) + (a + ff_i_defined_range)))" }, { "ast": [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0014", "PD0018" ], "id": "PD0023", "name": "Factorial", "parameters": [ "n", "z" ], "source": "exists ff_b_defined_factorial ff_c_defined_factorial. ((forall ff_i_defined_factorial_range. (exists ff_lt_defined_factorial_range_bound. ff_lt_defined_factorial_range_bound + S ff_i_defined_factorial_range = n) -> (((exists ff_h_defined_factorial_range_decoded. ff_h_defined_factorial_range_decoded + S (1 + ff_i_defined_factorial_range) = S ((S (ff_i_defined_factorial_range)) * ff_c_defined_factorial)) /\\ exists ff_q_defined_factorial_range_decoded. ff_b_defined_factorial = ff_q_defined_factorial_range_decoded * S ((S (ff_i_defined_factorial_range)) * ff_c_defined_factorial) + (1 + ff_i_defined_factorial_range)))) /\\ (exists ff_u_defined_factorial_product ff_v_defined_factorial_product. ((((exists ff_h_defined_factorial_product_start. ff_h_defined_factorial_product_start + S (1) = S ((S (0)) * ff_v_defined_factorial_product)) /\\ exists ff_q_defined_factorial_product_start. ff_u_defined_factorial_product = ff_q_defined_factorial_product_start * S ((S (0)) * ff_v_defined_factorial_product) + (1))) /\\ ((((exists ff_h_defined_factorial_product_terminal. ff_h_defined_factorial_product_terminal + S (z) = S ((S (n)) * ff_v_defined_factorial_product)) /\\ exists ff_q_defined_factorial_product_terminal. ff_u_defined_factorial_product = ff_q_defined_factorial_product_terminal * S ((S (n)) * ff_v_defined_factorial_product) + (z))) /\\ forall ff_i_defined_factorial_product. (exists ff_lt_defined_factorial_product_bound. ff_lt_defined_factorial_product_bound + S ff_i_defined_factorial_product = n) -> exists ff_p_defined_factorial_product ff_r_defined_factorial_product ff_s_defined_factorial_product. ((((exists ff_h_defined_factorial_product_factor. ff_h_defined_factorial_product_factor + S (ff_p_defined_factorial_product) = S ((S (ff_i_defined_factorial_product)) * ff_c_defined_factorial)) /\\ exists ff_q_defined_factorial_product_factor. ff_b_defined_factorial = ff_q_defined_factorial_product_factor * S ((S (ff_i_defined_factorial_product)) * ff_c_defined_factorial) + (ff_p_defined_factorial_product))) /\\ ((((exists ff_h_defined_factorial_product_partial. ff_h_defined_factorial_product_partial + S (ff_r_defined_factorial_product) = S ((S (ff_i_defined_factorial_product)) * ff_v_defined_factorial_product)) /\\ exists ff_q_defined_factorial_product_partial. ff_u_defined_factorial_product = ff_q_defined_factorial_product_partial * S ((S (ff_i_defined_factorial_product)) * ff_v_defined_factorial_product) + (ff_r_defined_factorial_product))) /\\ ((((exists ff_h_defined_factorial_product_successor. ff_h_defined_factorial_product_successor + S (ff_s_defined_factorial_product) = S ((S (S ff_i_defined_factorial_product)) * ff_v_defined_factorial_product)) /\\ exists ff_q_defined_factorial_product_successor. ff_u_defined_factorial_product = ff_q_defined_factorial_product_successor * S ((S (S ff_i_defined_factorial_product)) * ff_v_defined_factorial_product) + (ff_s_defined_factorial_product))) /\\ ff_s_defined_factorial_product = ff_r_defined_factorial_product * ff_p_defined_factorial_product)))))))" }, { "ast": [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "PD0015", "name": "Sum", "parameters": [ "b", "c", "l", "z" ], "source": "exists ff_u_defined_sum ff_v_defined_sum. ((((exists ff_h_defined_sum_start. ff_h_defined_sum_start + S (0) = S ((S (0)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_start. ff_u_defined_sum = ff_q_defined_sum_start * S ((S (0)) * ff_v_defined_sum) + (0))) /\\ ((((exists ff_h_defined_sum_terminal. ff_h_defined_sum_terminal + S (z) = S ((S (l)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_terminal. ff_u_defined_sum = ff_q_defined_sum_terminal * S ((S (l)) * ff_v_defined_sum) + (z))) /\\ forall ff_i_defined_sum. (exists ff_lt_defined_sum_bound. ff_lt_defined_sum_bound + S ff_i_defined_sum = l) -> exists ff_a_defined_sum ff_r_defined_sum ff_s_defined_sum. ((((exists ff_h_defined_sum_summand. ff_h_defined_sum_summand + S (ff_a_defined_sum) = S ((S (ff_i_defined_sum)) * c)) /\\ exists ff_q_defined_sum_summand. b = ff_q_defined_sum_summand * S ((S (ff_i_defined_sum)) * c) + (ff_a_defined_sum))) /\\ ((((exists ff_h_defined_sum_partial. ff_h_defined_sum_partial + S (ff_r_defined_sum) = S ((S (ff_i_defined_sum)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_partial. ff_u_defined_sum = ff_q_defined_sum_partial * S ((S (ff_i_defined_sum)) * ff_v_defined_sum) + (ff_r_defined_sum))) /\\ ((((exists ff_h_defined_sum_successor. ff_h_defined_sum_successor + S (ff_s_defined_sum) = S ((S (S ff_i_defined_sum)) * ff_v_defined_sum)) /\\ exists ff_q_defined_sum_successor. ff_u_defined_sum = ff_q_defined_sum_successor * S ((S (S ff_i_defined_sum)) * ff_v_defined_sum) + (ff_s_defined_sum))) /\\ ff_s_defined_sum = ff_r_defined_sum + ff_a_defined_sum)))))" }, { "ast": [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 8 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 7 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "mul", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ], [ "var", 0 ] ], [ "mul", [ "var", 4 ], [ "var", 1 ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0443", "name": "NatScaledExpRecurrence", "parameters": [ "d", "x", "tb", "tc", "N" ], "source": "∀ y. ∀ z. ∀ n. S y ≤ N → S z ≤ S (S y · tc) ∧ (∃ m. tb = m · S (S y · tc) + z) → S n ≤ S (S S y · tc) ∧ (∃ m. tb = m · S (S S y · tc) + n) → S y · d · n = x · z" }, { "ast": [ "and", [ "and", [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 6 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "var", 3 ], [ "var", 3 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 2 ], [ "var", 2 ] ] ] ] ], [ "mul", [ "succ", [ "var", 3 ] ], [ "succ", [ "var", 3 ] ] ] ] ] ] ], [ "and", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ] ], [ "eq", [ "mul", [ "var", 2 ], [ "var", 1 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 6 ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ], [ "add", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 1 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 1 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 10 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "add", [ "var", 4 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 13 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 1 ], [ "mul", [ "mul", [ "succ", [ "var", 2 ] ], [ "mul", [ "mul", [ "var", 6 ], [ "var", 10 ] ], [ "var", 5 ] ] ], [ "var", 0 ] ] ] ] ] ] ] ] ] ], [ "and", [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "add", [ "var", 5 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 14 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 14 ] ], [ "var", 9 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "mul", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 14 ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 9 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 16 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 13 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ], [ "var", 13 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 13 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 12 ] ] ] ], [ "var", 13 ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "var", 4 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 18 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 5 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 5 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 18 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "var", 6 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "var", 4 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "mul", [ "var", 7 ], [ "var", 11 ] ], [ "var", 6 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "mul", [ "mul", [ "var", 7 ], [ "var", 11 ] ], [ "var", 6 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 19 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 5 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 5 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 19 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "var", 6 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "eq", [ "var", 12 ], [ "mul", [ "var", 13 ], [ "var", 14 ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 18 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "add", [ "var", 3 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "add", [ "var", 3 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 18 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "add", [ "var", 4 ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 18 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0447", "ND0423", "ND0442", "PD0013", "PD0023", "PD0020", "PD0015" ], "id": "ND0448", "name": "NatSqrt2ExpSample", "parameters": [ "k", "d", "s", "D", "ab", "ac", "x", "p", "wb", "wc", "tb", "tc", "w0", "f", "h", "y" ], "source": "(∃ z. ∃ n. (∀ m. S m ≤ k → 3 ≤ S (S m · n) ∧ (∃ i. z = i · S (S m · n) + 2)) ∧ (∃ m. ∃ i. 2 ≤ S (1 · i) ∧ (∃ j. m = j · S (1 · i) + 1) ∧ (S d ≤ S (S k · i) ∧ (∃ j. m = j · S (S k · i) + d) ∧ (∀ j. S j ≤ k → ∃ u. ∃ v. ∃ w. S u ≤ S (S j · n) ∧ (∃ x0. z = x0 · S (S j · n) + u) ∧ (S v ≤ S (S j · i) ∧ (∃ x0. m = x0 · S (S j · i) + v) ∧ (S w ≤ S (S S j · i) ∧ (∃ x0. m = x0 · S (S S j · i) + w) ∧ w = v · u)))))) ∧ (s · s ≤ 2 · (d · d) ∧ S (2 · (d · d)) ≤ S s · S s) ∧ ((∀ z. ∀ n. S z ≤ S (2 · k) → S n ≤ S (S z · ac) ∧ (∃ m. ab = m · S (S z · ac) + n) → (∃ m. z = 2 · m ∧ n = 0) ∨ (∃ m. z = S (2 · m) ∧ z · n = 2 · D)) ∧ ((∃ z. ∃ n. 1 ≤ S (1 · n) ∧ (∃ m. z = m · S (1 · n) + 0) ∧ (∀ m. S m ≤ S (2 · k) → ∃ i. ∃ j. S i ≤ S (S m · n) ∧ (∃ u. z = u · S (S m · n) + i) ∧ (S j ≤ S (S m · ac) ∧ (∃ u. ab = u · S (S m · ac) + j) ∧ (S (3 · i + j) ≤ S (S S m · n) ∧ (∃ u. z = u · S (S S m · n) + (3 · i + j))))) ∧ (S x ≤ S (S S (2 · k) · n) ∧ (∃ m. z = m · S (S S (2 · k) · n) + x))) ∧ (∃ z. ∃ n. (∀ m. S m ≤ S (2 · k) → 4 ≤ S (S m · n) ∧ (∃ i. z = i · S (S m · n) + 3)) ∧ (∃ m. ∃ i. 2 ≤ S (1 · i) ∧ (∃ j. m = j · S (1 · i) + 1) ∧ (S p ≤ S (S S (2 · k) · i) ∧ (∃ j. m = j · S (S S (2 · k) · i) + p) ∧ (∀ j. S j ≤ S (2 · k) → ∃ u. ∃ v. ∃ w. S u ≤ S (S j · n) ∧ (∃ x0. z = x0 · S (S j · n) + u) ∧ (S v ≤ S (S j · i) ∧ (∃ x0. m = x0 · S (S j · i) + v) ∧ (S w ≤ S (S S j · i) ∧ (∃ x0. m = x0 · S (S S j · i) + w) ∧ w = v · u)))))))) ∧ (2 ≤ S (S (k + 2) · wc) ∧ (∃ z. wb = z · S (S (k + 2) · wc) + 1) ∧ (∀ z. ∀ n. ∀ m. S z ≤ k + 2 → S n ≤ S (S z · wc) ∧ (∃ i. wb = i · S (S z · wc) + n) → S m ≤ S (S S z · wc) ∧ (∃ i. wb = i · S (S S z · wc) + m) → n = S z · (D · p · s) · m) ∧ ((∀ z. ∀ n. ∀ m. ∀ i. S z ≤ S (k + 2) → S n ≤ S (S z · wc) ∧ (∃ j. wb = j · S (S z · wc) + n) → (∃ j. ∃ u. (∀ v. S v ≤ z → S (3 · x · d) ≤ S (S v · u) ∧ (∃ w. j = w · S (S v · u) + 3 · x · d)) ∧ (∃ v. ∃ w. 2 ≤ S (1 · w) ∧ (∃ x0. v = x0 · S (1 · w) + 1) ∧ (S m ≤ S (S z · w) ∧ (∃ x0. v = x0 · S (S z · w) + m) ∧ (∀ x0. S x0 ≤ z → ∃ x1. ∃ x2. ∃ x3. S x1 ≤ S (S x0 · u) ∧ (∃ x4. j = x4 · S (S x0 · u) + x1) ∧ (S x2 ≤ S (S x0 · w) ∧ (∃ x4. v = x4 · S (S x0 · w) + x2) ∧ (S x3 ≤ S (S S x0 · w) ∧ (∃ x4. v = x4 · S (S S x0 · w) + x3) ∧ x3 = x2 · x1)))))) → S i ≤ S (S z · tc) ∧ (∃ j. tb = j · S (S z · tc) + i) → i = n · m) ∧ (S w0 ≤ S (1 · wc) ∧ (∃ z. wb = z · S (1 · wc) + w0) ∧ (S w0 ≤ S (1 · tc) ∧ (∃ z. tb = z · S (1 · tc) + w0) ∧ ((∃ z. ∃ n. (∀ m. S m ≤ k + 2 → S (1 + m) ≤ S (S m · n) ∧ (∃ i. z = i · S (S m · n) + (1 + m))) ∧ (∃ m. ∃ i. 2 ≤ S (1 · i) ∧ (∃ j. m = j · S (1 · i) + 1) ∧ (S f ≤ S (S (k + 2) · i) ∧ (∃ j. m = j · S (S (k + 2) · i) + f) ∧ (∀ j. S j ≤ k + 2 → ∃ u. ∃ v. ∃ w. S u ≤ S (S j · n) ∧ (∃ x0. z = x0 · S (S j · n) + u) ∧ (S v ≤ S (S j · i) ∧ (∃ x0. m = x0 · S (S j · i) + v) ∧ (S w ≤ S (S S j · i) ∧ (∃ x0. m = x0 · S (S S j · i) + w) ∧ w = v · u)))))) ∧ ((∃ z. ∃ n. (∀ m. S m ≤ k + 2 → S (D · p · s) ≤ S (S m · n) ∧ (∃ i. z = i · S (S m · n) + D · p · s)) ∧ (∃ m. ∃ i. 2 ≤ S (1 · i) ∧ (∃ j. m = j · S (1 · i) + 1) ∧ (S h ≤ S (S (k + 2) · i) ∧ (∃ j. m = j · S (S (k + 2) · i) + h) ∧ (∀ j. S j ≤ k + 2 → ∃ u. ∃ v. ∃ w. S u ≤ S (S j · n) ∧ (∃ x0. z = x0 · S (S j · n) + u) ∧ (S v ≤ S (S j · i) ∧ (∃ x0. m = x0 · S (S j · i) + v) ∧ (S w ≤ S (S S j · i) ∧ (∃ x0. m = x0 · S (S S j · i) + w) ∧ w = v · u)))))) ∧ (w0 = f · h ∧ (∃ z. ∃ n. 1 ≤ S (1 · n) ∧ (∃ m. z = m · S (1 · n) + 0) ∧ (S y ≤ S (S S (k + 2) · n) ∧ (∃ m. z = m · S (S S (k + 2) · n) + y) ∧ (∀ m. S m ≤ S (k + 2) → ∃ i. ∃ j. ∃ u. S i ≤ S (S m · tc) ∧ (∃ v. tb = v · S (S m · tc) + i) ∧ (S j ≤ S (S m · n) ∧ (∃ v. z = v · S (S m · n) + j) ∧ (S u ≤ S (S S m · n) ∧ (∃ v. z = v · S (S S m · n) + u) ∧ u = j + i))))))))))))" }, { "ast": [ "imp", [ "eq", [ "var", 2 ], [ "zero" ] ], [ "bot" ] ], "dependencies": [], "id": "ND0382", "name": "IRatValid", "parameters": [ "p", "m", "d" ], "source": "¬d = 0" }, { "ast": [ "and", [ "imp", [ "eq", [ "var", 2 ], [ "zero" ] ], [ "bot" ] ], [ "and", [ "imp", [ "eq", [ "var", 5 ], [ "zero" ] ], [ "bot" ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "add", [ "mul", [ "var", 1 ], [ "var", 6 ] ], [ "mul", [ "var", 5 ], [ "var", 3 ] ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 6 ] ], [ "mul", [ "var", 4 ], [ "var", 3 ] ] ] ] ] ] ], "dependencies": [ "ND0382", "PD0001" ], "id": "ND0444", "name": "IRatLe", "parameters": [ "p", "m", "d", "P", "M", "D" ], "source": "¬d = 0 ∧ (¬D = 0 ∧ p · D + M · d ≤ m · D + P · d)" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 9 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 5 ], [ "mul", [ "var", 6 ], [ "var", 2 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 5 ], [ "mul", [ "var", 6 ], [ "var", 2 ] ] ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0411", "name": "NatAffineSlice", "parameters": [ "ab", "ac", "o", "t", "ub", "uc", "N" ], "source": "∀ x. ∀ y. S x ≤ N → S y ≤ S (S x · uc) ∧ (∃ z. ub = z · S (S x · uc) + y) → S y ≤ S (S (o + t · x) · ac) ∧ (∃ z. ab = z · S (S (o + t · x) · ac) + y)" }, { "ast": [ "and", [ "imp", [ "eq", [ "var", 2 ], [ "zero" ] ], [ "bot" ] ], [ "and", [ "imp", [ "eq", [ "var", 5 ], [ "zero" ] ], [ "bot" ] ], [ "eq", [ "add", [ "mul", [ "var", 0 ], [ "var", 5 ] ], [ "mul", [ "var", 4 ], [ "var", 2 ] ] ], [ "add", [ "mul", [ "var", 1 ], [ "var", 5 ] ], [ "mul", [ "var", 3 ], [ "var", 2 ] ] ] ] ] ], "dependencies": [ "ND0382" ], "id": "ND0383", "name": "IRatEq", "parameters": [ "p", "m", "d", "P", "M", "D" ], "source": "¬d = 0 ∧ (¬D = 0 ∧ p · D + M · d = m · D + P · d)" }, { "ast": [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 5 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 7 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 1 ], [ "mul", [ "var", 3 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0013", "PD0002" ], "id": "ND0428", "name": "NatGeometricWeightTable", "parameters": [ "t", "N", "wb", "wc" ], "source": "2 ≤ S (S N · wc) ∧ (∃ x. wb = x · S (S N · wc) + 1) ∧ (∀ x. ∀ y. ∀ z. S x ≤ N → S y ≤ S (S x · wc) ∧ (∃ n. wb = n · S (S x · wc) + y) → S z ≤ S (S S x · wc) ∧ (∃ n. wb = n · S (S S x · wc) + z) → y = t · z)" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 11 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 8 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0013", "PD0002" ], "id": "ND0429", "name": "NatHornerWeightedPrefix", "parameters": [ "ab", "ac", "wb", "wc", "qb", "qc", "N" ], "source": "∀ x. ∀ y. ∀ z. ∀ n. S x ≤ N → S y ≤ S (S x · ac) ∧ (∃ m. ab = m · S (S x · ac) + y) → S z ≤ S (S S x · wc) ∧ (∃ m. wb = m · S (S S x · wc) + z) → S n ≤ S (S x · qc) ∧ (∃ m. qb = m · S (S x · qc) + n) → n = y · z" }, { "ast": [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 13 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 18 ] ] ] ], [ "var", 13 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 18 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 9 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 8 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 8 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 16 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 16 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 15 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 18 ] ] ] ], [ "var", 15 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 15 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 15 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 19 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "imp", [ "eq", [ "var", 10 ], [ "zero" ] ], [ "bot" ] ], [ "and", [ "and", [ "and", [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 17 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 17 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "var", 19 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "var", 12 ], [ "var", 12 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 13 ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "mul", [ "var", 13 ], [ "var", 13 ] ] ] ] ], [ "mul", [ "succ", [ "var", 12 ] ], [ "succ", [ "var", 12 ] ] ] ] ] ] ], [ "and", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 16 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 0 ] ] ] ], [ "eq", [ "mul", [ "var", 2 ], [ "var", 1 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 13 ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 14 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ], [ "add", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 16 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 16 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 10 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 19 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 14 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 14 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 5 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "add", [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 1 ], [ "mul", [ "mul", [ "succ", [ "var", 2 ] ], [ "mul", [ "mul", [ "var", 13 ], [ "var", 9 ] ], [ "var", 14 ] ] ], [ "var", 0 ] ] ] ] ] ] ] ] ] ], [ "and", [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "add", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 15 ] ], [ "var", 20 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "mul", [ "mul", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ], [ "var", 15 ] ], [ "var", 20 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 9 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 16 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 5 ] ] ] ], [ "var", 16 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 16 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 3 ] ] ] ], [ "var", 16 ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "add", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "add", [ "var", 19 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "mul", [ "var", 14 ], [ "var", 10 ] ], [ "var", 15 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "mul", [ "mul", [ "var", 14 ], [ "var", 10 ] ], [ "var", 15 ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 18 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "add", [ "add", [ "var", 19 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "eq", [ "var", 15 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 17 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "add", [ "add", [ "var", 16 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "add", [ "add", [ "var", 16 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ], [ "var", 1 ] ] ] ], [ "var", 17 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "add", [ "add", [ "var", 17 ], [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ] ] ] ], [ "succ", [ "succ", [ "zero" ] ] ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], [ "imp", [ "eq", [ "var", 15 ], [ "zero" ] ], [ "bot" ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0013", "PD0023", "ND0448" ], "id": "ND0449", "name": "NatSqrt2ApproxRecord", "parameters": [ "n", "up", "ud", "rb", "rc" ], "source": "∃ x. ∃ y. ∃ z. ∃ m. ∃ k. ∃ i. ∃ j. ∃ u. ∃ v. ∃ w. ∃ x0. ∃ x1. ∃ x2. S x ≤ S (1 · rc) ∧ (∃ x3. rb = x3 · S (1 · rc) + x) ∧ (S y ≤ S (2 · rc) ∧ (∃ x3. rb = x3 · S (2 · rc) + y) ∧ (S z ≤ S (3 · rc) ∧ (∃ x3. rb = x3 · S (3 · rc) + z) ∧ (S m ≤ S (4 · rc) ∧ (∃ x3. rb = x3 · S (4 · rc) + m) ∧ (S k ≤ S (5 · rc) ∧ (∃ x3. rb = x3 · S (5 · rc) + k) ∧ (S i ≤ S (6 · rc) ∧ (∃ x3. rb = x3 · S (6 · rc) + i) ∧ (S j ≤ S (7 · rc) ∧ (∃ x3. rb = x3 · S (7 · rc) + j) ∧ (S u ≤ S (8 · rc) ∧ (∃ x3. rb = x3 · S (8 · rc) + u) ∧ (S v ≤ S (9 · rc) ∧ (∃ x3. rb = x3 · S (9 · rc) + v) ∧ (S w ≤ S (10 · rc) ∧ (∃ x3. rb = x3 · S (10 · rc) + w) ∧ (S x0 ≤ S (11 · rc) ∧ (∃ x3. rb = x3 · S (11 · rc) + x0) ∧ (S ud ≤ S (12 · rc) ∧ (∃ x3. rb = x3 · S (12 · rc) + ud) ∧ (S x1 ≤ S (13 · rc) ∧ (∃ x3. rb = x3 · S (13 · rc) + x1) ∧ (S x2 ≤ S (14 · rc) ∧ (∃ x3. rb = x3 · S (14 · rc) + x2) ∧ (S up ≤ S (15 · rc) ∧ (∃ x3. rb = x3 · S (15 · rc) + up))))))))))))))) ∧ ((∃ x3. ∃ x4. (∀ x5. S x5 ≤ S (2 · (n + 8)) → S (1 + x5) ≤ S (S x5 · x4) ∧ (∃ x6. x3 = x6 · S (S x5 · x4) + (1 + x5))) ∧ (∃ x5. ∃ x6. 2 ≤ S (1 · x6) ∧ (∃ x7. x5 = x7 · S (1 · x6) + 1) ∧ (S z ≤ S (S S (2 · (n + 8)) · x6) ∧ (∃ x7. x5 = x7 · S (S S (2 · (n + 8)) · x6) + z) ∧ (∀ x7. S x7 ≤ S (2 · (n + 8)) → ∃ x8. ∃ x9. ∃ x10. S x8 ≤ S (S x7 · x4) ∧ (∃ x11. x3 = x11 · S (S x7 · x4) + x8) ∧ (S x9 ≤ S (S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S x7 · x6) + x9) ∧ (S x10 ≤ S (S S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S S x7 · x6) + x10) ∧ x10 = x9 · x8)))))) ∧ (¬z = 0 ∧ ((∃ x3. ∃ x4. (∀ x5. S x5 ≤ n + 8 → 3 ≤ S (S x5 · x4) ∧ (∃ x6. x3 = x6 · S (S x5 · x4) + 2)) ∧ (∃ x5. ∃ x6. 2 ≤ S (1 · x6) ∧ (∃ x7. x5 = x7 · S (1 · x6) + 1) ∧ (S x ≤ S (S (n + 8) · x6) ∧ (∃ x7. x5 = x7 · S (S (n + 8) · x6) + x) ∧ (∀ x7. S x7 ≤ n + 8 → ∃ x8. ∃ x9. ∃ x10. S x8 ≤ S (S x7 · x4) ∧ (∃ x11. x3 = x11 · S (S x7 · x4) + x8) ∧ (S x9 ≤ S (S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S x7 · x6) + x9) ∧ (S x10 ≤ S (S S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S S x7 · x6) + x10) ∧ x10 = x9 · x8)))))) ∧ (y · y ≤ 2 · (x · x) ∧ S (2 · (x · x)) ≤ S y · S y) ∧ ((∀ x3. ∀ x4. S x3 ≤ S (2 · (n + 8)) → S x4 ≤ S (S x3 · k) ∧ (∃ x5. m = x5 · S (S x3 · k) + x4) → (∃ x5. x3 = 2 · x5 ∧ x4 = 0) ∨ (∃ x5. x3 = S (2 · x5) ∧ x3 · x4 = 2 · z)) ∧ ((∃ x3. ∃ x4. 1 ≤ S (1 · x4) ∧ (∃ x5. x3 = x5 · S (1 · x4) + 0) ∧ (∀ x5. S x5 ≤ S (2 · (n + 8)) → ∃ x6. ∃ x7. S x6 ≤ S (S x5 · x4) ∧ (∃ x8. x3 = x8 · S (S x5 · x4) + x6) ∧ (S x7 ≤ S (S x5 · k) ∧ (∃ x8. m = x8 · S (S x5 · k) + x7) ∧ (S (3 · x6 + x7) ≤ S (S S x5 · x4) ∧ (∃ x8. x3 = x8 · S (S S x5 · x4) + (3 · x6 + x7))))) ∧ (S i ≤ S (S S (2 · (n + 8)) · x4) ∧ (∃ x5. x3 = x5 · S (S S (2 · (n + 8)) · x4) + i))) ∧ (∃ x3. ∃ x4. (∀ x5. S x5 ≤ S (2 · (n + 8)) → 4 ≤ S (S x5 · x4) ∧ (∃ x6. x3 = x6 · S (S x5 · x4) + 3)) ∧ (∃ x5. ∃ x6. 2 ≤ S (1 · x6) ∧ (∃ x7. x5 = x7 · S (1 · x6) + 1) ∧ (S j ≤ S (S S (2 · (n + 8)) · x6) ∧ (∃ x7. x5 = x7 · S (S S (2 · (n + 8)) · x6) + j) ∧ (∀ x7. S x7 ≤ S (2 · (n + 8)) → ∃ x8. ∃ x9. ∃ x10. S x8 ≤ S (S x7 · x4) ∧ (∃ x11. x3 = x11 · S (S x7 · x4) + x8) ∧ (S x9 ≤ S (S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S x7 · x6) + x9) ∧ (S x10 ≤ S (S S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S S x7 · x6) + x10) ∧ x10 = x9 · x8)))))))) ∧ (2 ≤ S (S (n + 8 + 2) · v) ∧ (∃ x3. u = x3 · S (S (n + 8 + 2) · v) + 1) ∧ (∀ x3. ∀ x4. ∀ x5. S x3 ≤ n + 8 + 2 → S x4 ≤ S (S x3 · v) ∧ (∃ x6. u = x6 · S (S x3 · v) + x4) → S x5 ≤ S (S S x3 · v) ∧ (∃ x6. u = x6 · S (S S x3 · v) + x5) → x4 = S x3 · (z · j · y) · x5) ∧ ((∀ x3. ∀ x4. ∀ x5. ∀ x6. S x3 ≤ S (n + 8 + 2) → S x4 ≤ S (S x3 · v) ∧ (∃ x7. u = x7 · S (S x3 · v) + x4) → (∃ x7. ∃ x8. (∀ x9. S x9 ≤ x3 → S (3 · i · x) ≤ S (S x9 · x8) ∧ (∃ x10. x7 = x10 · S (S x9 · x8) + 3 · i · x)) ∧ (∃ x9. ∃ x10. 2 ≤ S (1 · x10) ∧ (∃ x11. x9 = x11 · S (1 · x10) + 1) ∧ (S x5 ≤ S (S x3 · x10) ∧ (∃ x11. x9 = x11 · S (S x3 · x10) + x5) ∧ (∀ x11. S x11 ≤ x3 → ∃ x12. ∃ x13. ∃ x14. S x12 ≤ S (S x11 · x8) ∧ (∃ x15. x7 = x15 · S (S x11 · x8) + x12) ∧ (S x13 ≤ S (S x11 · x10) ∧ (∃ x15. x9 = x15 · S (S x11 · x10) + x13) ∧ (S x14 ≤ S (S S x11 · x10) ∧ (∃ x15. x9 = x15 · S (S S x11 · x10) + x14) ∧ x14 = x13 · x12)))))) → S x6 ≤ S (S x3 · x0) ∧ (∃ x7. w = x7 · S (S x3 · x0) + x6) → x6 = x4 · x5) ∧ (S ud ≤ S (1 · v) ∧ (∃ x3. u = x3 · S (1 · v) + ud) ∧ (S ud ≤ S (1 · x0) ∧ (∃ x3. w = x3 · S (1 · x0) + ud) ∧ ((∃ x3. ∃ x4. (∀ x5. S x5 ≤ n + 8 + 2 → S (1 + x5) ≤ S (S x5 · x4) ∧ (∃ x6. x3 = x6 · S (S x5 · x4) + (1 + x5))) ∧ (∃ x5. ∃ x6. 2 ≤ S (1 · x6) ∧ (∃ x7. x5 = x7 · S (1 · x6) + 1) ∧ (S x1 ≤ S (S (n + 8 + 2) · x6) ∧ (∃ x7. x5 = x7 · S (S (n + 8 + 2) · x6) + x1) ∧ (∀ x7. S x7 ≤ n + 8 + 2 → ∃ x8. ∃ x9. ∃ x10. S x8 ≤ S (S x7 · x4) ∧ (∃ x11. x3 = x11 · S (S x7 · x4) + x8) ∧ (S x9 ≤ S (S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S x7 · x6) + x9) ∧ (S x10 ≤ S (S S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S S x7 · x6) + x10) ∧ x10 = x9 · x8)))))) ∧ ((∃ x3. ∃ x4. (∀ x5. S x5 ≤ n + 8 + 2 → S (z · j · y) ≤ S (S x5 · x4) ∧ (∃ x6. x3 = x6 · S (S x5 · x4) + z · j · y)) ∧ (∃ x5. ∃ x6. 2 ≤ S (1 · x6) ∧ (∃ x7. x5 = x7 · S (1 · x6) + 1) ∧ (S x2 ≤ S (S (n + 8 + 2) · x6) ∧ (∃ x7. x5 = x7 · S (S (n + 8 + 2) · x6) + x2) ∧ (∀ x7. S x7 ≤ n + 8 + 2 → ∃ x8. ∃ x9. ∃ x10. S x8 ≤ S (S x7 · x4) ∧ (∃ x11. x3 = x11 · S (S x7 · x4) + x8) ∧ (S x9 ≤ S (S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S x7 · x6) + x9) ∧ (S x10 ≤ S (S S x7 · x6) ∧ (∃ x11. x5 = x11 · S (S S x7 · x6) + x10) ∧ x10 = x9 · x8)))))) ∧ (ud = x1 · x2 ∧ (∃ x3. ∃ x4. 1 ≤ S (1 · x4) ∧ (∃ x5. x3 = x5 · S (1 · x4) + 0) ∧ (S up ≤ S (S S (n + 8 + 2) · x4) ∧ (∃ x5. x3 = x5 · S (S S (n + 8 + 2) · x4) + up) ∧ (∀ x5. S x5 ≤ S (n + 8 + 2) → ∃ x6. ∃ x7. ∃ x8. S x6 ≤ S (S x5 · x0) ∧ (∃ x9. w = x9 · S (S x5 · x0) + x6) ∧ (S x7 ≤ S (S x5 · x4) ∧ (∃ x9. x3 = x9 · S (S x5 · x4) + x7) ∧ (S x8 ≤ S (S S x5 · x4) ∧ (∃ x9. x3 = x9 · S (S S x5 · x4) + x8) ∧ x8 = x7 + x6)))))))))))) ∧ ¬ud = 0)))" }, { "ast": [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 10 ], [ "var", 2 ] ], [ "var", 9 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "var", 6 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 6 ] ], [ "var", 11 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 9 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 9 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 11 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0291" ], "id": "ND0292", "name": "PolynomialDiagonalTerm", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "i", "j", "t" ], "source": "exists pfc_complement_lowercontinuation pfc_left_lowercontinuation pfc_right_lowercontinuation. ((((j))+pfc_complement_lowercontinuation=((i))) /\\ ((((((exists pfa_gap_lowercontinuationleftinside. pfa_gap_lowercontinuationleftinside + S ((j)) = ((L))) /\\ ((((exists ff_h_pfp_lowercontinuationleftentry. ff_h_pfp_lowercontinuationleftentry + S (pfc_left_lowercontinuation) = S ((S ((j))) * (ac))) /\\ exists ff_q_pfp_lowercontinuationleftentry. (ab) = ff_q_pfp_lowercontinuationleftentry * S ((S ((j))) * (ac)) + (pfc_left_lowercontinuation)))))) \\/ (((exists pfc_gap_lowercontinuationleftoutside. pfc_gap_lowercontinuationleftoutside+((L))=((j))) /\\ (((pfc_left_lowercontinuation)=0))))) /\\ ((((((exists pfa_gap_lowercontinuationrightinside. pfa_gap_lowercontinuationrightinside + S (pfc_complement_lowercontinuation) = ((M))) /\\ ((((exists ff_h_pfp_lowercontinuationrightentry. ff_h_pfp_lowercontinuationrightentry + S (pfc_right_lowercontinuation) = S ((S (pfc_complement_lowercontinuation)) * (bc))) /\\ exists ff_q_pfp_lowercontinuationrightentry. (bb) = ff_q_pfp_lowercontinuationrightentry * S ((S (pfc_complement_lowercontinuation)) * (bc)) + (pfc_right_lowercontinuation)))))) \\/ (((exists pfc_gap_lowercontinuationrightoutside. pfc_gap_lowercontinuationrightoutside+((M))=(pfc_complement_lowercontinuation)) /\\ (((pfc_right_lowercontinuation)=0))))) /\\ ((((t))=pfc_left_lowercontinuation*pfc_right_lowercontinuation)))))))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 11 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 11 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 8 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 8 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 11 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 11 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0292" ], "id": "ND0293", "name": "PolynomialDiagonalPrefix", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "i", "db", "dc", "l" ], "source": "forall pfc_index_lowercontinuation. (exists pfa_gap_lowercontinuationbound. pfa_gap_lowercontinuationbound + S (pfc_index_lowercontinuation) = ((l))) -> exists pfc_value_lowercontinuation. ((((exists ff_h_pfp_lowercontinuationentry. ff_h_pfp_lowercontinuationentry + S (pfc_value_lowercontinuation) = S ((S (pfc_index_lowercontinuation)) * (dc))) /\\ exists ff_q_pfp_lowercontinuationentry. (db) = ff_q_pfp_lowercontinuationentry * S ((S (pfc_index_lowercontinuation)) * (dc)) + (pfc_value_lowercontinuation))) /\\ ((exists pfc_complement_lowercontinuationterm pfc_left_lowercontinuationterm pfc_right_lowercontinuationterm. (((pfc_index_lowercontinuation)+pfc_complement_lowercontinuationterm=((i))) /\\ ((((((exists pfa_gap_lowercontinuationtermleftinside. pfa_gap_lowercontinuationtermleftinside + S (pfc_index_lowercontinuation) = ((L))) /\\ ((((exists ff_h_pfp_lowercontinuationtermleftentry. ff_h_pfp_lowercontinuationtermleftentry + S (pfc_left_lowercontinuationterm) = S ((S (pfc_index_lowercontinuation)) * (ac))) /\\ exists ff_q_pfp_lowercontinuationtermleftentry. (ab) = ff_q_pfp_lowercontinuationtermleftentry * S ((S (pfc_index_lowercontinuation)) * (ac)) + (pfc_left_lowercontinuationterm)))))) \\/ (((exists pfc_gap_lowercontinuationtermleftoutside. pfc_gap_lowercontinuationtermleftoutside+((L))=(pfc_index_lowercontinuation)) /\\ (((pfc_left_lowercontinuationterm)=0))))) /\\ ((((((exists pfa_gap_lowercontinuationtermrightinside. pfa_gap_lowercontinuationtermrightinside + S (pfc_complement_lowercontinuationterm) = ((M))) /\\ ((((exists ff_h_pfp_lowercontinuationtermrightentry. ff_h_pfp_lowercontinuationtermrightentry + S (pfc_right_lowercontinuationterm) = S ((S (pfc_complement_lowercontinuationterm)) * (bc))) /\\ exists ff_q_pfp_lowercontinuationtermrightentry. (bb) = ff_q_pfp_lowercontinuationtermrightentry * S ((S (pfc_complement_lowercontinuationterm)) * (bc)) + (pfc_right_lowercontinuationterm)))))) \\/ (((exists pfc_gap_lowercontinuationtermrightoutside. pfc_gap_lowercontinuationtermrightoutside+((M))=(pfc_complement_lowercontinuationterm)) /\\ (((pfc_right_lowercontinuationterm)=0))))) /\\ (((pfc_value_lowercontinuation)=pfc_left_lowercontinuationterm*pfc_right_lowercontinuationterm))))))))))" }, { "ast": [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 10 ] ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 13 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 10 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 9 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 10 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 13 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 13 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 11 ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 11 ] ] ], [ "var", 1 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 12 ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0293", "PD0015" ], "id": "ND0401", "name": "NatConvolutionCoefficient", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "i", "n" ], "source": "∃ x. ∃ y. (∀ z. S z ≤ S i → ∃ m. S m ≤ S (S z · y) ∧ (∃ k. x = k · S (S z · y) + m) ∧ (∃ k. ∃ j. ∃ u. z + k = i ∧ ((S z ≤ L ∧ (S j ≤ S (S z · ac) ∧ (∃ v. ab = v · S (S z · ac) + j)) ∨ L ≤ z ∧ j = 0) ∧ ((S k ≤ M ∧ (S u ≤ S (S k · bc) ∧ (∃ v. bb = v · S (S k · bc) + u)) ∨ M ≤ k ∧ u = 0) ∧ m = j · u)))) ∧ (∃ z. ∃ m. 1 ≤ S (1 · m) ∧ (∃ k. z = k · S (1 · m) + 0) ∧ (S n ≤ S (S S i · m) ∧ (∃ k. z = k · S (S S i · m) + n) ∧ (∀ k. S k ≤ S i → ∃ j. ∃ u. ∃ v. S j ≤ S (S k · y) ∧ (∃ w. x = w · S (S k · y) + j) ∧ (S u ≤ S (S k · m) ∧ (∃ w. z = w · S (S k · m) + u) ∧ (S v ≤ S (S S k · m) ∧ (∃ w. z = w · S (S S k · m) + v) ∧ v = u + j)))))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 10 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 5 ] ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 8 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 12 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 12 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 15 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 14 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 15 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 6 ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 6 ] ] ], [ "var", 1 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 7 ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0401" ], "id": "ND0402", "name": "NatConvolutionPrefix", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "cb", "cc", "N" ], "source": "∀ x. S x ≤ N → ∃ y. S y ≤ S (S x · cc) ∧ (∃ z. cb = z · S (S x · cc) + y) ∧ (∃ z. ∃ n. (∀ m. S m ≤ S x → ∃ k. S k ≤ S (S m · n) ∧ (∃ i. z = i · S (S m · n) + k) ∧ (∃ i. ∃ j. ∃ u. m + i = x ∧ ((S m ≤ L ∧ (S j ≤ S (S m · ac) ∧ (∃ v. ab = v · S (S m · ac) + j)) ∨ L ≤ m ∧ j = 0) ∧ ((S i ≤ M ∧ (S u ≤ S (S i · bc) ∧ (∃ v. bb = v · S (S i · bc) + u)) ∨ M ≤ i ∧ u = 0) ∧ k = j · u)))) ∧ (∃ m. ∃ k. 1 ≤ S (1 · k) ∧ (∃ i. m = i · S (1 · k) + 0) ∧ (S y ≤ S (S S x · k) ∧ (∃ i. m = i · S (S S x · k) + y) ∧ (∀ i. S i ≤ S x → ∃ j. ∃ u. ∃ v. S j ≤ S (S i · n) ∧ (∃ w. z = w · S (S i · n) + j) ∧ (S u ≤ S (S i · k) ∧ (∃ w. m = w · S (S i · k) + u) ∧ (S v ≤ S (S S i · k) ∧ (∃ w. m = w · S (S S i · k) + v) ∧ v = u + j))))))" }, { "ast": [ "or", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 10 ], [ "var", 2 ] ], [ "var", 9 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "var", 6 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 6 ] ], [ "var", 11 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 9 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 9 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 11 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "var", 8 ] ] ], [ "eq", [ "var", 8 ], [ "zero" ] ] ] ], "dependencies": [ "ND0292", "PD0002" ], "id": "ND0430", "name": "NatCauchyCell", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "k", "j", "z" ], "source": "(∃ x. ∃ y. ∃ n. j + x = k ∧ ((S j ≤ L ∧ (S y ≤ S (S j · ac) ∧ (∃ m. ab = m · S (S j · ac) + y)) ∨ L ≤ j ∧ y = 0) ∧ ((S x ≤ M ∧ (S n ≤ S (S x · bc) ∧ (∃ m. bb = m · S (S x · bc) + n)) ∨ M ≤ x ∧ n = 0) ∧ z = y · n))) ∨ S k ≤ j ∧ z = 0" }, { "ast": [ "exists", [ "exists", [ "and", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 1 ] ], [ "var", 0 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 9 ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 3 ], [ "var", 2 ] ], [ "var", 4 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 8 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 8 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 11 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 11 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 13 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 1 ] ] ], [ "eq", [ "var", 10 ], [ "zero" ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "ND0430" ], "id": "ND0431", "name": "NatCauchyFlatCell", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "N", "v", "z" ], "source": "∃ x. ∃ y. v = S N · x + y ∧ (S y ≤ S N ∧ ((∃ n. ∃ m. ∃ k. y + n = x ∧ ((S y ≤ L ∧ (S m ≤ S (S y · ac) ∧ (∃ i. ab = i · S (S y · ac) + m)) ∨ L ≤ y ∧ m = 0) ∧ ((S n ≤ M ∧ (S k ≤ S (S n · bc) ∧ (∃ i. bb = i · S (S n · bc) + k)) ∨ M ≤ n ∧ k = 0) ∧ z = m · k))) ∨ S x ≤ y ∧ z = 0))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 11 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "succ", [ "var", 10 ] ], [ "var", 1 ] ], [ "var", 0 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 11 ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 3 ], [ "var", 2 ] ], [ "var", 4 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 10 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 10 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 13 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 13 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 5 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 1 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0431" ], "id": "ND0432", "name": "NatCauchyFlatPrefix", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "N", "gb", "gc", "T" ], "source": "∀ x. S x ≤ T → ∃ y. S y ≤ S (S x · gc) ∧ (∃ z. gb = z · S (S x · gc) + y) ∧ (∃ z. ∃ n. x = S N · z + n ∧ (S n ≤ S N ∧ ((∃ m. ∃ k. ∃ i. n + m = z ∧ ((S n ≤ L ∧ (S k ≤ S (S n · ac) ∧ (∃ j. ab = j · S (S n · ac) + k)) ∨ L ≤ n ∧ k = 0) ∧ ((S m ≤ M ∧ (S i ≤ S (S m · bc) ∧ (∃ j. bb = j · S (S m · bc) + i)) ∨ M ≤ m ∧ i = 0) ∧ y = k · i))) ∨ S z ≤ n ∧ y = 0)))" }, { "ast": [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "var", 10 ] ] ] ], [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "var", 10 ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "mul", [ "succ", [ "var", 10 ] ], [ "var", 3 ] ], [ "var", 2 ] ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "mul", [ "succ", [ "var", 10 ] ], [ "var", 3 ] ], [ "var", 2 ] ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 5 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 9 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 8 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 9 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 12 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 12 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 2 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0430" ], "id": "ND0433", "name": "NatCauchyGrid", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "N", "gb", "gc" ], "source": "∀ x. ∀ y. ∀ z. S x ≤ S N → S y ≤ S N → S z ≤ S (S (S N · x + y) · gc) ∧ (∃ n. gb = n · S (S (S N · x + y) · gc) + z) → (∃ n. ∃ m. ∃ k. y + n = x ∧ ((S y ≤ L ∧ (S m ≤ S (S y · ac) ∧ (∃ i. ab = i · S (S y · ac) + m)) ∨ L ≤ y ∧ m = 0) ∧ ((S n ≤ M ∧ (S k ≤ S (S n · bc) ∧ (∃ i. bb = i · S (S n · bc) + k)) ∨ M ≤ n ∧ k = 0) ∧ z = m · k))) ∨ S x ≤ y ∧ z = 0" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 14 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "mul", [ "var", 6 ], [ "var", 0 ] ], [ "add", [ "mul", [ "var", 4 ], [ "var", 2 ] ], [ "mul", [ "var", 5 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0409", "name": "NatLinearTriple", "parameters": [ "alpha", "beta", "gamma", "ab", "ac", "bb", "bc", "cb", "cc", "L" ], "source": "∀ x. ∀ y. ∀ z. ∀ n. S x ≤ L → S y ≤ S (S x · ac) ∧ (∃ m. ab = m · S (S x · ac) + y) → S z ≤ S (S x · bc) ∧ (∃ m. bb = m · S (S x · bc) + z) → S n ≤ S (S x · cc) ∧ (∃ m. cb = m · S (S x · cc) + n) → gamma · n = alpha · y + beta · z" }, { "ast": [ "exists", [ "exists", [ "and", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 9 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 7 ], [ "mul", [ "var", 8 ], [ "var", 2 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "var", 7 ], [ "mul", [ "var", 8 ], [ "var", 2 ] ] ] ], [ "var", 6 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 1 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 10 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0411", "PD0015" ], "id": "ND0412", "name": "NatAffineSum", "parameters": [ "ab", "ac", "o", "t", "N", "z" ], "source": "∃ x. ∃ y. (∀ n. ∀ m. S n ≤ N → S m ≤ S (S n · y) ∧ (∃ k. x = k · S (S n · y) + m) → S m ≤ S (S (o + t · n) · ac) ∧ (∃ k. ab = k · S (S (o + t · n) · ac) + m)) ∧ (∃ n. ∃ m. 1 ≤ S (1 · m) ∧ (∃ k. n = k · S (1 · m) + 0) ∧ (S z ≤ S (S N · m) ∧ (∃ k. n = k · S (S N · m) + z) ∧ (∀ k. S k ≤ N → ∃ i. ∃ j. ∃ u. S i ≤ S (S k · y) ∧ (∃ v. x = v · S (S k · y) + i) ∧ (S j ≤ S (S k · m) ∧ (∃ v. n = v · S (S k · m) + j) ∧ (S u ≤ S (S S k · m) ∧ (∃ v. n = v · S (S S k · m) + u) ∧ u = j + i)))))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 13 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 9 ], [ "mul", [ "var", 10 ], [ "var", 6 ] ] ], [ "mul", [ "var", 11 ], [ "var", 2 ] ] ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "add", [ "var", 9 ], [ "mul", [ "var", 10 ], [ "var", 6 ] ] ], [ "mul", [ "var", 11 ], [ "var", 2 ] ] ] ], [ "var", 8 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 1 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 14 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0412" ], "id": "ND0413", "name": "NatAffineRowSums", "parameters": [ "ab", "ac", "o", "s", "t", "m", "n", "rb", "rc" ], "source": "∀ x. S x ≤ m → ∃ y. S y ≤ S (S x · rc) ∧ (∃ z. rb = z · S (S x · rc) + y) ∧ (∃ z. ∃ k. (∀ i. ∀ j. S i ≤ n → S j ≤ S (S i · k) ∧ (∃ u. z = u · S (S i · k) + j) → S j ≤ S (S (o + s · x + t · i) · ac) ∧ (∃ u. ab = u · S (S (o + s · x + t · i) · ac) + j)) ∧ (∃ i. ∃ j. 1 ≤ S (1 · j) ∧ (∃ u. i = u · S (1 · j) + 0) ∧ (S y ≤ S (S n · j) ∧ (∃ u. i = u · S (S n · j) + y) ∧ (∀ u. S u ≤ n → ∃ v. ∃ w. ∃ x0. S v ≤ S (S u · k) ∧ (∃ x1. z = x1 · S (S u · k) + v) ∧ (S w ≤ S (S u · j) ∧ (∃ x1. i = x1 · S (S u · j) + w) ∧ (S x0 ≤ S (S S u · j) ∧ (∃ x1. i = x1 · S (S S u · j) + x0) ∧ x0 = w + v))))))" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 12 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 11 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 8 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 8 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 11 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 11 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "var", 2 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0430" ], "id": "ND0434", "name": "NatCauchyRow", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "k", "ub", "uc", "T" ], "source": "∀ x. ∀ y. S x ≤ T → S y ≤ S (S x · uc) ∧ (∃ z. ub = z · S (S x · uc) + y) → (∃ z. ∃ n. ∃ m. x + z = k ∧ ((S x ≤ L ∧ (S n ≤ S (S x · ac) ∧ (∃ i. ab = i · S (S x · ac) + n)) ∨ L ≤ x ∧ n = 0) ∧ ((S z ≤ M ∧ (S m ≤ S (S z · bc) ∧ (∃ i. bb = i · S (S z · bc) + m)) ∨ M ≤ z ∧ m = 0) ∧ y = n · m))) ∨ S k ≤ x ∧ y = 0" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 12 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 11 ], [ "var", 2 ] ], [ "var", 4 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "var", 8 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 12 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 12 ] ], [ "var", 7 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 8 ] ], [ "var", 12 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 11 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 11 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 9 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0430" ], "id": "ND0435", "name": "NatCauchyColumn", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "j", "ub", "uc", "T" ], "source": "∀ x. ∀ y. S x ≤ T → S y ≤ S (S x · uc) ∧ (∃ z. ub = z · S (S x · uc) + y) → (∃ z. ∃ n. ∃ m. j + z = x ∧ ((S j ≤ L ∧ (S n ≤ S (S j · ac) ∧ (∃ k. ab = k · S (S j · ac) + n)) ∨ L ≤ j ∧ n = 0) ∧ ((S z ≤ M ∧ (S m ≤ S (S z · bc) ∧ (∃ k. bb = k · S (S z · bc) + m)) ∨ M ≤ z ∧ m = 0) ∧ y = n · m))) ∨ S x ≤ j ∧ y = 0" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 6 ] ], [ "var", 2 ] ] ], [ "imp", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 5 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 5 ] ], [ "var", 2 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ], "dependencies": [ "PD0001", "ND0291" ], "id": "ND0436", "name": "NatZeroAfter", "parameters": [ "b", "c", "L", "r" ], "source": "∀ x. ∀ y. r ≤ x → S x ≤ L ∧ (S y ≤ S (S x · c) ∧ (∃ z. b = z · S (S x · c) + y)) ∨ L ≤ x ∧ y = 0 → y = 0" }, { "ast": [ "or", [ "and", [ "eq", [ "var", 0 ], [ "zero" ] ], [ "eq", [ "var", 1 ], [ "succ", [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 1 ], [ "succ", [ "var", 0 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ] ], "dependencies": [], "id": "ND0403", "name": "NatUnitCoefficient", "parameters": [ "i", "a" ], "source": "i = 0 ∧ a = 1 ∨ (∃ x. i = S x ∧ a = 0)" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "and", [ "eq", [ "var", 1 ], [ "zero" ] ], [ "eq", [ "var", 0 ], [ "succ", [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "var", 0 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0403" ], "id": "ND0404", "name": "NatUnitPrefix", "parameters": [ "ub", "uc", "H" ], "source": "∀ x. S x ≤ H → ∃ y. S y ≤ S (S x · uc) ∧ (∃ z. ub = z · S (S x · uc) + y) ∧ (x = 0 ∧ y = 1 ∨ (∃ z. x = S z ∧ y = 0))" }, { "ast": [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 2 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 4 ] ] ] ], [ "var", 7 ] ] ] ] ] ], "dependencies": [ "PD0013" ], "id": "ND0405", "name": "NatPowerRow", "parameters": [ "pb", "pc", "qb", "qc", "k", "ub", "uc" ], "source": "S ub ≤ S (S k · pc) ∧ (∃ x. pb = x · S (S k · pc) + ub) ∧ (S uc ≤ S (S k · qc) ∧ (∃ x. qb = x · S (S k · qc) + uc))" }, { "ast": [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 10 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 12 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 10 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 9 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 5 ] ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 8 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 16 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 15 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 16 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 17 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 17 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 6 ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 6 ] ] ], [ "var", 1 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 7 ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0405", "ND0402" ], "id": "ND0406", "name": "NatPowerStep", "parameters": [ "ab", "ac", "L", "H", "pb", "pc", "qb", "qc", "k" ], "source": "∃ x. ∃ y. ∃ z. ∃ n. S x ≤ S (S k · pc) ∧ (∃ m. pb = m · S (S k · pc) + x) ∧ (S y ≤ S (S k · qc) ∧ (∃ m. qb = m · S (S k · qc) + y)) ∧ (S z ≤ S (S S k · pc) ∧ (∃ m. pb = m · S (S S k · pc) + z) ∧ (S n ≤ S (S S k · qc) ∧ (∃ m. qb = m · S (S S k · qc) + n)) ∧ (∀ m. S m ≤ H → ∃ i. S i ≤ S (S m · n) ∧ (∃ j. z = j · S (S m · n) + i) ∧ (∃ j. ∃ u. (∀ v. S v ≤ S m → ∃ w. S w ≤ S (S v · u) ∧ (∃ x0. j = x0 · S (S v · u) + w) ∧ (∃ x0. ∃ x1. ∃ x2. v + x0 = m ∧ ((S v ≤ L ∧ (S x1 ≤ S (S v · ac) ∧ (∃ x3. ab = x3 · S (S v · ac) + x1)) ∨ L ≤ v ∧ x1 = 0) ∧ ((S x0 ≤ H ∧ (S x2 ≤ S (S x0 · y) ∧ (∃ x3. x = x3 · S (S x0 · y) + x2)) ∨ H ≤ x0 ∧ x2 = 0) ∧ w = x1 · x2)))) ∧ (∃ v. ∃ w. 1 ≤ S (1 · w) ∧ (∃ x0. v = x0 · S (1 · w) + 0) ∧ (S i ≤ S (S S m · w) ∧ (∃ x0. v = x0 · S (S S m · w) + i) ∧ (∀ x0. S x0 ≤ S m → ∃ x1. ∃ x2. ∃ x3. S x1 ≤ S (S x0 · u) ∧ (∃ x4. j = x4 · S (S x0 · u) + x1) ∧ (S x2 ≤ S (S x0 · w) ∧ (∃ x4. v = x4 · S (S x0 · w) + x2) ∧ (S x3 ≤ S (S S x0 · w) ∧ (∃ x4. v = x4 · S (S S x0 · w) + x3) ∧ x3 = x2 + x1))))))))" }, { "ast": [ "and", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 8 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 7 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "and", [ "eq", [ "var", 1 ], [ "zero" ] ], [ "eq", [ "var", 0 ], [ "succ", [ "zero" ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 2 ], [ "succ", [ "var", 0 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 10 ] ] ], [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 13 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 5 ] ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 5 ] ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 5 ] ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 5 ] ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 10 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 5 ] ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 3 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 4 ], [ "var", 2 ] ], [ "var", 8 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 17 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 16 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 17 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "var", 18 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 18 ] ], [ "var", 3 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 3 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 6 ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 6 ] ] ], [ "var", 1 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 7 ] ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 7 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "var", 1 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0405", "ND0404", "PD0002", "ND0406" ], "id": "ND0407", "name": "NatPowerTrace", "parameters": [ "ab", "ac", "L", "H", "pb", "pc", "qb", "qc", "K" ], "source": "(∃ x. ∃ y. S x ≤ S (1 · pc) ∧ (∃ z. pb = z · S (1 · pc) + x) ∧ (S y ≤ S (1 · qc) ∧ (∃ z. qb = z · S (1 · qc) + y)) ∧ (∀ z. S z ≤ H → ∃ n. S n ≤ S (S z · y) ∧ (∃ m. x = m · S (S z · y) + n) ∧ (z = 0 ∧ n = 1 ∨ (∃ m. z = S m ∧ n = 0)))) ∧ (∀ x. S x ≤ K → ∃ y. ∃ z. ∃ n. ∃ m. S y ≤ S (S x · pc) ∧ (∃ k. pb = k · S (S x · pc) + y) ∧ (S z ≤ S (S x · qc) ∧ (∃ k. qb = k · S (S x · qc) + z)) ∧ (S n ≤ S (S S x · pc) ∧ (∃ k. pb = k · S (S S x · pc) + n) ∧ (S m ≤ S (S S x · qc) ∧ (∃ k. qb = k · S (S S x · qc) + m)) ∧ (∀ k. S k ≤ H → ∃ i. S i ≤ S (S k · m) ∧ (∃ j. n = j · S (S k · m) + i) ∧ (∃ j. ∃ u. (∀ v. S v ≤ S k → ∃ w. S w ≤ S (S v · u) ∧ (∃ x0. j = x0 · S (S v · u) + w) ∧ (∃ x0. ∃ x1. ∃ x2. v + x0 = k ∧ ((S v ≤ L ∧ (S x1 ≤ S (S v · ac) ∧ (∃ x3. ab = x3 · S (S v · ac) + x1)) ∨ L ≤ v ∧ x1 = 0) ∧ ((S x0 ≤ H ∧ (S x2 ≤ S (S x0 · z) ∧ (∃ x3. y = x3 · S (S x0 · z) + x2)) ∨ H ≤ x0 ∧ x2 = 0) ∧ w = x1 · x2)))) ∧ (∃ v. ∃ w. 1 ≤ S (1 · w) ∧ (∃ x0. v = x0 · S (1 · w) + 0) ∧ (S i ≤ S (S S k · w) ∧ (∃ x0. v = x0 · S (S S k · w) + i) ∧ (∀ x0. S x0 ≤ S k → ∃ x1. ∃ x2. ∃ x3. S x1 ≤ S (S x0 · u) ∧ (∃ x4. j = x4 · S (S x0 · u) + x1) ∧ (S x2 ≤ S (S x0 · w) ∧ (∃ x4. v = x4 · S (S x0 · w) + x2) ∧ (S x3 ≤ S (S S x0 · w) ∧ (∃ x4. v = x4 · S (S S x0 · w) + x3) ∧ x3 = x2 + x1)))))))))" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 7 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 6 ] ] ] ], [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0410", "name": "NatEulerPrefix", "parameters": [ "ab", "ac", "db", "dc", "L" ], "source": "∀ x. ∀ y. S x ≤ L → S y ≤ S (S x · ac) ∧ (∃ z. ab = z · S (S x · ac) + y) → S (x · y) ≤ S (S x · dc) ∧ (∃ z. db = z · S (S x · dc) + x · y)" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 10 ] ] ], [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 1 ] ] ] ], [ "zero" ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 13 ] ] ], [ "exists", [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 4 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 7 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "add", [ "mul", [ "var", 14 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 4 ] ] ] ], [ "add", [ "mul", [ "var", 14 ], [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 12 ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 12 ] ], [ "var", 1 ] ] ] ], [ "var", 3 ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "ND0405", "ND0437", "PD0013" ], "id": "ND0438", "name": "NatRowHornerPrefix", "parameters": [ "pb", "pc", "qb", "qc", "t", "H", "vb", "vc", "K" ], "source": "∀ x. S x ≤ K → ∃ y. ∃ z. ∃ n. S y ≤ S (S x · pc) ∧ (∃ m. pb = m · S (S x · pc) + y) ∧ (S z ≤ S (S x · qc) ∧ (∃ m. qb = m · S (S x · qc) + z)) ∧ ((∃ m. ∃ k. 1 ≤ S (1 · k) ∧ (∃ i. m = i · S (1 · k) + 0) ∧ (∀ i. S i ≤ H → ∃ j. ∃ u. S j ≤ S (S i · k) ∧ (∃ v. m = v · S (S i · k) + j) ∧ (S u ≤ S (S i · z) ∧ (∃ v. y = v · S (S i · z) + u) ∧ (S (t · j + u) ≤ S (S S i · k) ∧ (∃ v. m = v · S (S S i · k) + (t · j + u))))) ∧ (S n ≤ S (S H · k) ∧ (∃ i. m = i · S (S H · k) + n))) ∧ (S n ≤ S (S x · vc) ∧ (∃ m. vb = m · S (S x · vc) + n)))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ] ], [ "exists", [ "eq", [ "var", 2 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 3 ] ] ] ], [ "zero" ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0419", "name": "NatZeroPrefix", "parameters": [ "ub", "uc", "H" ], "source": "∀ x. S x ≤ H → 1 ≤ S (S x · uc) ∧ (∃ y. ub = y · S (S x · uc) + 0)" }, { "ast": [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 12 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 14 ] ] ] ], [ "var", 6 ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 16 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 18 ] ] ] ], [ "var", 4 ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 19 ] ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 19 ] ] ], [ "var", 16 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 19 ] ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 19 ] ] ], [ "var", 18 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 19 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 12 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "mul", [ "succ", [ "zero" ] ], [ "var", 0 ] ], [ "add", [ "mul", [ "succ", [ "zero" ] ], [ "var", 2 ] ], [ "mul", [ "var", 4 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0405", "PD0013", "ND0409" ], "id": "ND0420", "name": "NatWeightedSeriesStep", "parameters": [ "H", "wb", "wc", "pb", "pc", "qb", "qc", "tb", "tc", "vb", "vc", "k" ], "source": "∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ i. ∃ j. S x ≤ S (S k · pc) ∧ (∃ u. pb = u · S (S k · pc) + x) ∧ (S y ≤ S (S k · qc) ∧ (∃ u. qb = u · S (S k · qc) + y)) ∧ (S z ≤ S (S k · tc) ∧ (∃ u. tb = u · S (S k · tc) + z) ∧ (S n ≤ S (S k · vc) ∧ (∃ u. vb = u · S (S k · vc) + n)) ∧ (S m ≤ S (S S k · tc) ∧ (∃ u. tb = u · S (S S k · tc) + m) ∧ (S i ≤ S (S S k · vc) ∧ (∃ u. vb = u · S (S S k · vc) + i)) ∧ (S j ≤ S (S k · wc) ∧ (∃ u. wb = u · S (S k · wc) + j) ∧ (∀ u. ∀ v. ∀ w. ∀ x0. S u ≤ H → S v ≤ S (S u · n) ∧ (∃ x1. z = x1 · S (S u · n) + v) → S w ≤ S (S u · y) ∧ (∃ x1. x = x1 · S (S u · y) + w) → S x0 ≤ S (S u · i) ∧ (∃ x1. m = x1 · S (S u · i) + x0) → 1 · x0 = 1 · v + j · w))))" }, { "ast": [ "and", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 4 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 1 ] ], [ "var", 2 ] ] ] ], [ "zero" ] ] ] ] ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 13 ] ] ], [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 13 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 15 ] ] ] ], [ "var", 6 ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 17 ] ] ] ] ], [ "exists", [ "eq", [ "var", 16 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 17 ] ] ] ], [ "var", 5 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 19 ] ] ] ] ], [ "exists", [ "eq", [ "var", 18 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 19 ] ] ] ], [ "var", 4 ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 8 ] ] ], [ "var", 17 ] ] ] ] ], [ "exists", [ "eq", [ "var", 16 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 8 ] ] ], [ "var", 17 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 8 ] ] ], [ "var", 19 ] ] ] ] ], [ "exists", [ "eq", [ "var", 18 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 8 ] ] ], [ "var", 19 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 8 ] ], [ "var", 11 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 13 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "mul", [ "succ", [ "zero" ] ], [ "var", 0 ] ], [ "add", [ "mul", [ "succ", [ "zero" ] ], [ "var", 2 ] ], [ "mul", [ "var", 4 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0405", "ND0419", "PD0002", "ND0420" ], "id": "ND0421", "name": "NatWeightedSeriesTrace", "parameters": [ "H", "wb", "wc", "pb", "pc", "qb", "qc", "tb", "tc", "vb", "vc", "K" ], "source": "(∃ x. ∃ y. S x ≤ S (1 · tc) ∧ (∃ z. tb = z · S (1 · tc) + x) ∧ (S y ≤ S (1 · vc) ∧ (∃ z. vb = z · S (1 · vc) + y)) ∧ (∀ z. S z ≤ H → 1 ≤ S (S z · y) ∧ (∃ n. x = n · S (S z · y) + 0))) ∧ (∀ x. S x ≤ K → ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ i. ∃ j. S y ≤ S (S x · pc) ∧ (∃ u. pb = u · S (S x · pc) + y) ∧ (S z ≤ S (S x · qc) ∧ (∃ u. qb = u · S (S x · qc) + z)) ∧ (S n ≤ S (S x · tc) ∧ (∃ u. tb = u · S (S x · tc) + n) ∧ (S m ≤ S (S x · vc) ∧ (∃ u. vb = u · S (S x · vc) + m)) ∧ (S k ≤ S (S S x · tc) ∧ (∃ u. tb = u · S (S S x · tc) + k) ∧ (S i ≤ S (S S x · vc) ∧ (∃ u. vb = u · S (S S x · vc) + i)) ∧ (S j ≤ S (S x · wc) ∧ (∃ u. wb = u · S (S x · wc) + j) ∧ (∀ u. ∀ v. ∀ w. ∀ x0. S u ≤ H → S v ≤ S (S u · m) ∧ (∃ x1. n = x1 · S (S u · m) + v) → S w ≤ S (S u · z) ∧ (∃ x1. y = x1 · S (S u · z) + w) → S x0 ≤ S (S u · i) ∧ (∃ x1. k = x1 · S (S u · i) + x0) → 1 · x0 = 1 · v + j · w)))))" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 7 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 2 ] ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 2 ] ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 1 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 6 ] ] ] ], [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0422", "name": "NatShiftedEulerPrefix", "parameters": [ "ab", "ac", "db", "dc", "N" ], "source": "∀ x. ∀ y. S x ≤ N → S y ≤ S (S S x · ac) ∧ (∃ z. ab = z · S (S S x · ac) + y) → S (S x · y) ≤ S (S x · dc) ∧ (∃ z. db = z · S (S x · dc) + S x · y)" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 11 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 8 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 10 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "mul", [ "var", 2 ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013" ], "id": "ND0439", "name": "NatPointwiseProductPrefix", "parameters": [ "wb", "wc", "vb", "vc", "zb", "zc", "K" ], "source": "∀ x. ∀ y. ∀ z. ∀ n. S x ≤ K → S y ≤ S (S x · wc) ∧ (∃ m. wb = m · S (S x · wc) + y) → S z ≤ S (S x · vc) ∧ (∃ m. vb = m · S (S x · vc) + z) → S n ≤ S (S x · zc) ∧ (∃ m. zb = m · S (S x · zc) + n) → n = y · z" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "var", 6 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 4 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 1 ] ], [ "var", 5 ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "PD0001" ], "id": "ND0440", "name": "NatBoundedPrefix", "parameters": [ "ab", "ac", "D", "N" ], "source": "∀ x. ∀ y. S x ≤ N → S y ≤ S (S x · ac) ∧ (∃ z. ab = z · S (S x · ac) + y) → y ≤ D" }, { "ast": [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "var", 3 ] ] ] ], [ "var", 7 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 3 ] ], [ "var", 5 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 5 ] ] ] ] ], [ "exists", [ "eq", [ "var", 4 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 5 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 1 ] ] ], [ "mul", [ "var", 6 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "PD0001" ], "id": "ND0441", "name": "NatRatioBoundPrefix", "parameters": [ "ab", "ac", "r", "N" ], "source": "∀ x. ∀ y. ∀ z. S S x ≤ N → S y ≤ S (S x · ac) ∧ (∃ n. ab = n · S (S x · ac) + y) → S z ≤ S (S S x · ac) ∧ (∃ n. ab = n · S (S S x · ac) + z) → S r · z ≤ r · y" }, { "ast": [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 2 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 4 ] ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 2 ] ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 4 ] ] ] ] ] ], [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "succ", [ "var", 4 ] ] ] ] ], [ "var", 7 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 6 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 4 ] ] ], [ "var", 6 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "var", 4 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "succ", [ "var", 4 ] ] ] ], [ "var", 6 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "eq", [ "mul", [ "succ", [ "succ", [ "var", 3 ] ] ], [ "var", 0 ] ], [ "add", [ "mul", [ "var", 3 ], [ "var", 2 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "var", 1 ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0013", "PD0002" ], "id": "ND0400", "name": "NatExpAtanhRecurrence", "parameters": [ "b", "c", "L", "D" ], "source": "S D ≤ S (1 · c) ∧ (∃ x. b = x · S (1 · c) + D) ∧ (S (2 · D) ≤ S (2 · c) ∧ (∃ x. b = x · S (2 · c) + 2 · D) ∧ (∀ x. ∀ y. ∀ z. ∀ n. S S S x ≤ L → S y ≤ S (S x · c) ∧ (∃ m. b = m · S (S x · c) + y) → S z ≤ S (S S x · c) ∧ (∃ m. b = m · S (S S x · c) + z) → S n ≤ S (S S S x · c) ∧ (∃ m. b = m · S (S S S x · c) + n) → S S x · n = x · y + 2 · z))" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 14 ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "mul", [ "var", 7 ], [ "var", 1 ] ] ], [ "add", [ "mul", [ "var", 5 ], [ "var", 3 ] ], [ "mul", [ "var", 6 ], [ "var", 2 ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "PD0001" ], "id": "ND0445", "name": "NatLinearBound", "parameters": [ "alpha", "beta", "gamma", "ab", "ac", "bb", "bc", "cb", "cc", "N" ], "source": "∀ x. ∀ y. ∀ z. ∀ n. S x ≤ N → S y ≤ S (S x · ac) ∧ (∃ m. ab = m · S (S x · ac) + y) → S z ≤ S (S x · bc) ∧ (∃ m. bb = m · S (S x · bc) + z) → S n ≤ S (S x · cc) ∧ (∃ m. cb = m · S (S x · cc) + n) → gamma · n ≤ alpha · y + beta · z" }, { "ast": [ "or", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 14 ], [ "add", [ "var", 3 ], [ "var", 15 ] ] ], [ "var", 13 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 15 ] ] ], [ "var", 7 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 15 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 15 ] ], [ "var", 6 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 7 ] ], [ "var", 15 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 10 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 9 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 10 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 16 ] ] ], [ "var", 13 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 16 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 16 ] ], [ "var", 12 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 13 ] ], [ "var", 16 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 16 ], [ "mul", [ "var", 2 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "add", [ "var", 11 ], [ "var", 12 ] ] ] ], [ "eq", [ "var", 12 ], [ "zero" ] ] ] ], "dependencies": [ "ND0291", "PD0002" ], "id": "ND0414", "name": "NatTripleCell", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "cb", "cc", "K", "i", "j", "l", "z" ], "source": "(∃ x. ∃ y. ∃ n. ∃ m. j + (x + l) = i ∧ ((S j ≤ L ∧ (S y ≤ S (S j · ac) ∧ (∃ k. ab = k · S (S j · ac) + y)) ∨ L ≤ j ∧ y = 0) ∧ ((S x ≤ M ∧ (S n ≤ S (S x · bc) ∧ (∃ k. bb = k · S (S x · bc) + n)) ∨ M ≤ x ∧ n = 0) ∧ ((S l ≤ K ∧ (S m ≤ S (S l · cc) ∧ (∃ k. cb = k · S (S l · cc) + m)) ∨ K ≤ l ∧ m = 0) ∧ z = y · (n · m))))) ∨ S i ≤ j + l ∧ z = 0" }, { "ast": [ "exists", [ "exists", [ "and", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "succ", [ "var", 11 ] ], [ "var", 1 ] ], [ "var", 0 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 12 ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 5 ], [ "add", [ "var", 3 ], [ "var", 4 ] ] ], [ "var", 15 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "var", 9 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 9 ] ], [ "var", 6 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 12 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 12 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 15 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 14 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 15 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 17 ], [ "mul", [ "var", 2 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "add", [ "var", 2 ], [ "var", 1 ] ] ] ], [ "eq", [ "var", 13 ], [ "zero" ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "ND0414" ], "id": "ND0415", "name": "NatTripleFlatCell", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "cb", "cc", "K", "i", "t", "z" ], "source": "∃ x. ∃ y. t = S i · x + y ∧ (S y ≤ S i ∧ ((∃ n. ∃ m. ∃ k. ∃ j. x + (n + y) = i ∧ ((S x ≤ L ∧ (S m ≤ S (S x · ac) ∧ (∃ u. ab = u · S (S x · ac) + m)) ∨ L ≤ x ∧ m = 0) ∧ ((S n ≤ M ∧ (S k ≤ S (S n · bc) ∧ (∃ u. bb = u · S (S n · bc) + k)) ∨ M ≤ n ∧ k = 0) ∧ ((S y ≤ K ∧ (S j ≤ S (S y · cc) ∧ (∃ u. cb = u · S (S y · cc) + j)) ∨ K ≤ y ∧ j = 0) ∧ z = m · (k · j))))) ∨ S i ≤ x + y ∧ z = 0))" }, { "ast": [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 14 ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 14 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "exists", [ "exists", [ "and", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 1 ] ], [ "var", 0 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 14 ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 5 ], [ "add", [ "var", 3 ], [ "var", 4 ] ] ], [ "var", 17 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "var", 11 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 10 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 11 ] ], [ "var", 6 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 14 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 13 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 14 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 17 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 16 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 17 ] ], [ "var", 5 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 6 ], [ "mul", [ "var", 2 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 14 ] ] ], [ "add", [ "var", 2 ], [ "var", 1 ] ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0415" ], "id": "ND0416", "name": "NatTripleFlatPrefix", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "cb", "cc", "K", "i", "gb", "gc", "N" ], "source": "∀ x. S x ≤ N → ∃ y. S y ≤ S (S x · gc) ∧ (∃ z. gb = z · S (S x · gc) + y) ∧ (∃ z. ∃ n. x = S i · z + n ∧ (S n ≤ S i ∧ ((∃ m. ∃ k. ∃ j. ∃ u. z + (m + n) = i ∧ ((S z ≤ L ∧ (S k ≤ S (S z · ac) ∧ (∃ v. ab = v · S (S z · ac) + k)) ∨ L ≤ z ∧ k = 0) ∧ ((S m ≤ M ∧ (S j ≤ S (S m · bc) ∧ (∃ v. bb = v · S (S m · bc) + j)) ∨ M ≤ m ∧ j = 0) ∧ ((S n ≤ K ∧ (S u ≤ S (S n · cc) ∧ (∃ v. cb = v · S (S n · cc) + u)) ∨ K ≤ n ∧ u = 0) ∧ y = k · (j · u))))) ∨ S i ≤ z + n ∧ y = 0)))" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "var", 12 ] ] ] ], [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "var", 12 ] ] ] ], [ "exists", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "add", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 3 ] ], [ "var", 2 ] ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "add", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 3 ] ], [ "var", 2 ] ] ], [ "var", 15 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 6 ], [ "add", [ "var", 3 ], [ "var", 5 ] ] ], [ "var", 16 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "var", 10 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 7 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 7 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 10 ] ], [ "var", 7 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 13 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 12 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 13 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "var", 16 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 15 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 16 ] ], [ "var", 6 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 4 ], [ "mul", [ "var", 2 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 13 ] ] ], [ "add", [ "var", 3 ], [ "var", 2 ] ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0414" ], "id": "ND0417", "name": "NatTripleGrid", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "cb", "cc", "K", "i", "gb", "gc" ], "source": "∀ x. ∀ y. S x ≤ S i → S y ≤ S i → ∃ z. S z ≤ S (S (S i · x + y) · gc) ∧ (∃ n. gb = n · S (S (S i · x + y) · gc) + z) ∧ ((∃ n. ∃ m. ∃ k. ∃ j. x + (n + y) = i ∧ ((S x ≤ L ∧ (S m ≤ S (S x · ac) ∧ (∃ u. ab = u · S (S x · ac) + m)) ∨ L ≤ x ∧ m = 0) ∧ ((S n ≤ M ∧ (S k ≤ S (S n · bc) ∧ (∃ u. bb = u · S (S n · bc) + k)) ∨ M ≤ n ∧ k = 0) ∧ ((S y ≤ K ∧ (S j ≤ S (S y · cc) ∧ (∃ u. cb = u · S (S y · cc) + j)) ∨ K ≤ y ∧ j = 0) ∧ z = m · (k · j))))) ∨ S i ≤ x + y ∧ z = 0)" }, { "ast": [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "var", 12 ] ] ] ], [ "imp", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 2 ] ], [ "var", 15 ] ] ] ], [ "var", 1 ] ] ] ] ], [ "or", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "eq", [ "add", [ "var", 16 ], [ "add", [ "var", 3 ], [ "var", 5 ] ] ], [ "var", 15 ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 17 ] ] ], [ "var", 9 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 17 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 17 ] ], [ "var", 8 ] ] ] ], [ "var", 3 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 9 ] ], [ "var", 17 ] ] ], [ "eq", [ "var", 2 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "var", 12 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 4 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 12 ] ], [ "var", 4 ] ] ], [ "eq", [ "var", 1 ], [ "zero" ] ] ] ], [ "and", [ "or", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "var", 15 ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 6 ] ], [ "var", 14 ] ] ] ], [ "var", 1 ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 15 ] ], [ "var", 6 ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ], [ "eq", [ "var", 4 ], [ "mul", [ "var", 2 ], [ "mul", [ "var", 1 ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "add", [ "var", 13 ], [ "var", 2 ] ] ] ], [ "eq", [ "var", 0 ], [ "zero" ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "PD0013", "ND0414" ], "id": "ND0418", "name": "NatTripleRowPrefix", "parameters": [ "ab", "ac", "L", "bb", "bc", "M", "cb", "cc", "K", "i", "j", "ub", "uc" ], "source": "∀ x. ∀ y. S x ≤ S i → S y ≤ S (S x · uc) ∧ (∃ z. ub = z · S (S x · uc) + y) → (∃ z. ∃ n. ∃ m. ∃ k. j + (z + x) = i ∧ ((S j ≤ L ∧ (S n ≤ S (S j · ac) ∧ (∃ u. ab = u · S (S j · ac) + n)) ∨ L ≤ j ∧ n = 0) ∧ ((S z ≤ M ∧ (S m ≤ S (S z · bc) ∧ (∃ u. bb = u · S (S z · bc) + m)) ∨ M ≤ z ∧ m = 0) ∧ ((S x ≤ K ∧ (S k ≤ S (S x · cc) ∧ (∃ u. cb = u · S (S x · cc) + k)) ∨ K ≤ x ∧ k = 0) ∧ y = n · (m · k))))) ∨ S i ≤ j + x ∧ y = 0" }, { "ast": [ "eq", [ "add", [ "var", 8 ], [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 5 ] ], [ "mul", [ "var", 1 ], [ "var", 4 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 7 ] ], [ "mul", [ "var", 3 ], [ "var", 6 ] ] ] ] ] ], [ "add", [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 4 ] ], [ "mul", [ "var", 1 ], [ "var", 5 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 6 ] ], [ "mul", [ "var", 3 ], [ "var", 7 ] ] ] ] ], [ "var", 9 ] ] ], "dependencies": [], "id": "ND0388", "name": "IQuadProductReal", "parameters": [ "ap", "an", "bp", "bn", "cp", "cn", "dp", "dn", "rp", "rn" ], "source": "rp + (ap · cn + an · cp + 2 · (bp · dn + bn · dp)) = ap · cp + an · cn + 2 · (bp · dp + bn · dn) + rn" }, { "ast": [ "eq", [ "add", [ "var", 8 ], [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 7 ] ], [ "mul", [ "var", 1 ], [ "var", 6 ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 5 ] ], [ "mul", [ "var", 3 ], [ "var", 4 ] ] ] ] ], [ "add", [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 6 ] ], [ "mul", [ "var", 1 ], [ "var", 7 ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 4 ] ], [ "mul", [ "var", 3 ], [ "var", 5 ] ] ] ], [ "var", 9 ] ] ], "dependencies": [], "id": "ND0389", "name": "IQuadProductRadical", "parameters": [ "ap", "an", "bp", "bn", "cp", "cn", "dp", "dn", "sp", "sn" ], "source": "sp + (ap · dn + an · dp + (bp · cn + bn · cp)) = ap · dp + an · dn + (bp · cp + bn · cn) + sn" }, { "ast": [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 2 ] ] ] ] ], [ "exists", [ "eq", [ "var", 1 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 2 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "var", 3 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 4 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 6 ] ] ] ] ], [ "exists", [ "eq", [ "var", 5 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 6 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 13 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 8 ] ] ] ] ], [ "exists", [ "eq", [ "var", 7 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 9 ] ], [ "var", 8 ] ] ] ], [ "var", 13 ] ] ] ] ] ] ] ], "dependencies": [ "PD0013" ], "id": "ND0390", "name": "IQuadAt", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "i", "ap", "an", "bp", "bn" ], "source": "S ap ≤ S (S i · ac) ∧ (∃ x. ab = x · S (S i · ac) + ap) ∧ (S an ≤ S (S i · bc) ∧ (∃ x. bb = x · S (S i · bc) + an) ∧ (S bp ≤ S (S i · cc) ∧ (∃ x. cb = x · S (S i · cc) + bp) ∧ (S bn ≤ S (S i · dc) ∧ (∃ x. db = x · S (S i · dc) + bn))))" }, { "ast": [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 8 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 14 ] ] ] ], [ "var", 8 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 16 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 18 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 20 ] ] ] ] ], [ "exists", [ "eq", [ "var", 19 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 20 ] ] ] ], [ "var", 5 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 22 ] ] ] ] ], [ "exists", [ "eq", [ "var", 21 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 22 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 24 ] ] ] ] ], [ "exists", [ "eq", [ "var", 23 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 24 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 26 ] ] ] ] ], [ "exists", [ "eq", [ "var", 25 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 26 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 28 ] ] ] ] ], [ "exists", [ "eq", [ "var", 27 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 28 ] ] ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 22 ] ] ] ] ], [ "exists", [ "eq", [ "var", 21 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 22 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 24 ] ] ] ] ], [ "exists", [ "eq", [ "var", 23 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 24 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 26 ] ] ] ] ], [ "exists", [ "eq", [ "var", 25 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 26 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 28 ] ] ] ] ], [ "exists", [ "eq", [ "var", 27 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 28 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "and", [ "eq", [ "add", [ "var", 3 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 6 ] ], [ "mul", [ "var", 10 ], [ "var", 7 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 4 ] ], [ "mul", [ "var", 8 ], [ "var", 5 ] ] ] ] ] ], [ "add", [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 7 ] ], [ "mul", [ "var", 10 ], [ "var", 6 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 5 ] ], [ "mul", [ "var", 8 ], [ "var", 4 ] ] ] ] ], [ "var", 2 ] ] ], [ "eq", [ "add", [ "var", 1 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 4 ] ], [ "mul", [ "var", 10 ], [ "var", 5 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 6 ] ], [ "mul", [ "var", 8 ], [ "var", 7 ] ] ] ] ], [ "add", [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 5 ] ], [ "mul", [ "var", 10 ], [ "var", 4 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 7 ] ], [ "mul", [ "var", 8 ], [ "var", 6 ] ] ] ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0390", "ND0388", "ND0389" ], "id": "ND0391", "name": "IQuadProductStep", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "ub", "uc", "vb", "vc", "wb", "wc", "xb", "xc", "i" ], "source": "∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ j. ∃ u. ∃ v. ∃ w. ∃ x0. ∃ x1. S m ≤ S (S i · ac) ∧ (∃ x2. ab = x2 · S (S i · ac) + m) ∧ (S k ≤ S (S i · bc) ∧ (∃ x2. bb = x2 · S (S i · bc) + k) ∧ (S j ≤ S (S i · cc) ∧ (∃ x2. cb = x2 · S (S i · cc) + j) ∧ (S u ≤ S (S i · dc) ∧ (∃ x2. db = x2 · S (S i · dc) + u)))) ∧ (S x ≤ S (S i · uc) ∧ (∃ x2. ub = x2 · S (S i · uc) + x) ∧ (S y ≤ S (S i · vc) ∧ (∃ x2. vb = x2 · S (S i · vc) + y) ∧ (S z ≤ S (S i · wc) ∧ (∃ x2. wb = x2 · S (S i · wc) + z) ∧ (S n ≤ S (S i · xc) ∧ (∃ x2. xb = x2 · S (S i · xc) + n)))) ∧ (S v ≤ S (S S i · uc) ∧ (∃ x2. ub = x2 · S (S S i · uc) + v) ∧ (S w ≤ S (S S i · vc) ∧ (∃ x2. vb = x2 · S (S S i · vc) + w) ∧ (S x0 ≤ S (S S i · wc) ∧ (∃ x2. wb = x2 · S (S S i · wc) + x0) ∧ (S x1 ≤ S (S S i · xc) ∧ (∃ x2. xb = x2 · S (S S i · xc) + x1)))) ∧ (v + (x · k + y · m + 2 · (z · u + n · j)) = x · m + y · k + 2 · (z · j + n · u) + w ∧ x0 + (x · u + y · j + (z · k + n · m)) = x · j + y · u + (z · m + n · k) + x1)))" }, { "ast": [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 12 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 14 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 16 ] ] ] ], [ "zero" ] ] ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 18 ] ] ], [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 8 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 15 ] ] ] ], [ "var", 8 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 17 ] ] ] ] ], [ "exists", [ "eq", [ "var", 16 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 17 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 19 ] ] ] ] ], [ "exists", [ "eq", [ "var", 18 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 19 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 21 ] ] ] ] ], [ "exists", [ "eq", [ "var", 20 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 21 ] ] ] ], [ "var", 5 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 23 ] ] ] ] ], [ "exists", [ "eq", [ "var", 22 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 23 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 25 ] ] ] ] ], [ "exists", [ "eq", [ "var", 24 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 25 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 27 ] ] ] ] ], [ "exists", [ "eq", [ "var", 26 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 27 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 29 ] ] ] ] ], [ "exists", [ "eq", [ "var", 28 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 29 ] ] ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 23 ] ] ] ] ], [ "exists", [ "eq", [ "var", 22 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 23 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 25 ] ] ] ] ], [ "exists", [ "eq", [ "var", 24 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 25 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 27 ] ] ] ] ], [ "exists", [ "eq", [ "var", 26 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 27 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 29 ] ] ] ] ], [ "exists", [ "eq", [ "var", 28 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 29 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "and", [ "eq", [ "add", [ "var", 3 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 6 ] ], [ "mul", [ "var", 10 ], [ "var", 7 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 4 ] ], [ "mul", [ "var", 8 ], [ "var", 5 ] ] ] ] ] ], [ "add", [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 7 ] ], [ "mul", [ "var", 10 ], [ "var", 6 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 5 ] ], [ "mul", [ "var", 8 ], [ "var", 4 ] ] ] ] ], [ "var", 2 ] ] ], [ "eq", [ "add", [ "var", 1 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 4 ] ], [ "mul", [ "var", 10 ], [ "var", 5 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 6 ] ], [ "mul", [ "var", 8 ], [ "var", 7 ] ] ] ] ], [ "add", [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 5 ] ], [ "mul", [ "var", 10 ], [ "var", 4 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 7 ] ], [ "mul", [ "var", 8 ], [ "var", 6 ] ] ] ], [ "var", 0 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0390", "PD0002", "ND0391" ], "id": "ND0392", "name": "IQuadProductTrace", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "ub", "uc", "vb", "vc", "wb", "wc", "xb", "xc", "L" ], "source": "2 ≤ S (1 · uc) ∧ (∃ x. ub = x · S (1 · uc) + 1) ∧ (1 ≤ S (1 · vc) ∧ (∃ x. vb = x · S (1 · vc) + 0) ∧ (1 ≤ S (1 · wc) ∧ (∃ x. wb = x · S (1 · wc) + 0) ∧ (1 ≤ S (1 · xc) ∧ (∃ x. xb = x · S (1 · xc) + 0)))) ∧ (∀ x. S x ≤ L → ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ i. ∃ j. ∃ u. ∃ v. ∃ w. ∃ x0. ∃ x1. S k ≤ S (S x · ac) ∧ (∃ x2. ab = x2 · S (S x · ac) + k) ∧ (S i ≤ S (S x · bc) ∧ (∃ x2. bb = x2 · S (S x · bc) + i) ∧ (S j ≤ S (S x · cc) ∧ (∃ x2. cb = x2 · S (S x · cc) + j) ∧ (S u ≤ S (S x · dc) ∧ (∃ x2. db = x2 · S (S x · dc) + u)))) ∧ (S y ≤ S (S x · uc) ∧ (∃ x2. ub = x2 · S (S x · uc) + y) ∧ (S z ≤ S (S x · vc) ∧ (∃ x2. vb = x2 · S (S x · vc) + z) ∧ (S n ≤ S (S x · wc) ∧ (∃ x2. wb = x2 · S (S x · wc) + n) ∧ (S m ≤ S (S x · xc) ∧ (∃ x2. xb = x2 · S (S x · xc) + m)))) ∧ (S v ≤ S (S S x · uc) ∧ (∃ x2. ub = x2 · S (S S x · uc) + v) ∧ (S w ≤ S (S S x · vc) ∧ (∃ x2. vb = x2 · S (S S x · vc) + w) ∧ (S x0 ≤ S (S S x · wc) ∧ (∃ x2. wb = x2 · S (S S x · wc) + x0) ∧ (S x1 ≤ S (S S x · xc) ∧ (∃ x2. xb = x2 · S (S S x · xc) + x1)))) ∧ (v + (y · i + z · k + 2 · (n · u + m · j)) = y · k + z · i + 2 · (n · j + m · u) + w ∧ x0 + (y · u + z · j + (n · i + m · k)) = y · j + z · u + (n · k + m · i) + x1))))" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 14 ] ] ], [ "imp", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "imp", [ "and", [ "eq", [ "var", 3 ], [ "var", 2 ] ], [ "eq", [ "var", 1 ], [ "var", 0 ] ] ], [ "bot" ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "ND0390" ], "id": "ND0393", "name": "IQuadNonzeroFactors", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "L" ], "source": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. S x ≤ L → S y ≤ S (S x · ac) ∧ (∃ k. ab = k · S (S x · ac) + y) ∧ (S z ≤ S (S x · bc) ∧ (∃ k. bb = k · S (S x · bc) + z) ∧ (S n ≤ S (S x · cc) ∧ (∃ k. cb = k · S (S x · cc) + n) ∧ (S m ≤ S (S x · dc) ∧ (∃ k. db = k · S (S x · dc) + m)))) → ¬(y = z ∧ n = m)" }, { "ast": [ "and", [ "eq", [ "var", 8 ], [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 4 ] ], [ "mul", [ "var", 1 ], [ "var", 5 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 6 ] ], [ "mul", [ "var", 3 ], [ "var", 7 ] ] ] ] ] ], [ "and", [ "eq", [ "var", 9 ], [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 5 ] ], [ "mul", [ "var", 1 ], [ "var", 4 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 7 ] ], [ "mul", [ "var", 3 ], [ "var", 6 ] ] ] ] ] ], [ "and", [ "eq", [ "var", 10 ], [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 6 ] ], [ "mul", [ "var", 1 ], [ "var", 7 ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 4 ] ], [ "mul", [ "var", 3 ], [ "var", 5 ] ] ] ] ], [ "eq", [ "var", 11 ], [ "add", [ "add", [ "mul", [ "var", 0 ], [ "var", 7 ] ], [ "mul", [ "var", 1 ], [ "var", 6 ] ] ], [ "add", [ "mul", [ "var", 2 ], [ "var", 5 ] ], [ "mul", [ "var", 3 ], [ "var", 4 ] ] ] ] ] ] ] ], "dependencies": [], "id": "ND0394", "name": "IQuadCanonicalProduct", "parameters": [ "ap", "an", "bp", "bn", "cp", "cn", "dp", "dn", "rp", "rn", "sp", "sn" ], "source": "rp = ap · cp + an · cn + 2 · (bp · dp + bn · dn) ∧ (rn = ap · cn + an · cp + 2 · (bp · dn + bn · dp) ∧ (sp = ap · dp + an · dn + (bp · cp + bn · cn) ∧ sn = ap · dn + an · dp + (bp · cn + bn · cp)))" }, { "ast": [ "eq", [ "var", 4 ], [ "add", [ "add", [ "var", 0 ], [ "var", 1 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 2 ], [ "var", 3 ] ] ] ] ], "dependencies": [], "id": "ND0395", "name": "IQuadHeight", "parameters": [ "ap", "an", "bp", "bn", "h" ], "source": "h = ap + an + 2 · (bp + bn)" }, { "ast": [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 8 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 14 ] ] ] ], [ "var", 8 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 16 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 18 ] ] ] ] ], [ "exists", [ "eq", [ "var", 17 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 18 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 20 ] ] ] ] ], [ "exists", [ "eq", [ "var", 19 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 20 ] ] ] ], [ "var", 5 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 22 ] ] ] ] ], [ "exists", [ "eq", [ "var", 21 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 22 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 24 ] ] ] ] ], [ "exists", [ "eq", [ "var", 23 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 24 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 26 ] ] ] ] ], [ "exists", [ "eq", [ "var", 25 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 26 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 28 ] ] ] ] ], [ "exists", [ "eq", [ "var", 27 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 29 ] ], [ "var", 28 ] ] ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 22 ] ] ] ] ], [ "exists", [ "eq", [ "var", 21 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 22 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 24 ] ] ] ] ], [ "exists", [ "eq", [ "var", 23 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 24 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 26 ] ] ] ] ], [ "exists", [ "eq", [ "var", 25 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 26 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 28 ] ] ] ] ], [ "exists", [ "eq", [ "var", 27 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 29 ] ] ], [ "var", 28 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "and", [ "eq", [ "var", 3 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 7 ] ], [ "mul", [ "var", 10 ], [ "var", 6 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 5 ] ], [ "mul", [ "var", 8 ], [ "var", 4 ] ] ] ] ] ], [ "and", [ "eq", [ "var", 2 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 6 ] ], [ "mul", [ "var", 10 ], [ "var", 7 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 4 ] ], [ "mul", [ "var", 8 ], [ "var", 5 ] ] ] ] ] ], [ "and", [ "eq", [ "var", 1 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 5 ] ], [ "mul", [ "var", 10 ], [ "var", 4 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 7 ] ], [ "mul", [ "var", 8 ], [ "var", 6 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 4 ] ], [ "mul", [ "var", 10 ], [ "var", 5 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 6 ] ], [ "mul", [ "var", 8 ], [ "var", 7 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0390", "ND0394" ], "id": "ND0396", "name": "IQuadCanonicalStep", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "ub", "uc", "vb", "vc", "wb", "wc", "xb", "xc", "i" ], "source": "∃ x. ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ j. ∃ u. ∃ v. ∃ w. ∃ x0. ∃ x1. S m ≤ S (S i · ac) ∧ (∃ x2. ab = x2 · S (S i · ac) + m) ∧ (S k ≤ S (S i · bc) ∧ (∃ x2. bb = x2 · S (S i · bc) + k) ∧ (S j ≤ S (S i · cc) ∧ (∃ x2. cb = x2 · S (S i · cc) + j) ∧ (S u ≤ S (S i · dc) ∧ (∃ x2. db = x2 · S (S i · dc) + u)))) ∧ (S x ≤ S (S i · uc) ∧ (∃ x2. ub = x2 · S (S i · uc) + x) ∧ (S y ≤ S (S i · vc) ∧ (∃ x2. vb = x2 · S (S i · vc) + y) ∧ (S z ≤ S (S i · wc) ∧ (∃ x2. wb = x2 · S (S i · wc) + z) ∧ (S n ≤ S (S i · xc) ∧ (∃ x2. xb = x2 · S (S i · xc) + n)))) ∧ (S v ≤ S (S S i · uc) ∧ (∃ x2. ub = x2 · S (S S i · uc) + v) ∧ (S w ≤ S (S S i · vc) ∧ (∃ x2. vb = x2 · S (S S i · vc) + w) ∧ (S x0 ≤ S (S S i · wc) ∧ (∃ x2. wb = x2 · S (S S i · wc) + x0) ∧ (S x1 ≤ S (S S i · xc) ∧ (∃ x2. xb = x2 · S (S S i · xc) + x1)))) ∧ (v = x · m + y · k + 2 · (z · j + n · u) ∧ (w = x · k + y · m + 2 · (z · u + n · j) ∧ (x0 = x · j + y · u + (z · m + n · k) ∧ x1 = x · u + y · j + (z · k + n · m))))))" }, { "ast": [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "succ", [ "zero" ] ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ] ], [ "exists", [ "eq", [ "var", 9 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 10 ] ] ] ], [ "succ", [ "zero" ] ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 12 ] ] ] ] ], [ "exists", [ "eq", [ "var", 11 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 12 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 14 ] ] ] ] ], [ "exists", [ "eq", [ "var", 13 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 14 ] ] ] ], [ "zero" ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "zero" ] ] ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 16 ] ] ] ] ], [ "exists", [ "eq", [ "var", 15 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "zero" ] ], [ "var", 16 ] ] ] ], [ "zero" ] ] ] ] ] ] ] ], [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "var", 18 ] ] ], [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "exists", [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 8 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 15 ] ] ] ] ], [ "exists", [ "eq", [ "var", 14 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 15 ] ] ] ], [ "var", 8 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 7 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 17 ] ] ] ] ], [ "exists", [ "eq", [ "var", 16 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 17 ] ] ] ], [ "var", 7 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 6 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 19 ] ] ] ] ], [ "exists", [ "eq", [ "var", 18 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 19 ] ] ] ], [ "var", 6 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 21 ] ] ] ] ], [ "exists", [ "eq", [ "var", 20 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 21 ] ] ] ], [ "var", 5 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 12 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 23 ] ] ] ] ], [ "exists", [ "eq", [ "var", 22 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 23 ] ] ] ], [ "var", 12 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 11 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 25 ] ] ] ] ], [ "exists", [ "eq", [ "var", 24 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 25 ] ] ] ], [ "var", 11 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 10 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 27 ] ] ] ] ], [ "exists", [ "eq", [ "var", 26 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 27 ] ] ] ], [ "var", 10 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 9 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 29 ] ] ] ] ], [ "exists", [ "eq", [ "var", 28 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 13 ] ], [ "var", 29 ] ] ] ], [ "var", 9 ] ] ] ] ] ] ] ], [ "and", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 23 ] ] ] ] ], [ "exists", [ "eq", [ "var", 22 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 23 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 25 ] ] ] ] ], [ "exists", [ "eq", [ "var", 24 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 25 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 27 ] ] ] ] ], [ "exists", [ "eq", [ "var", 26 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 27 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 29 ] ] ] ] ], [ "exists", [ "eq", [ "var", 28 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "succ", [ "var", 13 ] ] ], [ "var", 29 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "and", [ "eq", [ "var", 3 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 7 ] ], [ "mul", [ "var", 10 ], [ "var", 6 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 5 ] ], [ "mul", [ "var", 8 ], [ "var", 4 ] ] ] ] ] ], [ "and", [ "eq", [ "var", 2 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 6 ] ], [ "mul", [ "var", 10 ], [ "var", 7 ] ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 4 ] ], [ "mul", [ "var", 8 ], [ "var", 5 ] ] ] ] ] ], [ "and", [ "eq", [ "var", 1 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 5 ] ], [ "mul", [ "var", 10 ], [ "var", 4 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 7 ] ], [ "mul", [ "var", 8 ], [ "var", 6 ] ] ] ] ], [ "eq", [ "var", 0 ], [ "add", [ "add", [ "mul", [ "var", 11 ], [ "var", 4 ] ], [ "mul", [ "var", 10 ], [ "var", 5 ] ] ], [ "add", [ "mul", [ "var", 9 ], [ "var", 6 ] ], [ "mul", [ "var", 8 ], [ "var", 7 ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "ND0390", "PD0002", "ND0396" ], "id": "ND0397", "name": "IQuadCanonicalTrace", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "ub", "uc", "vb", "vc", "wb", "wc", "xb", "xc", "L" ], "source": "2 ≤ S (1 · uc) ∧ (∃ x. ub = x · S (1 · uc) + 1) ∧ (1 ≤ S (1 · vc) ∧ (∃ x. vb = x · S (1 · vc) + 0) ∧ (1 ≤ S (1 · wc) ∧ (∃ x. wb = x · S (1 · wc) + 0) ∧ (1 ≤ S (1 · xc) ∧ (∃ x. xb = x · S (1 · xc) + 0)))) ∧ (∀ x. S x ≤ L → ∃ y. ∃ z. ∃ n. ∃ m. ∃ k. ∃ i. ∃ j. ∃ u. ∃ v. ∃ w. ∃ x0. ∃ x1. S k ≤ S (S x · ac) ∧ (∃ x2. ab = x2 · S (S x · ac) + k) ∧ (S i ≤ S (S x · bc) ∧ (∃ x2. bb = x2 · S (S x · bc) + i) ∧ (S j ≤ S (S x · cc) ∧ (∃ x2. cb = x2 · S (S x · cc) + j) ∧ (S u ≤ S (S x · dc) ∧ (∃ x2. db = x2 · S (S x · dc) + u)))) ∧ (S y ≤ S (S x · uc) ∧ (∃ x2. ub = x2 · S (S x · uc) + y) ∧ (S z ≤ S (S x · vc) ∧ (∃ x2. vb = x2 · S (S x · vc) + z) ∧ (S n ≤ S (S x · wc) ∧ (∃ x2. wb = x2 · S (S x · wc) + n) ∧ (S m ≤ S (S x · xc) ∧ (∃ x2. xb = x2 · S (S x · xc) + m)))) ∧ (S v ≤ S (S S x · uc) ∧ (∃ x2. ub = x2 · S (S S x · uc) + v) ∧ (S w ≤ S (S S x · vc) ∧ (∃ x2. vb = x2 · S (S S x · vc) + w) ∧ (S x0 ≤ S (S S x · wc) ∧ (∃ x2. wb = x2 · S (S S x · wc) + x0) ∧ (S x1 ≤ S (S S x · xc) ∧ (∃ x2. xb = x2 · S (S S x · xc) + x1)))) ∧ (v = y · k + z · i + 2 · (n · j + m · u) ∧ (w = y · i + z · k + 2 · (n · u + m · j) ∧ (x0 = y · j + z · u + (n · k + m · i) ∧ x1 = y · u + z · j + (n · i + m · k)))))))" }, { "ast": [ "exists", [ "and", [ "eq", [ "var", 0 ], [ "add", [ "add", [ "var", 1 ], [ "var", 2 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 3 ], [ "var", 4 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 1 ] ], [ "var", 6 ] ] ] ] ], "dependencies": [ "ND0395", "PD0001" ], "id": "ND0398", "name": "IQuadHeightBound", "parameters": [ "ap", "an", "bp", "bn", "H" ], "source": "∃ x. x = ap + an + 2 · (bp + bn) ∧ x ≤ H" }, { "ast": [ "forall", [ "forall", [ "forall", [ "forall", [ "forall", [ "imp", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 5 ] ] ], [ "var", 14 ] ] ], [ "imp", [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 4 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ] ], [ "exists", [ "eq", [ "var", 6 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 7 ] ] ] ], [ "var", 4 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 3 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 9 ] ] ] ] ], [ "exists", [ "eq", [ "var", 8 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 9 ] ] ] ], [ "var", 3 ] ] ] ] ], [ "and", [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 2 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ] ], [ "exists", [ "eq", [ "var", 10 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 11 ] ] ] ], [ "var", 2 ] ] ] ] ], [ "and", [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "succ", [ "var", 1 ] ] ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 13 ] ] ] ] ], [ "exists", [ "eq", [ "var", 12 ], [ "add", [ "mul", [ "var", 0 ], [ "succ", [ "mul", [ "succ", [ "var", 5 ] ], [ "var", 13 ] ] ] ], [ "var", 1 ] ] ] ] ] ] ] ], [ "exists", [ "and", [ "eq", [ "var", 0 ], [ "add", [ "add", [ "var", 4 ], [ "var", 3 ] ], [ "mul", [ "succ", [ "succ", [ "zero" ] ] ], [ "add", [ "var", 2 ], [ "var", 1 ] ] ] ] ], [ "exists", [ "eq", [ "add", [ "var", 0 ], [ "var", 1 ] ], [ "var", 16 ] ] ] ] ] ] ] ] ] ] ] ], "dependencies": [ "PD0002", "ND0390", "ND0398" ], "id": "ND0399", "name": "IQuadFactorHeightBound", "parameters": [ "ab", "ac", "bb", "bc", "cb", "cc", "db", "dc", "L", "H" ], "source": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. S x ≤ L → S y ≤ S (S x · ac) ∧ (∃ k. ab = k · S (S x · ac) + y) ∧ (S z ≤ S (S x · bc) ∧ (∃ k. bb = k · S (S x · bc) + z) ∧ (S n ≤ S (S x · cc) ∧ (∃ k. cb = k · S (S x · cc) + n) ∧ (S m ≤ S (S x · dc) ∧ (∃ k. db = k · S (S x · dc) + m)))) → ∃ k. k = y + z + 2 · (n + m) ∧ k ≤ H" } ], "library_admissions": 0, "observation_count": 246, "rows": [ { "bundle_bytes": 331302, "bundle_sha256": "8097cee51aafd444321e288943d96cc527f06ab8e4f5dd4fd3b06e8366c9dd10", "case": "AP001", "contains_checked_statements": [ "CV025", "CV032", "CV086", "SQ001", "SQ002", "SQ003", "SQ004" ], "coverage": "construct_actual_dyadic_root_and_finite_odd_reciprocal_table_horner_value_and_power_prove_positive_quotient_denominator_and_upper_bound_one_no_analytic_logarithm_or_accuracy_assumption", "ha": "evidence/AP001-HA.json.gz", "internal_proof_nodes": [ 138, 143, 158, 128, 137, 9, 154, 159 ], "lean": "evidence/AP001-Lean.json.gz", "nodes": 161, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-root-argument-v1", "proof": "evidence/AP001-bundle.json.gz", "report_sha256": "118c3b1c22dd0d96a730f6f4e318f8d49f553113f8a2b6948bc00e759f722404", "root": 160, "statement": "forall k. exists d s D ab ac x p. (NatSqrt2LogArgument(k,d,s,D,ab,ac,x,p)) /\\ ((~d=0) /\\ ((~s=0) /\\ ((~D=0) /\\ ((~p=0) /\\ ((~((D*p)*s)=0) /\\ (Le((3*x)*d,(D*p)*s)))))))", "target_ast_sha256": "2b80a217cebb668c94e88b3a849c0738ad80d20aee1014489d6b8561839e89df", "title": "log root argument", "uses_definitions": [ "PD0001", "ND0447" ] }, { "bundle_bytes": 522625, "bundle_sha256": "254ee848b9237c568db4089d6f5460ffcf482cbd5b4d6aa34ec8937eaaa075e9", "case": "AP002", "contains_checked_statements": [ "AP001", "CV022", "CV025", "CV030", "CV032", "CV069", "CV070", "CV086", "SQ001", "SQ002", "SQ003", "SQ004" ], "coverage": "construct_actual_finite_exponential_sample_of_log_root_quotient_with_nonzero_normalization_factorial_power_weight_term_tables_sum_recurrence_and_all_index_normalization_no_analytic_identity_or_accuracy_claim", "ha": "evidence/AP002-HA.json.gz", "internal_proof_nodes": [ 160, 184 ], "lean": "evidence/AP002-Lean.json.gz", "nodes": 186, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sample-v1", "proof": "evidence/AP002-bundle.json.gz", "report_sha256": "36be23f14739f0446c3b376ee3e76cf3e0c875bb303df9b947a7e26a16b3b26b", "root": 185, "statement": "forall k. exists d s D ab ac x p wb wc tb tc w0 f h y. (NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y)) /\\ ((~w0=0) /\\ ((~((D*p)*s)=0) /\\ ((Le(((3*x)*d),((D*p)*s))) /\\ ((NatScaledExpRecurrence(((D*p)*s),((3*x)*d),tb,tc,(k+2))) /\\ (forall j t f0 p0 q. Le(j,(k+2)) -> BetaAt(tb,tc,j,t) -> Factorial(j,f0) -> Pow(((D*p)*s),j,p0) -> Pow(((3*x)*d),j,q) -> (t*f0)*p0=w0*q)))))", "target_ast_sha256": "934fbe9c81ad757420933f697a47c0e82aa1b3e12a326f07d6080b94c7c3e24f", "title": "exp sample", "uses_definitions": [ "PD0013", "PD0020", "PD0001", "PD0023", "ND0443", "ND0448" ] }, { "bundle_bytes": 338239, "bundle_sha256": "f4e8128b5ee924559e644bcabad76116cb70ba4ab8b2c5a9f60e54c38b765a5e", "case": "AP003", "contains_checked_statements": [ "CV092", "SQ001", "SQ002", "SQ003", "SQ004", "SQ009" ], "coverage": "supplied_finite_log_root_quotient_graphs_at_same_precision_with_independent_positive_coefficient_scales_have_identical_dyadic_data_and_powers_positive_denominators_and_exact_rational_value_identity_without_equating_table_codes", "ha": "evidence/AP003-HA.json.gz", "internal_proof_nodes": [ 138, 149, 152, 136, 137, 9, 38, 153, 154 ], "lean": "evidence/AP003-Lean.json.gz", "nodes": 156, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-root-independence-v1", "proof": "evidence/AP003-bundle.json.gz", "report_sha256": "bfbfceeeda9fd2b49bd684a90c1e6a75976d7e0f5c1a98ad4675bb9696afe3a1", "root": 155, "statement": "forall k d s D ab ac x p e t E bb bc z h. ~D=0 -> ~E=0 -> NatSqrt2LogArgument(k,d,s,D,ab,ac,x,p) -> NatSqrt2LogArgument(k,e,t,E,bb,bc,z,h) -> (d=e) /\\ ((s=t) /\\ ((p=h) /\\ ((~((D*p)*s)=0) /\\ ((~((E*h)*t)=0) /\\ (((3*x)*d)*((E*h)*t)=((3*z)*e)*((D*p)*s))))))", "target_ast_sha256": "a257baf8204aa80c57093525ce2915cb7d327803c89e4469be4206000abc6665", "title": "log root independence", "uses_definitions": [ "ND0447" ] }, { "bundle_bytes": 376642, "bundle_sha256": "07e3c3a0d8f95c3c80461832be66d79d54ccd6da2c213b6995af7c451fc22669", "case": "AP004", "contains_checked_statements": [ "CV086", "SQ002", "SQ009" ], "coverage": "every_supplied_finite_exponential_sample_with_positive_coefficient_scale_has_nonzero_input_and_output_denominators_input_at_most_one_exact_same_table_recurrence_and_all_index_factorial_normalization_no_replacement_computation", "ha": "evidence/AP004-HA.json.gz", "internal_proof_nodes": [ 150, 149, 125, 32, 148, 142, 143, 144 ], "lean": "evidence/AP004-Lean.json.gz", "nodes": 152, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supplied-exp-sample-v1", "proof": "evidence/AP004-bundle.json.gz", "report_sha256": "9a0404f677f1a3efc3d8c86856c5683b827287c733c578cd4b0f5e618f670db7", "root": 151, "statement": "forall k d s D ab ac x p wb wc tb tc w0 f h y. ~D=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> (~w0=0) /\\ ((~((D*p)*s)=0) /\\ ((Le(((3*x)*d),((D*p)*s))) /\\ ((NatScaledExpRecurrence(((D*p)*s),((3*x)*d),tb,tc,(k+2))) /\\ (forall j t f0 p0 q. Le(j,(k+2)) -> BetaAt(tb,tc,j,t) -> Factorial(j,f0) -> Pow(((D*p)*s),j,p0) -> Pow(((3*x)*d),j,q) -> (t*f0)*p0=w0*q))))", "target_ast_sha256": "2579d5bd54a8bf1f91831ce86386ca747605c6022da544548553dc22186cb198", "title": "supplied exp sample", "uses_definitions": [ "PD0013", "PD0020", "PD0001", "PD0023", "ND0443", "ND0448" ] }, { "bundle_bytes": 816597, "bundle_sha256": "65f5079b7765f94e219fa59d809d727dfde3de1071327445909b47c1fa4a2ab8", "case": "AP005", "contains_checked_statements": [ "AP003", "AP004", "CV009", "CV086", "CV092", "CV094", "CV095", "CV096", "CV097", "CV098", "SQ001", "SQ002", "SQ003", "SQ004", "SQ009" ], "coverage": "two_supplied_complete_finite_exponential_samples_at_same_precision_with_independent_positive_coefficient_scales_have_positive_output_denominators_and_exact_rational_value_identity_without_equating_codes_or_scales", "ha": "evidence/AP005-HA.json.gz", "internal_proof_nodes": [ 151, 175, 226 ], "lean": "evidence/AP005-Lean.json.gz", "nodes": 228, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sample-independence-v1", "proof": "evidence/AP005-bundle.json.gz", "report_sha256": "6bc059e711a7c7d6da5720346a63979ac6833335e845ddb562dfaffa9fd7be3a", "root": 227, "statement": "forall k d s D ab ac x p wb wc tb tc w0 f h y e t E bb bc z q vb vc ub uc v0 g r o. ~D=0 -> ~E=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> NatSqrt2ExpSample(k,e,t,E,bb,bc,z,q,vb,vc,ub,uc,v0,g,r,o) -> (~w0=0) /\\ ((~v0=0) /\\ (v0*y=w0*o))", "target_ast_sha256": "7b3f310db54974c295475b7f400dc4530ba3125902894517b92112b35c6d7383", "title": "exp sample independence", "uses_definitions": [ "ND0448" ] }, { "bundle_bytes": 595548, "bundle_sha256": "bf75787641e08251020beebea70fd3b959a6a124299bc6912608463dd6d8bb41", "case": "AP006", "contains_checked_statements": [ "CV032", "CV062", "CV086", "CV089", "CV090", "CV092", "CV093", "SQ002", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009", "SQ010" ], "coverage": "supplied_log_root_quotients_at_ordered_precisions_have_positive_denominators_and_two_sided_asymmetric_finite_error_from_logarithm_tail_and_reciprocal_root_modulus_without_monotonicity_assumption", "ha": "evidence/AP006-HA.json.gz", "internal_proof_nodes": [ 184, 185, 186, 148, 177, 145, 183, 134, 181, 8 ], "lean": "evidence/AP006-Lean.json.gz", "nodes": 188, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-argument-modulus-v1", "proof": "evidence/AP006-bundle.json.gz", "report_sha256": "12c9d6e8fd5097ccc694e05d27247605538d62a9a4ec52b7049620a74febd429", "root": 187, "statement": "forall k l d s D ab ac x p q t E bb bc z h. ~D=0 -> ~E=0 -> NatSqrt2LogArgument(k,d,s,D,ab,ac,x,p) -> NatSqrt2LogArgument(k+l,q,t,E,bb,bc,z,h) -> (~((D*p)*s)=0) /\\ ((~((E*h)*t)=0) /\\ ((Le(((D*p)*s)*((3*z)*q),((E*h)*t)*((3*x)*d)+((3*D)*(E*h))*(s*t))) /\\ (Le(d*(((E*h)*t)*((3*x)*d)),d*(((D*p)*s)*((3*z)*q))+((D*p)*s)*((E*h)*t)))))", "target_ast_sha256": "cbc032a1e559ba87536c09544a5f7e321cc43378460e87bddebc0cb65a85584c", "title": "argument modulus", "uses_definitions": [ "PD0001", "ND0447" ] }, { "bundle_bytes": 483848, "bundle_sha256": "147d7fa32b4cefb244f3a689bf9241c30b0170bc7eb867723c690751b82423bc", "case": "AP007", "contains_checked_statements": [ "CV009", "CV094", "CV095", "CV096", "CV097", "CV098" ], "coverage": "supplied_same_degree_finite_exponential_sums_have_two_sided_scaled_error_bounds_without_input_monotonicity_by_constructive_natural_order_and_actual_additive_gap_with_independent_denominator_and_head_scales", "ha": "evidence/AP007-HA.json.gz", "internal_proof_nodes": [ 14, 157, 162, 8, 6 ], "lean": "evidence/AP007-Lean.json.gz", "nodes": 164, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-two-sided-continuity-v1", "proof": "evidence/AP007-bundle.json.gz", "report_sha256": "54f2f405515eee992308e392e67c9794318c33bd2a06d7d8df93b2a0bc28bf60", "root": 163, "statement": "forall d e x y C R H N ab ac bb bc a0 b0 s t. ~d=0 -> ~e=0 -> Le(x,d) -> Le(y,e) -> Le(C*(d*y),C*(e*x)+R) -> Le(C*(e*x),C*(d*y)+H) -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Le(C*((d*e)*(a0*t)),C*((d*e)*(b0*s))+9*(R*(a0*b0))) /\\ Le(C*((d*e)*(b0*s)),C*((d*e)*(a0*t))+9*(H*(a0*b0)))", "target_ast_sha256": "d6c797957cd58e4b92514bcc10063e71045c1cdbe995aeeef3bff5020d434a11", "title": "exp two sided continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 657992, "bundle_sha256": "bc30f35fc3fbab167f5bd5d629e8cc410adc89a0b3f0300f74702335b96ef99e", "case": "AP008", "contains_checked_statements": [ "AP007", "CV009", "CV068", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100" ], "coverage": "supplied_finite_exponential_sums_at_unequal_horizons_have_two_sided_scaled_error_bounds_from_actual_constructed_prefix_and_supplied_cutoff_power_without_input_monotonicity_or_analytic_limit_assumptions", "ha": "evidence/AP008-HA.json.gz", "internal_proof_nodes": [ 163, 193, 194, 119, 195, 2, 1 ], "lean": "evidence/AP008-Lean.json.gz", "nodes": 197, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-two-sided-horizons-v1", "proof": "evidence/AP008-bundle.json.gz", "report_sha256": "b33facc312f1c042c1f192e524b82092b47be1fe950179a18469063908f63f8a", "root": 196, "statement": "forall d e x y C R H m N ab ac bb bc a0 b0 s t p. ~d=0 -> ~e=0 -> Le(x,d) -> Le(y,e) -> Le(C*(d*y),C*(e*x)+R) -> Le(C*(e*x),C*(d*y)+H) -> NatScaledExpRecurrence(d,x,ab,ac,m) -> NatScaledExpRecurrence(e,y,bb,bc,(S m)+N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S m,s) -> Sum(bb,bc,S((S m)+N),t) -> Pow(2,m,p) -> Le(p*(C*((d*e)*(a0*t))),p*(C*((d*e)*(b0*s)))+(9*(p*R)+2*(C*(d*e)))*(a0*b0)) /\\ Le(C*((d*e)*(b0*s)),C*((d*e)*(a0*t))+9*(H*(a0*b0)))", "target_ast_sha256": "345ef6c15b5813f3423a35a13c658fa76dd43b9bc8fc56f2aed0330e254c788a", "title": "exp two sided horizons", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 1257796, "bundle_sha256": "b30e8f3596e922ae6f3391b2bab3b75574459487def0a796d30dba6d283933df", "case": "AP009", "contains_checked_statements": [ "AP004", "AP006", "AP007", "AP008", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "SQ002", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009", "SQ010" ], "coverage": "actual_supplied_samples_at_k_and_k_plus_positive_increment_have_scale_free_two_sided_finite_errors_with_positive_output_and_cutoff_denominators_without_input_monotonicity_or_analytic_limit_claims", "ha": "evidence/AP009-HA.json.gz", "internal_proof_nodes": [ 270, 271, 187, 259, 269, 145, 129, 26, 21, 146, 7, 18, 6 ], "lean": "evidence/AP009-Lean.json.gz", "nodes": 273, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sample-modulus-v1", "proof": "evidence/AP009-bundle.json.gz", "report_sha256": "331a74ad24aeed0caa03e5d7ad60f16f5940681f45a4cbaaae69e37873900c23", "root": 272, "statement": "forall k l d s D ab ac x p wb wc tb tc w0 f h y q t E bb bc z v xb xc ub uc v0 g j o r. ~D=0 -> ~E=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> NatSqrt2ExpSample(k+S l,q,t,E,bb,bc,z,v,xb,xc,ub,uc,v0,g,j,o) -> Pow(2,k+2,r) -> (~d=0) /\\ ((~p=0) /\\ ((~r=0) /\\ ((~w0=0) /\\ ((~v0=0) /\\ ((Le((p*r)*(w0*o),(p*r)*(v0*y)+((9*3)*r+2*p)*(w0*v0))) /\\ (Le(d*(v0*y),d*(w0*o)+9*(w0*v0))))))))", "target_ast_sha256": "073e1e33119841ca820f40504b4be88839cec838afc45b6028bda6807fa464dd", "title": "exp sample modulus", "uses_definitions": [ "PD0020", "PD0001", "ND0448" ] }, { "bundle_bytes": 258080, "bundle_sha256": "340b616e44a6822279a5bef8dc9bc4e8a15d748592e8b2870b82f50ffa1268fe", "case": "AP010", "contains_checked_statements": [], "coverage": "actual_supplied_natural_powers_are_jointly_monotone_in_ordered_bases_and_additive_exponents_when_the_larger_base_is_nonzero_with_zero_smaller_base_and_zero_exponents_allowed", "ha": "evidence/AP010-HA.json.gz", "internal_proof_nodes": [ 123, 124, 110, 119, 120, 9 ], "lean": "evidence/AP010-Lean.json.gz", "nodes": 126, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-positive-power-comparison-v1", "proof": "evidence/AP010-bundle.json.gz", "report_sha256": "6b2f57c805c6d8be2ffd7ad8f9cf7f2c6e2b1b44eef6ec094f881564c5f5fc7a", "root": 125, "statement": "forall a b k l x y. Le(a,b) -> ~b=0 -> Pow(a,k,x) -> Pow(b,k+l,y) -> Le(x,y)", "target_ast_sha256": "4b0cf1709336ebd3b4aa2546b4ad7107701157fb692b6db7205344456192cd25", "title": "positive power comparison", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 270071, "bundle_sha256": "59a7b1470dfeb2ff50255f79fa96efc0cb349ddcb36ba6ffd8fcc0643ee68c02", "case": "AP011", "contains_checked_statements": [ "AP010" ], "coverage": "actual_supplied_dyadic_logarithm_and_exponential_cutoff_powers_are_positive_and_both_cutoff_powers_dominate_the_actual_dyadic_denominator_including_zero_precision", "ha": "evidence/AP011-HA.json.gz", "internal_proof_nodes": [ 126, 125, 120, 18, 33, 17 ], "lean": "evidence/AP011-Lean.json.gz", "nodes": 128, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-approximation-power-bounds-v1", "proof": "evidence/AP011-bundle.json.gz", "report_sha256": "6a3e73ef662d504a450863c92f65cd745f8d853acb7d66e60c229c4292a42724", "root": 127, "statement": "forall k d p r. Pow(2,k,d) -> Pow(3,S(2*k),p) -> Pow(2,k+2,r) -> (~d=0) /\\ ((~p=0) /\\ ((~r=0) /\\ ((Le(d,p)) /\\ (Le(d,r)))))", "target_ast_sha256": "a41d4293a571408ac987bb69b06a49ec1afd6527e75e6d2404265714b00fb69e", "title": "approximation power bounds", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 1339338, "bundle_sha256": "9de719a1127b72d5a0df0d7779b1ebf2906193ecac0502489e9b1830fd49cbe5", "case": "AP012", "contains_checked_statements": [ "AP004", "AP006", "AP007", "AP008", "AP009", "AP010", "AP011", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "SQ002", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009", "SQ010" ], "coverage": "actual_supplied_samples_at_strictly_finer_precision_have_uniform_two_sided_dyadic_error_with_positive_output_denominators_and_internally_constructed_cutoff_power_without_real_accuracy_or_irrationality_claims", "ha": "evidence/AP012-HA.json.gz", "internal_proof_nodes": [ 272, 278, 119, 279, 280 ], "lean": "evidence/AP012-Lean.json.gz", "nodes": 282, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sample-uniform-error-v1", "proof": "evidence/AP012-bundle.json.gz", "report_sha256": "f080cb330952e0c051c55a7bad88db0fcc6c22c6a05958c818ad2332cd07a5da", "root": 281, "statement": "forall k l d s D ab ac x p wb wc tb tc w0 f h y q t E bb bc z v xb xc ub uc v0 g j o. ~D=0 -> ~E=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> NatSqrt2ExpSample(k+S l,q,t,E,bb,bc,z,v,xb,xc,ub,uc,v0,g,j,o) -> (~d=0) /\\ ((~w0=0) /\\ ((~v0=0) /\\ ((Le(d*(w0*o),d*(v0*y)+((9*3)+2)*(w0*v0))) /\\ (Le(d*(v0*y),d*(w0*o)+((9*3)+2)*(w0*v0))))))", "target_ast_sha256": "309286b62c9ba3f6357861cfacebc829369cbcc7919b6c039df4852a2f7bfb99", "title": "exp sample uniform error", "uses_definitions": [ "PD0001", "ND0448" ] }, { "bundle_bytes": 1549613, "bundle_sha256": "e02a1654a4fa32bfa0c52da1c745d22083dc69a1eaf362ee3950df341d422a73", "case": "AP013", "contains_checked_statements": [ "AP003", "AP004", "AP005", "AP006", "AP007", "AP008", "AP009", "AP010", "AP011", "AP012", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "SQ001", "SQ002", "SQ003", "SQ004", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009", "SQ010" ], "coverage": "actual_supplied_samples_have_positive_output_denominators_and_uniform_two_sided_dyadic_error_for_every_precision_increment_including_zero_by_constructive_case_split_and_exact_value_independence", "ha": "evidence/AP013-HA.json.gz", "internal_proof_nodes": [ 281, 294, 145, 30, 295 ], "lean": "evidence/AP013-Lean.json.gz", "nodes": 297, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sample-all-precision-v1", "proof": "evidence/AP013-bundle.json.gz", "report_sha256": "e27a4798546228baa3c1c974c377b5c843d5611d40aa43a4b06fa12c377ac119", "root": 296, "statement": "forall k l d s D ab ac x p wb wc tb tc w0 f h y q t E bb bc z v xb xc ub uc v0 g j o. ~D=0 -> ~E=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> NatSqrt2ExpSample(k+l,q,t,E,bb,bc,z,v,xb,xc,ub,uc,v0,g,j,o) -> (~d=0) /\\ ((~w0=0) /\\ ((~v0=0) /\\ ((Le(d*(w0*o),d*(v0*y)+((9*3)+2)*(w0*v0))) /\\ (Le(d*(v0*y),d*(w0*o)+((9*3)+2)*(w0*v0))))))", "target_ast_sha256": "2ea71d464e47c52e625a986dcb55ac5c92ff24daf2fd7b95c71d36d58aad8634", "title": "exp sample all precision", "uses_definitions": [ "PD0001", "ND0448" ] }, { "bundle_bytes": 300966, "bundle_sha256": "79b74784f51a096ec9525ac54fbe9fce921049df06b0579fef8d1a44d2695eed", "case": "AP014", "contains_checked_statements": [ "AP010" ], "coverage": "actual_supplied_eighth_power_of_two_is_positive_and_dominates_the_finite_error_constant_via_constructed_fifth_power_and_small_ordinary_HA_arithmetic_without_unary_256_or_new_induction", "ha": "evidence/AP014-HA.json.gz", "internal_proof_nodes": [ 126, 127, 128, 125, 110, 120, 18, 9, 17 ], "lean": "evidence/AP014-Lean.json.gz", "nodes": 130, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-fixed-precision-threshold-v2", "proof": "evidence/AP014-bundle.json.gz", "report_sha256": "5e24641e7716e3c64f12e59e467da290dbb4a50960ae07cbebf43855f04c0b8b", "root": 129, "statement": "forall v. Pow(2,8,v) -> ~v=0 /\\ Le(((9*3)+2),v)", "target_ast_sha256": "2ec3a5c39030d975d8ddf79a41d7793afd4f5fdfd25bcd49657f7335af0fba3c", "title": "fixed precision threshold", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 1651429, "bundle_sha256": "0cd67027532d8498c60bbdb9cb930631e58a7ee3b16e60982e14bbdaaf415afd", "case": "AP015", "contains_checked_statements": [ "AP003", "AP004", "AP005", "AP006", "AP007", "AP008", "AP009", "AP010", "AP011", "AP012", "AP013", "AP014", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "SQ001", "SQ002", "SQ003", "SQ004", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009", "SQ010" ], "coverage": "actual_supplied_samples_at_n_plus_eight_and_any_finer_precision_have_two_sided_finite_error_at_most_one_over_a_constructed_positive_two_to_n_using_the_actual_coarse_power_and_a_constructed_positive_shift_power", "ha": "evidence/AP015-HA.json.gz", "internal_proof_nodes": [ 301, 296, 300, 119, 128, 129, 26 ], "lean": "evidence/AP015-Lean.json.gz", "nodes": 303, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sample-dyadic-error-v1", "proof": "evidence/AP015-bundle.json.gz", "report_sha256": "d4cb8729f69ab79fdaf2546cdd7eb06319410f67d07a6ed7bb93da1a711a28a7", "root": 302, "statement": "forall n l d s D ab ac x p wb wc tb tc w0 f h y q t E bb bc z v xb xc ub uc v0 g j o. ~D=0 -> ~E=0 -> NatSqrt2ExpSample(n+8,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> NatSqrt2ExpSample((n+8)+l,q,t,E,bb,bc,z,v,xb,xc,ub,uc,v0,g,j,o) -> exists r. (Pow(2,n,r)) /\\ ((~r=0) /\\ ((~w0=0) /\\ ((~v0=0) /\\ ((Le(r*(w0*o),r*(v0*y)+(w0*v0))) /\\ (Le(r*(v0*y),r*(w0*o)+(w0*v0)))))))", "target_ast_sha256": "98633f6b0d2144188f824e0f5f1f367d8113fe1999e66199a3b54eb28f615703", "title": "exp sample dyadic error", "uses_definitions": [ "PD0020", "PD0001", "ND0448" ] }, { "bundle_bytes": 167248, "bundle_sha256": "6623977ef1c4aba8612a0450d2e0fb1484d6cb00f31a5b7d14b70187c5660c00", "case": "AP016", "contains_checked_statements": [], "coverage": "construct_the_actual_odd_index_affine_slice_of_a_supplied_atanh_prefix_and_prove_all_odd_quotient_equations_and_all_even_zeros_including_empty_slice_and_zero_scale_without_analytic_identification", "ha": "evidence/AP016-HA.json.gz", "internal_proof_nodes": [ 105, 107, 108, 109, 110, 111 ], "lean": "evidence/AP016-Lean.json.gz", "nodes": 113, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-odd-slice-v1", "proof": "evidence/AP016-bundle.json.gz", "report_sha256": "4611295aeca54c32c28865a3715419178d2a51d935080291f23352d15dce4bfe", "root": 112, "statement": "forall d ab ac k. NatAtanhPrefix(d,ab,ac,S(2*k)) -> exists bb bc. NatAffineSlice(ab,ac,1,2,bb,bc,k) /\\ ((forall j a. Lt(j,k) -> BetaAt(bb,bc,j,a) -> (S(2*j))*a=2*d) /\\ (forall j a. Le(j,k) -> BetaAt(ab,ac,2*j,a) -> a=0))", "target_ast_sha256": "ebf68fd1ca1f37cd10b7e77ef28a69acf4aa4c4995cd8c215a6bc3d8a9f1d894", "title": "atanh odd slice", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0411", "ND0425" ] }, { "bundle_bytes": 62767, "bundle_sha256": "74390410057e6f52c8976706ee47906b16fb7699670d7602f3ac336bd25b2854", "case": "AP017", "contains_checked_statements": [], "coverage": "compare_actual_supplied_horner_histories_by_explicit_pairwise_induction_to_prove_long_value_equals_base_times_odd_slice_value_at_squared_base_with_every_even_coefficient_zero_including_the_terminal_entry", "ha": "evidence/AP017-HA.json.gz", "internal_proof_nodes": [ 38, 11 ], "lean": "evidence/AP017-Lean.json.gz", "nodes": 40, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-paired-horner-v1", "proof": "evidence/AP017-bundle.json.gz", "report_sha256": "90f892cbbf00036db456512f70a5c1d7578f0207091d953b2e9b037410523e05", "root": 39, "statement": "forall ab ac bb bc t K x y. NatAffineSlice(ab,ac,1,2,bb,bc,K) -> (forall j a. Le(j,K) -> BetaAt(ab,ac,2*j,a) -> a=0) -> NatHornerValue(ab,ac,t,S(2*K),x) -> NatHornerValue(bb,bc,t*t,K,y) -> x=t*y", "target_ast_sha256": "5235c154de5a45ae307e0f3e33b64ea3ba043dd2efe3fdc302aa824d100fe5d9", "title": "paired horner", "uses_definitions": [ "PD0001", "PD0013", "ND0411", "ND0437" ] }, { "bundle_bytes": 74072, "bundle_sha256": "84a656f636badc245aa0408832eadba4f4cdc185f95f94d4ba2e540ccdf4113d", "case": "AP018", "contains_checked_statements": [], "coverage": "normalize_both_actual_supplied_power_computations_to_the_squared_base_by_explicit_doubled_exponent_induction_and_a_final_successor_allowing_zero_base_and_zero_index_without_cancellation", "ha": "evidence/AP018-HA.json.gz", "internal_proof_nodes": [ 25, 23, 6 ], "lean": "evidence/AP018-Lean.json.gz", "nodes": 27, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-paired-power-v1", "proof": "evidence/AP018-bundle.json.gz", "report_sha256": "4888891fa1d9a7a3fcb7145d4fd5be439e701a0642970a99ece20786e2a29ebe", "root": 26, "statement": "forall t k x y. Pow(t,S(2*k),x) -> Pow(t*t,k,y) -> x=t*y", "target_ast_sha256": "1c4010443924fa4b57d39c7f9952916050880d32d433ee211b5b5cef8a1a4f0f", "title": "paired power", "uses_definitions": [ "PD0020" ] }, { "bundle_bytes": 310851, "bundle_sha256": "fd6d43b7cb41ce4722c57ebffffdde004e26d8fd5a289ffd74679af508db224a", "case": "AP019", "contains_checked_statements": [ "AP016", "AP017", "AP018", "CV032" ], "coverage": "construct_actual_odd_slice_horner_value_and_power_at_base_nine_then_prove_the_original_finite_log_quotient_equals_its_odd_horner_quotient_with_every_denominator_nonzero_without_infinite_analytic_identification", "ha": "evidence/AP019-HA.json.gz", "internal_proof_nodes": [ 138, 140, 112, 118, 125, 136, 132, 137, 12 ], "lean": "evidence/AP019-Lean.json.gz", "nodes": 142, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-odd-horner-quotient-v1", "proof": "evidence/AP019-bundle.json.gz", "report_sha256": "efa31a49a8ab206de111946310f0a11e1ed34d9a78893eec1dbf1dca369b22e9", "root": 141, "statement": "forall d ab ac k x p. ~d=0 -> NatAtanhPrefix(d,ab,ac,S(2*k)) -> NatHornerValue(ab,ac,3,S(2*k),x) -> Pow(3,S(2*k),p) -> exists bb bc y q. (NatAffineSlice(ab,ac,1,2,bb,bc,k) /\\ ((forall j a. Lt(j,k) -> BetaAt(bb,bc,j,a) -> (S(2*j))*a=2*d) /\\ (NatHornerValue(bb,bc,9,k,y) /\\ (Pow(9,k,q) /\\ (~q=0 /\\ (~p=0 /\\ (x=3*y /\\ (p=3*q /\\ IRatEq(3*x,0,d*p,3*y,0,d*q)))))))))", "target_ast_sha256": "ef40bb286d66b9b7abd7a8521c7ef114b6ccba8d8780e5d0574178f985096c33", "title": "odd horner quotient", "uses_definitions": [ "PD0002", "PD0013", "ND0411", "ND0437", "ND0425", "PD0020", "ND0383" ] }, { "bundle_bytes": 460882, "bundle_sha256": "b32cdcbcfc160830a6b0f97c0b6d132c5c05de4c5fef88f07c57ac3d0a7cb520", "case": "AP020", "contains_checked_statements": [ "AP016", "AP017", "AP018", "AP019", "CV032", "CV037", "CV038", "CV039" ], "coverage": "preserve_the_original_finite_log_computations_and_construct_an_actual_sum_table_identifying_every_normalized_term_with_its_odd_reciprocal_using_constructed_positive_denominator_powers_without_assuming_an_infinite_logarithm_identity", "ha": "evidence/AP020-HA.json.gz", "internal_proof_nodes": [ 141, 159, 165, 164, 167 ], "lean": "evidence/AP020-Lean.json.gz", "nodes": 169, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-finite-odd-log-series-v1", "proof": "evidence/AP020-bundle.json.gz", "report_sha256": "c4ac117578ec480aa814025ad3a79a0fda72d3ee0842550efd8c97ac5f649e8e", "root": 168, "statement": "forall d ab ac k x p. ~d=0 -> NatAtanhPrefix(d,ab,ac,S(2*k)) -> NatHornerValue(ab,ac,3,S(2*k),x) -> Pow(3,S(2*k),p) -> exists bb bc y q tb tc. (NatAffineSlice(ab,ac,1,2,bb,bc,k) /\\ ((forall j a. Lt(j,k) -> BetaAt(bb,bc,j,a) -> (S(2*j))*a=2*d) /\\ (NatHornerValue(bb,bc,9,k,y) /\\ (Pow(9,k,q) /\\ (~q=0 /\\ (~p=0 /\\ (x=3*y /\\ (p=3*q /\\ (IRatEq(3*x,0,d*p,3*y,0,d*q) /\\ (Sum(tb,tc,k,y) /\\ (forall j c. Lt(j,k) -> BetaAt(tb,tc,j,c) -> exists r. Pow(3,S(2*j),r) /\\ IRatEq(3*c,0,d*q,2,0,(S(2*j))*r))))))))))))", "target_ast_sha256": "1d918aad2d021a46dc9e5a2a09b838014a15d6b30db745a7c4c5c0f2f8c34c20", "title": "finite odd log series", "uses_definitions": [ "PD0002", "PD0013", "ND0411", "ND0437", "ND0425", "PD0020", "ND0383", "PD0015" ] }, { "bundle_bytes": 427389, "bundle_sha256": "54f2e7c6dafc48993d063b9a2d83baf25bd914591746ff2119806debafd152dd", "case": "AP021", "contains_checked_statements": [ "AP004", "CV086", "SQ002", "SQ009" ], "coverage": "preserve_the_actual_supplied_exponential_sample_table_and_sum_and_identify_each_normalized_term_with_its_factorial_series_value_using_constructed_factorials_and_both_powers_with_positive_denominators_without_infinite_exponential_identification", "ha": "evidence/AP021-HA.json.gz", "internal_proof_nodes": [ 151, 158 ], "lean": "evidence/AP021-Lean.json.gz", "nodes": 160, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-finite-exp-series-v1", "proof": "evidence/AP021-bundle.json.gz", "report_sha256": "709c7813625c32e4d5b41576753e2d072d6333ec9e0a0d5154650454fd71376a", "root": 159, "statement": "forall k d s D ab ac x p wb wc tb tc w0 f h y. ~D=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> (~w0=0) /\\ ((~((D*p)*s)=0) /\\ ((Sum(tb,tc,S (k+2),y)) /\\ ((forall j t. Le(j,(k+2)) -> BetaAt(tb,tc,j,t) -> exists f0 p0 q0. (Factorial(j,f0)) /\\ ((Pow(((D*p)*s),j,p0)) /\\ ((Pow(((3*x)*d),j,q0)) /\\ ((~f0=0) /\\ ((~p0=0) /\\ (IRatEq(t,0,w0,q0,0,f0*p0))))))))))", "target_ast_sha256": "8c30669eb1c317280a38fcade33d35967d3427cb940f0753752a02da1460daee", "title": "finite exp series", "uses_definitions": [ "PD0013", "PD0020", "PD0001", "PD0023", "PD0015", "ND0448", "ND0383" ] }, { "bundle_bytes": 227171, "bundle_sha256": "e60cea590e8fc2dcd2f130c163423b166de275b0d0af5036748f9e0bd034d9f5", "case": "AP022", "contains_checked_statements": [], "coverage": "construct_the_actual_positive_factorial_N_as_the_prescribed_logarithm_coefficient_scale_and_the_actual_coefficient_prefix_using_a_proved_positive_divisor_induction_and_existing_prefix_construction_without_claiming_the_whole_canonical_PR_approximation_trace", "ha": "evidence/AP022-HA.json.gz", "internal_proof_nodes": [ 116, 120, 122, 105, 106, 13 ], "lean": "evidence/AP022-Lean.json.gz", "nodes": 124, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-factorial-atanh-prefix-v1", "proof": "evidence/AP022-bundle.json.gz", "report_sha256": "c63e59bf54b7ea213a42b95f382fda816160578a994197fe983c1eefbcb4ed4f", "root": 123, "statement": "forall N. exists d ab ac. Factorial(N,d) /\\ (~d=0 /\\ (NatAtanhPrefix(d,ab,ac,N) /\\ NatZeroBefore(ab,ac,N,1)))", "target_ast_sha256": "717987b66df365b93849f2f28931c0d8b1ef892d210f4590b9c62997aee7989a", "title": "factorial atanh prefix", "uses_definitions": [ "ND0425", "ND0408", "PD0023" ] }, { "bundle_bytes": 543376, "bundle_sha256": "c16e77ecbdceec882bf43df4a345bd3a875fc7d0f4225e50983785e982294a61", "case": "AP023", "contains_checked_statements": [ "AP022", "CV022", "CV030", "CV032", "CV069", "CV070", "CV086", "SQ001", "SQ002", "SQ003", "SQ004" ], "coverage": "construct_the_complete_finite_log_root_exponential_sample_with_prescribed_coefficient_scale_factorial_two_k_plus_one_preserving_positive_scale_all_actual_computations_recurrence_and_normalization_without_claiming_the_whole_PR_trace_or_analytic_identification", "ha": "evidence/AP023-HA.json.gz", "internal_proof_nodes": [ 188, 187 ], "lean": "evidence/AP023-Lean.json.gz", "nodes": 190, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-factorial-exp-sample-v1", "proof": "evidence/AP023-bundle.json.gz", "report_sha256": "e47eaafdac84c9a2fcd9e6540e25ec38f60ed4828ecdcd3970ede8ceddeaf8ed", "root": 189, "statement": "forall k. exists d s D ab ac x p wb wc tb tc w0 f h y. (Factorial(S(2*k),D)) /\\ ((~D=0) /\\ ((NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y)) /\\ ((~w0=0) /\\ ((~((D*p)*s)=0) /\\ ((Le(((3*x)*d),((D*p)*s))) /\\ ((NatScaledExpRecurrence(((D*p)*s),((3*x)*d),tb,tc,(k+2))) /\\ (forall j t f0 p0 q. Le(j,(k+2)) -> BetaAt(tb,tc,j,t) -> Factorial(j,f0) -> Pow(((D*p)*s),j,p0) -> Pow(((3*x)*d),j,q) -> (t*f0)*p0=w0*q)))))))", "target_ast_sha256": "a49dec85075c8e059c999c262ce992200a03c7c1611ff0d9003643f5237b1fcf", "title": "factorial exp sample", "uses_definitions": [ "PD0013", "PD0020", "PD0001", "PD0023", "ND0443", "ND0448" ] }, { "bundle_bytes": 310665, "bundle_sha256": "e2e83ec51ce4d3cfc724c3e68e01de0d41efde85b51707031d85ee9d7b537c8c", "case": "AP024", "contains_checked_statements": [ "SQ001" ], "coverage": "constructively_decide_natural_equality_order_beta_decoding_powers_factorials_sums_and_integer_square_roots_from_replayed_totality_and_value_uniqueness_without_classical_axioms_or_a_claim_of_a_complete_CApprox_verifier", "ha": "evidence/AP024-HA.json.gz", "internal_proof_nodes": [ 143, 144, 145, 147, 148, 149, 150, 151 ], "lean": "evidence/AP024-Lean.json.gz", "nodes": 153, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atomic-computation-decisions-v1", "proof": "evidence/AP024-bundle.json.gz", "report_sha256": "3584028fa75df2b312651840fcb4ad222173dbf8f43a1c018e6f6f2e5290c8e3", "root": 152, "statement": "((forall a b. (a=b) \\/ ~(a=b))) /\\ (((forall a b. (Le(a,b)) \\/ ~(Le(a,b)))) /\\ (((forall a b. (Lt(a,b)) \\/ ~(Lt(a,b)))) /\\ (((forall b c i v. (BetaAt(b,c,i,v)) \\/ ~(BetaAt(b,c,i,v)))) /\\ (((forall a n v. (Pow(a,n,v)) \\/ ~(Pow(a,n,v)))) /\\ (((forall n v. (Factorial(n,v)) \\/ ~(Factorial(n,v)))) /\\ (((forall b c n v. (Sum(b,c,n,v)) \\/ ~(Sum(b,c,n,v)))) /\\ ((forall n v. (FloorSqrt(n,v)) \\/ ~(FloorSqrt(n,v))))))))))", "target_ast_sha256": "ed18ed2d5be19d785c821571ea8789461ac6dcc996c0e26364d9149e77d9d6f0", "title": "atomic computation decisions", "uses_definitions": [ "PD0002", "PD0013", "PD0020", "PD0001", "PD0051", "PD0023", "PD0015" ] }, { "bundle_bytes": 266977, "bundle_sha256": "541015fd0e99e2c907df19bcb07a173646ac30354ddda32986fab3b496091d71", "case": "AP025", "contains_checked_statements": [ "CV032", "SQ001" ], "coverage": "constructively_decide_the_actual_individual_logarithm_coefficient_Horner_value_and_dyadic_square_root_computation_graphs_for_all_inputs_including_invalid_candidates_without_new_induction_or_a_complete_CApprox_verifier_claim", "ha": "evidence/AP025-HA.json.gz", "internal_proof_nodes": [ 142, 143, 144 ], "lean": "evidence/AP025-Lean.json.gz", "nodes": 146, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-approximation-component-decisions-v1", "proof": "evidence/AP025-bundle.json.gz", "report_sha256": "145a7def8daf0f8ec83de78a9258e21ece53e3bda785bc79b429aff612cbce18", "root": 145, "statement": "((forall d i a. (NatAtanhCoefficient(d,i,a)) \\/ ~(NatAtanhCoefficient(d,i,a)))) /\\ (((forall ab ac t N v. (NatHornerValue(ab,ac,t,N,v)) \\/ ~(NatHornerValue(ab,ac,t,N,v)))) /\\ ((forall k d s. (DyadicSqrt2(k,d,s)) \\/ ~(DyadicSqrt2(k,d,s)))))", "target_ast_sha256": "73143d6dbbcad9e7c4558786a63cf9abc8870be8c71abd9d934090ddb9501c21", "title": "approximation component decisions", "uses_definitions": [ "ND0446", "ND0424", "ND0437" ] }, { "bundle_bytes": 251449, "bundle_sha256": "602f3a841c85f0210c2c54d4c23d406e6539f247b720acc6a07b04b76c495bb2", "case": "AP026", "contains_checked_statements": [], "coverage": "compile_and_replay_actual_beta_power_row_decisions_and_bounded_induction_decisions_for_supplied_coefficient_weight_and_term_tables_including_incorrect_tables_empty_prefixes_and_zero_scales_without_claiming_the_complete_CApprox_record", "ha": "evidence/AP026-HA.json.gz", "internal_proof_nodes": [ 128, 131, 130 ], "lean": "evidence/AP026-Lean.json.gz", "nodes": 133, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-finite-table-decisions-v1", "proof": "evidence/AP026-bundle.json.gz", "report_sha256": "d2b19e3ebef6d4b82e2c427f100a48a6dafe611de0e0cac9ba4db2310e269b37", "root": 132, "statement": "((forall d ab ac N. (NatAtanhPrefix(d,ab,ac,N)) \\/ ~(NatAtanhPrefix(d,ab,ac,N)))) /\\ (((forall d N wb wc. (NatExpWeightTable(d,N,wb,wc)) \\/ ~(NatExpWeightTable(d,N,wb,wc)))) /\\ ((forall x wb wc tb tc N. (NatPowerWeightedPrefix(x,wb,wc,tb,tc,N)) \\/ ~(NatPowerWeightedPrefix(x,wb,wc,tb,tc,N)))))", "target_ast_sha256": "6dbc10516866227c027792d1af728a7e88461da6be67734ad5f93540f85317ce", "title": "finite table decisions", "uses_definitions": [ "ND0425", "ND0423", "ND0442" ] }, { "bundle_bytes": 487296, "bundle_sha256": "8a300a0f1234e476a3ad2e12a5d7bea7320b757d76fb8138689c7b2eff25a6ce", "case": "AP027", "contains_checked_statements": [ "CV032", "SQ001" ], "coverage": "constructively_decide_the_entire_supplied_prescribed_factorial_sample_and_positive_scale_output_denominator_by_sequentially_checking_every_actual_computation_field_without_assuming_correctness_accuracy_or_canonical_record_construction", "ha": "evidence/AP027-HA.json.gz", "internal_proof_nodes": [ 168, 170, 172 ], "lean": "evidence/AP027-Lean.json.gz", "nodes": 174, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-prescribed-sample-decision-v1", "proof": "evidence/AP027-bundle.json.gz", "report_sha256": "12cd89dc2d78b08762c1b9999704e837d48f0ebafde18eb1d8deb7d3940083c0", "root": 173, "statement": "forall k d s D ab ac x p wb wc tb tc w0 f h y. ((Factorial(S(2*k),D)) /\\ ((~D=0) /\\ ((NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y)) /\\ (~w0=0)))) \\/ ~((Factorial(S(2*k),D)) /\\ ((~D=0) /\\ ((NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y)) /\\ (~w0=0))))", "target_ast_sha256": "0bfc201f0143e0b3fb6e1b1b1a23ea925cf898cbfb266cbfb272ce264b3275cd", "title": "prescribed sample decision", "uses_definitions": [ "PD0023", "ND0448" ] }, { "bundle_bytes": 734434, "bundle_sha256": "118016ea5a5b20b9116df0cdd125fcd0aff8a5d906b152f62b680d9b44db4199", "case": "AP028", "contains_checked_statements": [ "AP022", "AP023", "CV022", "CV030", "CV032", "CV069", "CV070", "CV086", "SQ001", "SQ002", "SQ003", "SQ004" ], "coverage": "construct_an_actual_fifteen_field_beta_record_of_the_positive_fixed_factorial_finite_sample_at_precision_n_plus_eight_with_both_code_components_explicit_and_all_decoded_fields_consumed_without_accuracy_or_analytic_clauses", "ha": "evidence/AP028-HA.json.gz", "internal_proof_nodes": [ 189, 204 ], "lean": "evidence/AP028-Lean.json.gz", "nodes": 206, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-approximation-record-exists-v1", "proof": "evidence/AP028-bundle.json.gz", "report_sha256": "0ae8a36423a952f711e51e96f569e627e62276fa76b4d17f0bb73b006feabeac", "root": 205, "statement": "forall n. exists up ud rb rc. NatSqrt2ApproxRecord(n,up,ud,rb,rc)", "target_ast_sha256": "6ff27609504236182a615ce1dd07fddf1b3d1c7d0161d869bb756ac824c4ef8e", "title": "approximation record exists", "uses_definitions": [ "ND0449" ] }, { "bundle_bytes": 1025060, "bundle_sha256": "e51a07de3eaf87cd53ec90045924966c992a365365a5643f872ffa483d7c78b7", "case": "AP029", "contains_checked_statements": [ "AP027", "CV032", "SQ001" ], "coverage": "constructively_decide_any_supplied_fifteen_field_approximation_record_by_actual_beta_decoding_complete_sample_decision_and_equality_transport_of_all_competing_internal_fields_without_unbounded_search_or_classical_axioms", "ha": "evidence/AP029-HA.json.gz", "internal_proof_nodes": [ 73, 175, 176 ], "lean": "evidence/AP029-Lean.json.gz", "nodes": 178, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-approximation-record-decision-v1", "proof": "evidence/AP029-bundle.json.gz", "report_sha256": "4550d87730efcd6984cef58cbf480a0f0a6ab2aac94b1a1bfaab57e8a761ce44", "root": 177, "statement": "forall n up ud rb rc. (NatSqrt2ApproxRecord(n,up,ud,rb,rc)) \\/ ~(NatSqrt2ApproxRecord(n,up,ud,rb,rc))", "target_ast_sha256": "b08c1a483e177d9fa80cb2c787464ff0ad2f2a27dc52543517056b1184e2896e", "title": "approximation record decision", "uses_definitions": [ "ND0449" ] }, { "bundle_bytes": 890587, "bundle_sha256": "589f5d1b6e439b368cc8bab9965ee1d543a55a8fe4ca62fc1330cae61785a23f", "case": "AP030", "contains_checked_statements": [ "AP003", "AP004", "AP005", "CV009", "CV086", "CV092", "CV094", "CV095", "CV096", "CV097", "CV098", "SQ001", "SQ002", "SQ003", "SQ004", "SQ009" ], "coverage": "prove_positive_output_denominators_and_cross_multiplied_rational_equality_for_two_actual_records_at_the_same_precision_by_replaying_AP005_without_requiring_equal_code_numbers", "ha": "evidence/AP030-HA.json.gz", "internal_proof_nodes": [ 227 ], "lean": "evidence/AP030-Lean.json.gz", "nodes": 229, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-approximation-record-independence-v1", "proof": "evidence/AP030-bundle.json.gz", "report_sha256": "628cf265cedc15c9f171c3ab34120ada6cb163f58ef2a5524e7680c90449063b", "root": 228, "statement": "forall n up ud rb rc vp vd sb sc. NatSqrt2ApproxRecord(n,up,ud,rb,rc) -> NatSqrt2ApproxRecord(n,vp,vd,sb,sc) -> ~ud=0 /\\ (~vd=0 /\\ vd*up=ud*vp)", "target_ast_sha256": "a2ccad96f8bb3ab29d434fafb17dcb326788759f0a66e60ec0707f9212c8f99b", "title": "approximation record independence", "uses_definitions": [ "ND0449" ] }, { "bundle_bytes": 1739726, "bundle_sha256": "ae901ee94df076839e47c0d5eb3a905c1d0f40061d415fa896dd7f318050657b", "case": "AP031", "contains_checked_statements": [ "AP003", "AP004", "AP005", "AP006", "AP007", "AP008", "AP009", "AP010", "AP011", "AP012", "AP013", "AP014", "AP015", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "SQ001", "SQ002", "SQ003", "SQ004", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009", "SQ010" ], "coverage": "transport_the_actual_all_finer_finite_dyadic_error_to_two_complete_records_after_proving_the_fixed_precision_shift_alignment_preserving_the_actual_power_and_positive_denominators_without_an_analytic_limit_claim", "ha": "evidence/AP031-HA.json.gz", "internal_proof_nodes": [ 302, 303 ], "lean": "evidence/AP031-Lean.json.gz", "nodes": 305, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-approximation-record-error-v2", "proof": "evidence/AP031-bundle.json.gz", "report_sha256": "2cccf9d2b66cf757d405b45506cb1ebf5804cea4c4e8d8d7806c803b9d4b2145", "root": 304, "statement": "forall n l up ud rb rc vp vd sb sc. NatSqrt2ApproxRecord(n,up,ud,rb,rc) -> NatSqrt2ApproxRecord(n+l,vp,vd,sb,sc) -> exists r. (Pow(2,n,r)) /\\ ((~r=0) /\\ ((~ud=0) /\\ ((~vd=0) /\\ ((Le(r*(ud*vp),r*(vd*up)+(ud*vd))) /\\ (Le(r*(vd*up),r*(ud*vp)+(ud*vd)))))))", "target_ast_sha256": "237174f38843f6146e5511de3aa4f49bd784a08738da6ca90da7734f1b08a057", "title": "approximation record error", "uses_definitions": [ "PD0020", "PD0001", "ND0449" ] }, { "bundle_bytes": 2718087, "bundle_sha256": "9762342212b1e76b6e2d3f640b5a1086fcd5ea3957daf4e257d97d987ef5923a", "case": "AP032", "contains_checked_statements": [ "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086" ], "coverage": "construct_actual_finite_logarithm_coefficients_and_exponential_terms_and_connect_the_same_product_table_to_its_positive_head_recurrence_sum_and_both_errors_near_two_without_asserting_the_infinite_logarithm_identity", "ha": "evidence/AP032-HA.json.gz", "internal_proof_nodes": [ 432, 433, 400, 436, 47, 437 ], "lean": "evidence/AP032-Lean.json.gz", "nodes": 439, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-exp-witness-v1", "proof": "evidence/AP032-bundle.json.gz", "report_sha256": "96db8e43f78cc722f4009585edaf186b948f2f9383616835856057bff451c429", "root": 438, "statement": "forall K. exists d ab ac x p tb tc h y r a. (~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S(S K))) /\\ ((NatHornerValue(ab,ac,3,S(S K),x)) /\\ ((Pow(3,S(S K),p)) /\\ ((Pow(3,S K,r)) /\\ ((Pow(2,S(S K),a)) /\\ ((~p=0) /\\ ((~r=0) /\\ ((~h=0) /\\ ((~(d*p)=0) /\\ ((Le(3*x,d*p)) /\\ ((NatScaledExpRecurrence(d*p,3*x,tb,tc,S(S K))) /\\ ((BetaAt(tb,tc,0,h)) /\\ ((Sum(tb,tc,S(S(S K)),y)) /\\ ((IRatLe(y,0,h,2*p+9*a,0,p)) /\\ (IRatLe(2,0,1,y*r+h,0,h*r))))))))))))))))", "target_ast_sha256": "f184312da8aa97085684bf27c6b090fe30ff3df7d6afcd251bbc4309ddb1c715", "title": "log exp witness", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0437", "ND0425", "ND0443", "ND0444" ] }, { "bundle_bytes": 139855, "bundle_sha256": "36caf73a72f7b03407395d92914e1fe5d1638cd8a9a23306b08ca8ec1d4d591b", "case": "AP033", "contains_checked_statements": [], "coverage": "prove_two_actual_power_inductions_and_the_explicit_cutoff_N_equals_eighteen_r_plus_two_budget_for_both_finite_log_exponential_errors_without_materializing_large_exponentials_or_altering_the_fixed_approximation_program", "ha": "evidence/AP033-HA.json.gz", "internal_proof_nodes": [ 41, 42, 43, 44, 33, 16, 9 ], "lean": "evidence/AP033-Lean.json.gz", "nodes": 46, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-exp-budget-v1", "proof": "evidence/AP033-bundle.json.gz", "report_sha256": "cac8ca68b7da6545a82b178c2f55635623555c78a8f87702564758f9fa145f35", "root": 45, "statement": "forall r a p q. Pow(2,S(S(18*r)),a) -> Pow(3,S(S(18*r)),p) -> Pow(3,S(18*r),q) -> ~p=0 /\\ (~q=0 /\\ (Le(r*(9*a),p) /\\ (Le(r,q))))", "target_ast_sha256": "f9b1341a43e515bdb6a1c4fc11f68e15deefecb21d6e6e05e93c1be5d19f8c43", "title": "log exp budget", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 2812849, "bundle_sha256": "99ef8cd7a0cdd76a9c39d828620e1c8858565b70d202b79dfb36df034bddb63e", "case": "AP034", "contains_checked_statements": [ "AP032", "AP033", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086" ], "coverage": "construct_actual_two_to_n_and_the_explicit_finite_log_exponential_computations_with_two_integer_error_bounds_at_most_two_to_minus_n_from_two_retained_stage_proofs_and_positive_comparison_powers_without_infinite_analytic_identification", "ha": "evidence/AP034-HA.json.gz", "internal_proof_nodes": [ 438, 447, 266, 280, 12, 448, 449, 450, 6, 9 ], "lean": "evidence/AP034-Lean.json.gz", "nodes": 452, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-exp-dyadic-witness-v1", "proof": "evidence/AP034-bundle.json.gz", "report_sha256": "b1945e84c98cf37d8d792b4aae5d6359f805f072b2a199821565a0a6bb30572d", "root": 451, "statement": "forall n. exists r d ab ac x p tb tc h y. (Pow(2,n,r)) /\\ ((~r=0) /\\ ((~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S(S(18*r)))) /\\ ((NatHornerValue(ab,ac,3,S(S(18*r)),x)) /\\ ((Pow(3,S(S(18*r)),p)) /\\ ((~p=0) /\\ ((~h=0) /\\ ((~(d*p)=0) /\\ ((Le(3*x,d*p)) /\\ ((NatScaledExpRecurrence(d*p,3*x,tb,tc,S(S(18*r)))) /\\ ((BetaAt(tb,tc,0,h)) /\\ ((Sum(tb,tc,S(S(S(18*r))),y)) /\\ ((Le(r*y,(r*2)*h+h)) /\\ (Le((r*2)*h,r*y+h)))))))))))))))", "target_ast_sha256": "c7f28f686ec9b7eb26f152b6aed3eecb9aa4b32a686ba022e3ba80d363e5f620", "title": "log exp dyadic witness", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0437", "ND0425", "ND0443" ] }, { "bundle_bytes": 108842, "bundle_sha256": "9cca6b66aa1a938f46ea6ec4dc47503a586239318bc48d616d5d68f48b8ee4bc", "case": "AP035", "contains_checked_statements": [], "coverage": "derive_all_index_factorial_power_normalization_from_any_supplied_actual_finite_exponential_recurrence_and_head_by_ordinary_induction_without_cancellation_including_zero_denominator_numerator_head_and_horizon", "ha": "evidence/AP035-HA.json.gz", "internal_proof_nodes": [ 30, 31, 32, 33, 24, 25, 11, 17, 12, 9, 8, 6, 34 ], "lean": "evidence/AP035-Lean.json.gz", "nodes": 36, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-recurrence-normalization-v1", "proof": "evidence/AP035-bundle.json.gz", "report_sha256": "befe42b7a71b91854a699831be4ed92aae3780044d8c745ca2a6fe9a9f6cf574", "root": 35, "statement": "forall d x tb tc N h. NatScaledExpRecurrence(d,x,tb,tc,N) -> BetaAt(tb,tc,0,h) -> forall j t f p q. Le(j,N) -> BetaAt(tb,tc,j,t) -> Factorial(j,f) -> Pow(d,j,p) -> Pow(x,j,q) -> (t*f)*p=h*q", "target_ast_sha256": "9b47638db02d23a2bbc549793da14c71bc39b811db179fadbf3410e4723bbdf6", "title": "exp recurrence normalization", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "PD0023", "ND0443" ] }, { "bundle_bytes": 282927, "bundle_sha256": "c15e4cf76dd83de836f3bbd457e0e5a21ae6779ee70dc926cd75fcecbb7e1aa6", "case": "AP036", "contains_checked_statements": [ "AP035" ], "coverage": "identify_every_actual_supplied_exponential_recurrence_term_with_its_valid_rational_factorial_series_value_constructing_the_factorial_and_both_powers_from_retained_generic_proofs_without_an_infinite_series_claim", "ha": "evidence/AP036-HA.json.gz", "internal_proof_nodes": [ 35, 128 ], "lean": "evidence/AP036-Lean.json.gz", "nodes": 130, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-recurrence-series-v1", "proof": "evidence/AP036-bundle.json.gz", "report_sha256": "e5fb766eb66c359b840a13a97ac2c651f0d9db9f08d2e7a94b77492f47965280", "root": 129, "statement": "forall d x tb tc N h. ~d=0 -> ~h=0 -> NatScaledExpRecurrence(d,x,tb,tc,N) -> BetaAt(tb,tc,0,h) -> forall j t. Le(j,N) -> BetaAt(tb,tc,j,t) -> exists f0 p0 q0. (Factorial(j,f0)) /\\ ((Pow(d,j,p0)) /\\ ((Pow(x,j,q0)) /\\ ((~f0=0) /\\ ((~p0=0) /\\ (IRatEq(t,0,h,q0,0,f0*p0))))))", "target_ast_sha256": "88e50501342eaa8afeb980d49d5c37f6d456c6f32acfa9cabd11f610e93e959c", "title": "exp recurrence series", "uses_definitions": [ "PD0013", "PD0020", "PD0001", "PD0023", "ND0443", "ND0383" ] }, { "bundle_bytes": 300980, "bundle_sha256": "66e3363a9ed58930f10db6803f0c605ec5bae24396b0b633a582fd2e26f462b3", "case": "AP037", "contains_checked_statements": [], "coverage": "transport_a_denominator_cleared_one_place_shift_through_actual_finite_convolution_coefficients_using_the_zero_head_and_only_required_local_indices_with_all_input_lengths_and_zero_scalars_allowed", "ha": "evidence/AP037-HA.json.gz", "internal_proof_nodes": [ 127, 128, 129, 76, 24, 30, 31, 114 ], "lean": "evidence/AP037-Lean.json.gz", "nodes": 131, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-shift-v1", "proof": "evidence/AP037-bundle.json.gz", "report_sha256": "b7a3515b423d30585bbaf2acdf41811180740154deeb0e6456e750394018c6ef", "root": 130, "statement": "forall alpha gamma ab ac L bb bc M cb cc K i u v. BetaZeroExtend(cb,cc,K,0,0) -> (forall j a c. Le(j,i) -> BetaZeroExtend(ab,ac,L,j,a) -> BetaZeroExtend(cb,cc,K,S j,c) -> gamma*c=alpha*a) -> NatConvolutionCoefficient(ab,ac,L,bb,bc,M,i,u) -> NatConvolutionCoefficient(cb,cc,K,bb,bc,M,S i,v) -> gamma*v=alpha*u", "target_ast_sha256": "d3eb18cd8e8c76b563ea0fbc13a70b6de3140d1c177e036fa23a2411b01980b2", "title": "convolution shift", "uses_definitions": [ "PD0001", "ND0291", "ND0401" ] }, { "bundle_bytes": 392744, "bundle_sha256": "eb3ee35d22bb87a6c5ca3963644da32ed3fc677db0ee176185b032bb6c3910e2", "case": "AP038", "contains_checked_statements": [ "AP037" ], "coverage": "derive_the_added_argument_exponential_convolution_recurrence_below_the_shared_finite_cutoff_using_actual_euler_tables_product_rule_and_local_denominator_cleared_shift_without_cancellation_or_an_infinite_identity", "ha": "evidence/AP038-HA.json.gz", "internal_proof_nodes": [ 136, 135, 138, 141, 130, 143, 144, 145 ], "lean": "evidence/AP038-Lean.json.gz", "nodes": 147, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-convolution-recurrence-v1", "proof": "evidence/AP038-bundle.json.gz", "report_sha256": "6f38c15290ec1680f3727eb6dd2aee62c68dd680a9fe84b160f16b97c6dfa8fd", "root": 146, "statement": "forall d x y ab ac bb bc N i u v. NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(d,y,bb,bc,N) -> Lt(i,N) -> NatConvolutionCoefficient(ab,ac,S N,bb,bc,S N,i,u) -> NatConvolutionCoefficient(ab,ac,S N,bb,bc,S N,S i,v) -> ((S i)*d)*v=(x+y)*u", "target_ast_sha256": "1d0487d27421420580ddbc9e31618cfe3a90ef13648ea8c02a25c5be0e87f7e0", "title": "exp convolution recurrence", "uses_definitions": [ "PD0002", "ND0401", "ND0443" ] }, { "bundle_bytes": 521875, "bundle_sha256": "eb3be34e7edbee2ca6fb1e7702208c35ab377dde25d37f3e7ea8225883911543", "case": "AP039", "contains_checked_statements": [ "AP035", "AP037", "AP038" ], "coverage": "construct_the_actual_finite_convolution_prefix_and_sum_prove_its_product_head_added_argument_recurrence_and_all_index_factorial_normalization_without_equating_the_truncated_sum_with_the_full_product_of_input_sums", "ha": "evidence/AP039-HA.json.gz", "internal_proof_nodes": [ 158, 107, 160, 161, 156 ], "lean": "evidence/AP039-Lean.json.gz", "nodes": 163, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-convolution-prefix-v1", "proof": "evidence/AP039-bundle.json.gz", "report_sha256": "03889fa9ca3d14aa60e4dfeedf1828254b5db083df8d44b540d098bcfb4a4f58", "root": 162, "statement": "forall d x y ab ac bb bc N a0 b0. NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(d,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> exists cb cc s. (NatConvolutionPrefix(ab,ac,S N,bb,bc,S N,cb,cc,S N)) /\\ ((NatScaledExpRecurrence(d,x+y,cb,cc,N)) /\\ ((BetaAt(cb,cc,0,a0*b0)) /\\ ((Sum(cb,cc,S N,s)) /\\ ((forall j t f p q. Le(j,N) -> BetaAt(cb,cc,j,t) -> Factorial(j,f) -> Pow(d,j,p) -> Pow(x+y,j,q) -> (t*f)*p=(a0*b0)*q)))))", "target_ast_sha256": "b2b010de8d9310bb2658728077ab0ed46dd768c452df5ea955fad36a1c6b40aa", "title": "exp convolution prefix", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0402", "PD0020", "PD0023", "ND0443" ] }, { "bundle_bytes": 105513, "bundle_sha256": "aaac37a0c2c53902ca053f77467100e9b542536f01288e2ba1d04e158e54a5d2", "case": "AP040", "contains_checked_statements": [ "CV095" ], "coverage": "prove_order_of_actual_convolution_coefficients_under_both_input_prefix_length_extensions_with_fixed_beta_codes_by_padding_product_and_finite_sum_order_including_empty_lengths_and_all_indices", "ha": "evidence/AP040-HA.json.gz", "internal_proof_nodes": [ 33, 44, 43 ], "lean": "evidence/AP040-Lean.json.gz", "nodes": 46, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-prefix-order-v1", "proof": "evidence/AP040-bundle.json.gz", "report_sha256": "007fb7a20257bb937151462f8801385d67cc7640714fd5896e764c7bbf96f27a", "root": 45, "statement": "forall ab ac L bb bc M N O i u v. Le(L,N) -> Le(M,O) -> NatConvolutionCoefficient(ab,ac,L,bb,bc,M,i,u) -> NatConvolutionCoefficient(ab,ac,N,bb,bc,O,i,v) -> Le(u,v)", "target_ast_sha256": "a4ddc6ed1dd2d269caed60e2cc1647676c11767abf060d3c1641e5d50ed1adb0", "title": "convolution prefix order", "uses_definitions": [ "PD0001", "ND0401" ] }, { "bundle_bytes": 693200, "bundle_sha256": "384a22dea45672963e1b1d54ae1c6273adc3f4abb76f484684a0c11b110f6222", "case": "AP041", "contains_checked_statements": [ "AP040", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV095" ], "coverage": "construct_the_actual_convolution_of_extended_inputs_and_sandwich_the_rectangular_product_of_the_short_input_sums_between_its_short_and_double_length_sums_by_finite_locality_prefix_order_and_the_full_product_identity", "ha": "evidence/AP041-HA.json.gz", "internal_proof_nodes": [ 178, 185, 123, 189 ], "lean": "evidence/AP041-Lean.json.gz", "nodes": 191, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-sum-sandwich-v1", "proof": "evidence/AP041-bundle.json.gz", "report_sha256": "b84bc6ea1b23b2b080520cc2337c20732ffc31fb4cbcb4d4b80c478023a4ef42", "root": 190, "statement": "forall ab ac bb bc L M. Le(L,M) -> exists cb cc u v s t. (NatConvolutionPrefix(ab,ac,M,bb,bc,M,cb,cc,L+L)) /\\ ((Sum(ab,ac,L,u)) /\\ ((Sum(bb,bc,L,v)) /\\ ((Sum(cb,cc,L,s)) /\\ ((Sum(cb,cc,L+L,t)) /\\ ((Le(s,u*v)) /\\ (Le(u*v,t)))))))", "target_ast_sha256": "f7b106d5f635ac96c28ef758a3a64c6f827875f97e9103f80b6f48ce84f679c5", "title": "convolution sum sandwich", "uses_definitions": [ "PD0001", "PD0015", "ND0402" ] }, { "bundle_bytes": 997802, "bundle_sha256": "69283fcae5315a56746e333c6d6b9e3162d423ea3c9dc974a3218d0776975369", "case": "AP042", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV087", "CV088", "CV095", "CV100" ], "coverage": "construct_actual_added_argument_exponential_convolution_and_bound_the_rectangular_degree_n_product_error_by_two_times_the_product_head_over_two_to_n_using_actual_finite_tails_for_x_plus_y_at_most_d_without_an_infinite_identity", "ha": "evidence/AP042-HA.json.gz", "internal_proof_nodes": [ 190, 213, 214, 241, 38, 12 ], "lean": "evidence/AP042-Lean.json.gz", "nodes": 243, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-product-error-v1", "proof": "evidence/AP042-bundle.json.gz", "report_sha256": "663fb46786ef1ce6d6cbdbe803acbd94effec27d8da89304c14b9f5310fef356", "root": 242, "statement": "forall d x y ab ac bb bc n a0 b0 p. ~d=0 -> Le(x+y,d) -> NatScaledExpRecurrence(d,x,ab,ac,(S n)+n) -> NatScaledExpRecurrence(d,y,bb,bc,(S n)+n) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Pow(2,n,p) -> exists cb cc u v s t. (NatConvolutionPrefix(ab,ac,S((S n)+n),bb,bc,S((S n)+n),cb,cc,S((S n)+n))) /\\ ((NatScaledExpRecurrence(d,x+y,cb,cc,(S n)+n)) /\\ ((BetaAt(cb,cc,0,a0*b0)) /\\ ((Sum(ab,ac,S n,u)) /\\ ((Sum(bb,bc,S n,v)) /\\ ((Sum(cb,cc,S n,s)) /\\ ((Sum(cb,cc,S((S n)+n),t)) /\\ ((Le(s,u*v)) /\\ (Le(p*(u*v),p*s+2*(a0*b0))))))))))", "target_ast_sha256": "d0f60d98de139b481df7189728d38390769ed3a1530b9f2346887861df4da53a", "title": "exp product error", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0402", "PD0020", "ND0443" ] }, { "bundle_bytes": 1182066, "bundle_sha256": "7528721c11078fdd7e425f270ff142f1629aeb9c0086f3426714067128d33823", "case": "AP043", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP042", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100" ], "coverage": "transport_the_finite_exponential_product_error_to_three_actual_supplied_recurrences_heads_and_short_sums_with_independent_output_code_and_normalization_using_checked_cross_value_identity_and_no_infinite_addition", "ha": "evidence/AP043-HA.json.gz", "internal_proof_nodes": [ 242, 274, 275, 125, 276 ], "lean": "evidence/AP043-Lean.json.gz", "nodes": 278, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-supplied-product-v1", "proof": "evidence/AP043-bundle.json.gz", "report_sha256": "cd31acb266c53cf3886d54dd8b69f2e2996d909cf64747ec30e6b53c2bde6b8a", "root": 277, "statement": "forall d x y ab ac bb bc cb cc n a0 b0 c0 u v w p. ~d=0 -> Le(x+y,d) -> NatScaledExpRecurrence(d,x,ab,ac,(S n)+n) -> NatScaledExpRecurrence(d,y,bb,bc,(S n)+n) -> NatScaledExpRecurrence(d,x+y,cb,cc,n) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> BetaAt(cb,cc,0,c0) -> Sum(ab,ac,S n,u) -> Sum(bb,bc,S n,v) -> Sum(cb,cc,S n,w) -> Pow(2,n,p) -> Le((a0*b0)*w,c0*(u*v)) /\\ Le(p*(c0*(u*v)),p*((a0*b0)*w)+2*(c0*(a0*b0)))", "target_ast_sha256": "926558364bf250779cdac1389ddfb0966bc77936da82f1e1a24e511ccefdcc45", "title": "exp supplied product", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 1357563, "bundle_sha256": "f65fd8094f9b078f62b6ee80b550b4bb62cff1c7a7728edc179d6215ae3fb413", "case": "AP044", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV069", "CV070", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100" ], "coverage": "construct_positive_factorial_weighted_half_argument_exponential_terms_and_both_actual_sums_and_bound_the_short_sum_square_against_the_supplied_whole_argument_sum_without_logarithm_or_square_root_identification", "ha": "evidence/AP044-HA.json.gz", "internal_proof_nodes": [ 298, 123, 301, 10, 66 ], "lean": "evidence/AP044-Lean.json.gz", "nodes": 303, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-half-square-v1", "proof": "evidence/AP044-bundle.json.gz", "report_sha256": "552e3ad7445b1ddf0032451873d6507a3031b47f12eeb46ea6dd20df44231683", "root": 302, "statement": "forall d x bb bc n b0 v p. ~d=0 -> Le(x,d) -> NatScaledExpRecurrence(d,x,bb,bc,n) -> BetaAt(bb,bc,0,b0) -> Sum(bb,bc,S n,v) -> Pow(2,n,p) -> exists wb wc ab ac h f q z u. (~h=0) /\\ ((NatExpWeightTable(2*d,(S n)+n,wb,wc)) /\\ ((NatPowerWeightedPrefix(x,wb,wc,ab,ac,S((S n)+n))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(ab,ac,0,h)) /\\ ((Factorial((S n)+n,f)) /\\ ((Pow(2*d,(S n)+n,q)) /\\ ((h=f*q) /\\ ((Sum(ab,ac,S((S n)+n),z)) /\\ ((NatScaledExpRecurrence(2*d,x,ab,ac,(S n)+n)) /\\ ((Sum(ab,ac,S n,u)) /\\ (Le((h*h)*v,b0*(u*u)) /\\ Le(p*(b0*(u*u)),p*((h*h)*v)+2*(b0*(h*h))))))))))))))", "target_ast_sha256": "a0ae3b4a0d82774e120a7703228bf12483ea7f606b04303d09fac179c58075e7", "title": "exp half square", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0442", "ND0443" ] }, { "bundle_bytes": 49023, "bundle_sha256": "7291b21a18f894888444ab9878c95369fecca20d89d7797c733d11b451fd8d09", "case": "AP045", "contains_checked_statements": [], "coverage": "combine_supplied_finite_cross_product_error_two_over_p_and_reference_error_one_over_r_with_r_at_most_p_by_positive_cancellation_to_obtain_upper_error_three_over_r_and_lower_error_one_over_r_without_constructing_or_identifying_any_analytic_object", "ha": "evidence/AP045-HA.json.gz", "internal_proof_nodes": [ 24, 25, 26 ], "lean": "evidence/AP045-Lean.json.gz", "nodes": 28, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-square-error-transfer-v1", "proof": "evidence/AP045-bundle.json.gz", "report_sha256": "9f74d512df78e44db7a237da7d925a8bda4caafe661771cad59e1c25b72332a6", "root": 27, "statement": "forall r p b a s v. ~b=0 -> ~p=0 -> Le(r,p) -> Le(a*v,b*s) -> Le(p*(b*s),p*(a*v)+2*(b*a)) -> Le(r*v,(r*2)*b+b) -> Le((r*2)*b,r*v+b) -> Le(r*s,(r*2)*a+3*a) /\\ Le((r*2)*a,r*s+a)", "target_ast_sha256": "3d7f11a02582d2bf7b63377cc8ec2bbabf74c584db7e7e1f5627256c208e7c84", "title": "half square error transfer", "uses_definitions": [ "PD0001" ] }, { "bundle_bytes": 78373, "bundle_sha256": "96815002d9cbada491bb67202df76d032ac56ed7b8f0c85b32d92a1406e3368a", "case": "AP046", "contains_checked_statements": [], "coverage": "prove_generic_binary_growth_n_plus_one_at_most_two_to_n_by_ordinary_induction_then_bound_the_actual_two_to_eighteen_r_plus_two_cutoff_power_below_by_r_without_materializing_large_ground_powers", "ha": "evidence/AP046-HA.json.gz", "internal_proof_nodes": [ 35, 32, 16, 21, 9, 31, 15 ], "lean": "evidence/AP046-Lean.json.gz", "nodes": 37, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-precision-v1", "proof": "evidence/AP046-bundle.json.gz", "report_sha256": "dc98d2f28c9086eaeaa631cbc76d1fda3683c343f67dd92658789f47a03ae49a", "root": 36, "statement": "forall r p. Pow(2,S(S(18*r)),p) -> ~p=0 /\\ Le(r,p)", "target_ast_sha256": "936a988b2a6fae61ebb3206a2a33746d5c8ad2e23501f642b0528b6b9e0e7877", "title": "half log precision", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 3394200, "bundle_sha256": "bcfc3286e107e3bdd1183f92793abd07c53a65c28f685c9f8f147cde6b08dc08", "case": "AP047", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "AP044", "AP045", "AP046", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100" ], "coverage": "construct_the_actual_atanh_horner_half_logarithm_exponential_and_prove_its_short_sum_square_has_upper_error_three_over_two_to_n_and_lower_error_one_over_two_to_n_with_positive_normalization_without_identifying_an_infinite_exponential_or_sqrt_two", "ha": "evidence/AP047-HA.json.gz", "internal_proof_nodes": [ 451, 524, 529, 533, 266 ], "lean": "evidence/AP047-Lean.json.gz", "nodes": 535, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-square-v2", "proof": "evidence/AP047-bundle.json.gz", "report_sha256": "50b81dc589245262b646e5691b38bd9b6bab02a0204f4931c1e06f85ec71abeb", "root": 534, "statement": "forall n. exists r d lb lc x q wb wc ab ac h f t z u. (Pow(2,n,r)) /\\ ((~r=0) /\\ ((~d=0) /\\ ((NatAtanhPrefix(d,lb,lc,S(S(18*r)))) /\\ ((NatHornerValue(lb,lc,3,S(S(18*r)),x)) /\\ ((Pow(3,S(S(18*r)),q)) /\\ ((~q=0) /\\ ((~(d*q)=0) /\\ ((Le(3*x,d*q)) /\\ ((~h=0) /\\ ((NatExpWeightTable(2*(d*q),(S(S(S(18*r))))+(S(S(18*r))),wb,wc)) /\\ ((NatPowerWeightedPrefix(3*x,wb,wc,ab,ac,S((S(S(S(18*r))))+(S(S(18*r)))))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(ab,ac,0,h)) /\\ ((Factorial((S(S(S(18*r))))+(S(S(18*r))),f)) /\\ ((Pow(2*(d*q),(S(S(S(18*r))))+(S(S(18*r))),t)) /\\ ((h=f*t) /\\ ((Sum(ab,ac,S((S(S(S(18*r))))+(S(S(18*r)))),z)) /\\ ((NatScaledExpRecurrence(2*(d*q),3*x,ab,ac,(S(S(S(18*r))))+(S(S(18*r))))) /\\ ((Sum(ab,ac,S(S(S(18*r))),u)) /\\ ((Le(r*(u*u),(r*2)*(h*h)+3*(h*h))) /\\ (Le((r*2)*(h*h),r*(u*u)+(h*h)))))))))))))))))))))))", "target_ast_sha256": "b66c87c593359c085b44249ad7621de4058ef1ab24fbba8de8d70e15ef26af40", "title": "half log square", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0437", "ND0425", "ND0442", "ND0443" ] }, { "bundle_bytes": 36655, "bundle_sha256": "a69169cdb3b811afc5b43e104aae815589b7f6c60381eb67edd17d5543986487", "case": "AP048", "contains_checked_statements": [], "coverage": "convert_two_supplied_natural_square_errors_to_linear_errors_by_decidable_order_actual_additive_gaps_difference_of_squares_and_positive_scale_cancellation_without_an_analytic_square_root_assumption", "ha": "evidence/AP048-HA.json.gz", "internal_proof_nodes": [ 29, 2 ], "lean": "evidence/AP048-Lean.json.gz", "nodes": 31, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-square-error-linear-v1", "proof": "evidence/AP048-bundle.json.gz", "report_sha256": "de16da97cc5f6a7e0005d8525e412f6f042537b24117ed77d256ac82c7f71360", "root": 30, "statement": "forall r x y k a b. ~k=0 -> Le(k,x+y) -> Le(r*(x*x),r*(y*y)+a*(k*k)) -> Le(r*(y*y),r*(x*x)+b*(k*k)) -> Le(r*x,r*y+a*k) /\\ Le(r*y,r*x+b*k)", "target_ast_sha256": "f1e75bcb47244d65e8f878a65bf2a5d7d7328ac4d6b03d2211b360a44946610c", "title": "square error linear", "uses_definitions": [ "PD0001" ] }, { "bundle_bytes": 105365, "bundle_sha256": "04b9f79d32692a6e9a82255c8dc6f5f969d9e7b7ebe1b4776e22906e159a5f3a", "case": "AP049", "contains_checked_statements": [ "AP048" ], "coverage": "compare_supplied_positive_half_log_and_dyadic_root_square_witnesses_at_r_at_most_d_with_forward_error_seven_over_r_and_reverse_error_one_over_r_by_natural_cross_products_without_constructing_witnesses_or_assuming_a_real_identity", "ha": "evidence/AP049-HA.json.gz", "internal_proof_nodes": [ 34, 35, 36, 30, 18, 31 ], "lean": "evidence/AP049-Lean.json.gz", "nodes": 38, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-root-comparison-v1", "proof": "evidence/AP049-bundle.json.gz", "report_sha256": "9bd5c7c91fd5d24e80e3af104ab93b11efb7a80ce2a5f6beccf6e3165ad6dfe7", "root": 37, "statement": "forall r u h s d. ~h=0 -> ~d=0 -> Le(r,d) -> Le(d,s) -> Le(s*s,2*(d*d)) -> Le(2*(d*d),s*s+2*(2*d)) -> Le(r*(u*u),(r*2)*(h*h)+3*(h*h)) -> Le((r*2)*(h*h),r*(u*u)+(h*h)) -> Le(r*(u*d),r*(s*h)+7*(h*d)) /\\ Le(r*(s*h),r*(u*d)+(h*d))", "target_ast_sha256": "ca484cb2209e8e2ed81d29e1903feb31e830a531b0e621cf00ce722956568c9b", "title": "half log root comparison", "uses_definitions": [ "PD0001" ] }, { "bundle_bytes": 3534091, "bundle_sha256": "7c36ebfc20bc30b67500cb3c89192873f059af2f30b2cf4f26baeb80c3d822b6", "case": "AP050", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "AP044", "AP045", "AP046", "AP047", "AP048", "AP049", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "SQ002", "SQ009" ], "coverage": "construct_at_every_precision_the_actual_half_log_exponential_and_same_denominator_dyadic_floor_root_preserving_all_computations_and_square_bounds_and_proving_both_finite_cross_product_errors_without_an_infinite_identity", "ha": "evidence/AP050-HA.json.gz", "internal_proof_nodes": [ 534, 546, 555, 558, 13 ], "lean": "evidence/AP050-Lean.json.gz", "nodes": 560, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-root-witness-v1", "proof": "evidence/AP050-bundle.json.gz", "report_sha256": "a0ad6bc4c4c5d6f97cc05855a9a8fb9fe7cad4f85ba820e86b0300858fe0c326", "root": 559, "statement": "forall n. exists r d lb lc x q wb wc ab ac h f t z u s. (Pow(2,n,r)) /\\ ((~r=0) /\\ ((~d=0) /\\ ((NatAtanhPrefix(d,lb,lc,S(S(18*r)))) /\\ ((NatHornerValue(lb,lc,3,S(S(18*r)),x)) /\\ ((Pow(3,S(S(18*r)),q)) /\\ ((~q=0) /\\ ((~(d*q)=0) /\\ ((Le(3*x,d*q)) /\\ ((~h=0) /\\ ((NatExpWeightTable(2*(d*q),(S(S(S(18*r))))+(S(S(18*r))),wb,wc)) /\\ ((NatPowerWeightedPrefix(3*x,wb,wc,ab,ac,S((S(S(S(18*r))))+(S(S(18*r)))))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(ab,ac,0,h)) /\\ ((Factorial((S(S(S(18*r))))+(S(S(18*r))),f)) /\\ ((Pow(2*(d*q),(S(S(S(18*r))))+(S(S(18*r))),t)) /\\ ((h=f*t) /\\ ((Sum(ab,ac,S((S(S(S(18*r))))+(S(S(18*r)))),z)) /\\ ((NatScaledExpRecurrence(2*(d*q),3*x,ab,ac,(S(S(S(18*r))))+(S(S(18*r))))) /\\ ((Sum(ab,ac,S(S(S(18*r))),u)) /\\ ((Le(r*(u*u),(r*2)*(h*h)+3*(h*h))) /\\ ((Le((r*2)*(h*h),r*(u*u)+(h*h))) /\\ ((DyadicSqrt2(n,r,s)) /\\ ((Le(r*(u*r),r*(s*h)+7*(h*r))) /\\ (Le(r*(s*h),r*(u*r)+(h*r))))))))))))))))))))))))))", "target_ast_sha256": "4110740f69dfccac4fd2f8dfa027de161e05948d42f010fe33bdf3b03c1bcbca", "title": "half log root witness", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0437", "ND0425", "ND0442", "ND0443", "ND0446" ] }, { "bundle_bytes": 27070, "bundle_sha256": "8c573ecc91ad5f0f76f2f09bf876aa3d6c442aaab055155a442382b3dc4ae9e4", "case": "AP051", "contains_checked_statements": [], "coverage": "prove_halving_of_each_supplied_exponential_recurrence_step_beyond_twice_an_arbitrary_natural_argument_bound_by_positive_full_coefficient_cancellation_including_zero_argument_without_a_finite_sum_or_analytic_assumption", "ha": "evidence/AP051-HA.json.gz", "internal_proof_nodes": [ 21, 19, 15, 11, 8, 10, 16 ], "lean": "evidence/AP051-Lean.json.gz", "nodes": 23, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-large-argument-ratio-v1", "proof": "evidence/AP051-bundle.json.gz", "report_sha256": "073812f7f882fb31123f92dea019dd8899639070439ea1bf18db1d77f900dd2d", "root": 22, "statement": "forall d x L j a b. ~d=0 -> Le(x,L*d) -> Le(2*L,S j) -> ((S j)*d)*b=x*a -> Le(2*b,a)", "target_ast_sha256": "0ed9c296638e7d978dbfb8e18e70dba5fbd546cda5f0e856d746cddb49f5ba40", "title": "exp large argument ratio", "uses_definitions": [ "PD0001" ] }, { "bundle_bytes": 220057, "bundle_sha256": "911984292a5e6478f05be815332b1ea99357351be9c5150149a0930b1da3293a", "case": "AP052", "contains_checked_statements": [ "AP051", "CV068" ], "coverage": "construct_the_actual_shifted_tail_head_halving_relation_and_sum_of_a_supplied_exponential_recurrence_at_any_cutoff_at_least_twice_its_natural_argument_bound_with_finite_sum_at_most_twice_the_head_including_zero_cutoffs_without_an_infinite_tail", "ha": "evidence/AP052-HA.json.gz", "internal_proof_nodes": [ 115, 78, 120, 121, 118 ], "lean": "evidence/AP052-Lean.json.gz", "nodes": 123, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-large-argument-tail-v1", "proof": "evidence/AP052-bundle.json.gz", "report_sha256": "5a2bbf26da579cfe62a21c3a8e24c496ff6c7b65fb252b3f2fca319d32d8aae3", "root": 122, "statement": "forall d x L tb tc M N. ~d=0 -> Le(x,L*d) -> Le(2*L,M) -> NatScaledExpRecurrence(d,x,tb,tc,M+N) -> exists ub uc a s. (NatAffineSlice(tb,tc,M,1,ub,uc,S N)) /\\ ((NatRatioBoundPrefix(ub,uc,1,S N)) /\\ ((BetaAt(tb,tc,M,a)) /\\ ((BetaAt(ub,uc,0,a)) /\\ ((Sum(ub,uc,S N,s)) /\\ (Le(s,2*a))))))", "target_ast_sha256": "bb49801153c9bc3821d1f494707843d81d2f5e20181f6172563fe79531ae95eb", "title": "exp large argument tail", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0411", "ND0441", "ND0443" ] }, { "bundle_bytes": 75671, "bundle_sha256": "8ab91a86da1d7a0fb5ffb4d711d6723b98695277ef3042587d475462c3bf0bbf", "case": "AP053", "contains_checked_statements": [], "coverage": "prove_symbolic_binary_decay_at_every_bounded_index_of_an_actual_supplied_halving_prefix_by_one_ordinary_HA_induction_including_zero_head_index_and_horizon_without_division_or_an_infinite_sequence", "ha": "evidence/AP053-HA.json.gz", "internal_proof_nodes": [ 32, 33, 27, 28, 14, 20, 9, 19, 12, 34, 11 ], "lean": "evidence/AP053-Lean.json.gz", "nodes": 36, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-halving-prefix-decay-v1", "proof": "evidence/AP053-bundle.json.gz", "report_sha256": "328c654e28213375c7007ede52a24d45e7ca5382eea3ee25e946de774389132f", "root": 35, "statement": "forall k ab ac N a0 a p. Le(k,N) -> NatRatioBoundPrefix(ab,ac,1,S N) -> BetaAt(ab,ac,0,a0) -> BetaAt(ab,ac,k,a) -> Pow(2,k,p) -> Le(p*a,a0)", "target_ast_sha256": "07ae73101d211aba3dfdb13a23c81410079b1c0ab2cde23e58d19ff2b5e0d6d8", "title": "halving prefix decay", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0441" ] }, { "bundle_bytes": 293917, "bundle_sha256": "1390d3a7c2b97fdc38e34ee664d4941d9aa338b669daa1d77b5e68f7e22d6fe5", "case": "AP054", "contains_checked_statements": [ "AP051", "AP052", "AP053", "CV068" ], "coverage": "construct_actual_powers_heads_shifted_slices_and_sums_with_dyadic_decay_after_every_extra_cutoff_beyond_twice_a_general_natural_argument_bound_including_all_zero_margins_and_without_an_infinite_remainder_claim", "ha": "evidence/AP054-HA.json.gz", "internal_proof_nodes": [ 122, 134, 78, 136, 135, 2, 11 ], "lean": "evidence/AP054-Lean.json.gz", "nodes": 138, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-large-argument-decay-v1", "proof": "evidence/AP054-bundle.json.gz", "report_sha256": "4135a21d234b275186533dce617318529e4351a7e116f9b0d5a943ca82f0afb5", "root": 137, "statement": "forall d x L tb tc K m N. ~d=0 -> Le(x,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,tb,tc,(K+m)+N) -> exists p ub uc a0 a s. (Pow(2,m,p)) /\\ ((BetaAt(tb,tc,K,a0)) /\\ ((BetaAt(tb,tc,K+m,a)) /\\ ((NatAffineSlice(tb,tc,K+m,1,ub,uc,S N)) /\\ ((NatRatioBoundPrefix(ub,uc,1,S N)) /\\ ((BetaAt(ub,uc,0,a)) /\\ ((Sum(ub,uc,S N,s)) /\\ ((Le(p*a,a0)) /\\ (Le(p*s,2*a0)))))))))", "target_ast_sha256": "a11bcab301bcc1baa600ebeb7741038a4ddea68d8a679317bf9afb96a127d859", "title": "exp large argument decay", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0411", "PD0020", "ND0441", "ND0443" ] }, { "bundle_bytes": 396287, "bundle_sha256": "4b2704a69fe125f681a594b0efd0eb04b37c832be124a46f148afc31eb798a53", "case": "AP055", "contains_checked_statements": [ "AP051", "AP052", "AP053", "AP054", "CV009", "CV068" ], "coverage": "identify_the_constructed_large_argument_tail_with_two_supplied_sum_values_and_prove_both_truncation_orders_with_the_supplied_head_and_binary_power_including_an_empty_short_prefix_without_infinite_analysis", "ha": "evidence/AP055-HA.json.gz", "internal_proof_nodes": [ 137, 155, 79, 153, 154 ], "lean": "evidence/AP055-Lean.json.gz", "nodes": 157, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-large-argument-truncation-v1", "proof": "evidence/AP055-bundle.json.gz", "report_sha256": "01d8e4ad1b57b29d625fccdb72ab4828af49ac5ba27480c3c161aa54ca089a1f", "root": 156, "statement": "forall d x L ab ac K m N a0 s t p. ~d=0 -> Le(x,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,(K+m)+N) -> BetaAt(ab,ac,K,a0) -> Sum(ab,ac,K+m,s) -> Sum(ab,ac,S((K+m)+N),t) -> Pow(2,m,p) -> Le(s,t) /\\ Le(p*t,p*s+2*a0)", "target_ast_sha256": "6d652892aa498fb4573642b7316fc9a30f17cc0102198e2d0ca31291e5c299b0", "title": "exp large argument truncation", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 87280, "bundle_sha256": "6d40a0ff19af1e096df1647183ec59ca894502ec46e796c50b4652362eb3ee33", "case": "AP056", "contains_checked_statements": [], "coverage": "prove_every_bounded_actual_exponential_term_is_at_most_the_actual_natural_argument_bound_power_times_its_original_head_by_one_ordinary_induction_and_positive_denominator_cancellation_including_zero_argument_bound", "ha": "evidence/AP056-HA.json.gz", "internal_proof_nodes": [ 42, 37, 38, 32, 33, 17, 24, 9, 15, 39, 11, 6 ], "lean": "evidence/AP056-Lean.json.gz", "nodes": 44, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-geometric-majorant-v1", "proof": "evidence/AP056-bundle.json.gz", "report_sha256": "e9b75c0ac842801ae630457cf30463e39c37f9f99720103fbc608265ac5b4e94", "root": 43, "statement": "forall k d x L tb tc N h a b. ~d=0 -> Le(x,L*d) -> Le(k,N) -> NatScaledExpRecurrence(d,x,tb,tc,N) -> BetaAt(tb,tc,0,h) -> BetaAt(tb,tc,k,a) -> Pow(L,k,b) -> Le(a,b*h)", "target_ast_sha256": "e86ad6b58635a4c4ece5af53c0cf336a09fc69caf323c09e6c79fea863077e79", "title": "exp geometric majorant", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0443" ] }, { "bundle_bytes": 422324, "bundle_sha256": "cc59b8f6360dec4a08b2e261e02420319cf7924020589c9285163adb763d294c", "case": "AP057", "contains_checked_statements": [ "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "CV009", "CV068" ], "coverage": "prove_the_large_argument_truncation_error_relative_to_the_original_head_while_retaining_the_supplied_table_sums_and_two_actual_powers_including_zero_head_and_empty_prefix_without_an_analytic_limit", "ha": "evidence/AP057-HA.json.gz", "internal_proof_nodes": [ 156, 158, 78, 2, 11, 19, 113 ], "lean": "evidence/AP057-Lean.json.gz", "nodes": 160, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-normalized-truncation-v1", "proof": "evidence/AP057-bundle.json.gz", "report_sha256": "4621db290d8697ac9bc7653d2166eb9d841bab5f10d500905a55a4cf1c038cb4", "root": 159, "statement": "forall d x L ab ac K m N h s t p B. ~d=0 -> Le(x,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,(K+m)+N) -> BetaAt(ab,ac,0,h) -> Sum(ab,ac,K+m,s) -> Sum(ab,ac,S((K+m)+N),t) -> Pow(2,m,p) -> Pow(L,K,B) -> Le(s,t) /\\ Le(p*t,p*s+2*(B*h))", "target_ast_sha256": "f74987c7317766d697ca0d5792985ab700e38691f10bd6149d8c1ca141fc33b9", "title": "exp normalized truncation", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 606966, "bundle_sha256": "df0dfd944205f1586fc319d7674976003be31c2c08bfe39b5a678294a9277e16", "case": "AP058", "contains_checked_statements": [ "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "CV009", "CV022", "CV030", "CV068", "CV069", "CV070" ], "coverage": "construct_actual_factorial_weighted_exponential_tables_positive_original_normalization_both_powers_and_sums_with_original_head_normalized_finite_tail_error_for_arbitrary_bounded_arguments_without_an_analytic_exponential", "ha": "evidence/AP058-HA.json.gz", "internal_proof_nodes": [ 159, 181, 134, 182, 10, 110 ], "lean": "evidence/AP058-Lean.json.gz", "nodes": 184, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-normalized-witness-v1", "proof": "evidence/AP058-bundle.json.gz", "report_sha256": "e07d7baedfffc64379fee86275d0e9926c310982a31ea0d321c2f4c88ac9f2cc", "root": 183, "statement": "forall d x L K m N. ~d=0 -> Le(x,L*d) -> Le(2*L,K) -> exists wb wc tb tc h f q t p B s. (NatExpWeightTable(d,(K+m)+N,wb,wc)) /\\ ((NatPowerWeightedPrefix(x,wb,wc,tb,tc,S((K+m)+N))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(tb,tc,0,h)) /\\ ((~h=0) /\\ ((Factorial((K+m)+N,f)) /\\ ((Pow(d,(K+m)+N,q)) /\\ ((h=f*q) /\\ ((Sum(tb,tc,S((K+m)+N),t)) /\\ ((NatScaledExpRecurrence(d,x,tb,tc,(K+m)+N)) /\\ ((forall j v fj dj xj. Le(j,(K+m)+N) -> BetaAt(tb,tc,j,v) -> Factorial(j,fj) -> Pow(d,j,dj) -> Pow(x,j,xj) -> (v*fj)*dj=h*xj) /\\ ((Pow(2,m,p)) /\\ ((~p=0) /\\ ((Pow(L,K,B)) /\\ ((Sum(tb,tc,K+m,s)) /\\ ((Le(s,t)) /\\ (Le(p*t,p*s+2*(B*h))))))))))))))))))", "target_ast_sha256": "d7fe383cf8b88b0d32538bcdd03ef48992ec2b80666f1188f153a402dde188e5", "title": "exp normalized witness", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0442", "ND0443" ] }, { "bundle_bytes": 1042764, "bundle_sha256": "67a11f71c4bf3e2627a1a9312bcd6c03448719f660047da1f33d10d257b59445", "case": "AP059", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV095" ], "coverage": "construct_the_actual_finite_exponential_Cauchy_product_recurrence_head_and_four_sums_with_an_original_head_normalized_quantitative_product_error_for_arbitrarily_bounded_arguments_without_an_exponential_addition_axiom", "ha": "evidence/AP059-HA.json.gz", "internal_proof_nodes": [ 190, 213, 214, 246, 38, 12 ], "lean": "evidence/AP059-Lean.json.gz", "nodes": 248, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-large-argument-product-v1", "proof": "evidence/AP059-bundle.json.gz", "report_sha256": "ac0e8f4de0ae114c56dfc73503566ae831eaa78220195028e38b266abecfdb20", "root": 247, "statement": "forall d x y L ab ac bb bc K m a0 b0 p B. ~d=0 -> Le(x+y,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,S(K+m)+(K+m)) -> NatScaledExpRecurrence(d,y,bb,bc,S(K+m)+(K+m)) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Pow(2,S m,p) -> Pow(L,K,B) -> exists cb cc u v s t. (NatConvolutionPrefix(ab,ac,S(S(K+m)+(K+m)),bb,bc,S(S(K+m)+(K+m)),cb,cc,S(S(K+m)+(K+m)))) /\\ ((NatScaledExpRecurrence(d,x+y,cb,cc,S(K+m)+(K+m))) /\\ ((BetaAt(cb,cc,0,a0*b0)) /\\ ((Sum(ab,ac,S(K+m),u)) /\\ ((Sum(bb,bc,S(K+m),v)) /\\ ((Sum(cb,cc,S(K+m),s)) /\\ ((Sum(cb,cc,S(S(K+m)+(K+m)),t)) /\\ ((Le(s,u*v)) /\\ (Le(p*(u*v),p*s+2*(B*(a0*b0)))))))))))", "target_ast_sha256": "7b046134336250ad84a04def0d99f230641a6586ae277d5538980db4b66aa271", "title": "exp large argument product", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0402", "PD0020", "ND0443" ] }, { "bundle_bytes": 94980, "bundle_sha256": "87bd710ef99048e7cafb38dc72a681d8ea931d8054616880a6d04f87f4ecc748", "case": "AP060", "contains_checked_statements": [], "coverage": "prove_cross_value_equality_of_two_actual_supplied_exponential_recurrences_and_sums_at_equal_rational_arguments_without_any_argument_bound_or_positive_head_assumption_by_one_ordinary_induction_and_existing_sum_linearity", "ha": "evidence/AP060-HA.json.gz", "internal_proof_nodes": [ 35, 32, 4 ], "lean": "evidence/AP060-Lean.json.gz", "nodes": 37, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-recurrence-independence-v1", "proof": "evidence/AP060-bundle.json.gz", "report_sha256": "2a334ccd7e4565c8f77832a2f7cc94a2d153d2e2ee41fdf000eb6910fef140db", "root": 36, "statement": "forall d e x y N ab ac bb bc h g s t. ~d=0 -> ~e=0 -> x*e=y*d -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,h) -> BetaAt(bb,bc,0,g) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> g*s=h*t", "target_ast_sha256": "7beddbf0b9c56535b34cc9097c02c1dd441a5ba9cb163ee87131584651dfc897", "title": "exp recurrence independence", "uses_definitions": [ "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 1093896, "bundle_sha256": "bd5cfc30ebecd494e85593efcf0c03cd77f252288449cc975a0d04d79cc5c861", "case": "AP061", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP059", "AP060", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV095" ], "coverage": "retain_three_independently_supplied_exponential_tables_heads_and_sums_with_general_argument_product_error_using_unrestricted_recurrence_value_independence_and_no_analytic_addition_axiom", "ha": "evidence/AP061-HA.json.gz", "internal_proof_nodes": [ 247, 251, 252, 125, 253 ], "lean": "evidence/AP061-Lean.json.gz", "nodes": 255, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-supplied-large-product-v1", "proof": "evidence/AP061-bundle.json.gz", "report_sha256": "a653f6737ce9a07e8a42217b9cfee29acd9a8e68925df17aa9f193730e76a27f", "root": 254, "statement": "forall d x y L ab ac bb bc cb cc K m a0 b0 c0 u v w p B. ~d=0 -> Le(x+y,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,S(K+m)+(K+m)) -> NatScaledExpRecurrence(d,y,bb,bc,S(K+m)+(K+m)) -> NatScaledExpRecurrence(d,x+y,cb,cc,K+m) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> BetaAt(cb,cc,0,c0) -> Sum(ab,ac,S(K+m),u) -> Sum(bb,bc,S(K+m),v) -> Sum(cb,cc,S(K+m),w) -> Pow(2,S m,p) -> Pow(L,K,B) -> Le((a0*b0)*w,c0*(u*v)) /\\ Le(p*(c0*(u*v)),p*((a0*b0)*w)+2*(B*(c0*(a0*b0))))", "target_ast_sha256": "5b7c88d4d5aebd8dea844bd2f45f5a1966eacf6503b969bc711e7b8e1bc1163d", "title": "exp supplied large product", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 271322, "bundle_sha256": "39d7d7a5d5dbff58bd1267471f3e1b6de19b268d0688a7806e0e8c278173e0ad", "case": "AP062", "contains_checked_statements": [], "coverage": "prove_an_explicit_symbolic_dyadic_budget_for_arbitrary_natural_argument_majorants_by_one_ordinary_power_induction_and_actual_power_addition_including_zero_base_exponent_and_precision_without_large_ground_numerals", "ha": "evidence/AP062-HA.json.gz", "internal_proof_nodes": [ 126, 127, 110, 113, 119, 120, 17 ], "lean": "evidence/AP062-Lean.json.gz", "nodes": 129, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-binary-budget-v1", "proof": "evidence/AP062-bundle.json.gz", "report_sha256": "1e19ea09fa69f6d3b47ebbbd512a64788b5a09bb0511778edc1816108b2837e3", "root": 128, "statement": "forall L K n B r p. Pow(L,K,B) -> Pow(2,n,r) -> Pow(2,S(L*K+n),p) -> ~r=0 /\\ (~p=0 /\\ Le(2*(r*B),p))", "target_ast_sha256": "b028bfcd3503ce1d69225b9daa26335368cc2a1d1388058cffbf5bd29ca99c55", "title": "exp binary budget", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 486928, "bundle_sha256": "b057dee9e4adb11a462fbfd234cd9eb053c0e7275d2c3171ed29df76a3870023", "case": "AP063", "contains_checked_statements": [ "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP062", "CV009", "CV068" ], "coverage": "prove_requested_dyadic_accuracy_for_two_supplied_finite_exponential_sums_at_an_explicit_large_argument_cutoff_uniformly_in_the_extra_finite_horizon_with_auxiliary_powers_constructed_and_no_analytic_limit", "ha": "evidence/AP063-HA.json.gz", "internal_proof_nodes": [ 159, 168, 134, 169 ], "lean": "evidence/AP063-Lean.json.gz", "nodes": 171, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-requested-truncation-v1", "proof": "evidence/AP063-bundle.json.gz", "report_sha256": "be99fcbf3be1f17356e4c0f1a1c2f2d4b3c6a74abf13a9e5fd936fb93248ecd4", "root": 170, "statement": "forall d x L K n N ab ac h s t r. ~d=0 -> Le(x,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,(K+S(L*K+n))+N) -> BetaAt(ab,ac,0,h) -> Sum(ab,ac,K+S(L*K+n),s) -> Sum(ab,ac,S((K+S(L*K+n))+N),t) -> Pow(2,n,r) -> ~r=0 /\\ (Le(s,t) /\\ Le(r*t,r*s+h))", "target_ast_sha256": "87d46a5afea3d3cb839af399daa22055d469c3506826aa7404168a3b15983873", "title": "exp requested truncation", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 1163417, "bundle_sha256": "cd072e20381973dfe744997e4f8b40cfb61735f27a0be2bd40f78caa95769495", "case": "AP064", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP059", "AP060", "AP061", "AP062", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV095" ], "coverage": "prove_requested_dyadic_accuracy_for_the_product_of_two_independently_supplied_finite_exponential_sums_against_a_third_supplied_sum_at_the_added_argument_with_explicit_symbolic_cutoffs_and_no_analytic_addition_axiom", "ha": "evidence/AP064-HA.json.gz", "internal_proof_nodes": [ 254, 263, 264, 232 ], "lean": "evidence/AP064-Lean.json.gz", "nodes": 266, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-requested-product-v1", "proof": "evidence/AP064-bundle.json.gz", "report_sha256": "d6974f561a1eeaa77f200ad8043a48879052484944ebcd27a6423fbff87083af", "root": 265, "statement": "forall d x y L ab ac bb bc cb cc K n a0 b0 c0 u v w r. ~d=0 -> Le(x+y,L*d) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,S(K+(L*K+n))+(K+(L*K+n))) -> NatScaledExpRecurrence(d,y,bb,bc,S(K+(L*K+n))+(K+(L*K+n))) -> NatScaledExpRecurrence(d,x+y,cb,cc,K+(L*K+n)) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> BetaAt(cb,cc,0,c0) -> Sum(ab,ac,S(K+(L*K+n)),u) -> Sum(bb,bc,S(K+(L*K+n)),v) -> Sum(cb,cc,S(K+(L*K+n)),w) -> Pow(2,n,r) -> ~r=0 /\\ (Le((a0*b0)*w,c0*(u*v)) /\\ Le(r*(c0*(u*v)),r*((a0*b0)*w)+c0*(a0*b0)))", "target_ast_sha256": "f0434acec36b8372e936d5c8e5f08cb7bacbc68bca03a222d3af6dfa45e2d636", "title": "exp requested product", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 576250, "bundle_sha256": "2ad83c07ef353332351dba47fd70922980c533807a82e0ab7c479479d09f1e80", "case": "AP065", "contains_checked_statements": [ "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP060", "AP062", "AP063", "CV009", "CV068" ], "coverage": "prove_requested_dyadic_Cauchy_accuracy_across_two_independently_supplied_exponential_codes_denominators_and_heads_at_equal_rational_arguments_and_different_finite_horizons_by_constructing_the_actual_fine_short_sum", "ha": "evidence/AP065-HA.json.gz", "internal_proof_nodes": [ 36, 175, 176, 117, 10, 19, 11, 2, 177 ], "lean": "evidence/AP065-Lean.json.gz", "nodes": 179, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-independent-horizon-v1", "proof": "evidence/AP065-bundle.json.gz", "report_sha256": "e6518179c582f996af69e453fa536f6909ccebae194dfa2ecc3f1bcaff497d34", "root": 178, "statement": "forall d e x y L K n N ab ac bb bc h g s t r. ~d=0 -> ~e=0 -> x*e=y*d -> Le(y,L*e) -> Le(2*L,K) -> NatScaledExpRecurrence(d,x,ab,ac,K+(L*K+n)) -> NatScaledExpRecurrence(e,y,bb,bc,S(K+(L*K+n))+N) -> BetaAt(ab,ac,0,h) -> BetaAt(bb,bc,0,g) -> Sum(ab,ac,S(K+(L*K+n)),s) -> Sum(bb,bc,S(S(K+(L*K+n))+N),t) -> Pow(2,n,r) -> ~r=0 /\\ (Le(g*s,h*t) /\\ Le(r*(h*t),r*(g*s)+h*g))", "target_ast_sha256": "650e729c29cf242dce7af6b0b54b08b896036d8256a6ec458c80134305b3ee39", "title": "exp independent horizon", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 786488, "bundle_sha256": "21e999a7af380b2fa76fc861ba96979e5942ec7acfcb115ec655a9b91b167beb", "case": "AP066", "contains_checked_statements": [ "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP060", "AP062", "AP063", "AP065", "CV009", "CV022", "CV030", "CV068", "CV069", "CV070" ], "coverage": "construct_one_positive_factorial_normalized_coarse_exponential_table_and_precision_power_before_the_universal_extra_horizon_then_actual_positive_fine_tables_with_requested_dyadic_error_without_an_analytic_limit", "ha": "evidence/AP066-HA.json.gz", "internal_proof_nodes": [ 178, 200, 139, 166, 9 ], "lean": "evidence/AP066-Lean.json.gz", "nodes": 202, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-uniform-witness-v1", "proof": "evidence/AP066-bundle.json.gz", "report_sha256": "ff32a056a97c38a9531beafffc66028aa8c10bb62f8c27fa1de3f1e49006dce1", "root": 201, "statement": "forall d x L K n. ~d=0 -> Le(x,L*d) -> Le(2*L,K) -> exists wb wc ab ac h f q s r. (NatExpWeightTable(d,K+(L*K+n),wb,wc)) /\\ ((NatPowerWeightedPrefix(x,wb,wc,ab,ac,S(K+(L*K+n)))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(ab,ac,0,h)) /\\ ((~h=0) /\\ ((Factorial(K+(L*K+n),f)) /\\ ((Pow(d,K+(L*K+n),q)) /\\ ((h=f*q) /\\ ((Sum(ab,ac,S(K+(L*K+n)),s)) /\\ ((NatScaledExpRecurrence(d,x,ab,ac,K+(L*K+n))) /\\ ((forall j v fj dj xj. Le(j,K+(L*K+n)) -> BetaAt(ab,ac,j,v) -> Factorial(j,fj) -> Pow(d,j,dj) -> Pow(x,j,xj) -> (v*fj)*dj=h*xj) /\\ ((Pow(2,n,r)) /\\ ((~r=0) /\\ (forall N. exists vb vc bb bc g F Q t. (NatExpWeightTable(d,S(K+(L*K+n))+N,vb,vc)) /\\ ((NatPowerWeightedPrefix(x,vb,vc,bb,bc,S(S(K+(L*K+n))+N))) /\\ ((BetaAt(vb,vc,0,g)) /\\ ((BetaAt(bb,bc,0,g)) /\\ ((~g=0) /\\ ((Factorial(S(K+(L*K+n))+N,F)) /\\ ((Pow(d,S(K+(L*K+n))+N,Q)) /\\ ((g=F*Q) /\\ ((Sum(bb,bc,S(S(K+(L*K+n))+N),t)) /\\ ((NatScaledExpRecurrence(d,x,bb,bc,S(K+(L*K+n))+N)) /\\ ((forall j v fj dj xj. Le(j,S(K+(L*K+n))+N) -> BetaAt(bb,bc,j,v) -> Factorial(j,fj) -> Pow(d,j,dj) -> Pow(x,j,xj) -> (v*fj)*dj=g*xj) /\\ ((Le(g*s,h*t)) /\\ (Le(r*(h*t),r*(g*s)+h*g))))))))))))))))))))))))))", "target_ast_sha256": "1271edf53171c0ca044ba475b8805112ebf82d4ac21ba32f38e69819c32da8c8", "title": "exp uniform witness", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0442", "ND0443" ] }, { "bundle_bytes": 1418631, "bundle_sha256": "d685c940c575f292c72c45d963a0be7d367a9f6cc3be8e83e5bac6b8fdb71cb6", "case": "AP067", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP059", "AP060", "AP061", "AP062", "AP064", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV069", "CV070", "CV095" ], "coverage": "construct_three_positive_factorial_normalized_exponential_tables_and_actual_input_prefix_sums_with_requested_dyadic_product_error_retaining_all_indexwise_normalizations_without_an_analytic_addition_axiom", "ha": "evidence/AP067-HA.json.gz", "internal_proof_nodes": [ 265, 287, 232, 123 ], "lean": "evidence/AP067-Lean.json.gz", "nodes": 289, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-product-witness-v1", "proof": "evidence/AP067-bundle.json.gz", "report_sha256": "1e56428f0daca4b82cadcdcf43de278b8f1509edadd252899ad59800d90812b3", "root": 288, "statement": "forall d x y L K n. ~d=0 -> Le(x+y,L*d) -> Le(2*L,K) -> exists aw az ab ac a0 af aq aFull bw bz bb bc b0 bf bq bFull cw cz cb cc c0 cf cq w u v r. (NatExpWeightTable(d,S(K+(L*K+n))+(K+(L*K+n)),aw,az)) /\\ ((NatPowerWeightedPrefix(x,aw,az,ab,ac,S(S(K+(L*K+n))+(K+(L*K+n))))) /\\ ((BetaAt(aw,az,0,a0)) /\\ ((BetaAt(ab,ac,0,a0)) /\\ ((~a0=0) /\\ ((Factorial(S(K+(L*K+n))+(K+(L*K+n)),af)) /\\ ((Pow(d,S(K+(L*K+n))+(K+(L*K+n)),aq)) /\\ ((a0=af*aq) /\\ ((Sum(ab,ac,S(S(K+(L*K+n))+(K+(L*K+n))),aFull)) /\\ ((NatScaledExpRecurrence(d,x,ab,ac,S(K+(L*K+n))+(K+(L*K+n)))) /\\ ((forall j a fj dj xj. Le(j,S(K+(L*K+n))+(K+(L*K+n))) -> BetaAt(ab,ac,j,a) -> Factorial(j,fj) -> Pow(d,j,dj) -> Pow(x,j,xj) -> (a*fj)*dj=a0*xj) /\\ ((NatExpWeightTable(d,S(K+(L*K+n))+(K+(L*K+n)),bw,bz)) /\\ ((NatPowerWeightedPrefix(y,bw,bz,bb,bc,S(S(K+(L*K+n))+(K+(L*K+n))))) /\\ ((BetaAt(bw,bz,0,b0)) /\\ ((BetaAt(bb,bc,0,b0)) /\\ ((~b0=0) /\\ ((Factorial(S(K+(L*K+n))+(K+(L*K+n)),bf)) /\\ ((Pow(d,S(K+(L*K+n))+(K+(L*K+n)),bq)) /\\ ((b0=bf*bq) /\\ ((Sum(bb,bc,S(S(K+(L*K+n))+(K+(L*K+n))),bFull)) /\\ ((NatScaledExpRecurrence(d,y,bb,bc,S(K+(L*K+n))+(K+(L*K+n)))) /\\ ((forall j a fj dj xj. Le(j,S(K+(L*K+n))+(K+(L*K+n))) -> BetaAt(bb,bc,j,a) -> Factorial(j,fj) -> Pow(d,j,dj) -> Pow(y,j,xj) -> (a*fj)*dj=b0*xj) /\\ ((NatExpWeightTable(d,K+(L*K+n),cw,cz)) /\\ ((NatPowerWeightedPrefix(x+y,cw,cz,cb,cc,S(K+(L*K+n)))) /\\ ((BetaAt(cw,cz,0,c0)) /\\ ((BetaAt(cb,cc,0,c0)) /\\ ((~c0=0) /\\ ((Factorial(K+(L*K+n),cf)) /\\ ((Pow(d,K+(L*K+n),cq)) /\\ ((c0=cf*cq) /\\ ((Sum(cb,cc,S(K+(L*K+n)),w)) /\\ ((NatScaledExpRecurrence(d,x+y,cb,cc,K+(L*K+n))) /\\ ((forall j a fj dj xj. Le(j,K+(L*K+n)) -> BetaAt(cb,cc,j,a) -> Factorial(j,fj) -> Pow(d,j,dj) -> Pow(x+y,j,xj) -> (a*fj)*dj=c0*xj) /\\ ((Sum(ab,ac,S(K+(L*K+n)),u)) /\\ ((Sum(bb,bc,S(K+(L*K+n)),v)) /\\ ((Pow(2,n,r)) /\\ ((~r=0) /\\ ((Le((a0*b0)*w,c0*(u*v))) /\\ (Le(r*(c0*(u*v)),r*((a0*b0)*w)+c0*(a0*b0))))))))))))))))))))))))))))))))))))))))", "target_ast_sha256": "90a7f53271dfd7883b4edc4850e14370009306488126a38310f44e94983de3fd", "title": "exp product witness", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0442", "ND0443" ] }, { "bundle_bytes": 331452, "bundle_sha256": "0fa098dd5ca136b1341a37b5c0e3ef5211e1d8316588a5601d47c9d8d43f68df", "case": "AP068", "contains_checked_statements": [ "CV009", "CV094", "CV095" ], "coverage": "prove_relative_finite_exponential_sum_continuity_for_arbitrary_nonnegative_arguments_from_actual_tail_slices_without_any_small_argument_bound_or_analytic_limit", "ha": "evidence/AP068-HA.json.gz", "internal_proof_nodes": [ 132, 134, 44, 119, 133, 135, 9, 4 ], "lean": "evidence/AP068-Lean.json.gz", "nodes": 137, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-relative-continuity-v1", "proof": "evidence/AP068-bundle.json.gz", "report_sha256": "8864cd6250e08325acc33cca5cd6e4ed9b58ddd2ec6fda98aa7beffbc50b33b5", "root": 136, "statement": "forall d x y g N ab ac bb bc a0 s t. ~d=0 -> y=x+g -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(d,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,a0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Le(s,t) /\\ Le(d*t,d*s+g*t)", "target_ast_sha256": "44c446356566e235233dd3597393197285525996f1312f828627e5f6c621ea1d", "title": "exp relative continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 384083, "bundle_sha256": "1f05df465a0a247f69b287823dcc92b3d2c728cb3808e5b3aa561dc6edcec249", "case": "AP069", "contains_checked_statements": [ "AP068", "CV009", "CV094", "CV095", "CV097" ], "coverage": "prove_unrestricted_relative_finite_exponential_continuity_across_independent_positive_argument_denominators_arbitrary_heads_and_actual_codes_by_constructing_rescaled_common_head_tables", "ha": "evidence/AP069-HA.json.gz", "internal_proof_nodes": [ 136, 142, 27, 6 ], "lean": "evidence/AP069-Lean.json.gz", "nodes": 144, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-relative-cross-continuity-v1", "proof": "evidence/AP069-bundle.json.gz", "report_sha256": "7eedbdac946e172b9e8b9ba99bc1173cc54e887f804076c07d9c2347477d98f7", "root": 143, "statement": "forall d e x y g N ab ac bb bc a0 b0 s t. ~d=0 -> ~e=0 -> d*y=e*x+g -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Le(b0*s,a0*t) /\\ Le((d*e)*(a0*t),(d*e)*(b0*s)+g*(a0*t))", "target_ast_sha256": "bd0a751f36b51993e4a8a1c8d3802afe6e72a7304960b031618cbee1c6e66f62", "title": "exp relative cross continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 398512, "bundle_sha256": "1804aac88411201a7aba240f21ec51eb677f7c532748c43fd7c5ae9bd547a9a8", "case": "AP070", "contains_checked_statements": [ "AP068", "AP069", "CV009", "CV094", "CV095", "CV097" ], "coverage": "prove_requested_absolute_finite_exponential_continuity_from_two_actual_independent_computations_an_explicit_sum_majorant_and_symbolic_precision_budget_without_assuming_the_majorant_or_precision_power_as_an_analytic_axiom", "ha": "evidence/AP070-HA.json.gz", "internal_proof_nodes": [ 143, 27, 35, 8, 6, 144 ], "lean": "evidence/AP070-Lean.json.gz", "nodes": 146, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-absolute-continuity-v1", "proof": "evidence/AP070-bundle.json.gz", "report_sha256": "eaee105a2a0fbc204f139c5228e1795ceba665b54c24c2d5df8ea9301b1a2c08", "root": 145, "statement": "forall d e x y g N ab ac bb bc a0 b0 s t C r. ~d=0 -> ~e=0 -> d*y=e*x+g -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Le(t,C*b0) -> Le((r*g)*C,d*e) -> Le(b0*s,a0*t) /\\ Le(r*(a0*t),r*(b0*s)+a0*b0)", "target_ast_sha256": "a085e6805bd448394a0ee2f1cd547da010b34b51bebbfe4b683fe338d87aeba9", "title": "exp absolute continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 861106, "bundle_sha256": "9e82458eb38f49caf7d705ed9f288ba9de5be9b6e3884670b80be8a2c1b61d11", "case": "AP071", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP060", "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047", "CV048", "CV095" ], "coverage": "prove_unrestricted_submultiplicativity_for_three_actual_finite_exponential_sums_at_the_same_degree_by_constructing_a_Cauchy_table_and_identifying_its_cross_normalized_value_without_extra_horizons_or_analytic_addition", "ha": "evidence/AP071-HA.json.gz", "internal_proof_nodes": [ 190, 208, 209, 214, 125, 11, 12, 38 ], "lean": "evidence/AP071-Lean.json.gz", "nodes": 216, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-submultiplicative-v1", "proof": "evidence/AP071-bundle.json.gz", "report_sha256": "01a8d9cfa59d3bab2db561bd0fe8f86e1022799e1c29e5ba1bf315921c1052a1", "root": 215, "statement": "forall d x y N ab ac bb bc cb cc a0 b0 c0 u v w. ~d=0 -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(d,y,bb,bc,N) -> NatScaledExpRecurrence(d,x+y,cb,cc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> BetaAt(cb,cc,0,c0) -> Sum(ab,ac,S N,u) -> Sum(bb,bc,S N,v) -> Sum(cb,cc,S N,w) -> Le((a0*b0)*w,c0*(u*v))", "target_ast_sha256": "172dd2d034e151140ec0f7866fbfea14a1de644c3bc4f01e38d61a48f66c9873", "title": "exp submultiplicative", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 1139220, "bundle_sha256": "9211e4b07e0c954f47f5621b318226c957dbe6d4b59a960dfe098b91fcefa839", "case": "AP072", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP060", "AP071", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV069", "CV070", "CV095" ], "coverage": "prove_a_horizon_independent_power_majorant_at_every_integer_argument_by_ordinary_induction_and_actual_positive_successor_factor_tables_without_assuming_positivity_of_the_supplied_head_or_an_analytic_exponential", "ha": "evidence/AP072-HA.json.gz", "internal_proof_nodes": [ 215, 253, 264, 240, 241, 8, 4, 17, 5, 265 ], "lean": "evidence/AP072-Lean.json.gz", "nodes": 267, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-integer-majorant-v1", "proof": "evidence/AP072-bundle.json.gz", "report_sha256": "9b77029833f49ff076cd26cf08ada33331ae779befdb223d0e6de1ba97c32e20", "root": 266, "statement": "forall L d N tb tc h s C. ~d=0 -> NatScaledExpRecurrence(d,L*d,tb,tc,N) -> BetaAt(tb,tc,0,h) -> Sum(tb,tc,S N,s) -> Pow(9,S L,C) -> Le(s,C*h)", "target_ast_sha256": "0c8801ad08d671c34b8ab3b48766c6133267c4b5abfd1b796eb18c6422b8d237", "title": "exp integer majorant", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 1238445, "bundle_sha256": "9b94bb19dbf67b5b77d0f8e7315675f73673189d0c87d1989f2e93ee18c07d2f", "case": "AP073", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP060", "AP068", "AP069", "AP071", "AP072", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV069", "CV070", "CV094", "CV095", "CV097" ], "coverage": "prove_a_horizon_independent_power_majorant_for_every_actual_bounded_rational_argument_by_constructing_a_positive_integer_argument_reference_and_cancelling_only_its_head_without_an_analytic_axiom", "ha": "evidence/AP073-HA.json.gz", "internal_proof_nodes": [ 266, 253, 279, 38, 46, 6, 7, 2, 12, 254 ], "lean": "evidence/AP073-Lean.json.gz", "nodes": 281, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-bounded-majorant-v1", "proof": "evidence/AP073-bundle.json.gz", "report_sha256": "fcb56622a8478ee1403a483dbf8f7bcfa4c4320894ddcc2c6dc110469bff525a", "root": 280, "statement": "forall L d x N ab ac h s C. ~d=0 -> Le(x,L*d) -> NatScaledExpRecurrence(d,x,ab,ac,N) -> BetaAt(ab,ac,0,h) -> Sum(ab,ac,S N,s) -> Pow(9,S L,C) -> Le(s,C*h)", "target_ast_sha256": "b50deb3120bcff57e9f7bb87d27d0227041c2cae062f4de207339d31f41a23ff", "title": "exp bounded majorant", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 1281896, "bundle_sha256": "e8deaebfa9c117104743c0ae0298c1d8cffe8f8ccf5f713f332a46ffeb48873e", "case": "AP074", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP060", "AP068", "AP069", "AP070", "AP071", "AP072", "AP073", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV069", "CV070", "CV094", "CV095", "CV097" ], "coverage": "prove_two_sided_absolute_continuity_of_actual_independent_bounded_finite_exponential_computations_by_a_checked_power_majorant_decidable_order_split_and_explicit_precision_budget_without_an_analytic_limit_claim", "ha": "evidence/AP074-HA.json.gz", "internal_proof_nodes": [ 13, 283, 286, 6 ], "lean": "evidence/AP074-Lean.json.gz", "nodes": 288, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-bounded-continuity-v1", "proof": "evidence/AP074-bundle.json.gz", "report_sha256": "0b4f2969b0619e43c43865cf86410534121c1fd9e6d62e6cd9ce72d39014cdbd", "root": 287, "statement": "forall L d e x y g N ab ac bb bc a0 b0 s t C r. ~d=0 -> ~e=0 -> Le(x,L*d) -> Le(y,L*e) -> Le(d*y,e*x+g) -> Le(e*x,d*y+g) -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Pow(9,S L,C) -> Le((r*g)*C,d*e) -> Le(r*(b0*s),r*(a0*t)+a0*b0) /\\ Le(r*(a0*t),r*(b0*s)+a0*b0)", "target_ast_sha256": "33db4b325e45a75d8927db006e7cc1ae244b680af38c9eda0005ba8149899723", "title": "exp bounded continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 1349580, "bundle_sha256": "619b8df88cde20ce381fc9c7a79e48d8c81e09e3824a99bd36b97584bee770af", "case": "AP075", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP060", "AP062", "AP068", "AP069", "AP070", "AP071", "AP072", "AP073", "AP074", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV069", "CV070", "CV094", "CV095", "CV097" ], "coverage": "prove_an_explicit_linear_dyadic_input_precision_modulus_for_two_actual_bounded_finite_exponential_computations_by_constructing_the_majorant_power_and_reusing_the_checked_symbolic_binary_budget_without_changing_the_original_approximation_program", "ha": "evidence/AP075-HA.json.gz", "internal_proof_nodes": [ 287, 296, 221, 297 ], "lean": "evidence/AP075-Lean.json.gz", "nodes": 299, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-dyadic-continuity-v1", "proof": "evidence/AP075-bundle.json.gz", "report_sha256": "2f9812b2f2dc166a6b73ca75d38b584b6df9b36dd8b7e3aa56e8b1c2f6963952", "root": 298, "statement": "forall L d e x y g n N ab ac bb bc a0 b0 s t r p. ~d=0 -> ~e=0 -> Le(x,L*d) -> Le(y,L*e) -> Le(d*y,e*x+g) -> Le(e*x,d*y+g) -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Pow(2,n,r) -> Pow(2,S(9*(S L)+n),p) -> Le(p*g,d*e) -> ~r=0 /\\ (~p=0 /\\ (Le(r*(b0*s),r*(a0*t)+a0*b0) /\\ Le(r*(a0*t),r*(b0*s)+a0*b0)))", "target_ast_sha256": "52a66a0660b914cf13ff19d92ee7d59293e729e8ef8813555714e427ec0646c4", "title": "exp dyadic continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 95110, "bundle_sha256": "f0ca341bf0363808fabc242631285ec8ced1c63c25f227a557e94235f202c4f7", "case": "AP076", "contains_checked_statements": [], "coverage": "prove_the_ordered_natural_power_difference_bound_for_three_actual_supplied_powers_by_ordinary_induction_including_zero_bases_without_division_or_analytic_axioms", "ha": "evidence/AP076-HA.json.gz", "internal_proof_nodes": [ 33, 26, 27, 30, 34, 13, 9, 2, 12, 6 ], "lean": "evidence/AP076-Lean.json.gz", "nodes": 36, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-ordered-power-difference-v1", "proof": "evidence/AP076-bundle.json.gz", "report_sha256": "7fd5d6da7b594ee9b1ac3e410f77b73ad8000573788a3114abe68ca4285bde63", "root": 35, "statement": "forall a b M g k u v w. b=a+g -> Le(b,M) -> Pow(a,S k,u) -> Pow(b,S k,v) -> Pow(M,k,w) -> Le(v,u+(S k)*(g*w))", "target_ast_sha256": "d4d5beb6c20b976f94c1c1e219e65c4389e0dddce29a5b2f642ed9c3014a7b04", "title": "ordered power difference", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 113033, "bundle_sha256": "b720ef339bcbf9658387bf1a1dabbffa5358cdb0d304477734ed87996d1644aa", "case": "AP077", "contains_checked_statements": [ "AP076" ], "coverage": "prove_two_sided_error_propagation_for_three_actual_bounded_natural_powers_by_decidable_input_order_and_the_extracted_gap_without_positive_base_assumptions_or_new_induction", "ha": "evidence/AP077-HA.json.gz", "internal_proof_nodes": [ 14, 36, 39 ], "lean": "evidence/AP077-Lean.json.gz", "nodes": 41, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-symmetric-power-difference-v1", "proof": "evidence/AP077-bundle.json.gz", "report_sha256": "80398d8305beae75e2b415ddca7099aa1fa76decccf5e5ac077649af70265ea7", "root": 40, "statement": "forall a b M g k u v w. Le(a,M) -> Le(b,M) -> Le(b,a+g) -> Le(a,b+g) -> Pow(a,S k,u) -> Pow(b,S k,v) -> Pow(M,k,w) -> Le(v,u+(S k)*(g*w)) /\\ Le(u,v+(S k)*(g*w))", "target_ast_sha256": "9f376fc95073195bee675f73105660651b69555fbeb652e7cd0ed1e8550cdc98", "title": "symmetric power difference", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 237676, "bundle_sha256": "8ad49de00359af56d8e961c650663dca0440bcb9e90492d10df23c9cf564327a", "case": "AP078", "contains_checked_statements": [], "coverage": "prove_the_product_of_actual_natural_power_values_by_constructing_previous_powers_and_identifying_supplied_successors_with_uniqueness_including_zero_bases_and_exponent", "ha": "evidence/AP078-HA.json.gz", "internal_proof_nodes": [ 112, 110, 119, 11, 6, 15 ], "lean": "evidence/AP078-Lean.json.gz", "nodes": 121, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-product-power-v1", "proof": "evidence/AP078-bundle.json.gz", "report_sha256": "d73a1c687848c8df034741e2b3c96cfadfd69ed580dc2fbce272b18ff53ab332", "root": 120, "statement": "forall k a b u v w. Pow(a,k,u) -> Pow(b,k,v) -> Pow(a*b,k,w) -> w=u*v", "target_ast_sha256": "5f37e55e15971ec0ef5bcf1a9435c0ed369dbe1f3b53b3fa089c6e8ee4a2294c", "title": "product power", "uses_definitions": [ "PD0020" ] }, { "bundle_bytes": 316595, "bundle_sha256": "3d6d57fcdb4790e5b501149babd64eeace241bc7c05a6c2f072a4ea02e463edb", "case": "AP079", "contains_checked_statements": [ "AP076", "AP077", "AP078" ], "coverage": "prove_symmetric_error_propagation_for_independently_normalized_nonnegative_rational_powers_by_constructing_actual_cross_numerator_powers_preserving_all_six_supplied_powers_and_proving_output_denominators_nonzero", "ha": "evidence/AP079-HA.json.gz", "internal_proof_nodes": [ 40, 133, 126, 134, 10, 11, 6 ], "lean": "evidence/AP079-Lean.json.gz", "nodes": 136, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rational-power-difference-v1", "proof": "evidence/AP079-bundle.json.gz", "report_sha256": "ace86adb19f36817c276215871cb5b139fbcb3b1e2dea6346570eedebb0dc78f", "root": 135, "statement": "forall d e a b M g k u v D E C Q. ~d=0 -> ~e=0 -> Le(a,M*d) -> Le(b,M*e) -> Le(b*d,a*e+g) -> Le(a*e,b*d+g) -> Pow(a,S k,u) -> Pow(b,S k,v) -> Pow(d,S k,D) -> Pow(e,S k,E) -> Pow(M,k,C) -> Pow(d*e,k,Q) -> ~D=0 /\\ (~E=0 /\\ (Le(v*D,u*E+(S k)*(g*(C*Q))) /\\ Le(u*E,v*D+(S k)*(g*(C*Q)))))", "target_ast_sha256": "0be1195ec265753f6216218bc22c6933b86df205ce422c95f0c25df112074b61", "title": "rational power difference", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 333958, "bundle_sha256": "b99a77859fb3f803516deef4349cb3c9bbf9345efccbe959770537b6f782c1e6", "case": "AP080", "contains_checked_statements": [ "AP076", "AP077", "AP078", "AP079" ], "coverage": "prove_requested_rational_power_precision_under_an_explicit_input_budget_by_constructing_denominator_product_powers_and_replaying_actual_successor_and_product_identities_without_cancellation_or_analytic_axioms", "ha": "evidence/AP080-HA.json.gz", "internal_proof_nodes": [ 135, 133, 132, 126, 136, 6 ], "lean": "evidence/AP080-Lean.json.gz", "nodes": 138, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-requested-power-precision-v1", "proof": "evidence/AP080-bundle.json.gz", "report_sha256": "42f93fabc238c28cac8df95988f5d2c6fff228fc0f79abaddd79728daba75945", "root": 137, "statement": "forall d e a b M g k u v D E C r. ~d=0 -> ~e=0 -> Le(a,M*d) -> Le(b,M*e) -> Le(b*d,a*e+g) -> Le(a*e,b*d+g) -> Pow(a,S k,u) -> Pow(b,S k,v) -> Pow(d,S k,D) -> Pow(e,S k,E) -> Pow(M,k,C) -> Le((r*(S k))*(g*C),d*e) -> ~D=0 /\\ (~E=0 /\\ (Le(r*(v*D),r*(u*E)+D*E) /\\ Le(r*(u*E),r*(v*D)+D*E)))", "target_ast_sha256": "486fc0a6d7574427552d0fefc51c3ea3d4176a018a649275113340d0cf60d1d1", "title": "requested power precision", "uses_definitions": [ "PD0020", "PD0001" ] }, { "bundle_bytes": 1522170, "bundle_sha256": "59ca12ab08a8f656d9d6503f675d5d770dd16a330430746c7c00077f8aa32f25", "case": "AP081", "contains_checked_statements": [ "AP037", "AP038", "AP040", "AP041", "AP051", "AP052", "AP053", "AP054", "AP055", "AP056", "AP057", "AP060", "AP062", "AP063", "AP068", "AP069", "AP070", "AP071", "AP072", "AP073", "AP074", "AP075", "CV008", "CV009", "CV011", "CV022", "CV030", "CV043", "CV044", "CV046", "CV047", "CV048", "CV068", "CV069", "CV070", "CV094", "CV095", "CV097" ], "coverage": "prove_explicit_combined_input_and_truncation_errors_for_independent_bounded_finite_exponential_computations_by_constructing_the_actual_middle_prefix_preserving_the_two_error_budgets_without_analytic_limit_assumptions", "ha": "evidence/AP081-HA.json.gz", "internal_proof_nodes": [ 298, 319, 320, 123, 11, 18, 12, 2, 321 ], "lean": "evidence/AP081-Lean.json.gz", "nodes": 323, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-joint-error-v1", "proof": "evidence/AP081-bundle.json.gz", "report_sha256": "21c4408bd7dd2217cafe48c881d97b352356d23f2a06813a13c44ba243740c93", "root": 322, "statement": "forall L K d e x y g n N ab ac bb bc a0 b0 s t r p. ~d=0 -> ~e=0 -> Le(x,L*d) -> Le(y,L*e) -> Le(2*L,K) -> Le(d*y,e*x+g) -> Le(e*x,d*y+g) -> NatScaledExpRecurrence(d,x,ab,ac,K+(L*K+n)) -> NatScaledExpRecurrence(e,y,bb,bc,S(K+(L*K+n))+N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S(K+(L*K+n)),s) -> Sum(bb,bc,S(S(K+(L*K+n))+N),t) -> Pow(2,n,r) -> Pow(2,S(9*(S L)+n),p) -> Le(p*g,d*e) -> ~r=0 /\\ (~p=0 /\\ (Le(r*(b0*s),r*(a0*t)+a0*b0) /\\ Le(r*(a0*t),r*(b0*s)+2*(a0*b0))))", "target_ast_sha256": "7cd2d99c4a00cc142b8b0811c8a227ab0423f1aaad7a8f745af23c75e8b7b638", "title": "exp joint error", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 892431, "bundle_sha256": "0261db5cc15244a1345941dd847fc8bf283af92dbb74cf19ff6a9f22168eff88", "case": "AP082", "contains_checked_statements": [ "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "prove_supplied_logarithm_and_exponential_refinement_across_independent_scales_codes_and_horizons_with_explicit_27_over_ternary_power_plus_2_over_binary_power_error_without_limit_or_code_equality_assumptions", "ha": "evidence/AP082-HA.json.gz", "internal_proof_nodes": [ 154, 218, 222, 223, 224, 225, 226, 144, 11, 60, 8, 6, 10, 1, 227 ], "lean": "evidence/AP082-Lean.json.gz", "nodes": 229, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-exp-refinement-v1", "proof": "evidence/AP082-bundle.json.gz", "report_sha256": "343e0251d185427fbb52af1a8c03a045451880dc744d5bae86e910c76eb3291e", "root": 228, "statement": "forall d e ab ac bb bc k l x z p q m N cb cc eb ec a0 b0 u v P. ~d=0 -> ~e=0 -> NatAtanhPrefix(d,ab,ac,S k) -> NatAtanhPrefix(e,bb,bc,(S k)+l) -> NatHornerValue(ab,ac,3,S k,x) -> NatHornerValue(bb,bc,3,(S k)+l,z) -> Pow(3,S k,p) -> Pow(3,(S k)+l,q) -> NatScaledExpRecurrence(d*p,3*x,cb,cc,m) -> NatScaledExpRecurrence(e*q,3*z,eb,ec,(S m)+N) -> BetaAt(cb,cc,0,a0) -> BetaAt(eb,ec,0,b0) -> Sum(cb,cc,S m,u) -> Sum(eb,ec,S((S m)+N),v) -> Pow(2,m,P) -> ~p=0 /\\ (~q=0 /\\ (~P=0 /\\ (~(d*p)=0 /\\ (~(e*q)=0 /\\ (Le(b0*u,a0*v) /\\ Le((p*P)*(a0*v),(p*P)*(b0*u)+((9*3)*P+2*p)*(a0*b0)))))))", "target_ast_sha256": "bccfdfa515c324f3b34a5a205185f1a81d590d8d27d1b41dcba91fbbd65883cc", "title": "log exp refinement", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015" ] }, { "bundle_bytes": 995536, "bundle_sha256": "18275d48a04f78840e82c3d6f549a632496ea202cafc32b10de83ef2be0979fa", "case": "AP083", "contains_checked_statements": [ "AP082", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "prove_the_existing_auxiliary_18_r_plus_2_cutoff_pays_both_logarithm_and_exponential_refinement_errors_from_actual_power_computations_without_assuming_the_budgets_or_changing_the_fixed_approximation_program", "ha": "evidence/AP083-HA.json.gz", "internal_proof_nodes": [ 228, 235, 236 ], "lean": "evidence/AP083-Lean.json.gz", "nodes": 238, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-exp-cutoff-refinement-v1", "proof": "evidence/AP083-bundle.json.gz", "report_sha256": "0e73c2b8fe639ebf1b28dadd4b7cb15faf3ae58fc4e8ebd885274bd4700de09c", "root": 237, "statement": "forall r d e ab ac bb bc l x z p q N cb cc eb ec a0 b0 u v P. ~d=0 -> ~e=0 -> NatAtanhPrefix(d,ab,ac,S(S(18*r))) -> NatAtanhPrefix(e,bb,bc,(S(S(18*r)))+l) -> NatHornerValue(ab,ac,3,S(S(18*r)),x) -> NatHornerValue(bb,bc,3,(S(S(18*r)))+l,z) -> Pow(3,S(S(18*r)),p) -> Pow(3,(S(S(18*r)))+l,q) -> NatScaledExpRecurrence(d*p,3*x,cb,cc,S(S(18*r))) -> NatScaledExpRecurrence(e*q,3*z,eb,ec,(S(S(S(18*r))))+N) -> BetaAt(cb,cc,0,a0) -> BetaAt(eb,ec,0,b0) -> Sum(cb,cc,S(S(S(18*r))),u) -> Sum(eb,ec,S((S(S(S(18*r))))+N),v) -> Pow(2,S(S(18*r)),P) -> ~p=0 /\\ (~q=0 /\\ (~P=0 /\\ (~(d*p)=0 /\\ (~(e*q)=0 /\\ (Le(b0*u,a0*v) /\\ Le(r*(a0*v),r*(b0*u)+a0*b0))))))", "target_ast_sha256": "97d070681313d49334933a292e949acae3f61cc484c18b5a122d05060269d1c5", "title": "log exp cutoff refinement", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015" ] }, { "bundle_bytes": 3313343, "bundle_sha256": "0bf21cd1c9bfc2162b766da7d48f010422e725cddc665f08f8d2b12460ef34ce", "case": "AP084", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP082", "AP083", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "construct_one_actual_dyadic_reference_before_arbitrary_independent_later_logarithm_and_exponential_tables_and_prove_uniform_near_two_bounds_by_positive_reference_head_cancellation_without_assuming_an_analytic_limit", "ha": "evidence/AP084-HA.json.gz", "internal_proof_nodes": [ 451, 508, 266, 509 ], "lean": "evidence/AP084-Lean.json.gz", "nodes": 511, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-exp-uniform-two-v2", "proof": "evidence/AP084-bundle.json.gz", "report_sha256": "ae9be38cdafe98c7394acd93ff829bafacc2496b36859b5b1fa306efa9d75758", "root": 510, "statement": "forall n. exists r. Pow(2,n,r) /\\ (~r=0 /\\ (forall e bb bc l z q N eb ec b v. ~e=0 -> NatAtanhPrefix(e,bb,bc,(S(S(18*r)))+l) -> NatHornerValue(bb,bc,3,(S(S(18*r)))+l,z) -> Pow(3,(S(S(18*r)))+l,q) -> NatScaledExpRecurrence(e*q,3*z,eb,ec,(S(S(S(18*r))))+N) -> BetaAt(eb,ec,0,b) -> Sum(eb,ec,S((S(S(S(18*r))))+N),v) -> ~q=0 /\\ (~(e*q)=0 /\\ (Le(r*v,(r*2)*b+2*b) /\\ Le((r*2)*b,r*v+b)))))", "target_ast_sha256": "018632b34699a2f0f8feb879b56eb195d89cfb1304ffe2f5da46ef3ed3607898", "title": "log exp uniform two", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015" ] }, { "bundle_bytes": 3330729, "bundle_sha256": "68e17a66cae013163f697f1a85a48dc2d9a05b8f84a2c621b5a3ca15547cf2a7", "case": "AP085", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP082", "AP083", "AP084", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "prove_uniform_finite_near_two_bounds_for_the_fixed_two_k_plus_one_logarithm_prefix_and_k_plus_two_exponential_degree_by_constructing_the_actual_cutoff_gap_and_proving_both_index_alignments_without_changing_the_computations", "ha": "evidence/AP085-HA.json.gz", "internal_proof_nodes": [ 510, 499, 511 ], "lean": "evidence/AP085-Lean.json.gz", "nodes": 513, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-fixed-log-exp-convergence-v1", "proof": "evidence/AP085-bundle.json.gz", "report_sha256": "3a17cb7d4c35bd136c48fe040a46136decfabf65b692ef136ec0245ba6469296", "root": 512, "statement": "forall n. exists r. Pow(2,n,r) /\\ (~r=0 /\\ (forall k d ab ac x p tb tc h y. Le(S(S(18*r)),k) -> ~d=0 -> NatAtanhPrefix(d,ab,ac,S(2*k)) -> NatHornerValue(ab,ac,3,S(2*k),x) -> Pow(3,S(2*k),p) -> NatScaledExpRecurrence(d*p,3*x,tb,tc,S(S k)) -> BetaAt(tb,tc,0,h) -> Sum(tb,tc,S(S(S k)),y) -> ~p=0 /\\ (~(d*p)=0 /\\ (Le(r*y,(r*2)*h+2*h) /\\ Le((r*2)*h,r*y+h)))))", "target_ast_sha256": "c49cb9e52773c0bae1157f7b1eedbc0f29391c3163cb80a3c9c28445c69c37b1", "title": "fixed log exp convergence", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015" ] }, { "bundle_bytes": 3448335, "bundle_sha256": "64101f309bab0e6494249ca21315df0e64d1a96bd905228e70a880181c50af25", "case": "AP086", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP082", "AP083", "AP084", "AP085", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "construct_positive_actual_logarithm_and_factorial_normalized_exponential_computations_at_every_eligible_fixed_schedule_index_after_one_uniform_dyadic_threshold_preserving_all_index_normalization_and_both_near_two_bounds", "ha": "evidence/AP086-HA.json.gz", "internal_proof_nodes": [ 512, 228, 520, 358, 266, 280, 12, 47 ], "lean": "evidence/AP086-Lean.json.gz", "nodes": 522, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-fixed-log-exp-witness-v1", "proof": "evidence/AP086-bundle.json.gz", "report_sha256": "19547f31eb776a5b16e98dcad7bcfc832dab3c73684b4aa29e1ee7ccef02d0e3", "root": 521, "statement": "forall n. exists r. Pow(2,n,r) /\\ (~r=0 /\\ (forall k. Le(S(S(18*r)),k) -> exists d ab ac x p wb wc tb tc h f q y. (~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S(2*k))) /\\ ((NatHornerValue(ab,ac,3,S(2*k),x)) /\\ ((Pow(3,S(2*k),p)) /\\ ((~p=0) /\\ ((~(d*p)=0) /\\ ((NatExpWeightTable(d*p,S(S k),wb,wc)) /\\ ((NatPowerWeightedPrefix(3*x,wb,wc,tb,tc,S(S(S k)))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(tb,tc,0,h)) /\\ ((~h=0) /\\ ((Factorial(S(S k),f)) /\\ ((Pow(d*p,S(S k),q)) /\\ ((h=f*q) /\\ ((Sum(tb,tc,S(S(S k)),y)) /\\ ((NatScaledExpRecurrence(d*p,3*x,tb,tc,S(S k))) /\\ ((forall j a fj dj xj. Le(j,S(S k)) -> BetaAt(tb,tc,j,a) -> Factorial(j,fj) -> Pow(d*p,j,dj) -> Pow(3*x,j,xj) -> (a*fj)*dj=h*xj) /\\ ((Le(r*y,(r*2)*h+2*h)) /\\ (Le((r*2)*h,r*y+h)))))))))))))))))))))", "target_ast_sha256": "3ff68dfaeccdb0e61f146f12e87cf0d13b773fc609d14b0ad64a489af6b11d33", "title": "fixed log exp witness", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015", "ND0423", "ND0442", "PD0023" ] }, { "bundle_bytes": 927210, "bundle_sha256": "df2878ab2be432ad492885406eec7d5bdff383560a2e4934e947062bdc7b9160", "case": "AP087", "contains_checked_statements": [ "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "prove_independent_half_logarithm_exponential_refinement_by_explicit_doubled_argument_denominators_and_scaled_gap_transport_while_retaining_the_unchanged_original_logarithm_prefixes_values_and_powers", "ha": "evidence/AP087-HA.json.gz", "internal_proof_nodes": [ 154, 218, 222, 223, 224, 226, 227, 144, 11, 60, 8, 6, 10, 1, 228, 229 ], "lean": "evidence/AP087-Lean.json.gz", "nodes": 231, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-exp-refinement-v1", "proof": "evidence/AP087-bundle.json.gz", "report_sha256": "8fe08e252f7f1838cec1b1835e9f6ef0541dce3bc471ebadac274231f17b9da0", "root": 230, "statement": "forall d e ab ac bb bc k l x z p q m N cb cc eb ec a0 b0 u v P. ~d=0 -> ~e=0 -> NatAtanhPrefix(d,ab,ac,S k) -> NatAtanhPrefix(e,bb,bc,(S k)+l) -> NatHornerValue(ab,ac,3,S k,x) -> NatHornerValue(bb,bc,3,(S k)+l,z) -> Pow(3,S k,p) -> Pow(3,(S k)+l,q) -> NatScaledExpRecurrence(2*(d*p),3*x,cb,cc,m) -> NatScaledExpRecurrence(2*(e*q),3*z,eb,ec,(S m)+N) -> BetaAt(cb,cc,0,a0) -> BetaAt(eb,ec,0,b0) -> Sum(cb,cc,S m,u) -> Sum(eb,ec,S((S m)+N),v) -> Pow(2,m,P) -> ~p=0 /\\ (~q=0 /\\ (~P=0 /\\ (~(2*(d*p))=0 /\\ (~(2*(e*q))=0 /\\ (Le(b0*u,a0*v) /\\ Le((p*P)*(a0*v),(p*P)*(b0*u)+((9*3)*P+2*p)*(a0*b0)))))))", "target_ast_sha256": "74d9571876bc3c082af4e6ab56b2f18a0e49718f24ef4ea8fe3ddf85f58dcf49", "title": "half log exp refinement", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015" ] }, { "bundle_bytes": 1030455, "bundle_sha256": "2c3fe08cb08b747f0bfb835ea226855277aebb872f9f587736be3e74c22da8cc", "case": "AP088", "contains_checked_statements": [ "AP087", "CV009", "CV032", "CV062", "CV068", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "apply_checked_actual_power_cutoff_budgets_and_positive_cancellation_to_independent_half_logarithm_exponential_refinement_without_assuming_the_budgets_changing_the_original_logarithm_computations_or_adding_arithmetic_helpers", "ha": "evidence/AP088-HA.json.gz", "internal_proof_nodes": [ 230, 237, 238 ], "lean": "evidence/AP088-Lean.json.gz", "nodes": 240, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-exp-cutoff-v1", "proof": "evidence/AP088-bundle.json.gz", "report_sha256": "bc6d27fe3c772c90d742aaf81bf033f838058bcfd48965269a8db60c84a2f3ec", "root": 239, "statement": "forall r d e ab ac bb bc l x z p q N cb cc eb ec a0 b0 u v P. ~d=0 -> ~e=0 -> NatAtanhPrefix(d,ab,ac,S(S(18*r))) -> NatAtanhPrefix(e,bb,bc,(S(S(18*r)))+l) -> NatHornerValue(ab,ac,3,S(S(18*r)),x) -> NatHornerValue(bb,bc,3,(S(S(18*r)))+l,z) -> Pow(3,S(S(18*r)),p) -> Pow(3,(S(S(18*r)))+l,q) -> NatScaledExpRecurrence(2*(d*p),3*x,cb,cc,S(S(18*r))) -> NatScaledExpRecurrence(2*(e*q),3*z,eb,ec,(S(S(S(18*r))))+N) -> BetaAt(cb,cc,0,a0) -> BetaAt(eb,ec,0,b0) -> Sum(cb,cc,S(S(S(18*r))),u) -> Sum(eb,ec,S((S(S(S(18*r))))+N),v) -> Pow(2,S(S(18*r)),P) -> ~p=0 /\\ (~q=0 /\\ (~P=0 /\\ (~(2*(d*p))=0 /\\ (~(2*(e*q))=0 /\\ (Le(b0*u,a0*v) /\\ Le(r*(a0*v),r*(b0*u)+a0*b0))))))", "target_ast_sha256": "da6645f80e599d951709a835b4f07996fb457ad7c848fe65a66fb2ed76dbf5b0", "title": "half log exp cutoff", "uses_definitions": [ "PD0001", "PD0013", "ND0425", "ND0437", "PD0020", "ND0443", "PD0015" ] }, { "bundle_bytes": 3848765, "bundle_sha256": "40a9356ff0ed241d51a10a9948e73f850ff7b8cd979cb9ca1531a692fa238b49", "case": "AP089", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "AP044", "AP045", "AP046", "AP047", "AP048", "AP049", "AP050", "AP087", "AP088", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101", "SQ002", "SQ009" ], "coverage": "choose_one_actual_dyadic_root_before_all_later_independent_half_logarithm_exponential_tables_and_prove_both_finite_cross_errors_by_checked_reference_restriction_refinement_and_positive_scale_cancellation", "ha": "evidence/AP089-HA.json.gz", "internal_proof_nodes": [ 559, 580, 510, 266, 581 ], "lean": "evidence/AP089-Lean.json.gz", "nodes": 583, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-half-log-root-uniform-v3", "proof": "evidence/AP089-bundle.json.gz", "report_sha256": "a5eb1412a33abb5f92bd79dc4085c9a76888e5d6af1a183556c79f3860c04800", "root": 582, "statement": "forall n. exists r s. Pow(2,n,r) /\\ (~r=0 /\\ (DyadicSqrt2(n,r,s) /\\ (forall e bb bc l z q N eb ec b v. ~e=0 -> NatAtanhPrefix(e,bb,bc,(S(S(18*r)))+l) -> NatHornerValue(bb,bc,3,(S(S(18*r)))+l,z) -> Pow(3,(S(S(18*r)))+l,q) -> NatScaledExpRecurrence(2*(e*q),3*z,eb,ec,(S(S(S(18*r))))+N) -> BetaAt(eb,ec,0,b) -> Sum(eb,ec,S((S(S(S(18*r))))+N),v) -> ~q=0 /\\ (~(2*(e*q))=0 /\\ (Le(r*v,s*b+8*b) /\\ Le(s*b,r*v+b))))))", "target_ast_sha256": "bdd05acec456f38c8dbd660093564668cdcb6a6207fd285396940740dcf42ff8", "title": "half log root uniform", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0446", "ND0425", "ND0437", "ND0443", "PD0015" ] }, { "bundle_bytes": 3868304, "bundle_sha256": "603a7c00633d251a1f6b4476247eb6e48a8d13d47b47b4dbf75b5eab5747ef8c", "case": "AP090", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "AP044", "AP045", "AP046", "AP047", "AP048", "AP049", "AP050", "AP087", "AP088", "AP089", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101", "SQ002", "SQ009" ], "coverage": "align_the_independent_half_logarithm_root_comparison_with_the_original_fixed_logarithm_length_and_exponential_degree_using_checked_gap_and_index_equalities_without_changing_any_computation_or_witness_quantifier_order", "ha": "evidence/AP090-HA.json.gz", "internal_proof_nodes": [ 582, 535, 583 ], "lean": "evidence/AP090-Lean.json.gz", "nodes": 585, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-fixed-half-log-root-v1", "proof": "evidence/AP090-bundle.json.gz", "report_sha256": "d97afbf151e451450e0551b8d49e8c0e0f6ac074c43c42d44b716d6f685ebf86", "root": 584, "statement": "forall n. exists r s. Pow(2,n,r) /\\ (~r=0 /\\ (DyadicSqrt2(n,r,s) /\\ (forall k e bb bc z q eb ec b v. Le(S(S(18*r)),k) -> ~e=0 -> NatAtanhPrefix(e,bb,bc,S(2*k)) -> NatHornerValue(bb,bc,3,S(2*k),z) -> Pow(3,S(2*k),q) -> NatScaledExpRecurrence(2*(e*q),3*z,eb,ec,S(S k)) -> BetaAt(eb,ec,0,b) -> Sum(eb,ec,S(S(S k)),v) -> ~q=0 /\\ (~(2*(e*q))=0 /\\ (Le(r*v,s*b+8*b) /\\ Le(s*b,r*v+b))))))", "target_ast_sha256": "41651cb7c7afc07ceb0eaeab7bf02392001a704ec0891d7bf1db0ed4aa33211f", "title": "fixed half log root", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0446", "ND0425", "ND0437", "ND0443", "PD0015" ] }, { "bundle_bytes": 3913686, "bundle_sha256": "ca771f818bb607d30de80c5bc19a15ee2d05a0932355fc71c7800b0ef096d67b", "case": "AP091", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "AP044", "AP045", "AP046", "AP047", "AP048", "AP049", "AP050", "AP087", "AP088", "AP089", "AP090", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101", "SQ002", "SQ009" ], "coverage": "construct_positive_fixed_schedule_half_logarithm_exponential_tables_after_one_chosen_dyadic_root_and_retain_every_actual_factorial_normalization_and_both_checked_cross_errors", "ha": "evidence/AP091-HA.json.gz", "internal_proof_nodes": [ 584, 228, 520, 358, 266, 280, 12, 47 ], "lean": "evidence/AP091-Lean.json.gz", "nodes": 586, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-fixed-half-log-root-witness-v1", "proof": "evidence/AP091-bundle.json.gz", "report_sha256": "9a9889aa4290b7c3b769213620d4b2137eb01274a50fb994d7b66347d67adba3", "root": 585, "statement": "forall n. exists r s. Pow(2,n,r) /\\ (~r=0 /\\ (DyadicSqrt2(n,r,s) /\\ (forall k. Le(S(S(18*r)),k) -> exists d ab ac x p wb wc tb tc h f q y. (~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S(2*k))) /\\ ((NatHornerValue(ab,ac,3,S(2*k),x)) /\\ ((Pow(3,S(2*k),p)) /\\ ((~p=0) /\\ ((~(2*(d*p))=0) /\\ ((NatExpWeightTable(2*(d*p),S(S k),wb,wc)) /\\ ((NatPowerWeightedPrefix(3*x,wb,wc,tb,tc,S(S(S k)))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(tb,tc,0,h)) /\\ ((~h=0) /\\ ((Factorial(S(S k),f)) /\\ ((Pow(2*(d*p),S(S k),q)) /\\ ((h=f*q) /\\ ((Sum(tb,tc,S(S(S k)),y)) /\\ ((NatScaledExpRecurrence(2*(d*p),3*x,tb,tc,S(S k))) /\\ ((forall j a fj dj xj. Le(j,S(S k)) -> BetaAt(tb,tc,j,a) -> Factorial(j,fj) -> Pow(2*(d*p),j,dj) -> Pow(3*x,j,xj) -> (a*fj)*dj=h*xj) /\\ ((Le(r*y,s*h+8*h)) /\\ (Le(s*h,r*y+h))))))))))))))))))))))", "target_ast_sha256": "9fba277808e08f9fc794a636d0be0242f1a8498dbd631870d0f88440174b708a", "title": "fixed half log root witness", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0446", "ND0425", "ND0437", "ND0443", "PD0015", "ND0423", "ND0442", "PD0023" ] }, { "bundle_bytes": 4000332, "bundle_sha256": "c9cabc5640c33f87f1fe31817ab71b04bfb0b43896488e79b8da6b69d62e232c", "case": "AP092", "contains_checked_statements": [ "AP032", "AP033", "AP034", "AP037", "AP038", "AP040", "AP041", "AP042", "AP043", "AP044", "AP045", "AP046", "AP047", "AP048", "AP049", "AP050", "AP087", "AP088", "AP089", "AP090", "AP091", "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV030", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV068", "CV069", "CV070", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084", "CV086", "CV087", "CV088", "CV089", "CV090", "CV092", "CV093", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101", "SQ002", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009" ], "coverage": "transfer_constructed_fixed_schedule_half_logarithm_samples_to_any_supplied_finer_dyadic_root_at_the_same_index_preserving_all_positive_normalizations_and_both_finite_errors", "ha": "evidence/AP092-HA.json.gz", "internal_proof_nodes": [ 585, 592, 555, 593 ], "lean": "evidence/AP092-Lean.json.gz", "nodes": 595, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-fixed-half-log-fine-root-v1", "proof": "evidence/AP092-bundle.json.gz", "report_sha256": "894f2e9fee8e423a655bb66d1414982836f7b910a6b46e518ea58909b107896a", "root": 594, "statement": "forall n. exists r. Pow(2,n,r) /\\ (~r=0 /\\ (forall l e t. DyadicSqrt2(n+l,e,t) -> Le(S(S(18*r)),n+l) -> exists d ab ac x p wb wc tb tc h f q y. (~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S(2*(n+l)))) /\\ ((NatHornerValue(ab,ac,3,S(2*(n+l)),x)) /\\ ((Pow(3,S(2*(n+l)),p)) /\\ ((~p=0) /\\ ((~(2*(d*p))=0) /\\ ((NatExpWeightTable(2*(d*p),S(S (n+l)),wb,wc)) /\\ ((NatPowerWeightedPrefix(3*x,wb,wc,tb,tc,S(S(S (n+l))))) /\\ ((BetaAt(wb,wc,0,h)) /\\ ((BetaAt(tb,tc,0,h)) /\\ ((~h=0) /\\ ((Factorial(S(S (n+l)),f)) /\\ ((Pow(2*(d*p),S(S (n+l)),q)) /\\ ((h=f*q) /\\ ((Sum(tb,tc,S(S(S (n+l))),y)) /\\ ((NatScaledExpRecurrence(2*(d*p),3*x,tb,tc,S(S (n+l)))) /\\ ((forall j a fj dj xj. Le(j,S(S (n+l))) -> BetaAt(tb,tc,j,a) -> Factorial(j,fj) -> Pow(2*(d*p),j,dj) -> Pow(3*x,j,xj) -> (a*fj)*dj=h*xj) /\\ ((~e=0) /\\ ((~t=0) /\\ ((Le(e,t)) /\\ ((Lt(t,2*e)) /\\ ((Le(r*(e*y),r*(t*h)+8*(e*h))) /\\ (Le(r*(t*h),r*(e*y)+2*(e*h))))))))))))))))))))))))))", "target_ast_sha256": "6d4b5804c59942bc029957cbb384a3a6d2235616d7c0e1cf4c0af32be3e12771", "title": "fixed half log fine root", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0020", "ND0446", "ND0425", "ND0437", "ND0443", "PD0015", "ND0423", "ND0442", "PD0023" ] }, { "bundle_bytes": 189044, "bundle_sha256": "d8a884f082830521daf08ad2866919f91894adbf5dcff76af374cdc6049f2ec5", "case": "AP093", "contains_checked_statements": [ "CV086", "SQ002", "SQ009" ], "coverage": "compare_supplied_original_log_root_quotient_with_half_root_argument_proving_both_denominators_positive_unit_interval_bounds_and_explicit_two_over_d_square_residual_without_an_exact_finite_or_analytic_identity", "ha": "evidence/AP093-HA.json.gz", "internal_proof_nodes": [ 63, 66, 52, 9, 31, 17, 11, 2, 67 ], "lean": "evidence/AP093-Lean.json.gz", "nodes": 69, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-log-root-half-argument-v1", "proof": "evidence/AP093-bundle.json.gz", "report_sha256": "b9086c95f0cf46fd14bf298917c39f179d29699d84010cf6d0391c8e76d5051f", "root": 68, "statement": "forall k d s D ab ac x p. ~D=0 -> NatSqrt2LogArgument(k,d,s,D,ab,ac,x,p) -> (~d=0) /\\ ((~s=0) /\\ ((~p=0) /\\ ((~(D*p)=0) /\\ ((~((D*p)*s)=0) /\\ ((~(2*((D*p)*d))=0) /\\ ((Le(((3*x)*d),((D*p)*s))) /\\ ((Le(((3*x)*s),(2*((D*p)*d)))) /\\ ((Le(((D*p)*s)*((3*x)*s),(2*((D*p)*d))*((3*x)*d))) /\\ (Le(d*((2*((D*p)*d))*((3*x)*d)),d*(((D*p)*s)*((3*x)*s))+2*(((D*p)*s)*(2*((D*p)*d)))))))))))))", "target_ast_sha256": "244e83e9bb6930441b55dd6bc755802920abede937100ccbd61c180210112cf3", "title": "log root half argument", "uses_definitions": [ "PD0001", "ND0447" ] }, { "bundle_bytes": 817481, "bundle_sha256": "b234e8883d2975dd2cb5ab32133703fb623196946051fffc74181f9e8a7d3f6c", "case": "AP094", "contains_checked_statements": [ "AP004", "AP007", "AP093", "CV009", "CV086", "CV094", "CV095", "CV096", "CV097", "CV098", "SQ002", "SQ009" ], "coverage": "transport_the_actual_original_finite_exponential_sample_to_an_independent_positive_half_root_recurrence_at_the_same_k_plus_two_degree_with_order_and_eighteen_over_d_error_cancelling_only_proved_nonzero_factors", "ha": "evidence/AP094-HA.json.gz", "internal_proof_nodes": [ 68, 210, 192, 56, 212 ], "lean": "evidence/AP094-Lean.json.gz", "nodes": 214, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-log-root-half-transport-v1", "proof": "evidence/AP094-bundle.json.gz", "report_sha256": "d3a7c127b6ac9f241009ab6faefeff6c80ba0342c01fbbf26582f9e8dfcc4864", "root": 213, "statement": "forall k d s D ab ac x p wb wc tb tc w0 f h y jb jc v z. ~D=0 -> NatSqrt2ExpSample(k,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> ~v=0 -> NatScaledExpRecurrence((2*((D*p)*d)),((3*x)*s),jb,jc,k+2) -> BetaAt(jb,jc,0,v) -> Sum(jb,jc,S(k+2),z) -> (~d=0) /\\ ((~((D*p)*s)=0) /\\ ((~(2*((D*p)*d))=0) /\\ ((~w0=0) /\\ ((~v=0) /\\ ((Le(w0*z,v*y)) /\\ (Le(d*(v*y),d*(w0*z)+18*(w0*v))))))))", "target_ast_sha256": "d987799164353f66ab8981951790a46ec16203c12f377c44a23181eb042ebe7a", "title": "exp log root half transport", "uses_definitions": [ "PD0013", "PD0001", "PD0015", "ND0443", "ND0448" ] }, { "bundle_bytes": 969777, "bundle_sha256": "07d48fca1276244b05d415dc9c46a505b1fd357dc8178abea55a624e7347ff7e", "case": "AP095", "contains_checked_statements": [ "AP004", "AP007", "AP010", "AP014", "AP093", "AP094", "CV009", "CV086", "CV094", "CV095", "CV096", "CV097", "CV098", "SQ002", "SQ009" ], "coverage": "derive_requested_dyadic_half_root_comparison_error_for_the_unchanged_n_plus_eight_sample_by_constructing_actual_powers_and_replaying_AP015_scale_cancellation_preserving_independent_positive_heads_and_original_degree", "ha": "evidence/AP095-HA.json.gz", "internal_proof_nodes": [ 213, 228, 227, 229, 198, 219, 52, 9, 11 ], "lean": "evidence/AP095-Lean.json.gz", "nodes": 231, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-requested-half-transport-v1", "proof": "evidence/AP095-bundle.json.gz", "report_sha256": "2706862cc72b4d7e85655b7b52acf1147d78d8940e6c099dbc57bd67985e96c6", "root": 230, "statement": "forall n d s D ab ac x p wb wc tb tc w0 f h y jb jc v z. ~D=0 -> NatSqrt2ExpSample(n+8,d,s,D,ab,ac,x,p,wb,wc,tb,tc,w0,f,h,y) -> ~v=0 -> NatScaledExpRecurrence((2*((D*p)*d)),((3*x)*s),jb,jc,(n+8)+2) -> BetaAt(jb,jc,0,v) -> Sum(jb,jc,S((n+8)+2),z) -> exists r. (Pow(2,n,r)) /\\ ((~r=0) /\\ ((~d=0) /\\ ((~w0=0) /\\ ((~v=0) /\\ ((Le(w0*z,v*y)) /\\ (Le(r*(v*y),r*(w0*z)+(w0*v))))))))", "target_ast_sha256": "0b7a36d2e72c832a07f9a5512667ba40107b5b367d5a8715c776889e56a485e5", "title": "requested half transport", "uses_definitions": [ "PD0013", "PD0020", "PD0001", "PD0015", "ND0443", "ND0448" ] }, { "bundle_bytes": 58402, "bundle_sha256": "93d3871e7c7f06fc5974b61bec48616a7a7f13e6f919bf7dbf34570624e918b6", "case": "CV001", "contains_checked_statements": [], "coverage": "universal_natural_convolution_vanishing_not_IR079", "ha": "evidence/CV001-HA.json.gz", "internal_proof_nodes": [ 17, 14, 15, 9, 18, 4, 26 ], "lean": "evidence/CV001-Lean.json.gz", "nodes": 28, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-vanishing-v3", "proof": "evidence/CV001-bundle.json.gz", "report_sha256": "73c5717191efdca89eeb93c5ce1df1065d82a5adba2444d6832ca9efd258df91", "root": 27, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. (∀ x1. ∀ x2. S x1 ≤ i → S x1 ≤ z ∧ (S x2 ≤ S (S x1 · y) ∧ (∃ x3. x = x3 · S (S x1 · y) + x2)) ∨ z ≤ x1 ∧ x2 = 0 → x2 = 0) → (∀ x1. ∀ x2. S x1 ≤ j → S x1 ≤ k ∧ (S x2 ≤ S (S x1 · m) ∧ (∃ x3. n = x3 · S (S x1 · m) + x2)) ∨ k ≤ x1 ∧ x2 = 0 → x2 = 0) → S u ≤ i + j → (∀ x1. S x1 ≤ S u → ∃ x2. S x2 ≤ S (S x1 · w) ∧ (∃ x3. v = x3 · S (S x1 · w) + x2) ∧ (∃ x3. ∃ x4. ∃ x5. x1 + x3 = u ∧ ((S x1 ≤ z ∧ (S x4 ≤ S (S x1 · y) ∧ (∃ x6. x = x6 · S (S x1 · y) + x4)) ∨ z ≤ x1 ∧ x4 = 0) ∧ ((S x3 ≤ k ∧ (S x5 ≤ S (S x3 · m) ∧ (∃ x6. n = x6 · S (S x3 · m) + x5)) ∨ k ≤ x3 ∧ x5 = 0) ∧ x2 = x4 · x5)))) → (∃ x1. ∃ x2. 1 ≤ S (1 · x2) ∧ (∃ x3. x1 = x3 · S (1 · x2) + 0) ∧ (S x0 ≤ S (S S u · x2) ∧ (∃ x3. x1 = x3 · S (S S u · x2) + x0) ∧ (∀ x3. S x3 ≤ S u → ∃ x4. ∃ x5. ∃ x6. S x4 ≤ S (S x3 · w) ∧ (∃ x7. v = x7 · S (S x3 · w) + x4) ∧ (S x5 ≤ S (S x3 · x2) ∧ (∃ x7. x1 = x7 · S (S x3 · x2) + x5) ∧ (S x6 ≤ S (S S x3 · x2) ∧ (∃ x7. x1 = x7 · S (S S x3 · x2) + x6) ∧ x6 = x5 + x4))))) → x0 = 0", "target_ast_sha256": "b5ca16c60cf9aff317a93603448843e717d17228380fb7c1a1287e31f12e5e22", "title": "convolution vanishing", "uses_definitions": [] }, { "bundle_bytes": 46824, "bundle_sha256": "3e66d595d6329dd0d7fd3cb22da62e34cce7ec4cf22f89624bec0f6917f94db1", "case": "CV002", "contains_checked_statements": [], "coverage": "all_indices_natural_logexp_recurrence_solution_not_composition_or_analytic_IR080", "ha": "evidence/CV002-HA.json.gz", "internal_proof_nodes": [ 15, 27, 28, 29 ], "lean": "evidence/CV002-Lean.json.gz", "nodes": 31, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-logexp-recurrence-v2", "proof": "evidence/CV002-bundle.json.gz", "report_sha256": "36b3ba79d914b600c6d6202234ea8fd11bc350699be56c0fd1ec55f988e8aadb", "root": 30, "statement": "forall b c L D i v. NatExpAtanhRecurrence(b,c,L,D) -> Lt(i,L) -> BetaAt(b,c,i,v) -> ((i=0 /\\ v=D) \\/ (exists k. i=S k /\\ v=2*D))", "target_ast_sha256": "798208f708ff6a1d8a5d5d279698ac3f19608a2c01a96bd5059fa4b582a9520a", "title": "logexp recurrence", "uses_definitions": [ "PD0002", "PD0013", "ND0400" ] }, { "bundle_bytes": 267044, "bundle_sha256": "ce973e6781fe1dd61db9e92cb368d4a86875740fa2a5cf0c744884a56486588b", "case": "CV003", "contains_checked_statements": [], "coverage": "constructed_unique_finite_natural_convolution_not_power_tables_or_IR079", "ha": "evidence/CV003-HA.json.gz", "internal_proof_nodes": [ 123, 124 ], "lean": "evidence/CV003-Lean.json.gz", "nodes": 126, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-natural-convolution-v1", "proof": "evidence/CV003-bundle.json.gz", "report_sha256": "f0da185bbf307eb72a9fed97b21c4a96cc04de0f9ea4d2ec7863cc6bd4bfe3cc", "root": 125, "statement": "forall ab ac L bb bc M N. exists cb cc. NatConvolutionPrefix(ab,ac,L,bb,bc,M,cb,cc,N) /\\ (forall eb ec. NatConvolutionPrefix(ab,ac,L,bb,bc,M,eb,ec,N) -> (forall j a. Lt(j,N) -> BetaAt(cb,cc,j,a) -> BetaAt(eb,ec,j,a)))", "target_ast_sha256": "b5ad2ef80b9ae8fd100f25b2a128eac7af5d8d58a783e89d2aab7f5bf9c4b0d8", "title": "natural convolution", "uses_definitions": [ "PD0002", "PD0013", "ND0402" ] }, { "bundle_bytes": 246245, "bundle_sha256": "2d6398dce2762252fbd18ade9897b03bd7b4d6c5502424ccb3f734712e9e789d", "case": "CV004", "contains_checked_statements": [], "coverage": "constructed_finite_natural_power_tables_not_vanishing_or_composition_IR079", "ha": "evidence/CV004-HA.json.gz", "internal_proof_nodes": [ 23, 37, 115, 117, 119 ], "lean": "evidence/CV004-Lean.json.gz", "nodes": 121, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-series-power-trace-v2", "proof": "evidence/CV004-bundle.json.gz", "report_sha256": "d38c24469668d6bcf7f5bef804eb8dd8b50199bb1e61df25519410bd7e428051", "root": 120, "statement": "forall ab ac L H K. exists pb pc qb qc. NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,K)", "target_ast_sha256": "d39fa99bbfda29fc077a02cf8d23cc3a5d36c8f829bb9b76988a537042145777", "title": "series power trace", "uses_definitions": [ "ND0407" ] }, { "bundle_bytes": 309480, "bundle_sha256": "364445bf362ce8746ce876026d69f22d8df264856ff4afda7b32a6940e196c0e", "case": "CV005", "contains_checked_statements": [ "CV001", "CV004" ], "coverage": "constructed_natural_power_tables_vanish_below_exponent_not_composition_IR079", "ha": "evidence/CV005-HA.json.gz", "internal_proof_nodes": [ 121, 129 ], "lean": "evidence/CV005-Lean.json.gz", "nodes": 131, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-series-power-vanishing-v1", "proof": "evidence/CV005-bundle.json.gz", "report_sha256": "8078e409a6f1a04a49b826a8d27c72806be6231fc64f0da8fd8469add4f6a6a8", "root": 130, "statement": "forall ab ac L H K. NatZeroBefore(ab,ac,L,1) -> exists pb pc qb qc. NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,K) /\\ (forall k ub uc. Le(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatZeroBefore(ub,uc,H,k))", "target_ast_sha256": "db647ab11de25c9cae8ffd94e24655fd15cc955113f9e4a7f12ee383e21e224d", "title": "series power vanishing", "uses_definitions": [ "PD0001", "ND0405", "ND0407", "ND0408" ] }, { "bundle_bytes": 192535, "bundle_sha256": "87db349766e9779113c3e954f52700942c6f15137bb7082e87497ce8c65d5c10", "case": "CV006", "contains_checked_statements": [], "coverage": "constructed_actual_weighted_natural_sum_linearity_not_formal_derivative_IR079", "ha": "evidence/CV006-HA.json.gz", "internal_proof_nodes": [ 104, 107 ], "lean": "evidence/CV006-Lean.json.gz", "nodes": 109, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-sum-linearity-v1", "proof": "evidence/CV006-bundle.json.gz", "report_sha256": "6041831a7029ac057a347ca1020fd958448c127b5db4b1de22e3d2336c85908e", "root": 108, "statement": "forall alpha beta gamma ab ac bb bc cb cc L. NatLinearTriple(alpha,beta,gamma,ab,ac,bb,bc,cb,cc,L) -> exists u v w. Sum(ab,ac,L,u) /\\ (Sum(bb,bc,L,v) /\\ (Sum(cb,cc,L,w) /\\ gamma*w=alpha*u+beta*v))", "target_ast_sha256": "bcf6a93be452d1bf0a424cda787eca7580644608eb69a236c5d1c99c3acea02a", "title": "sum linearity", "uses_definitions": [ "PD0015", "ND0409" ] }, { "bundle_bytes": 258239, "bundle_sha256": "140a7b41c2c895ce52fdcec8b9189af18def5d054be512c8a3dffef23bf2b169", "case": "CV007", "contains_checked_statements": [], "coverage": "constructed_coefficientwise_natural_euler_product_rule_not_composition_IR079", "ha": "evidence/CV007-HA.json.gz", "internal_proof_nodes": [ 119, 115, 121 ], "lean": "evidence/CV007-Lean.json.gz", "nodes": 123, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-euler-product-v1", "proof": "evidence/CV007-bundle.json.gz", "report_sha256": "67658e65b46c3da53eed2a9808569ca9081e1b3854a41860b6ed8bb9647dcc6e", "root": 122, "statement": "forall ab ac L bb bc M i. exists db dc eb ec p q r. NatEulerPrefix(ab,ac,db,dc,L) /\\ (NatEulerPrefix(bb,bc,eb,ec,M) /\\ (NatConvolutionCoefficient(ab,ac,L,bb,bc,M,i,p) /\\ (NatConvolutionCoefficient(db,dc,L,bb,bc,M,i,q) /\\ (NatConvolutionCoefficient(ab,ac,L,eb,ec,M,i,r) /\\ i*p=q+r))))", "target_ast_sha256": "f8bff474361b2be3a4a2a9c81b30cdb9c317dd510ffdaf1ac64fa5762046cbea", "title": "euler product", "uses_definitions": [ "ND0401", "ND0410" ] }, { "bundle_bytes": 303172, "bundle_sha256": "ee2157fd318e4de4909bf579ec1f3bae6efe8a8d4b56e9b75e938df87d5c2450", "case": "CV008", "contains_checked_statements": [], "coverage": "constructed_all_rectangles_natural_fubini_not_convolution_associativity", "ha": "evidence/CV008-HA.json.gz", "internal_proof_nodes": [ 119, 106, 124 ], "lean": "evidence/CV008-Lean.json.gz", "nodes": 126, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-natural-fubini-v1", "proof": "evidence/CV008-bundle.json.gz", "report_sha256": "b117982ce1c795e25e07ce6fe74ffa5ef4a3fcc32fedd4389888e42ecb8ed182", "root": 125, "statement": "forall ab ac o s t m n. exists rb rc cb cc z. NatAffineRowSums(ab,ac,o,s,t,m,n,rb,rc) /\\ (NatAffineRowSums(ab,ac,o,t,s,n,m,cb,cc) /\\ (Sum(rb,rc,m,z) /\\ Sum(cb,cc,n,z)))", "target_ast_sha256": "144c71f1f4627b2ab6b53db389f1fc57d00868e5daf3716c01c6858aa379fec7", "title": "natural fubini", "uses_definitions": [ "PD0015", "ND0413" ] }, { "bundle_bytes": 240857, "bundle_sha256": "10d2619e1cb26a945cb18471091aaa96331b1718c28ce736a669c001fbaa26ed", "case": "CV009", "contains_checked_statements": [], "coverage": "constructed_natural_prefix_tail_sum_splitting_not_convolution_associativity", "ha": "evidence/CV009-HA.json.gz", "internal_proof_nodes": [ 106, 113, 117, 118, 0, 9 ], "lean": "evidence/CV009-Lean.json.gz", "nodes": 120, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-sum-splitting-v1", "proof": "evidence/CV009-bundle.json.gz", "report_sha256": "146752db44bd26c8de066c4b831ba564560fb8766ffa4f511a0db4cd96cf94c7", "root": 119, "statement": "forall ab ac m n. exists x y z. Sum(ab,ac,m,x) /\\ (NatAffineSum(ab,ac,m,1,n,y) /\\ (Sum(ab,ac,m+n,z) /\\ z=x+y))", "target_ast_sha256": "2aa07f3f9d93b14be669dd9bcca9eb51364030fe16587b130f508419e396c462", "title": "sum splitting", "uses_definitions": [ "PD0015", "ND0412" ] }, { "bundle_bytes": 305383, "bundle_sha256": "35c67ca650e7222ffedb065f157777f40bd88d06acfb746314dcb8d6c6afb093", "case": "CV010", "contains_checked_statements": [], "coverage": "constructed_commutative_natural_convolution_coefficients_not_associativity", "ha": "evidence/CV010-HA.json.gz", "internal_proof_nodes": [ 119, 131 ], "lean": "evidence/CV010-Lean.json.gz", "nodes": 133, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-commutative-v1", "proof": "evidence/CV010-bundle.json.gz", "report_sha256": "a15e83240a14e601c965da0b957e411b79f8647ef60a4bc3f58e1db99a1dc263", "root": 132, "statement": "forall ab ac L bb bc M i. exists n. NatConvolutionCoefficient(ab,ac,L,bb,bc,M,i,n) /\\ NatConvolutionCoefficient(bb,bc,M,ab,ac,L,i,n)", "target_ast_sha256": "3e62febc17b41a970070ca88e057e05ef7888d821a65f9019ac15bfb8fc5a825", "title": "convolution commutative", "uses_definitions": [ "ND0401" ] }, { "bundle_bytes": 264420, "bundle_sha256": "bdd2e9754c7c3700e19d4a26426a490e34fabfb435f0020e9b99ab26d984ff55", "case": "CV011", "contains_checked_statements": [ "CV009" ], "coverage": "actual_zero_tail_finite_sum_transport_not_convolution_associativity", "ha": "evidence/CV011-HA.json.gz", "internal_proof_nodes": [ 119, 121, 108, 76, 9 ], "lean": "evidence/CV011-Lean.json.gz", "nodes": 123, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-sum-zero-tail-v1", "proof": "evidence/CV011-bundle.json.gz", "report_sha256": "0621ad61b52801e93fa953cdd0cc859dc57166ff888592725151d551c256d942", "root": 122, "statement": "forall ab ac m n x. Sum(ab,ac,m,x) -> (forall j. Lt(j,n) -> BetaAt(ab,ac,m+j,0)) -> Sum(ab,ac,m+n,x)", "target_ast_sha256": "a7f186fd8bb8fecb316c79650a437ffc8e7c8a66e971c9cdf9740c56e26fe28e", "title": "sum zero tail", "uses_definitions": [ "PD0002", "PD0013", "PD0015" ] }, { "bundle_bytes": 178964, "bundle_sha256": "77fdbd64eb98decdc1a310bb553b5c8bc5e675e5cdb1e252438f0538dad2b646", "case": "CV012", "contains_checked_statements": [], "coverage": "constructed_actual_finite_triple_product_grid_not_sum_identities_or_associativity", "ha": "evidence/CV012-HA.json.gz", "internal_proof_nodes": [ 108, 105, 43 ], "lean": "evidence/CV012-Lean.json.gz", "nodes": 110, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-triple-grid-v1", "proof": "evidence/CV012-bundle.json.gz", "report_sha256": "ed5c332d040adcd429eddbf84128cf63ae3ee018040881246ceb8be40892bba5", "root": 109, "statement": "forall ab ac L bb bc M cb cc K i. exists gb gc. NatTripleGrid(ab,ac,L,bb,bc,M,cb,cc,K,i,gb,gc)", "target_ast_sha256": "ac50bc2ab4e67d47f7b5eb88eead79f814654ca2b863006b38df9015214803c2", "title": "triple grid", "uses_definitions": [ "ND0417" ] }, { "bundle_bytes": 329542, "bundle_sha256": "39db448c4be4a9bee828be1d616a9fd4065555cd9fa7416c5ee57eb4019e41e7", "case": "CV013", "contains_checked_statements": [ "CV009", "CV011" ], "coverage": "constructed_actual_triple_row_sum_as_scaled_convolution_not_associativity", "ha": "evidence/CV013-HA.json.gz", "internal_proof_nodes": [ 118, 107, 137 ], "lean": "evidence/CV013-Lean.json.gz", "nodes": 139, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-triple-row-sum-v1", "proof": "evidence/CV013-bundle.json.gz", "report_sha256": "c9f09ea68dd16747d0ba0d025dc77332da1e9f51fa9960a2a442f3cc64c2077f", "root": 138, "statement": "forall ab ac L bb bc M cb cc K i j r a ub uc. j+r=i -> BetaZeroExtend(ab,ac,L,j,a) -> NatTripleRowPrefix(ab,ac,L,bb,bc,M,cb,cc,K,i,j,ub,uc) -> exists t s. NatConvolutionCoefficient(cb,cc,K,bb,bc,M,r,t) /\\ (Sum(ub,uc,S i,s) /\\ s=a*t)", "target_ast_sha256": "835ed495fdd7f286cacdd0a59c544660f11c1fac47fb70eac20dad374b982a57", "title": "triple row sum", "uses_definitions": [ "PD0015", "ND0291", "ND0401", "ND0418" ] }, { "bundle_bytes": 568920, "bundle_sha256": "82a2a015e265224a899337bc2b375693c9881f2774e68487d87420cdf909478a", "case": "CV014", "contains_checked_statements": [ "CV008", "CV009", "CV011", "CV012" ], "coverage": "constructed_natural_convolution_associativity_below_common_width_not_composition", "ha": "evidence/CV014-HA.json.gz", "internal_proof_nodes": [ 121, 166 ], "lean": "evidence/CV014-Lean.json.gz", "nodes": 168, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-associative-v1", "proof": "evidence/CV014-bundle.json.gz", "report_sha256": "b56eb70b8cc10cd74c4b2ba32d2c6f11159162170d5fb96247f6ad39f42c9c76", "root": 167, "statement": "forall ab ac L bb bc M cb cc K i W. Lt(i,W) -> exists pb pc qb qc z. NatConvolutionPrefix(ab,ac,L,bb,bc,M,pb,pc,W) /\\ (NatConvolutionPrefix(bb,bc,M,cb,cc,K,qb,qc,W) /\\ (NatConvolutionCoefficient(pb,pc,W,cb,cc,K,i,z) /\\ NatConvolutionCoefficient(ab,ac,L,qb,qc,W,i,z)))", "target_ast_sha256": "a455e5f4420f28921f898952ebc74012686b69749b15192af32727495f22413b", "title": "convolution associative", "uses_definitions": [ "PD0002", "ND0401", "ND0402" ] }, { "bundle_bytes": 261414, "bundle_sha256": "7f4a09006875aad735b0543e6386b8e508c35c8890e2a801ffaf91c14f6dbf84", "case": "CV015", "contains_checked_statements": [], "coverage": "constructed_weighted_natural_convolution_linearity_not_composition", "ha": "evidence/CV015-HA.json.gz", "internal_proof_nodes": [ 115, 121 ], "lean": "evidence/CV015-Lean.json.gz", "nodes": 123, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-linearity-v1", "proof": "evidence/CV015-bundle.json.gz", "report_sha256": "de228cab1f70c9620442c0a48f30212c9138abfc361243094d22808d0852cc91", "root": 122, "statement": "forall alpha beta gamma ab ac bb bc cb cc L db dc M i. NatLinearTriple(alpha,beta,gamma,ab,ac,bb,bc,cb,cc,L) -> exists u v w. NatConvolutionCoefficient(ab,ac,L,db,dc,M,i,u) /\\ (NatConvolutionCoefficient(bb,bc,L,db,dc,M,i,v) /\\ (NatConvolutionCoefficient(cb,cc,L,db,dc,M,i,w) /\\ gamma*w=alpha*u+beta*v))", "target_ast_sha256": "454dd9464f21bb867fd39a78eebb4d9eabe59aae256c134ab0624e05d8d17f91", "title": "convolution linearity", "uses_definitions": [ "ND0401", "ND0409" ] }, { "bundle_bytes": 331284, "bundle_sha256": "354eae0281744eaefdf71202d0e84fd1e9e1ad154f712f8f8969b942294c7c77", "case": "CV016", "contains_checked_statements": [ "CV009", "CV011" ], "coverage": "constructed_natural_convolution_unit_below_width_not_composition", "ha": "evidence/CV016-HA.json.gz", "internal_proof_nodes": [ 121, 112, 140 ], "lean": "evidence/CV016-Lean.json.gz", "nodes": 142, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-unit-v1", "proof": "evidence/CV016-bundle.json.gz", "report_sha256": "71fea13e1877002aa163194064bebce9bc7a3367fffcd5ef93206683d7491fde", "root": 141, "statement": "forall ab ac L H i. Lt(i,H) -> exists ub uc a. NatUnitPrefix(ub,uc,H) /\\ (BetaZeroExtend(ab,ac,L,i,a) /\\ NatConvolutionCoefficient(ub,uc,H,ab,ac,L,i,a))", "target_ast_sha256": "80e61d585bc00d1ae1a6ea2c87fb652ee553927823efe339412d1d7e39c7b5d4", "title": "convolution unit", "uses_definitions": [ "PD0002", "ND0291", "ND0401", "ND0404" ] }, { "bundle_bytes": 768657, "bundle_sha256": "d94cbf0c9fb9f42a645258b60f5d7ba325f8ec20a54a983922da9b13c990f208", "case": "CV017", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV012" ], "coverage": "constructed_all_exponents_natural_euler_power_rule_not_composition", "ha": "evidence/CV017-HA.json.gz", "internal_proof_nodes": [ 134, 130, 191 ], "lean": "evidence/CV017-Lean.json.gz", "nodes": 193, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-series-power-euler-v1", "proof": "evidence/CV017-bundle.json.gz", "report_sha256": "b20cb9415224063301ad55ca4cbb74e51df9044d67eb6cf9c63690db73136bd3", "root": 192, "statement": "forall ab ac L H K. exists db dc pb pc qb qc. NatEulerPrefix(ab,ac,db,dc,L) /\\ (NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,K) /\\ (forall k ub uc vb vc i p t. Lt(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatPowerRow(pb,pc,qb,qc,S k,vb,vc) -> Lt(i,H) -> BetaAt(vb,vc,i,p) -> NatConvolutionCoefficient(db,dc,L,ub,uc,H,i,t) -> i*p=(S k)*t))", "target_ast_sha256": "e71ae746c83206309479f65ede0e10c2e11d077b3ac2123df4942475e453d1e6", "title": "series power euler", "uses_definitions": [ "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0410" ] }, { "bundle_bytes": 277534, "bundle_sha256": "5b951b467a3ba0decc2263327cdb596869bc977f11ed2a58fabd53aa3a7bfdc6", "case": "CV018", "contains_checked_statements": [ "CV015" ], "coverage": "constructed_weighted_natural_prefix_and_convolution_values_not_composition", "ha": "evidence/CV018-HA.json.gz", "internal_proof_nodes": [ 123, 122, 9 ], "lean": "evidence/CV018-Lean.json.gz", "nodes": 125, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-series-linear-combination-v1", "proof": "evidence/CV018-bundle.json.gz", "report_sha256": "e959f2ad730fb108f3d38de1dec0989da8efec05e98dfc8ab1e3deab4df29eea", "root": 124, "statement": "forall alpha beta ab ac bb bc L. exists cb cc. NatLinearTriple(alpha,beta,1,ab,ac,bb,bc,cb,cc,L) /\\ (forall db dc M i. exists u v w. NatConvolutionCoefficient(ab,ac,L,db,dc,M,i,u) /\\ (NatConvolutionCoefficient(bb,bc,L,db,dc,M,i,v) /\\ (NatConvolutionCoefficient(cb,cc,L,db,dc,M,i,w) /\\ w=alpha*u+beta*v)))", "target_ast_sha256": "f7bac172808c99e0125de6a05eaa1bb261596422da0685f05dbeb18d5b9ac56b", "title": "series linear combination", "uses_definitions": [ "ND0401", "ND0409" ] }, { "bundle_bytes": 197098, "bundle_sha256": "b0631c667d51a9b670287b6589503801ca8d8c2fb482b4473a3ba866ca0cd9d8", "case": "CV019", "contains_checked_statements": [], "coverage": "constructed_all_lengths_weighted_series_accumulation_trace_not_chain_rule", "ha": "evidence/CV019-HA.json.gz", "internal_proof_nodes": [ 106, 104, 23, 36, 108 ], "lean": "evidence/CV019-Lean.json.gz", "nodes": 110, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-weighted-series-trace-v1", "proof": "evidence/CV019-bundle.json.gz", "report_sha256": "01f603902649ad3fcb13f9cc3effbc1d5fb71bf22b5dfb3b2d9663a347daabf7", "root": 109, "statement": "forall H wb wc pb pc qb qc K. exists tb tc vb vc. NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,K)", "target_ast_sha256": "911c376cb25c2cb18aefd0e0b77325ea871669c2faefda9024ec48e53e0a0bbf", "title": "weighted series trace", "uses_definitions": [ "ND0421" ] }, { "bundle_bytes": 164302, "bundle_sha256": "b0ec06a09c31f62884ee4243ba011cb43c4e99f95b1c064da395fdc4a57e3f1b", "case": "CV020", "contains_checked_statements": [], "coverage": "constructed_shifted_euler_coefficient_table_not_chain_rule_or_analytic_derivative", "ha": "evidence/CV020-HA.json.gz", "internal_proof_nodes": [ 103, 104, 105 ], "lean": "evidence/CV020-Lean.json.gz", "nodes": 107, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-shifted-euler-prefix-v1", "proof": "evidence/CV020-bundle.json.gz", "report_sha256": "e0f1b2777e8e367aea1dee7b0f0e5c6f0a6e7dc5f8b48b822e5675a8926c7e87", "root": 106, "statement": "forall ab ac N. exists db dc. NatShiftedEulerPrefix(ab,ac,db,dc,N)", "target_ast_sha256": "a653a577dc205366278791e1c5bb895d21bdb2ea525bea07c4494276a051b832", "title": "shifted euler prefix", "uses_definitions": [ "ND0422" ] }, { "bundle_bytes": 909502, "bundle_sha256": "7e48274c5a65c371d3cc536fd0d8fdb08596ba956b8795ea96c1b30349d08688", "case": "CV021", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020" ], "coverage": "constructed_finite_composition_euler_chain_rule_not_exponential_or_analytic_identity", "ha": "evidence/CV021-HA.json.gz", "internal_proof_nodes": [ 134, 130, 198, 196, 206, 13 ], "lean": "evidence/CV021-Lean.json.gz", "nodes": 208, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-finite-composition-euler-v1", "proof": "evidence/CV021-bundle.json.gz", "report_sha256": "bbe6e1eaef98659bac2b690498c43b3d5b57d51aaba56206112e887ee1c71ba8", "root": 207, "statement": "forall ab ac L H wb wc N. exists db dc pb pc qb qc yb yc tb tc vb vc rb rc sb sc. NatEulerPrefix(ab,ac,db,dc,L) /\\ (NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,N) /\\ (NatShiftedEulerPrefix(wb,wc,yb,yc,N) /\\ (NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\\ (NatWeightedSeriesTrace(H,yb,yc,pb,pc,qb,qc,rb,rc,sb,sc,N) /\\ (forall fb fc gb gc i p t. NatPowerRow(tb,tc,vb,vc,S N,fb,fc) -> NatPowerRow(rb,rc,sb,sc,N,gb,gc) -> Lt(i,H) -> BetaAt(fb,fc,i,p) -> NatConvolutionCoefficient(db,dc,L,gb,gc,H,i,t) -> i*p=t)))))", "target_ast_sha256": "b8aae36bc9df2d03bb7bd14cae145075883cbe46a5eb5d5881d71b46927b2a65", "title": "finite composition euler", "uses_definitions": [ "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0421", "ND0410", "ND0422" ] }, { "bundle_bytes": 175040, "bundle_sha256": "e94cd7dab31d292ec52a9fff9ca85a05f17b5ba0aac7c396fa8c25c449b7b998", "case": "CV022", "contains_checked_statements": [], "coverage": "constructed_terminal_one_exponential_weight_recurrence_and_nonvanishing_not_factorial_interpretation", "ha": "evidence/CV022-HA.json.gz", "internal_proof_nodes": [ 102, 74, 23, 36, 25, 12, 106 ], "lean": "evidence/CV022-Lean.json.gz", "nodes": 108, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exponential-weights-v1", "proof": "evidence/CV022-bundle.json.gz", "report_sha256": "5d89d025ff8e9e4b11586b25a6240a619a5d8b570b3f58837ab4185c2dcb9975", "root": 107, "statement": "forall d N. exists wb wc. NatExpWeightTable(d,N,wb,wc) /\\ (forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0)", "target_ast_sha256": "67f4d5cffffb0e5b5dd1f559eb6b6d8cd2dbb5ac23bdeb79d580326b4cf5a880", "title": "exponential weights", "uses_definitions": [ "PD0001", "PD0013", "ND0423" ] }, { "bundle_bytes": 269143, "bundle_sha256": "3bebdb7ac4f793a5bf43e6e100c0a1286483ea73b7fb8a87d482df3f8422e108", "case": "CV023", "contains_checked_statements": [ "CV019" ], "coverage": "constructed_weighted_traces_with_all_rows_scalar_transport_not_exp_or_analytic_identity", "ha": "evidence/CV023-HA.json.gz", "internal_proof_nodes": [ 109, 114, 9, 4 ], "lean": "evidence/CV023-Lean.json.gz", "nodes": 116, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-weighted-series-scaling-v1", "proof": "evidence/CV023-bundle.json.gz", "report_sha256": "0990db9cf57bc8d41d0b840959cc52c3eaff1885a3fba9a5729f9a5c10a605af", "root": 115, "statement": "forall H pb pc qb qc wb wc yb yc d K. NatLinearTriple(d,0,1,yb,yc,yb,yc,wb,wc,K) -> exists tb tc vb vc rb rc sb sc. NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,K) /\\ (NatWeightedSeriesTrace(H,yb,yc,pb,pc,qb,qc,rb,rc,sb,sc,K) /\\ (forall k ub uc eb ec. Le(k,K) -> NatPowerRow(tb,tc,vb,vc,k,ub,uc) -> NatPowerRow(rb,rc,sb,sc,k,eb,ec) -> NatLinearTriple(d,0,1,eb,ec,eb,ec,ub,uc,H)))", "target_ast_sha256": "942e7e78560841ddb311dc397deeb9086050bd51b7d759f6926caddefd3d230d", "title": "weighted series scaling", "uses_definitions": [ "PD0001", "ND0405", "ND0409", "ND0421" ] }, { "bundle_bytes": 1059704, "bundle_sha256": "e727513a9752a1f0ca45550447de9ad92c582a4d221c1c8380146aa1aae068c4", "case": "CV024", "contains_checked_statements": [ "CV001", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022" ], "coverage": "constructed_denominator_cleared_finite_euler_equation_not_factorial_or_analytic_identity", "ha": "evidence/CV024-HA.json.gz", "internal_proof_nodes": [ 213, 207, 126, 120, 221, 222, 9, 4 ], "lean": "evidence/CV024-Lean.json.gz", "nodes": 224, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-denominator-euler-v1", "proof": "evidence/CV024-bundle.json.gz", "report_sha256": "3e6da0c59671370955150b2a4be95e0bb56270addbc9575db5b997225604eede", "root": 223, "statement": "forall ab ac L d N. NatZeroBefore(ab,ac,L,1) -> exists wb wc db dc pb pc qb qc yb yc tb tc vb vc rb rc sb sc. NatExpWeightTable(d,N,wb,wc) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ (NatEulerPrefix(ab,ac,db,dc,L) /\\ (NatPowerTrace(ab,ac,L,N,pb,pc,qb,qc,N) /\\ (NatShiftedEulerPrefix(wb,wc,yb,yc,N) /\\ (NatWeightedSeriesTrace(N,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\\ (NatWeightedSeriesTrace(N,yb,yc,pb,pc,qb,qc,rb,rc,sb,sc,N) /\\ ((forall fb fc i p v. NatPowerRow(tb,tc,vb,vc,S N,fb,fc) -> Lt(i,N) -> BetaAt(fb,fc,i,p) -> NatConvolutionCoefficient(db,dc,L,fb,fc,N,i,v) -> d*(i*p)=v))))))))", "target_ast_sha256": "b3b638e846a53d3cee1e16e8f927226ab00da021924e9c98b2978fc3a534989f", "title": "denominator euler", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0421", "ND0410", "ND0422", "ND0423", "ND0408" ] }, { "bundle_bytes": 164433, "bundle_sha256": "13c0d9f6839971d10e598ca6718963146822e77ed913e89d536ffa541e01f441", "case": "CV025", "contains_checked_statements": [], "coverage": "constructed_denominator_cleared_odd_reciprocal_prefix_not_logarithm_identity", "ha": "evidence/CV025-HA.json.gz", "internal_proof_nodes": [ 85, 105, 13, 106 ], "lean": "evidence/CV025-Lean.json.gz", "nodes": 108, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-coefficients-v1", "proof": "evidence/CV025-bundle.json.gz", "report_sha256": "d3c7cbf034931d890979331790e5bb944e66e0541566adb57f762d044bfb8546", "root": 107, "statement": "forall N. exists d ab ac. ~d=0 /\\ (NatAtanhPrefix(d,ab,ac,N) /\\ NatZeroBefore(ab,ac,N,1))", "target_ast_sha256": "6a235e46d7ff57ccd23bfc381f5c48d024d739de8f78abded4fd58a37aea8dbd", "title": "atanh coefficients", "uses_definitions": [ "ND0408", "ND0425" ] }, { "bundle_bytes": 177857, "bundle_sha256": "9395c1f7fc2f6eb5e13d52f715f983964060b080a6743960bf1787349b809ca9", "case": "CV026", "contains_checked_statements": [], "coverage": "constructed_atanh_euler_table_with_initial_values_and_period_two_not_analytic_derivative", "ha": "evidence/CV026-HA.json.gz", "internal_proof_nodes": [ 104, 73, 107, 108, 109 ], "lean": "evidence/CV026-Lean.json.gz", "nodes": 111, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-euler-v1", "proof": "evidence/CV026-bundle.json.gz", "report_sha256": "5c84752ff249437240c074f9fc218f1642833360a6ef6fcb9da18f39245fa73e", "root": 110, "statement": "forall d ab ac N. NatAtanhPrefix(d,ab,ac,N) -> exists db dc. NatEulerPrefix(ab,ac,db,dc,N) /\\ ((Lt(0,N) -> BetaAt(db,dc,0,0)) /\\ ((Lt(1,N) -> BetaAt(db,dc,1,2*d)) /\\ (forall i a b. Lt(S(S i),N) -> BetaAt(db,dc,i,a) -> BetaAt(db,dc,S(S i),b) -> a=b)))", "target_ast_sha256": "5a9b991ec4dccaeb1d26ec04fa722f7dd24100442b583d871f9cb4b5887431b2", "title": "atanh euler", "uses_definitions": [ "PD0002", "PD0013", "ND0425", "ND0410" ] }, { "bundle_bytes": 309165, "bundle_sha256": "48a775c9d61abf6bb9adf64eb39d170b6694988e0a2ee2dc7b654a0cca714b2f", "case": "CV027", "contains_checked_statements": [], "coverage": "constructed_convolution_coefficients_of_period_two_prefix_not_exp_or_analytic_identity", "ha": "evidence/CV027-HA.json.gz", "internal_proof_nodes": [ 122, 119, 76, 130, 131, 24, 30, 134, 133, 135, 4, 0, 2 ], "lean": "evidence/CV027-Lean.json.gz", "nodes": 137, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-periodic-convolution-v1", "proof": "evidence/CV027-bundle.json.gz", "report_sha256": "2e402891fc2c4739e714d88e42ac54b62b7a82eaa44d74f64c251e8f9fd2191f", "root": 136, "statement": "forall ab ac cb cc N i h c. Lt(S(S i),N) -> BetaAt(ab,ac,0,0) -> BetaAt(ab,ac,1,h) -> (forall j x y. Lt(S(S j),N) -> BetaAt(ab,ac,j,x) -> BetaAt(ab,ac,S(S j),y) -> x=y) -> BetaAt(cb,cc,S i,c) -> exists u v. NatConvolutionCoefficient(ab,ac,N,cb,cc,N,i,u) /\\ (NatConvolutionCoefficient(ab,ac,N,cb,cc,N,S(S i),v) /\\ v=u+h*c)", "target_ast_sha256": "3850a7a9f34d3796dd4b8e08f3a80925758444d026b1c0ce93c108108af94dec", "title": "periodic convolution", "uses_definitions": [ "PD0002", "PD0013", "ND0401" ] }, { "bundle_bytes": 1149748, "bundle_sha256": "c29b4cd038b31a9cec9ead465584d49e015dc18993c8d71ddb849b591b070a89", "case": "CV028", "contains_checked_statements": [ "CV001", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027" ], "coverage": "constructed_actual_finite_composition_three_term_recurrence_not_initial_values_or_analytic_identity", "ha": "evidence/CV028-HA.json.gz", "internal_proof_nodes": [ 228, 223, 126, 242, 243 ], "lean": "evidence/CV028-Lean.json.gz", "nodes": 245, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-composition-recurrence-v1", "proof": "evidence/CV028-bundle.json.gz", "report_sha256": "bd15940c2264eaad104cd95e87b5fbf6274920fc0d5cb3ebda4cd1e2c5bb2c11", "root": 244, "statement": "forall N. exists d ab ac wb wc db dc pb pc qb qc tb tc vb vc cb cc. ~d=0 /\\ (NatAtanhPrefix(d,ab,ac,N) /\\ (NatExpWeightTable(d,N,wb,wc) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ (NatEulerPrefix(ab,ac,db,dc,N) /\\ (NatPowerTrace(ab,ac,N,N,pb,pc,qb,qc,N) /\\ (NatWeightedSeriesTrace(N,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\\ (NatPowerRow(tb,tc,vb,vc,S N,cb,cc) /\\ ((forall j x y. Lt(j,N) -> BetaAt(cb,cc,j,x) -> NatConvolutionCoefficient(db,dc,N,cb,cc,N,j,y) -> d*(j*x)=y) /\\ ((forall i a b c. Lt(S(S i),N) -> BetaAt(cb,cc,i,a) -> BetaAt(cb,cc,S i,b) -> BetaAt(cb,cc,S(S i),c) -> (S(S i))*c=i*a+2*b))))))))))", "target_ast_sha256": "52788665088b222b10bd5663fc56a1708052b6dff8d6bda63f6719cae2553927", "title": "composition recurrence", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0421", "ND0410", "ND0423", "ND0425" ] }, { "bundle_bytes": 1247644, "bundle_sha256": "401c29b8458ca2615288aa3d695825d8ecc57b021ecce5e677c0920c4df360d9", "case": "CV029", "contains_checked_statements": [ "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028" ], "coverage": "constructed_actual_finite_composition_initial_and_all_coefficients_not_factorial_or_analytic_identity", "ha": "evidence/CV029-HA.json.gz", "internal_proof_nodes": [ 244, 76, 250, 246 ], "lean": "evidence/CV029-Lean.json.gz", "nodes": 252, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-composition-coefficients-v1", "proof": "evidence/CV029-bundle.json.gz", "report_sha256": "f910e83539c2bb12668854d314ff3b74e6b3e3e765c3c2b6858bf3ebd5be3dce", "root": 251, "statement": "forall N. Lt(1,N) -> exists d ab ac wb wc db dc pb pc qb qc tb tc vb vc cb cc z. (~d=0 /\\ (NatAtanhPrefix(d,ab,ac,N) /\\ (NatExpWeightTable(d,N,wb,wc) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ (NatEulerPrefix(ab,ac,db,dc,N) /\\ (NatPowerTrace(ab,ac,N,N,pb,pc,qb,qc,N) /\\ (NatWeightedSeriesTrace(N,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\\ (NatPowerRow(tb,tc,vb,vc,S N,cb,cc) /\\ ((forall j x y. Lt(j,N) -> BetaAt(cb,cc,j,x) -> NatConvolutionCoefficient(db,dc,N,cb,cc,N,j,y) -> d*(j*x)=y) /\\ ((forall i a b c. Lt(S(S i),N) -> BetaAt(cb,cc,i,a) -> BetaAt(cb,cc,S i,b) -> BetaAt(cb,cc,S(S i),c) -> (S(S i))*c=i*a+2*b))))))))))) /\\ (BetaAt(wb,wc,0,z) /\\ (~z=0 /\\ (NatExpAtanhRecurrence(cb,cc,N,z) /\\ (forall i a. Lt(i,N) -> BetaAt(cb,cc,i,a) -> ((i=0 /\\ a=z) \\/ (exists k. i=S k /\\ a=2*z))))))", "target_ast_sha256": "88f055686ab9620a1835d315530f9884d4bbd0626a5b97d2524141ff1701f83b", "title": "composition coefficients", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0421", "ND0410", "ND0423", "ND0425", "ND0400" ] }, { "bundle_bytes": 281234, "bundle_sha256": "c7972044ee20c85a4f3cf7cff03b6217745c4a16071999769c4714bf6d940e10", "case": "CV030", "contains_checked_statements": [ "CV022" ], "coverage": "constructed_actual_weight_factorial_power_normalization_not_rational_or_analytic_identity", "ha": "evidence/CV030-HA.json.gz", "internal_proof_nodes": [ 126, 74, 112, 118, 128, 13, 9 ], "lean": "evidence/CV030-Lean.json.gz", "nodes": 130, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-weight-factorials-v1", "proof": "evidence/CV030-bundle.json.gz", "report_sha256": "a003cc0b42015600386efe1ee1fb8444cd0d710a8e13128799b04b08b1f7d5cb", "root": 129, "statement": "forall d N. exists wb wc w0 f p. NatExpWeightTable(d,N,wb,wc) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ (BetaAt(wb,wc,0,w0) /\\ (Factorial(N,f) /\\ (Pow(d,N,p) /\\ (w0=f*p /\\ (forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=w0))))))", "target_ast_sha256": "7416c0350bc9ea0e640eccd8e16eeb667dc7c4ae78dfb37bc2abd3442cdaa923", "title": "weight factorials", "uses_definitions": [ "PD0001", "PD0013", "ND0423", "PD0020", "PD0023" ] }, { "bundle_bytes": 1369285, "bundle_sha256": "37a4fbb48c282439ac816d0efea5198bc6905ad57e7736aad23242cd96f8be71", "case": "CV031", "contains_checked_statements": [ "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029" ], "coverage": "same_actual_composition_weights_normalized_by_factorial_powers_not_rational_evaluation_or_analytic_identity", "ha": "evidence/CV031-HA.json.gz", "internal_proof_nodes": [ 251, 270, 260, 266, 13, 9 ], "lean": "evidence/CV031-Lean.json.gz", "nodes": 272, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-normalized-composition-v1", "proof": "evidence/CV031-bundle.json.gz", "report_sha256": "dd770e7b37bb6d35b19efd727ac23c1734cc3c4de85f1d155952a272bc068441", "root": 271, "statement": "forall N. Lt(1,N) -> exists d ab ac wb wc db dc pb pc qb qc tb tc vb vc cb cc z fn pn. ((~d=0 /\\ (NatAtanhPrefix(d,ab,ac,N) /\\ (NatExpWeightTable(d,N,wb,wc) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ (NatEulerPrefix(ab,ac,db,dc,N) /\\ (NatPowerTrace(ab,ac,N,N,pb,pc,qb,qc,N) /\\ (NatWeightedSeriesTrace(N,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\\ (NatPowerRow(tb,tc,vb,vc,S N,cb,cc) /\\ ((forall j x y. Lt(j,N) -> BetaAt(cb,cc,j,x) -> NatConvolutionCoefficient(db,dc,N,cb,cc,N,j,y) -> d*(j*x)=y) /\\ ((forall i a b c. Lt(S(S i),N) -> BetaAt(cb,cc,i,a) -> BetaAt(cb,cc,S i,b) -> BetaAt(cb,cc,S(S i),c) -> (S(S i))*c=i*a+2*b))))))))))) /\\ (BetaAt(wb,wc,0,z) /\\ (~z=0 /\\ (NatExpAtanhRecurrence(cb,cc,N,z) /\\ (forall i a. Lt(i,N) -> BetaAt(cb,cc,i,a) -> ((i=0 /\\ a=z) \\/ (exists k. i=S k /\\ a=2*z))))))) /\\ (Factorial(N,fn) /\\ (Pow(d,N,pn) /\\ (z=fn*pn /\\ (forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=z))))", "target_ast_sha256": "255502b20ed9e685e90c18c8a121a614c14ffe23ae181d749f51ced7a170f19e", "title": "normalized composition", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0421", "ND0410", "ND0423", "ND0425", "ND0400", "PD0020", "PD0023" ] }, { "bundle_bytes": 175054, "bundle_sha256": "97f4656c45ac339b593d9cfb83632028e297d6ba9cfc7ef4bd2c7bfe7f066787", "case": "CV032", "contains_checked_statements": [], "coverage": "constructed_integer_horner_trace_with_all_in_range_value_uniqueness_not_rational_or_analytic_identity", "ha": "evidence/CV032-HA.json.gz", "internal_proof_nodes": [ 102, 23, 36, 104, 105 ], "lean": "evidence/CV032-Lean.json.gz", "nodes": 107, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-trace-v1", "proof": "evidence/CV032-bundle.json.gz", "report_sha256": "7052b93d2ee89bdf22292438f19bd877441281ecf74ee18e945136f5d2456bc9", "root": 106, "statement": "forall ab ac t N. exists hb hc. NatScaledHornerTrace(ab,ac,t,N,hb,hc) /\\ (forall tb tc. NatScaledHornerTrace(ab,ac,t,N,tb,tc) -> forall k h u. Le(k,N) -> BetaAt(hb,hc,k,h) -> BetaAt(tb,tc,k,u) -> h=u)", "target_ast_sha256": "ef5c41487529723c8676ac25e7e7972e81abf1f34730285cf1abfaae5b63ce80", "title": "horner trace", "uses_definitions": [ "PD0001", "PD0013", "ND0427" ] }, { "bundle_bytes": 245286, "bundle_sha256": "d21aa29f793a9af547b2a92db2057ac2e272fb688da7b44c6bf1226b24b72ca9", "case": "CV033", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_horner_value_and_nonzero_power_with_exact_geometric_remainder_given_all_coefficients_not_analytic_identity", "ha": "evidence/CV033-HA.json.gz", "internal_proof_nodes": [ 106, 73, 115, 124, 13, 123, 12 ], "lean": "evidence/CV033-Lean.json.gz", "nodes": 126, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-geometric-horner-v1", "proof": "evidence/CV033-bundle.json.gz", "report_sha256": "eb6807076d058409c3a007fcfb64dcb069b68c4436c4d2ee1a4a38ee8cf9847d", "root": 125, "statement": "forall ab ac z k. (forall i a. Lt(i,S k) -> BetaAt(ab,ac,i,a) -> ((i=0 /\\ a=z) \\/ (exists j. i=S j /\\ a=2*z))) -> exists hb hc r p. NatScaledHornerTrace(ab,ac,3,S k,hb,hc) /\\ (BetaAt(hb,hc,S k,r) /\\ (Pow(3,k,p) /\\ (~p=0 /\\ r+z=(2*z)*p)))", "target_ast_sha256": "fbb3d37960c61bdda6f6242ac8a1ea3bd5ea98c9d28b11ee26556b39dce16fd6", "title": "geometric horner", "uses_definitions": [ "PD0002", "PD0013", "ND0427", "PD0020" ] }, { "bundle_bytes": 1450354, "bundle_sha256": "258242852ea96fe63fafe958f98ae1478c0c38348948623ad9b83231e0fe1f23", "case": "CV034", "contains_checked_statements": [ "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033" ], "coverage": "constructed_normalized_composition_and_same_coefficients_horner_value_with_exact_geometric_remainder_not_analytic_identity", "ha": "evidence/CV034-HA.json.gz", "internal_proof_nodes": [ 271, 282, 24, 30, 47 ], "lean": "evidence/CV034-Lean.json.gz", "nodes": 284, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-composition-evaluation-v1", "proof": "evidence/CV034-bundle.json.gz", "report_sha256": "c323d2cedd2c4af90ab54bcae89ecba64b076e74f1b521749a71d3c6762290bf", "root": 283, "statement": "forall K. exists N d ab ac wb wc db dc pb pc qb qc tb tc vb vc cb cc z fn pn hb hc r p. N=S(S K) /\\ ((((~d=0 /\\ (NatAtanhPrefix(d,ab,ac,N) /\\ (NatExpWeightTable(d,N,wb,wc) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ (NatEulerPrefix(ab,ac,db,dc,N) /\\ (NatPowerTrace(ab,ac,N,N,pb,pc,qb,qc,N) /\\ (NatWeightedSeriesTrace(N,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) /\\ (NatPowerRow(tb,tc,vb,vc,S N,cb,cc) /\\ ((forall j x y. Lt(j,N) -> BetaAt(cb,cc,j,x) -> NatConvolutionCoefficient(db,dc,N,cb,cc,N,j,y) -> d*(j*x)=y) /\\ ((forall i a b c. Lt(S(S i),N) -> BetaAt(cb,cc,i,a) -> BetaAt(cb,cc,S i,b) -> BetaAt(cb,cc,S(S i),c) -> (S(S i))*c=i*a+2*b))))))))))) /\\ (BetaAt(wb,wc,0,z) /\\ (~z=0 /\\ (NatExpAtanhRecurrence(cb,cc,N,z) /\\ (forall i a. Lt(i,N) -> BetaAt(cb,cc,i,a) -> ((i=0 /\\ a=z) \\/ (exists k. i=S k /\\ a=2*z))))))) /\\ (Factorial(N,fn) /\\ (Pow(d,N,pn) /\\ (z=fn*pn /\\ (forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=z))))) /\\ (NatScaledHornerTrace(cb,cc,3,N,hb,hc) /\\ (BetaAt(hb,hc,N,r) /\\ (Pow(3,S K,p) /\\ (~p=0 /\\ (r+z=(2*z)*p /\\ (~(z*p=0))))))))", "target_ast_sha256": "11b40acf4f70651db35615b86f321bc4dc95ea05a8e711c8a3f41d9c2fd3ed34", "title": "composition evaluation", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0401", "ND0405", "ND0407", "ND0421", "ND0410", "ND0423", "ND0425", "ND0400", "PD0020", "PD0023", "ND0427" ] }, { "bundle_bytes": 200908, "bundle_sha256": "77477ae2e7d646455c9401f03bef0c0e798ef50fd7286954b938c74074793b65", "case": "CV035", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_three_horner_evaluations_with_weighted_linearity_from_pointwise_relation_not_multiplication_or_analytic_identity", "ha": "evidence/CV035-HA.json.gz", "internal_proof_nodes": [ 106, 73, 109, 13 ], "lean": "evidence/CV035-Lean.json.gz", "nodes": 111, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-linearity-v1", "proof": "evidence/CV035-bundle.json.gz", "report_sha256": "8f480d5f34c435637c2c084a43070e2b2e88ef235bd22d00e8f26b167b94c1ba", "root": 110, "statement": "forall alpha beta gamma ab ac bb bc cb cc t N. NatLinearTriple(alpha,beta,gamma,ab,ac,bb,bc,cb,cc,N) -> exists ub uc vb vc wb wc u v w. NatScaledHornerTrace(ab,ac,t,N,ub,uc) /\\ (NatScaledHornerTrace(bb,bc,t,N,vb,vc) /\\ (NatScaledHornerTrace(cb,cc,t,N,wb,wc) /\\ (BetaAt(ub,uc,N,u) /\\ (BetaAt(vb,vc,N,v) /\\ (BetaAt(wb,wc,N,w) /\\ gamma*w=alpha*u+beta*v)))))", "target_ast_sha256": "ad3d9e7f204d9e713be5a888404827c60c7a6e4a2ceca04adb2deb516e9682df", "title": "horner linearity", "uses_definitions": [ "PD0013", "ND0409", "ND0427" ] }, { "bundle_bytes": 237346, "bundle_sha256": "106c098a8896d97701ffa0444b2cadc8aafab9d4bbdab9c5603b903fb8963881", "case": "CV036", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_horner_endpoints_and_actual_power_with_exact_zero_tail_scaling_not_multiplication_or_analytic_identity", "ha": "evidence/CV036-HA.json.gz", "internal_proof_nodes": [ 106, 73, 115, 120, 13 ], "lean": "evidence/CV036-Lean.json.gz", "nodes": 122, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-zero-tail-v1", "proof": "evidence/CV036-bundle.json.gz", "report_sha256": "f6cc7d0a7e997c5b70d8579a944e6650a8fd43ce73b25612150c845d76243127", "root": 121, "statement": "forall ab ac t M N. (forall j a. Lt(j,N) -> BetaAt(ab,ac,M+j,a) -> a=0) -> exists hb hc u v p. NatScaledHornerTrace(ab,ac,t,M+N,hb,hc) /\\ (BetaAt(hb,hc,M,u) /\\ (BetaAt(hb,hc,M+N,v) /\\ (Pow(t,N,p) /\\ v=p*u)))", "target_ast_sha256": "ba1d797f495177c7c2b6f504d04e3cb81acb6460d86b60a22e46f3369a8a78d9", "title": "horner zero tail", "uses_definitions": [ "PD0002", "PD0013", "ND0427", "PD0020" ] }, { "bundle_bytes": 210816, "bundle_sha256": "1f1e394ceee77030f4f8b71ae091f587e9815c47546e2ee674f699de2c8189c6", "case": "CV037", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_horner_endpoint_equals_actual_sum_given_geometric_weights_and_pointwise_products_not_their_existence_or_analytic_identity", "ha": "evidence/CV037-HA.json.gz", "internal_proof_nodes": [ 106, 73, 109, 113, 13, 10 ], "lean": "evidence/CV037-Lean.json.gz", "nodes": 115, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-weighted-sum-v1", "proof": "evidence/CV037-bundle.json.gz", "report_sha256": "be0bd536ec550116889a900c406943586b47d053fa3f52a78f97c2afc762ca8e", "root": 114, "statement": "forall ab ac t N wb wc qb qc. NatGeometricWeightTable(t,N,wb,wc) -> NatHornerWeightedPrefix(ab,ac,wb,wc,qb,qc,N) -> exists hb hc r. NatScaledHornerTrace(ab,ac,t,N,hb,hc) /\\ (BetaAt(hb,hc,N,r) /\\ Sum(qb,qc,N,r))", "target_ast_sha256": "6f5222fbdb659a953b151731a5f8b309685813f395efda3e55af160b5757bfd8", "title": "horner weighted sum", "uses_definitions": [ "PD0013", "PD0015", "ND0427", "ND0428", "ND0429" ] }, { "bundle_bytes": 233337, "bundle_sha256": "18b213e913d77b5362b7f2b8bfe27f7969d2379affd7519141080407eb31caa8", "case": "CV038", "contains_checked_statements": [], "coverage": "constructed_terminal_one_geometric_weights_and_actual_power_with_all_weight_power_identities_not_weighted_term_existence_or_analytic_identity", "ha": "evidence/CV038-HA.json.gz", "internal_proof_nodes": [ 120, 73, 114, 121, 13, 9 ], "lean": "evidence/CV038-Lean.json.gz", "nodes": 123, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-geometric-weights-v1", "proof": "evidence/CV038-bundle.json.gz", "report_sha256": "93a1ff22739144a24ef50bb5fdd39fb5a64b4e6e93b3343ff8c54e40bbd86902", "root": 122, "statement": "forall t N. exists wb wc p. NatGeometricWeightTable(t,N,wb,wc) /\\ (Pow(t,N,p) /\\ (BetaAt(wb,wc,0,p) /\\ (forall j w q. Le(j,N) -> BetaAt(wb,wc,j,w) -> Pow(t,j,q) -> w*q=p)))", "target_ast_sha256": "917e753656a45bbae99c1856843859780ffcc5d52d3d25e45991093293a278be", "title": "geometric weights", "uses_definitions": [ "PD0001", "PD0013", "ND0428", "PD0020" ] }, { "bundle_bytes": 308879, "bundle_sha256": "8b1f8f7f6c20cc160fb79285ee98a4c22931aee8c7494f9e39d848055b15fc5c", "case": "CV039", "contains_checked_statements": [ "CV032", "CV037", "CV038" ], "coverage": "constructed_geometric_weights_actual_power_pointwise_products_horner_trace_and_actual_sum_for_same_endpoint_not_multiplication_or_analytic_identity", "ha": "evidence/CV039-HA.json.gz", "internal_proof_nodes": [ 122, 135, 133 ], "lean": "evidence/CV039-Lean.json.gz", "nodes": 137, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-constructed-horner-sum-v1", "proof": "evidence/CV039-bundle.json.gz", "report_sha256": "24e8a941ae58406d9161a134000e47611b7f28a42aabe3f5a3d6185f6582ce59", "root": 136, "statement": "forall ab ac t N. exists wb wc p qb qc hb hc r. (NatGeometricWeightTable(t,N,wb,wc) /\\ (Pow(t,N,p) /\\ (BetaAt(wb,wc,0,p) /\\ (forall j w q. Le(j,N) -> BetaAt(wb,wc,j,w) -> Pow(t,j,q) -> w*q=p)))) /\\ (NatHornerWeightedPrefix(ab,ac,wb,wc,qb,qc,N) /\\ (NatScaledHornerTrace(ab,ac,t,N,hb,hc) /\\ (BetaAt(hb,hc,N,r) /\\ Sum(qb,qc,N,r))))", "target_ast_sha256": "04e9780ee714dda23598933ee36bc81764c56ad8b2edc3632b84e8161529ddf8", "title": "constructed horner sum", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0427", "ND0428", "ND0429", "PD0020" ] }, { "bundle_bytes": 340745, "bundle_sha256": "c01db734918315570a4e104464a29d4f6932796b37d351bae01926e53c12e496", "case": "CV040", "contains_checked_statements": [ "CV032", "CV037", "CV038", "CV039" ], "coverage": "constructed_same_horner_sum_with_exact_complementary_power_and_monomial_terms_not_multiplication_or_analytic_identity", "ha": "evidence/CV040-HA.json.gz", "internal_proof_nodes": [ 136, 139, 140 ], "lean": "evidence/CV040-Lean.json.gz", "nodes": 142, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-monomials-v1", "proof": "evidence/CV040-bundle.json.gz", "report_sha256": "8701d9b99dfe58112bfaf468740580a7e3a77cf63626df55e16434982dfe80b4", "root": 141, "statement": "forall ab ac t N. exists wb wc p qb qc hb hc r. ((NatGeometricWeightTable(t,N,wb,wc) /\\ (Pow(t,N,p) /\\ (BetaAt(wb,wc,0,p) /\\ (forall j w q. Le(j,N) -> BetaAt(wb,wc,j,w) -> Pow(t,j,q) -> w*q=p)))) /\\ (NatHornerWeightedPrefix(ab,ac,wb,wc,qb,qc,N) /\\ (NatScaledHornerTrace(ab,ac,t,N,hb,hc) /\\ (BetaAt(hb,hc,N,r) /\\ Sum(qb,qc,N,r))))) /\\ ((forall e j w q. j+e=N -> BetaAt(wb,wc,j,w) -> Pow(t,e,q) -> w=q) /\\ (forall j e a q c. (S j)+e=N -> BetaAt(ab,ac,j,a) -> Pow(t,e,q) -> BetaAt(qb,qc,j,c) -> c=a*q))", "target_ast_sha256": "76c2e7135bb0294d18ddca7f6750fc378916725dd05a107878f9dc22c55483d1", "title": "horner monomials", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0427", "ND0428", "ND0429", "PD0020" ] }, { "bundle_bytes": 319346, "bundle_sha256": "9d6be9a12cba3dd86cda9629336da3aa6a5c55902974c818f3fddb9f2a5d4e23", "case": "CV041", "contains_checked_statements": [ "CV038" ], "coverage": "constructed_three_geometric_tables_with_convolution_weight_product_law_via_actual_archived_power_add_not_full_evaluation_or_analytic_identity", "ha": "evidence/CV041-HA.json.gz", "internal_proof_nodes": [ 122, 133 ], "lean": "evidence/CV041-Lean.json.gz", "nodes": 135, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-weights-v1", "proof": "evidence/CV041-bundle.json.gz", "report_sha256": "a85158d761ac1583ae6d978b36d0a20a7b060849a0bb00b6d5dc1e00f3701992", "root": 134, "statement": "forall t L M. exists lb lc pl mb mc pm nb nc pn. (NatGeometricWeightTable(t,L,lb,lc) /\\ (Pow(t,L,pl) /\\ (BetaAt(lb,lc,0,pl) /\\ (forall j w q. Le(j,L) -> BetaAt(lb,lc,j,w) -> Pow(t,j,q) -> w*q=pl)))) /\\ ((NatGeometricWeightTable(t,M,mb,mc) /\\ (Pow(t,M,pm) /\\ (BetaAt(mb,mc,0,pm) /\\ (forall j w q. Le(j,M) -> BetaAt(mb,mc,j,w) -> Pow(t,j,q) -> w*q=pm)))) /\\ ((NatGeometricWeightTable(t,L+M,nb,nc) /\\ (Pow(t,L+M,pn) /\\ (BetaAt(nb,nc,0,pn) /\\ (forall j w q. Le(j,L+M) -> BetaAt(nb,nc,j,w) -> Pow(t,j,q) -> w*q=pn)))) /\\ (forall i j k a b c. Lt(i,L) -> Lt(j,M) -> i+j=k -> BetaAt(lb,lc,S i,a) -> BetaAt(mb,mc,S j,b) -> BetaAt(nb,nc,S k,c) -> c=t*(a*b))))", "target_ast_sha256": "2673216a113a76804e456d1ac260a39e3d0707fa44828c7d4e784d90e55e6f87", "title": "convolution weights", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0428", "PD0020" ] }, { "bundle_bytes": 378143, "bundle_sha256": "fa0ff9d23182905616197f0cd573075710d2cd07a9c6f9fef7f31fe2bea7dc9f", "case": "CV042", "contains_checked_statements": [], "coverage": "actual_coefficient_weighting_identity_for_all_indices_given_three_weight_tables_and_two_product_prefixes_not_full_evaluation_or_analytic_identity", "ha": "evidence/CV042-HA.json.gz", "internal_proof_nodes": [ 137, 73, 146 ], "lean": "evidence/CV042-Lean.json.gz", "nodes": 148, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-weighted-convolution-v1", "proof": "evidence/CV042-bundle.json.gz", "report_sha256": "bb8dbc04a9a8268991ecfbacc41c073824c83660abbb13302b7ecc9399aa726c", "root": 147, "statement": "forall t L M ab ac bb bc lb lc mb mc nb nc ub uc vb vc k. NatGeometricWeightTable(t,L,lb,lc) -> NatGeometricWeightTable(t,M,mb,mc) -> NatGeometricWeightTable(t,L+M,nb,nc) -> NatHornerWeightedPrefix(ab,ac,lb,lc,ub,uc,L) -> NatHornerWeightedPrefix(bb,bc,mb,mc,vb,vc,M) -> exists u v w. NatConvolutionCoefficient(ab,ac,L,bb,bc,M,k,u) /\\ (NatConvolutionCoefficient(ub,uc,L,vb,vc,M,k,v) /\\ (BetaAt(nb,nc,S k,w) /\\ u*w=t*v))", "target_ast_sha256": "6537f190afaf524e595debbbbc01006bfd85ea9994906ff6b4aca84d7b4057af", "title": "weighted convolution", "uses_definitions": [ "PD0013", "ND0401", "ND0428", "ND0429" ] }, { "bundle_bytes": 277659, "bundle_sha256": "1568eaa83e40883b44bda2ac0fa02854792abb1da4e9f099e327e168e4fac1c1", "case": "CV043", "contains_checked_statements": [ "CV009", "CV011" ], "coverage": "constructed_sum_values_for_actual_scaled_shifted_block_with_leading_and_trailing_zeros_not_full_convolution_evaluation_or_analytic_identity", "ha": "evidence/CV043-HA.json.gz", "internal_proof_nodes": [ 106, 119, 121, 122, 123, 0 ], "lean": "evidence/CV043-Lean.json.gz", "nodes": 125, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-shifted-scaled-sum-v1", "proof": "evidence/CV043-bundle.json.gz", "report_sha256": "9c8cf804392a165131e52524c3b1ae9a10d364768712bc1d2ed2b0c8087a4439", "root": 124, "statement": "forall ab ac ub uc s N q a. (forall i u. Lt(i,s) -> BetaAt(ub,uc,i,u) -> u=0) -> (forall j b c. Lt(j,N) -> BetaAt(ab,ac,j,b) -> BetaAt(ub,uc,s+j,c) -> c=a*b) -> (forall j. Lt(j,q) -> BetaAt(ub,uc,(s+N)+j,0)) -> exists x y. Sum(ab,ac,N,x) /\\ (Sum(ub,uc,(s+N)+q,y) /\\ y=a*x)", "target_ast_sha256": "6ee048a7b7670833a6caa1ec7172b086096264b73542ae00b8122a9ea9f004fa", "title": "shifted scaled sum", "uses_definitions": [ "PD0002", "PD0013", "PD0015" ] }, { "bundle_bytes": 174288, "bundle_sha256": "9c96b3fa34227010da9bdd33d71361ce7d3edbae1ee2f1c230f90f46c5a74ad0", "case": "CV044", "contains_checked_statements": [], "coverage": "constructed_square_cauchy_grid_from_actual_zero_padded_antidiagonal_products_and_triangular_zeros_not_row_column_sum_or_evaluation_identity", "ha": "evidence/CV044-HA.json.gz", "internal_proof_nodes": [ 108, 105, 43, 75 ], "lean": "evidence/CV044-Lean.json.gz", "nodes": 110, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-cauchy-grid-v1", "proof": "evidence/CV044-bundle.json.gz", "report_sha256": "c9c8d860077a35ddf7c1e0132138fe4e26d78b456d11b7596612be974062ffcc", "root": 109, "statement": "forall ab ac L bb bc M N. exists gb gc. NatCauchyGrid(ab,ac,L,bb,bc,M,N,gb,gc)", "target_ast_sha256": "78d464d63eb29155ecd6adef39b240abfae271ca11d414e46e9ecc2be332a0a4", "title": "cauchy grid", "uses_definitions": [ "ND0433" ] }, { "bundle_bytes": 321100, "bundle_sha256": "0ca030a7524f9c287ebb556c1577d3b530d13d8d90e9f26e53c538ef10c1da6e", "case": "CV045", "contains_checked_statements": [ "CV009", "CV011", "CV044" ], "coverage": "constructed_cauchy_grid_and_all_actual_row_sums_identified_with_convolution_coefficients_using_proved_zero_tail_not_column_sums_or_evaluation_identity", "ha": "evidence/CV045-HA.json.gz", "internal_proof_nodes": [ 109, 134 ], "lean": "evidence/CV045-Lean.json.gz", "nodes": 136, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-cauchy-row-sum-v1", "proof": "evidence/CV045-bundle.json.gz", "report_sha256": "02d6e5c526f9064dd0a4e492711f297a73571a63485d5ef5e11e1188a974565d", "root": 135, "statement": "forall ab ac L bb bc M N. exists gb gc. NatCauchyGrid(ab,ac,L,bb,bc,M,N,gb,gc) /\\ (forall k. Lt(k,S N) -> exists u. NatConvolutionCoefficient(ab,ac,L,bb,bc,M,k,u) /\\ NatAffineSum(gb,gc,(S N)*k,1,S N,u))", "target_ast_sha256": "1e5bd50b1aebb4f3f9ccee2b1ed520e7c8faafdc3731a8285363a3811d4a23da", "title": "cauchy row sum", "uses_definitions": [ "PD0002", "ND0401", "ND0433", "ND0412" ] }, { "bundle_bytes": 307670, "bundle_sha256": "51a97eb9d305b9d355bd3f737745a5fb8736bf7c9d3423149e3f209882f7b742", "case": "CV046", "contains_checked_statements": [ "CV009", "CV011", "CV043" ], "coverage": "actual_complete_cauchy_column_sum_equals_padded_left_value_times_right_sum_under_explicit_span_equality_not_unconditional_grid_columns_or_evaluation", "ha": "evidence/CV046-HA.json.gz", "internal_proof_nodes": [ 131, 14, 125, 130, 123, 73, 2, 3, 127 ], "lean": "evidence/CV046-Lean.json.gz", "nodes": 133, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-cauchy-column-sum-v1", "proof": "evidence/CV046-bundle.json.gz", "report_sha256": "3ad140973f094212f1984e40206bf467abc5d6e5564a9bc84c7b036cdda032e2", "root": 132, "statement": "forall ab ac L bb bc M j a ub uc T q. BetaZeroExtend(ab,ac,L,j,a) -> NatCauchyColumn(ab,ac,L,bb,bc,M,j,ub,uc,T) -> (j+M)+q=T -> exists x y. Sum(bb,bc,M,x) /\\ (Sum(ub,uc,T,y) /\\ y=a*x)", "target_ast_sha256": "94eff8a4d0a0b19ee9a9c9a63604deabe5c06331fd7e9e1fd64593a4026f0b4f", "title": "cauchy column sum", "uses_definitions": [ "PD0015", "ND0291", "ND0435" ] }, { "bundle_bytes": 400130, "bundle_sha256": "3f86c41ea1d9b3740f12a7e74c131e5baf272e7b037b4bd54b45967ab9324f95", "case": "CV047", "contains_checked_statements": [ "CV009", "CV011", "CV043", "CV044", "CV046" ], "coverage": "constructed_single_cauchy_grid_and_all_full_length_row_column_sums_with_shared_right_sum_and_zero_columns_not_yet_fubini_sum_product_or_evaluation", "ha": "evidence/CV047-HA.json.gz", "internal_proof_nodes": [ 147 ], "lean": "evidence/CV047-Lean.json.gz", "nodes": 149, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-cauchy-grid-sums-v2", "proof": "evidence/CV047-bundle.json.gz", "report_sha256": "88b70ca8c4f455816dcbf1bef4cc049317ec06c5fc4ebb83596a0951a0502168", "root": 148, "statement": "forall ab ac L bb bc M. exists N gb gc x. N=L+M /\\ (NatCauchyGrid(ab,ac,L,bb,bc,M,N,gb,gc) /\\ (Sum(bb,bc,M,x) /\\ ((forall k. Lt(k,N) -> exists u. NatConvolutionCoefficient(ab,ac,L,bb,bc,M,k,u) /\\ NatAffineSum(gb,gc,(S N)*k,1,N,u)) /\\ (forall j a. Lt(j,N) -> BetaZeroExtend(ab,ac,L,j,a) -> exists y. NatAffineSum(gb,gc,j,S N,N,y) /\\ y=a*x))))", "target_ast_sha256": "f9641b30ca39287a97a195a21f4f444b93c513dc143c2f42166caff02b8f76a0", "title": "cauchy grid sums", "uses_definitions": [ "PD0002", "PD0015", "ND0291", "ND0401", "ND0433", "ND0412" ] }, { "bundle_bytes": 531668, "bundle_sha256": "e36b7cbb77811b8fac027866066f3e9f28703f247168785b2c003b12e2efc7d7", "case": "CV048", "contains_checked_statements": [ "CV008", "CV009", "CV011", "CV043", "CV044", "CV046", "CV047" ], "coverage": "constructed_full_length_natural_convolution_and_all_three_sums_with_output_sum_equal_to_product_of_input_sums_via_actual_fubini_not_horner_or_analytic_identity", "ha": "evidence/CV048-HA.json.gz", "internal_proof_nodes": [ 148, 156, 106, 157, 159, 160, 122, 108 ], "lean": "evidence/CV048-Lean.json.gz", "nodes": 162, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-total-v2", "proof": "evidence/CV048-bundle.json.gz", "report_sha256": "4b5b70b5a3996fb73e32ef010d2ede2a5daf6d9efc2a2cf64dd1607fd47abfb0", "root": 161, "statement": "forall ab ac L bb bc M. exists rb rc x y z. NatConvolutionPrefix(ab,ac,L,bb,bc,M,rb,rc,L+M) /\\ (Sum(ab,ac,L,x) /\\ (Sum(bb,bc,M,y) /\\ (Sum(rb,rc,L+M,z) /\\ z=x*y)))", "target_ast_sha256": "074045ab4db4d6936b3447f35096441e7ed3857f9eb93287ea1335d93cbad0dc", "title": "convolution total", "uses_definitions": [ "PD0015", "ND0402" ] }, { "bundle_bytes": 832029, "bundle_sha256": "07fee4d447429fe28b7c296c10327f9331046aef8d12be69574f73e21c1e09e6", "case": "CV049", "contains_checked_statements": [ "CV008", "CV009", "CV011", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048" ], "coverage": "constructed_full_length_convolution_and_actual_three_horner_traces_with_exact_extra_multiplier_product_law_including_zero_multiplier_not_truncated_evaluation_or_analytic_identity", "ha": "evidence/CV049-HA.json.gz", "internal_proof_nodes": [ 215, 213, 161, 216, 112, 108, 9, 4 ], "lean": "evidence/CV049-Lean.json.gz", "nodes": 218, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-product-v1", "proof": "evidence/CV049-bundle.json.gz", "report_sha256": "843b92f299f7d4cafdc5a61ce8180825aac30307fc78155f35453fe8cc6ee9f0", "root": 217, "statement": "forall ab ac L bb bc M t. exists cb cc ahb ahc x bhb bhc y chb chc z. NatConvolutionPrefix(ab,ac,L,bb,bc,M,cb,cc,L+M) /\\ (NatScaledHornerTrace(ab,ac,t,L,ahb,ahc) /\\ (BetaAt(ahb,ahc,L,x) /\\ (NatScaledHornerTrace(bb,bc,t,M,bhb,bhc) /\\ (BetaAt(bhb,bhc,M,y) /\\ (NatScaledHornerTrace(cb,cc,t,L+M,chb,chc) /\\ (BetaAt(chb,chc,L+M,z) /\\ z=t*(x*y)))))))", "target_ast_sha256": "396f5d28929411112a14eafdfa700edebdae4db2f0d4669afc4dec841105c464", "title": "horner product", "uses_definitions": [ "PD0013", "ND0402", "ND0427" ] }, { "bundle_bytes": 213329, "bundle_sha256": "eefa1a560700fd4767facdeeb2765dd01c5c6caed05d6d6b4653f75e84619e07", "case": "CV050", "contains_checked_statements": [], "coverage": "constructed_common_actual_coefficient_from_two_local_padded_input_transports_below_explicit_strict_cutoff_not_all_degree_or_analytic_equality", "ha": "evidence/CV050-HA.json.gz", "internal_proof_nodes": [ 112, 115 ], "lean": "evidence/CV050-Lean.json.gz", "nodes": 117, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-locality-v1", "proof": "evidence/CV050-bundle.json.gz", "report_sha256": "4657f5216a59b3ab9ffbf32ccf88948585c5f4e96e71b36650c079cbb25fa6d0", "root": 116, "statement": "forall ab ac L bb bc M cb cc N db dc O H i. (forall j a. Lt(j,H) -> BetaZeroExtend(ab,ac,L,j,a) -> BetaZeroExtend(cb,cc,N,j,a)) -> (forall j a. Lt(j,H) -> BetaZeroExtend(bb,bc,M,j,a) -> BetaZeroExtend(db,dc,O,j,a)) -> Lt(i,H) -> exists n. NatConvolutionCoefficient(ab,ac,L,bb,bc,M,i,n) /\\ NatConvolutionCoefficient(cb,cc,N,db,dc,O,i,n)", "target_ast_sha256": "750e75f60ad34a2d8fa423359399e8fcd60a9a1c90dbde559a2faf8a856457c0", "title": "convolution locality", "uses_definitions": [ "PD0002", "ND0291", "ND0401" ] }, { "bundle_bytes": 388057, "bundle_sha256": "82e79e5d443ffe702660ea79c2d7ee9acfe8005212bd913d69866558ff5954bb", "case": "CV051", "contains_checked_statements": [ "CV004" ], "coverage": "constructed_two_actual_power_traces_and_all_row_low_coefficient_transport_under_explicit_width_order_not_truncated_full_evaluation_equality", "ha": "evidence/CV051-HA.json.gz", "internal_proof_nodes": [ 130, 139 ], "lean": "evidence/CV051-Lean.json.gz", "nodes": 141, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-power-width-transport-v3", "proof": "evidence/CV051-bundle.json.gz", "report_sha256": "37637adc072742ab2f9cf98baf63397bf33ac6db2966d0c1f83a068ad2c5294f", "root": 140, "statement": "forall ab ac L H J K. Le(H,J) -> exists pb pc qb qc rb rc sb sc. NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,K) /\\ (NatPowerTrace(ab,ac,L,J,rb,rc,sb,sc,K) /\\ (forall k ub uc vb vc i a. Le(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatPowerRow(rb,rc,sb,sc,k,vb,vc) -> Lt(i,H) -> BetaAt(ub,uc,i,a) -> BetaAt(vb,vc,i,a)))", "target_ast_sha256": "d22452396ee9b966bdb17b27cd501a09a8cda4e1433e5d3885685cd54982e6eb", "title": "power width transport", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0405", "ND0407" ] }, { "bundle_bytes": 502507, "bundle_sha256": "6e941807907f0cf199d63cefee4e47b7d1726ad90e6c2dd9f20a55897bdad89f", "case": "CV052", "contains_checked_statements": [ "CV004", "CV019", "CV051" ], "coverage": "constructed_cross_width_powers_and_weighted_compositions_preserving_all_low_coefficients_with_same_input_and_weights_not_full_evaluation_or_tail_identity", "ha": "evidence/CV052-HA.json.gz", "internal_proof_nodes": [ 140, 145, 149 ], "lean": "evidence/CV052-Lean.json.gz", "nodes": 151, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-composition-width-transport-v1", "proof": "evidence/CV052-bundle.json.gz", "report_sha256": "545f497cd085b85b35a87a44399f44593c14232d7a02c7369e4a12e5e1554d6a", "root": 150, "statement": "forall ab ac L H J wb wc K. Le(H,J) -> exists pb pc qb qc rb rc sb sc tb tc vb vc db dc eb ec. NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,K) /\\ (NatPowerTrace(ab,ac,L,J,rb,rc,sb,sc,K) /\\ (NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,K) /\\ (NatWeightedSeriesTrace(J,wb,wc,rb,rc,sb,sc,db,dc,eb,ec,K) /\\ ((forall k ub uc fb fc i a. Le(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatPowerRow(rb,rc,sb,sc,k,fb,fc) -> Lt(i,H) -> BetaAt(ub,uc,i,a) -> BetaAt(fb,fc,i,a)) /\\ (forall k ub uc fb fc i a. Le(k,K) -> NatPowerRow(tb,tc,vb,vc,k,ub,uc) -> NatPowerRow(db,dc,eb,ec,k,fb,fc) -> Lt(i,H) -> BetaAt(ub,uc,i,a) -> BetaAt(fb,fc,i,a))))))", "target_ast_sha256": "47daedfb22c29d1e812581a080205cad3a0b0a394afbe6937e03886dcfd6c17e", "title": "composition width transport", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0405", "ND0407", "ND0421" ] }, { "bundle_bytes": 259719, "bundle_sha256": "eb53e85a3f7f6adb863962998a2b23b4c0e83382b1cec2a9e0e971829a777a45", "case": "CV053", "contains_checked_statements": [], "coverage": "constructed_actual_convolution_with_output_zero_after_sum_of_explicit_input_cutoffs_not_power_support_or_analytic_tail_estimate", "ha": "evidence/CV053-HA.json.gz", "internal_proof_nodes": [ 115, 125 ], "lean": "evidence/CV053-Lean.json.gz", "nodes": 127, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-convolution-upper-support-v1", "proof": "evidence/CV053-bundle.json.gz", "report_sha256": "b71f97bc57413ea91b41846267f6001850bac85cc78b9dde76696a88cd4e1a51", "root": 126, "statement": "forall ab ac L bb bc M r s N. NatZeroAfter(ab,ac,L,r) -> NatZeroAfter(bb,bc,M,s) -> exists cb cc. NatConvolutionPrefix(ab,ac,L,bb,bc,M,cb,cc,N) /\\ NatZeroAfter(cb,cc,N,r+s)", "target_ast_sha256": "10fb7e6300ce6b56bf6d231d7f357343629838d1ed95469a0309b42e521c276a", "title": "convolution upper support", "uses_definitions": [ "ND0402", "ND0436" ] }, { "bundle_bytes": 357879, "bundle_sha256": "d80b5d0ed46e168bb40518f1fadabb7efc0c20f30639f2560994f933a91320e3", "case": "CV054", "contains_checked_statements": [ "CV004" ], "coverage": "constructed_actual_power_trace_with_every_row_zero_at_and_above_one_plus_exponent_times_input_length_including_unit_and_empty_cases_not_evaluation_or_analytic_identity", "ha": "evidence/CV054-HA.json.gz", "internal_proof_nodes": [ 122, 139 ], "lean": "evidence/CV054-Lean.json.gz", "nodes": 141, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-power-upper-support-v1", "proof": "evidence/CV054-bundle.json.gz", "report_sha256": "05c33dee3cf3479c76d702c52a3de5db152e0bc84c15fc5ed70ea2e126f93045", "root": 140, "statement": "forall ab ac L H K. exists pb pc qb qc. NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,K) /\\ (forall k ub uc. Le(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatZeroAfter(ub,uc,H,S(k*L)))", "target_ast_sha256": "ef95e6269e88a485d4fd577edb6814ce45c9bf1f61fddfcc6aae499ce488e879", "title": "power upper support", "uses_definitions": [ "PD0001", "ND0405", "ND0407", "ND0436" ] }, { "bundle_bytes": 183156, "bundle_sha256": "6382c26066688b7d2c5cc25eb442a350ec02d7c876667e06e0b32e597dfa04f7", "case": "CV055", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_same_actual_horner_history_for_matching_shorter_input_prefix_and_its_endpoint_not_equality_of_different_length_endpoints", "ha": "evidence/CV055-HA.json.gz", "internal_proof_nodes": [ 106, 73, 107 ], "lean": "evidence/CV055-Lean.json.gz", "nodes": 109, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-prefix-transport-v1", "proof": "evidence/CV055-bundle.json.gz", "report_sha256": "14e5f17477591f607b69fe6cd181f41629c779948ddeee3c14faaf9522192e0d", "root": 108, "statement": "forall ab ac bb bc t M N. Le(M,N) -> (forall i a. Lt(i,M) -> BetaAt(ab,ac,i,a) -> BetaAt(bb,bc,i,a)) -> exists hb hc x. NatScaledHornerTrace(ab,ac,t,N,hb,hc) /\\ (NatScaledHornerTrace(bb,bc,t,M,hb,hc) /\\ BetaAt(hb,hc,M,x))", "target_ast_sha256": "6c091c8fbcbf99ab1176e5414d83f55bd66930b61f484841194d2dec7a45b4a4", "title": "horner prefix transport", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0427" ] }, { "bundle_bytes": 894119, "bundle_sha256": "7835488cb0d4b1045a1989d2979b9f5c6138e5b0fc49ed08155aa1e578decbb2", "case": "CV056", "contains_checked_statements": [ "CV008", "CV009", "CV011", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049" ], "coverage": "constructed_support_complete_shorter_convolution_horner_product_with_actual_power_scaling_and_extra_multiplier_no_cancellation_or_arbitrary_truncation_identity", "ha": "evidence/CV056-HA.json.gz", "internal_proof_nodes": [ 217, 73, 169, 226, 224, 227, 225, 2, 13 ], "lean": "evidence/CV056-Lean.json.gz", "nodes": 229, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supported-horner-product-v1", "proof": "evidence/CV056-bundle.json.gz", "report_sha256": "ae257257fb81660c536d88876d876a87838cde9a320e7ea389d5245d631d5779", "root": 228, "statement": "forall ab ac L bb bc M r s t N q. NatZeroAfter(ab,ac,L,r) -> NatZeroAfter(bb,bc,M,s) -> Le(r+s,N) -> N+q=L+M -> exists cb cc ahb ahc x bhb bhc y chb chc z p. NatConvolutionPrefix(ab,ac,L,bb,bc,M,cb,cc,N) /\\ (NatScaledHornerTrace(ab,ac,t,L,ahb,ahc) /\\ (BetaAt(ahb,ahc,L,x) /\\ (NatScaledHornerTrace(bb,bc,t,M,bhb,bhc) /\\ (BetaAt(bhb,bhc,M,y) /\\ (NatScaledHornerTrace(cb,cc,t,N,chb,chc) /\\ (BetaAt(chb,chc,N,z) /\\ (Pow(t,q,p) /\\ p*z=t*(x*y))))))))", "target_ast_sha256": "635fb0c8045c84285d3fe940d860ef76da985384f4e0dc4f4fbb19c8fe5c1b74", "title": "supported horner product", "uses_definitions": [ "PD0001", "PD0013", "ND0402", "ND0427", "PD0020", "ND0436" ] }, { "bundle_bytes": 930343, "bundle_sha256": "adf11d705133f6bc0082777b36d663241bdbf3204ef1262757ea59157b4bf639", "case": "CV057", "contains_checked_statements": [ "CV008", "CV009", "CV011", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056" ], "coverage": "support_complete_product_for_supplied_actual_convolution_horner_and_power_values_by_extensional_transport_not_code_equality_or_arbitrary_truncation_identity", "ha": "evidence/CV057-HA.json.gz", "internal_proof_nodes": [ 228, 206, 224, 231, 13, 180 ], "lean": "evidence/CV057-Lean.json.gz", "nodes": 233, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supported-horner-values-v1", "proof": "evidence/CV057-bundle.json.gz", "report_sha256": "48f992ca7069d3deec041394217d49fdc8a11113400f0d821586e2d222ace47b", "root": 232, "statement": "forall ab ac L bb bc M cb cc r s t N q ahb ahc bhb bhc chb chc x y z p. NatZeroAfter(ab,ac,L,r) -> NatZeroAfter(bb,bc,M,s) -> Le(r+s,N) -> N+q=L+M -> NatConvolutionPrefix(ab,ac,L,bb,bc,M,cb,cc,N) -> NatScaledHornerTrace(ab,ac,t,L,ahb,ahc) -> BetaAt(ahb,ahc,L,x) -> NatScaledHornerTrace(bb,bc,t,M,bhb,bhc) -> BetaAt(bhb,bhc,M,y) -> NatScaledHornerTrace(cb,cc,t,N,chb,chc) -> BetaAt(chb,chc,N,z) -> Pow(t,q,p) -> p*z=t*(x*y)", "target_ast_sha256": "77e93a327c7aa0811b3c17aa3152b979b62a19aa3ae70c957544ac810f17661a", "title": "supported horner values", "uses_definitions": [ "PD0001", "PD0013", "ND0402", "ND0427", "PD0020", "ND0436" ] }, { "bundle_bytes": 1100998, "bundle_sha256": "fd1e3c3606821577e5c30e547c3fd17f3f7408f9fdb3083854024f19f3b756a1", "case": "CV058", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057" ], "coverage": "constructed_finite_power_trace_with_all_exponent_horner_evaluation_under_explicit_whole_support_width_bound_retaining_unit_and_all_scaling_factors_not_analytic_composition", "ha": "evidence/CV058-HA.json.gz", "internal_proof_nodes": [ 239, 207, 73, 169, 251 ], "lean": "evidence/CV058-Lean.json.gz", "nodes": 253, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-finite-power-evaluation-v2", "proof": "evidence/CV058-bundle.json.gz", "report_sha256": "274367b21554e9b9bcfdb685495a2d6b87006b687013707faf8b3e18d2f20815", "root": 252, "statement": "forall ab ac L t J K. Le(S(K*L),S J) -> exists pb pc qb qc ahb ahc x p v. NatPowerTrace(ab,ac,L,S J,pb,pc,qb,qc,K) /\\ (NatScaledHornerTrace(ab,ac,t,L,ahb,ahc) /\\ (BetaAt(ahb,ahc,L,x) /\\ (Pow(t,L,p) /\\ (Pow(t,J,v) /\\ (forall k ub uc uhb uhc y a b c. Le(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatScaledHornerTrace(ub,uc,t,S J,uhb,uhc) -> BetaAt(uhb,uhc,S J,y) -> Pow(p,k,a) -> Pow(t,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c)))))", "target_ast_sha256": "5106a1fb454a028ea96ed82dcd34c652807316d25e5009ec9a5c5f7191df4689", "title": "finite power evaluation", "uses_definitions": [ "PD0001", "PD0013", "ND0427", "PD0020", "ND0405", "ND0407" ] }, { "bundle_bytes": 214811, "bundle_sha256": "e1a375ba40571463ff4d30a66216e348605613a70967c04b32c9402c9d9af06a", "case": "CV059", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_actual_horner_value_prefix_for_arbitrary_coded_rows_with_all_value_extensionality_not_weighted_accumulation_or_analytic_composition", "ha": "evidence/CV059-HA.json.gz", "internal_proof_nodes": [ 113, 111 ], "lean": "evidence/CV059-Lean.json.gz", "nodes": 115, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-row-horner-values-v1", "proof": "evidence/CV059-bundle.json.gz", "report_sha256": "933df44b0dfe20aa3ff86db6a7fa4c4d82efe76cf602915c4fda6031ca0431c2", "root": 114, "statement": "forall pb pc qb qc t H K. exists vb vc. NatRowHornerPrefix(pb,pc,qb,qc,t,H,vb,vc,K) /\\ (forall k ub uc y. Lt(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatHornerValue(ub,uc,t,H,y) -> BetaAt(vb,vc,k,y))", "target_ast_sha256": "b2b6e620f12b84c498f8ab5190550c6d41534be74a012313e03297548ed9eefc", "title": "row horner values", "uses_definitions": [ "PD0002", "PD0013", "ND0405", "ND0437", "ND0438" ] }, { "bundle_bytes": 364611, "bundle_sha256": "9689e059b662da007e17f7ce588f1b8b57f671eca5beac3f9842410bc00e0fe5", "case": "CV060", "contains_checked_statements": [ "CV019", "CV032", "CV059" ], "coverage": "constructed_actual_weighted_accumulation_row_values_pointwise_products_and_sum_with_every_terminal_row_evaluating_to_that_sum_not_analytic_composition", "ha": "evidence/CV060-HA.json.gz", "internal_proof_nodes": [ 114, 134, 125, 117, 108, 135, 13 ], "lean": "evidence/CV060-Lean.json.gz", "nodes": 137, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-weighted-accumulation-evaluation-v1", "proof": "evidence/CV060-bundle.json.gz", "report_sha256": "84ca307167880412d942eae7a731d35e48cc0b8c6a43c4ff1576acd411f4e5fa", "root": 136, "statement": "forall H t wb wc pb pc qb qc K. exists rb rc sb sc vb vc zb zc s. NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,K) /\\ (NatRowHornerPrefix(pb,pc,qb,qc,t,H,vb,vc,K) /\\ (NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,K) /\\ (Sum(zb,zc,K,s) /\\ (forall ub uc. NatPowerRow(rb,rc,sb,sc,K,ub,uc) -> NatHornerValue(ub,uc,t,H,s)))))", "target_ast_sha256": "f1e9ea342059786ba3af115a2f4d0613913410b92f8bc7dbd138144349c3b7b4", "title": "weighted accumulation evaluation", "uses_definitions": [ "PD0015", "ND0405", "ND0437", "ND0438", "ND0421", "ND0439" ] }, { "bundle_bytes": 1315237, "bundle_sha256": "101556adbd46ec693ecb3f630be9b7a5471bb14b5135ef20121d61b54b2161dd", "case": "CV061", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV019", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060" ], "coverage": "same_witness_whole_support_power_trace_weighted_accumulation_row_values_products_and_sum_with_exact_denominator_cleared_monomial_law_not_analytic_exponential_identity", "ha": "evidence/CV061-HA.json.gz", "internal_proof_nodes": [ 252, 273, 274 ], "lean": "evidence/CV061-Lean.json.gz", "nodes": 276, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-full-composition-evaluation-v1", "proof": "evidence/CV061-bundle.json.gz", "report_sha256": "b869e50e408dc3482d23e4212ce60efbb1294078766fe87b0dd5df9e8cd38004", "root": 275, "statement": "forall ab ac L t J K wb wc. Le(S(K*L),S J) -> exists pb pc qb qc ahb ahc x p v rb rc sb sc vb vc zb zc s. (NatPowerTrace(ab,ac,L,S J,pb,pc,qb,qc,K)) /\\ ((NatScaledHornerTrace(ab,ac,t,L,ahb,ahc)) /\\ ((BetaAt(ahb,ahc,L,x)) /\\ ((Pow(t,L,p)) /\\ ((Pow(t,J,v)) /\\ ((NatWeightedSeriesTrace(S J,wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,K)) /\\ ((NatRowHornerPrefix(pb,pc,qb,qc,t,S J,vb,vc,K)) /\\ ((NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,K)) /\\ ((Sum(zb,zc,K,s)) /\\ ((forall ub uc. NatPowerRow(rb,rc,sb,sc,K,ub,uc) -> NatHornerValue(ub,uc,t,S J,s)) /\\ (forall k y a b c. Lt(k,K) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(t,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c))))))))))", "target_ast_sha256": "574be3227d697f0533ee581f7baa664dcfca27230214e8992f1bab97fe4e3aaf", "title": "full composition evaluation", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0015", "ND0427", "PD0020", "ND0405", "ND0407", "ND0437", "ND0438", "ND0421", "ND0439" ] }, { "bundle_bytes": 256057, "bundle_sha256": "bfe45da726ddbbc6db9574b817d48471d27286387f9a56505fde8f0e6598baf9", "case": "CV062", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_actual_shifted_tail_and_prefix_tail_whole_horner_values_with_exact_power_scaled_splitting_including_zero_empty_cases_not_a_tail_estimate", "ha": "evidence/CV062-HA.json.gz", "internal_proof_nodes": [ 104, 120, 73, 113, 121, 2, 13, 122 ], "lean": "evidence/CV062-Lean.json.gz", "nodes": 124, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-splitting-v1", "proof": "evidence/CV062-bundle.json.gz", "report_sha256": "521ad56ec55b1d986acec99646e48a3fafa8e12f5361fc949cca2c2c9a03a394", "root": 123, "statement": "forall ab ac t M N. exists bb bc x y z p. NatAffineSlice(ab,ac,M,1,bb,bc,N) /\\ (NatHornerValue(ab,ac,t,M,x) /\\ (NatHornerValue(bb,bc,t,N,y) /\\ (NatHornerValue(ab,ac,t,M+N,z) /\\ (Pow(t,N,p) /\\ z=p*x+y))))", "target_ast_sha256": "9cde54938675644bbc152c11b3e0270bf4ac303df68a8275b2e5ab6341e8f467", "title": "horner splitting", "uses_definitions": [ "ND0411", "PD0020", "ND0437" ] }, { "bundle_bytes": 259517, "bundle_sha256": "de7052c68bfe61d50e1ecc04b1833a5040b0f84ed2385c881e5666c954485c4f", "case": "CV063", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_actual_horner_values_and_powers_for_ordered_bases_with_power_order_and_exact_length_minus_one_scaled_comparison_including_zero_bases_not_an_analytic_limit", "ha": "evidence/CV063-HA.json.gz", "internal_proof_nodes": [ 118, 73, 111, 123, 13 ], "lean": "evidence/CV063-Lean.json.gz", "nodes": 125, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-base-comparison-v1", "proof": "evidence/CV063-bundle.json.gz", "report_sha256": "69bad58189f80d547585a87bcd552c9f016c885ccba4c360101f770f408b0a7f", "root": 124, "statement": "forall ab ac a b r. Le(a,b) -> exists p q x y. NatHornerValue(ab,ac,b,S r,x) /\\ (NatHornerValue(ab,ac,a,S r,y) /\\ (Pow(a,r,p) /\\ (Pow(b,r,q) /\\ (Le(p,q) /\\ Le(p*x,q*y)))))", "target_ast_sha256": "9427a92cb26d4fcfea9d3160585b295d542e0c343f466f0326c7f8acbf25bbe0", "title": "horner base comparison", "uses_definitions": [ "PD0001", "PD0020", "ND0437" ] }, { "bundle_bytes": 305518, "bundle_sha256": "19b994fd3a6014407807961c23e963367e51814be7a6231a0e0fd337f8eb7d0f", "case": "CV064", "contains_checked_statements": [ "CV032", "CV062", "CV063" ], "coverage": "constructed_genuine_tail_and_whole_comparison_values_with_exact_integer_scaled_tail_majorant_for_ordered_bases_not_rational_division_transport_or_analytic_limit", "ha": "evidence/CV064-HA.json.gz", "internal_proof_nodes": [ 123, 129, 130, 126, 14 ], "lean": "evidence/CV064-Lean.json.gz", "nodes": 132, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-horner-tail-majorant-v1", "proof": "evidence/CV064-bundle.json.gz", "report_sha256": "e9874241cce54c98023c33caa4184dd60711daeec997cc836dbc56ac397ce453", "root": 131, "statement": "forall ab ac a b N r. Le(a,b) -> exists tb tc u v p q. NatAffineSlice(ab,ac,N,1,tb,tc,S r) /\\ (NatHornerValue(tb,tc,b,S r,u) /\\ (NatHornerValue(ab,ac,a,N+S r,v) /\\ (Pow(a,r,p) /\\ (Pow(b,r,q) /\\ Le(p*u,q*v)))))", "target_ast_sha256": "adec9c34b4e9fd8be3a611ebe168b24c3cfbd3ca6d6b96568180e2849705609f", "title": "horner tail majorant", "uses_definitions": [ "PD0001", "ND0411", "PD0020", "ND0437" ] }, { "bundle_bytes": 240999, "bundle_sha256": "015481deac61973d996c1014b3c0c9f76bbfd20614546338bc6b86a4be7a41b8", "case": "CV065", "contains_checked_statements": [ "CV032" ], "coverage": "constructed_horner_value_and_actual_power_with_integer_majorant_from_separate_bounded_coefficients_and_zero_head_including_zero_bound_not_an_analytic_limit", "ha": "evidence/CV065-HA.json.gz", "internal_proof_nodes": [ 118, 73, 111, 13, 123 ], "lean": "evidence/CV065-Lean.json.gz", "nodes": 125, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-bounded-horner-v1", "proof": "evidence/CV065-bundle.json.gz", "report_sha256": "d0c45e627760f03b6ce7eb19a97397a7cc497f99dadec1108397489f07b175c1", "root": 124, "statement": "forall ab ac D k. BetaAt(ab,ac,0,0) -> NatBoundedPrefix(ab,ac,D,S k) -> exists y p. NatHornerValue(ab,ac,2,S k,y) /\\ (Pow(2,k,p) /\\ Le(y+D,D*p))", "target_ast_sha256": "9014d749012734cecddb6aba75ab714b63794d50d28aef611ecc1a9018f2c091", "title": "bounded horner", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0437", "ND0440" ] }, { "bundle_bytes": 271081, "bundle_sha256": "775e8f82cf01abe8eecd1022c91ab0a44028d1b97d0beab802cb5f75c272f81e", "case": "CV066", "contains_checked_statements": [ "CV025", "CV032", "CV065" ], "coverage": "same_actual_atanh_coefficient_table_nonzero_denominator_zero_head_and_entry_bounds_with_horner_value_power_and_integer_majorant_not_logarithm_or_rational_division_transport", "ha": "evidence/CV066-HA.json.gz", "internal_proof_nodes": [ 129, 134, 133, 132 ], "lean": "evidence/CV066-Lean.json.gz", "nodes": 136, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-majorant-v1", "proof": "evidence/CV066-bundle.json.gz", "report_sha256": "19c91e05ad935db4b326fcc667cb2947dac3d75f66e0995d415189c9d46d976d", "root": 135, "statement": "forall k. exists d ab ac y p. ~d=0 /\\ (NatAtanhPrefix(d,ab,ac,S k) /\\ (NatZeroBefore(ab,ac,S k,1) /\\ (BetaAt(ab,ac,0,0) /\\ (NatBoundedPrefix(ab,ac,2*d,S k) /\\ (NatHornerValue(ab,ac,2,S k,y) /\\ (Pow(2,k,p) /\\ Le(y+2*d,(2*d)*p)))))))", "target_ast_sha256": "b21b3f67a0aa28eb1456681cada027f47d5323c8f0b7e3c66defa64552756e5d", "title": "atanh majorant", "uses_definitions": [ "PD0001", "PD0013", "PD0020", "ND0437", "ND0440", "ND0408", "ND0425" ] }, { "bundle_bytes": 246822, "bundle_sha256": "f99f5b94eb22287763f6a6bb48442810c51672f1c13147237cd61c8310601252", "case": "CV067", "contains_checked_statements": [], "coverage": "constructed_actual_factorial_k_plus_two_and_power_three_to_k_with_twice_power_lower_bound_not_yet_a_finite_exponential_sum_bound", "ha": "evidence/CV067-HA.json.gz", "internal_proof_nodes": [ 111, 117, 123 ], "lean": "evidence/CV067-Lean.json.gz", "nodes": 125, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-factorial-majorant-v1", "proof": "evidence/CV067-bundle.json.gz", "report_sha256": "eb1c2aa38054c1a6b666115a6f4f754183a4224bc3745411d06cdd65dc1e6a73", "root": 124, "statement": "forall k. exists f p. Factorial(S(S k),f) /\\ (Pow(3,k,p) /\\ Le(2*p,f))", "target_ast_sha256": "906291d44d802e2ac53f432d1d7ae6b05c1920dda811c18335f095def75f9293", "title": "factorial majorant", "uses_definitions": [ "PD0001", "PD0020", "PD0023" ] }, { "bundle_bytes": 188250, "bundle_sha256": "88d9a3d159a222530ecfb17ba3316739ad965970ac80dd427f3bbd98c8e94638", "case": "CV068", "contains_checked_statements": [], "coverage": "constructed_actual_finite_sum_bounded_by_ratio_head_with_empty_prefix_and_zero_ratio_cases_from_explicit_separate_adjacent_entry_bounds_not_an_analytic_limit", "ha": "evidence/CV068-HA.json.gz", "internal_proof_nodes": [ 104, 110 ], "lean": "evidence/CV068-Lean.json.gz", "nodes": 112, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-ratio-sum-bound-v1", "proof": "evidence/CV068-bundle.json.gz", "report_sha256": "677e16565bce11eaf9ef1d0fb6d970abcccec26df917cb3c9ad9a3dc048e8d4d", "root": 111, "statement": "forall ab ac r N a0. NatRatioBoundPrefix(ab,ac,r,N) -> BetaAt(ab,ac,0,a0) -> exists s. Sum(ab,ac,N,s) /\\ Le(s,(S r)*a0)", "target_ast_sha256": "fd9dbb392a8f45725f090de43a5e13f1e269a2e0b79293eeef87a1464dcd8f28", "title": "ratio sum bound", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0441" ] }, { "bundle_bytes": 219969, "bundle_sha256": "78c022fee7071b3379f26cdce3ad991097813fb284760516d9a31b50bd1d4c61", "case": "CV069", "contains_checked_statements": [], "coverage": "constructed_actual_weight_times_actual_power_prefix_and_same_prefix_sum_for_arbitrary_weights_including_zero_base_and_empty_prefix_not_factorial_normalization_or_analytic_bound", "ha": "evidence/CV069-HA.json.gz", "internal_proof_nodes": [ 119, 105 ], "lean": "evidence/CV069-Lean.json.gz", "nodes": 121, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-power-weighted-prefix-v1", "proof": "evidence/CV069-bundle.json.gz", "report_sha256": "c717bcfce7414da105f9ac8bf540dff0dcbd9c5d073cc30bf82ba381de4d7dc9", "root": 120, "statement": "forall x wb wc K. exists tb tc s. NatPowerWeightedPrefix(x,wb,wc,tb,tc,K) /\\ Sum(tb,tc,K,s)", "target_ast_sha256": "ed362b7903fc9933f4dc5185589dd926c667bfdd97db615b673296a1e094fd9c", "title": "power weighted prefix", "uses_definitions": [ "PD0015", "ND0442" ] }, { "bundle_bytes": 381155, "bundle_sha256": "2be0d10e36a67786ed5fd20800f6a2d52c10229fea219ea68edd70e832a8b075", "case": "CV070", "contains_checked_statements": [ "CV022", "CV030", "CV069" ], "coverage": "same_actual_factorial_weights_and_power_terms_with_nonzero_initial_value_exact_scaled_exponential_recurrence_and_factorial_normalization_not_finite_exp_bound_or_analytic_limit", "ha": "evidence/CV070-HA.json.gz", "internal_proof_nodes": [ 139, 120, 140, 141, 142, 23 ], "lean": "evidence/CV070-Lean.json.gz", "nodes": 144, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-scaled-exp-terms-v1", "proof": "evidence/CV070-bundle.json.gz", "report_sha256": "b98bd3cdc3dea62a15e3ef1f7471b5fff3765f27fa3443fe90fec64578ef5361", "root": 143, "statement": "forall d x N. ~d=0 -> exists wb wc tb tc w0 f p s. NatExpWeightTable(d,N,wb,wc) /\\ (NatPowerWeightedPrefix(x,wb,wc,tb,tc,S N) /\\ (BetaAt(wb,wc,0,w0) /\\ (BetaAt(tb,tc,0,w0) /\\ (~w0=0 /\\ (Factorial(N,f) /\\ (Pow(d,N,p) /\\ (w0=f*p /\\ (Sum(tb,tc,S N,s) /\\ (NatScaledExpRecurrence(d,x,tb,tc,N) /\\ (forall j t f0 p0 q. Le(j,N) -> BetaAt(tb,tc,j,t) -> Factorial(j,f0) -> Pow(d,j,p0) -> Pow(x,j,q) -> (t*f0)*p0=w0*q))))))))))", "target_ast_sha256": "7f9e9c6aa4b255463768b661ebe9736b133dd810525dfd27ff94dfbe3693bf1d", "title": "scaled exp terms", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0442", "ND0443" ] }, { "bundle_bytes": 462377, "bundle_sha256": "4d7e1dad4315320f229e167f800afa3ac7c13c3e5b338638e81bc06fb259ec32", "case": "CV071", "contains_checked_statements": [ "CV022", "CV030", "CV069", "CV070" ], "coverage": "all_actual_scaled_exponential_term_data_retained_with_finite_sum_at_most_nine_times_nonzero_initial_weight_under_x_le_twice_nonzero_denominator_not_rational_division_or_analytic_limit", "ha": "evidence/CV071-HA.json.gz", "internal_proof_nodes": [ 143, 161 ], "lean": "evidence/CV071-Lean.json.gz", "nodes": 163, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-scaled-exp-bound-v1", "proof": "evidence/CV071-bundle.json.gz", "report_sha256": "0f766852bcea8bb332ad51198a55142b445de48bb42a8ab992202de775fe15c5", "root": 162, "statement": "forall d x N. ~d=0 -> Le(x,2*d) -> exists wb wc tb tc w0 f p s. NatExpWeightTable(d,N,wb,wc) /\\ (NatPowerWeightedPrefix(x,wb,wc,tb,tc,S N) /\\ (BetaAt(wb,wc,0,w0) /\\ (BetaAt(tb,tc,0,w0) /\\ (~w0=0 /\\ (Factorial(N,f) /\\ (Pow(d,N,p) /\\ (w0=f*p /\\ (Sum(tb,tc,S N,s) /\\ (NatScaledExpRecurrence(d,x,tb,tc,N) /\\ ((forall j t f0 p0 q. Le(j,N) -> BetaAt(tb,tc,j,t) -> Factorial(j,f0) -> Pow(d,j,p0) -> Pow(x,j,q) -> (t*f0)*p0=w0*q) /\\ (Le(s,9*w0))))))))))))", "target_ast_sha256": "042906aaf5e07838dfb7be22391a4c26d6395af6e8d451e8c4322563fb810e26", "title": "scaled exp bound", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0442", "ND0443" ] }, { "bundle_bytes": 237052, "bundle_sha256": "8dbc3b54ee1dc9cca4b6031ae181390bd9d73fde574f8e0b26207e7f9615710f", "case": "CV072", "contains_checked_statements": [], "coverage": "exact_supplied_composition_monomial_law_implies_actual_head_and_adjacent_row_recurrence_with_nonzero_scale_and_actual_power_cancellation_not_composition_construction_or_analytic_limit", "ha": "evidence/CV072-HA.json.gz", "internal_proof_nodes": [ 116, 117 ], "lean": "evidence/CV072-Lean.json.gz", "nodes": 119, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-monomial-recurrence-v1", "proof": "evidence/CV072-bundle.json.gz", "report_sha256": "91097d7e17370fd750030d65dfeb15801e3216edecd6104b2136aac735be41d4", "root": 118, "statement": "forall p t x v K vb vc. ~p=0 -> (forall k y a b c. Lt(k,K) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(t,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c) -> (Lt(0,K) -> BetaAt(vb,vc,0,v)) /\\ (forall k y z. Lt(S k,K) -> BetaAt(vb,vc,k,y) -> BetaAt(vb,vc,S k,z) -> p*z=(t*x)*y)", "target_ast_sha256": "1cac9f24c8b7573372bdfab838b9e0b9248a42d1681fde0d65e7da2768df79c8", "title": "monomial recurrence", "uses_definitions": [ "PD0002", "PD0013", "PD0020" ] }, { "bundle_bytes": 261816, "bundle_sha256": "3365ad973b40bee6ffab11db7b582e17939db61a4768fd46c46adb6836b31d10", "case": "CV073", "contains_checked_statements": [ "CV072" ], "coverage": "same_supplied_factorial_weights_monomial_values_and_pointwise_products_obey_scaled_exponential_recurrence_at_denominator_d_times_p_and_numerator_t_times_x_with_head_w0_times_v_not_yet_constructed_composition_bound", "ha": "evidence/CV073-HA.json.gz", "internal_proof_nodes": [ 118, 119, 120, 23, 29 ], "lean": "evidence/CV073-Lean.json.gz", "nodes": 122, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-weighted-monomial-recurrence-v1", "proof": "evidence/CV073-bundle.json.gz", "report_sha256": "ad5d2926122e9c49295c9f5c0c5e798f83fbc267c1dd4dfb8676648d5509394a", "root": 121, "statement": "forall d p t x v N wb wc vb vc zb zc w0. ~p=0 -> (forall k y a b c. Lt(k,S N) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(t,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c) -> NatExpWeightTable(d,N,wb,wc) -> NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,S N) -> BetaAt(wb,wc,0,w0) -> BetaAt(zb,zc,0,w0*v) /\\ NatScaledExpRecurrence(d*p,t*x,zb,zc,N)", "target_ast_sha256": "e23d9c7bce72b64cec08978482e066d449fff7b4e18c3a1cfc5bc8dc55f9d6e5", "title": "weighted monomial recurrence", "uses_definitions": [ "PD0002", "PD0013", "ND0423", "PD0020", "ND0439", "ND0443" ] }, { "bundle_bytes": 1627627, "bundle_sha256": "1ec6611c5e094c392f09d2203b46e4ce9380c95ad0ab421a8c9f3d7c600aa221", "case": "CV074", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV019", "CV022", "CV025", "CV030", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV065", "CV066", "CV072", "CV073" ], "coverage": "constructed_same_atanh_table_factorial_weights_whole_support_composition_rows_products_and_sum_at_base_two_with_nonzero_normalization_and_sum_at_most_nine_times_w0_times_two_to_J_not_rational_division_or_analytic_limit", "ha": "evidence/CV074-HA.json.gz", "internal_proof_nodes": [ 290, 302, 275, 322, 315, 254, 323, 316, 45, 12, 23, 13 ], "lean": "evidence/CV074-Lean.json.gz", "nodes": 325, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-composition-majorant-v1", "proof": "evidence/CV074-bundle.json.gz", "report_sha256": "4b363664f108e036cf73b255e9c948d5b2d4e83d33d08699a7cefa6b0b393b60", "root": 324, "statement": "forall k N. exists d ab ac q wb wc w0 f dn pb pc qb qc ahb ahc x p v rb rc sb sc vb vc zb zc s. (~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S k)) /\\ ((NatZeroBefore(ab,ac,S k,1)) /\\ ((BetaAt(ab,ac,0,0)) /\\ ((NatBoundedPrefix(ab,ac,2*d,S k)) /\\ ((NatHornerValue(ab,ac,2,S k,x)) /\\ ((Pow(2,k,q)) /\\ ((Le(x+2*d,(2*d)*q)) /\\ ((NatExpWeightTable(d,N,wb,wc)) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ ((BetaAt(wb,wc,0,w0)) /\\ ((Factorial(N,f)) /\\ ((Pow(d,N,dn)) /\\ ((w0=f*dn) /\\ ((forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=w0) /\\ ((~w0=0) /\\ ((NatPowerTrace(ab,ac,S k,S ((S N)*(S k)),pb,pc,qb,qc,S N)) /\\ ((NatScaledHornerTrace(ab,ac,2,S k,ahb,ahc)) /\\ ((BetaAt(ahb,ahc,S k,x)) /\\ ((Pow(2,S k,p)) /\\ ((Pow(2,((S N)*(S k)),v)) /\\ ((NatWeightedSeriesTrace(S ((S N)*(S k)),wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,S N)) /\\ ((NatRowHornerPrefix(pb,pc,qb,qc,2,S ((S N)*(S k)),vb,vc,S N)) /\\ ((NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,S N)) /\\ ((Sum(zb,zc,S N,s)) /\\ ((forall ub uc. NatPowerRow(rb,rc,sb,sc,S N,ub,uc) -> NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\\ ((forall k y a b c. Lt(k,S N) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(2,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c) /\\ ((~p=0) /\\ ((~v=0) /\\ ((~(d*p)=0) /\\ ((~(w0*v)=0) /\\ ((p=q*2) /\\ ((Le(x,d*p)) /\\ ((Le(2*x,2*(d*p))) /\\ ((BetaAt(zb,zc,0,w0*v)) /\\ ((NatScaledExpRecurrence(d*p,2*x,zb,zc,N)) /\\ (Le(s,9*(w0*v))))))))))))))))))))))))))))))))))))))", "target_ast_sha256": "0a0ef5814cf4d9655010b47abf33ab8cbcb9f86f7d6f336cafcc1a6d7556273b", "title": "composition majorant", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0015", "ND0423", "ND0427", "PD0020", "PD0023", "ND0405", "ND0407", "ND0437", "ND0438", "ND0421", "ND0439", "ND0440", "ND0408", "ND0425", "ND0443" ] }, { "bundle_bytes": 389010, "bundle_sha256": "5e9943d53ae57de27108eb1c6c31783198b59e581ba08365f20e368274d9bbda", "case": "CV075", "contains_checked_statements": [ "CV032", "CV062", "CV063", "CV064" ], "coverage": "constructed_actual_nonempty_tail_and_base_three_value_with_actual_powers_nonzero_denominators_and_valid_rational_bound_B_times_two_over_three_to_M_from_supplied_whole_table_majorant_not_analytic_limit_or_empty_tail_case", "ha": "evidence/CV075-HA.json.gz", "internal_proof_nodes": [ 137, 130, 113, 122, 140, 45, 12, 13, 31, 142, 141 ], "lean": "evidence/CV075-Lean.json.gz", "nodes": 144, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rational-tail-majorant-v1", "proof": "evidence/CV075-bundle.json.gz", "report_sha256": "2513015677480e90acd8e7830f91ef1ff7c04811d9aade54c12a1e12e5b8f200", "root": 143, "statement": "forall ab ac D B M r y v. ~D=0 -> NatHornerValue(ab,ac,2,S(M+r),y) -> Pow(2,M+r,v) -> Le(y,(B*D)*v) -> exists tb tc u p q h. (NatAffineSlice(ab,ac,M,1,tb,tc,S r)) /\\ ((NatHornerValue(tb,tc,3,S r,u)) /\\ ((Pow(2,M,p)) /\\ ((Pow(3,M,q)) /\\ ((Pow(3,M+r,h)) /\\ ((~p=0) /\\ ((~q=0) /\\ ((~h=0) /\\ ((~(D*h)=0) /\\ ((Le(u*q,(B*p)*(D*h))) /\\ (IRatLe(u,0,D*h,B*p,0,q)))))))))))", "target_ast_sha256": "afeaa7ea69229597287202c3420e527d2b96f2c2bbd77a455ad620d128c7edff", "title": "rational tail majorant", "uses_definitions": [ "PD0001", "ND0411", "PD0020", "ND0437", "ND0444" ] }, { "bundle_bytes": 1824130, "bundle_sha256": "cf5c84b108e2db80fcaf846e709839e44402011c309e596c4b9c90c74560289e", "case": "CV076", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV019", "CV022", "CV025", "CV030", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV066", "CV072", "CV073", "CV074", "CV075" ], "coverage": "all_actual_composition_majorant_data_retained_with_one_decoded_terminal_row_and_every_in_range_actual_tail_certified_below_nine_times_two_over_three_to_cutoff_in_valid_rational_order_not_analytic_limit", "ha": "evidence/CV076-HA.json.gz", "internal_proof_nodes": [ 324, 235, 334 ], "lean": "evidence/CV076-Lean.json.gz", "nodes": 336, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-composition-tail-bound-v1", "proof": "evidence/CV076-bundle.json.gz", "report_sha256": "ce0a9c09ceb26c8fc6d0dabb323f8db1b97b7bea8673a2a3e2337917a74028ac", "root": 335, "statement": "forall k N. exists d ab ac q wb wc w0 f dn pb pc qb qc ahb ahc x p v rb rc sb sc vb vc zb zc s ub uc. (~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,S k)) /\\ ((NatZeroBefore(ab,ac,S k,1)) /\\ ((BetaAt(ab,ac,0,0)) /\\ ((NatBoundedPrefix(ab,ac,2*d,S k)) /\\ ((NatHornerValue(ab,ac,2,S k,x)) /\\ ((Pow(2,k,q)) /\\ ((Le(x+2*d,(2*d)*q)) /\\ ((NatExpWeightTable(d,N,wb,wc)) /\\ ((forall j a. Le(j,N) -> BetaAt(wb,wc,j,a) -> ~d=0 -> ~a=0) /\\ ((BetaAt(wb,wc,0,w0)) /\\ ((Factorial(N,f)) /\\ ((Pow(d,N,dn)) /\\ ((w0=f*dn) /\\ ((forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=w0) /\\ ((~w0=0) /\\ ((NatPowerTrace(ab,ac,S k,S ((S N)*(S k)),pb,pc,qb,qc,S N)) /\\ ((NatScaledHornerTrace(ab,ac,2,S k,ahb,ahc)) /\\ ((BetaAt(ahb,ahc,S k,x)) /\\ ((Pow(2,S k,p)) /\\ ((Pow(2,((S N)*(S k)),v)) /\\ ((NatWeightedSeriesTrace(S ((S N)*(S k)),wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,S N)) /\\ ((NatRowHornerPrefix(pb,pc,qb,qc,2,S ((S N)*(S k)),vb,vc,S N)) /\\ ((NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,S N)) /\\ ((Sum(zb,zc,S N,s)) /\\ ((forall ub uc. NatPowerRow(rb,rc,sb,sc,S N,ub,uc) -> NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\\ ((forall k y a b c. Lt(k,S N) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(2,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c) /\\ ((~p=0) /\\ ((~v=0) /\\ ((~(d*p)=0) /\\ ((~(w0*v)=0) /\\ ((p=q*2) /\\ ((Le(x,d*p)) /\\ ((Le(2*x,2*(d*p))) /\\ ((BetaAt(zb,zc,0,w0*v)) /\\ ((NatScaledExpRecurrence(d*p,2*x,zb,zc,N)) /\\ ((Le(s,9*(w0*v))) /\\ ((NatPowerRow(rb,rc,sb,sc,S N,ub,uc)) /\\ ((NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\\ (forall M. Le(M,((S N)*(S k))) -> exists r tb tc u p0 q0 h. (((S N)*(S k))=M+r) /\\ ((NatAffineSlice(ub,uc,M,1,tb,tc,S r)) /\\ ((NatHornerValue(tb,tc,3,S r,u)) /\\ ((Pow(2,M,p0)) /\\ ((Pow(3,M,q0)) /\\ ((Pow(3,((S N)*(S k)),h)) /\\ ((~p0=0) /\\ ((~q0=0) /\\ ((~h=0) /\\ ((~(w0*h)=0) /\\ ((Le(u*q0,(9*p0)*(w0*h))) /\\ (IRatLe(u,0,w0*h,9*p0,0,q0)))))))))))))))))))))))))))))))))))))))))))))))))))", "target_ast_sha256": "2baa8df68b336dc9d80d78e7f1d9797fdeb2859efc39377e00c1c0a18de9f83f", "title": "composition tail bound", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0015", "ND0411", "ND0423", "ND0427", "PD0020", "PD0023", "ND0405", "ND0407", "ND0437", "ND0438", "ND0421", "ND0439", "ND0440", "ND0408", "ND0425", "ND0443", "ND0444" ] }, { "bundle_bytes": 1661719, "bundle_sha256": "8b53d2ca3cedcb2b91b3752eb9b01fea4860ef11429582dc25a3b8a9efd78a41", "case": "CV077", "contains_checked_statements": [ "CV004", "CV008", "CV009", "CV011", "CV019", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV072", "CV073", "CV075" ], "coverage": "supplied_atanh_and_weight_codes_preserved_in_constructed_full_composition_majorant_and_every_in_range_actual_rational_tail_with_supplied_nonzero_head_not_table_identity_assumption_or_analytic_limit", "ha": "evidence/CV077-HA.json.gz", "internal_proof_nodes": [ 284, 305, 275, 304, 297, 315, 254, 235, 298, 45, 12, 13 ], "lean": "evidence/CV077-Lean.json.gz", "nodes": 317, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supplied-composition-bound-v1", "proof": "evidence/CV077-bundle.json.gz", "report_sha256": "1a8af331a6ca8bc7b7c7a4fab827d5b466b89ebe94a4cb54d30af4642efbe1ca", "root": 316, "statement": "forall d ab ac k N wb wc w0. ~d=0 -> NatAtanhPrefix(d,ab,ac,S k) -> NatExpWeightTable(d,N,wb,wc) -> BetaAt(wb,wc,0,w0) -> ~w0=0 -> exists q pb pc qb qc ahb ahc x p v rb rc sb sc vb vc zb zc s ub uc. (NatPowerTrace(ab,ac,S k,S ((S N)*(S k)),pb,pc,qb,qc,S N)) /\\ ((NatScaledHornerTrace(ab,ac,2,S k,ahb,ahc)) /\\ ((BetaAt(ahb,ahc,S k,x)) /\\ ((Pow(2,S k,p)) /\\ ((Pow(2,((S N)*(S k)),v)) /\\ ((NatWeightedSeriesTrace(S ((S N)*(S k)),wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,S N)) /\\ ((NatRowHornerPrefix(pb,pc,qb,qc,2,S ((S N)*(S k)),vb,vc,S N)) /\\ ((NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,S N)) /\\ ((Sum(zb,zc,S N,s)) /\\ ((forall ub uc. NatPowerRow(rb,rc,sb,sc,S N,ub,uc) -> NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\\ ((forall k y a b c. Lt(k,S N) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(2,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c) /\\ ((NatHornerValue(ab,ac,2,S k,x)) /\\ ((Pow(2,k,q)) /\\ ((Le(x+2*d,(2*d)*q)) /\\ ((p=q*2) /\\ ((~p=0) /\\ ((~v=0) /\\ ((~(d*p)=0) /\\ ((~(w0*v)=0) /\\ ((BetaAt(zb,zc,0,w0*v)) /\\ ((NatScaledExpRecurrence(d*p,2*x,zb,zc,N)) /\\ ((Le(s,9*(w0*v))) /\\ ((NatPowerRow(rb,rc,sb,sc,S N,ub,uc)) /\\ ((NatHornerValue(ub,uc,2,S ((S N)*(S k)),s)) /\\ (forall M. Le(M,((S N)*(S k))) -> exists r tb tc u p0 q0 h. (((S N)*(S k))=M+r) /\\ ((NatAffineSlice(ub,uc,M,1,tb,tc,S r)) /\\ ((NatHornerValue(tb,tc,3,S r,u)) /\\ ((Pow(2,M,p0)) /\\ ((Pow(3,M,q0)) /\\ ((Pow(3,((S N)*(S k)),h)) /\\ ((~p0=0) /\\ ((~q0=0) /\\ ((~h=0) /\\ ((~(w0*h)=0) /\\ ((Le(u*q0,(9*p0)*(w0*h))) /\\ (IRatLe(u,0,w0*h,9*p0,0,q0))))))))))))))))))))))))))))))))))))", "target_ast_sha256": "e21e4c6b538bc01cb88d592709a53809b0e7e69f1048918f9661e0965cd5e421", "title": "supplied composition bound", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0015", "ND0411", "ND0423", "ND0427", "PD0020", "ND0405", "ND0407", "ND0437", "ND0438", "ND0421", "ND0439", "ND0425", "ND0443", "ND0444" ] }, { "bundle_bytes": 85206, "bundle_sha256": "040652ba1ae46390151d5fd1023a97f06f598cf6976a6ff1faf81d528a44d7f4", "case": "CV078", "contains_checked_statements": [], "coverage": "supplied_weighted_accumulation_rows_through_K_preserve_low_coefficients_from_agreement_only_of_consumed_source_rows_below_K_with_same_weights_and_ordered_widths_not_code_equality", "ha": "evidence/CV078-HA.json.gz", "internal_proof_nodes": [ 10, 16, 11, 23, 24, 25, 26, 27, 28 ], "lean": "evidence/CV078-Lean.json.gz", "nodes": 30, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-consumed-row-transport-v1", "proof": "evidence/CV078-bundle.json.gz", "report_sha256": "d440d4daba9d5b46fac158fa381a40b5493fe9576857bebdb68fb76f1d1f9999", "root": 29, "statement": "forall H J wb wc pb pc qb qc rb rc sb sc tb tc vb vc db dc eb ec K. Le(H,J) -> (forall k ub uc fb fc i a. Lt(k,K) -> NatPowerRow(pb,pc,qb,qc,k,ub,uc) -> NatPowerRow(rb,rc,sb,sc,k,fb,fc) -> Lt(i,H) -> BetaAt(ub,uc,i,a) -> BetaAt(fb,fc,i,a)) -> NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,K) -> NatWeightedSeriesTrace(J,wb,wc,rb,rc,sb,sc,db,dc,eb,ec,K) -> forall k ub uc fb fc i a. Le(k,K) -> NatPowerRow(tb,tc,vb,vc,k,ub,uc) -> NatPowerRow(db,dc,eb,ec,k,fb,fc) -> Lt(i,H) -> BetaAt(ub,uc,i,a) -> BetaAt(fb,fc,i,a)", "target_ast_sha256": "2f33ca6e896147c1f9c2b66a154e0be06e18e77fd718ba4123744b81650ed78d", "title": "consumed row transport", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0405", "ND0421" ] }, { "bundle_bytes": 262829, "bundle_sha256": "84fb9a7149a02574f575743f16e812568cf10e8c19c03e451a785ffeac535d55", "case": "CV079", "contains_checked_statements": [ "CV078" ], "coverage": "supplied_power_depths_N_and_N_plus_one_transport_all_N_plus_one_weighted_accumulation_rows_by_explicit_large_trace_restriction_and_consumed_row_agreement_with_same_input_and_weights", "ha": "evidence/CV079-HA.json.gz", "internal_proof_nodes": [ 53, 51 ], "lean": "evidence/CV079-Lean.json.gz", "nodes": 55, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-successor-composition-transport-v1", "proof": "evidence/CV079-bundle.json.gz", "report_sha256": "da802ffff22f372f458be550737d41d1c095d7c7d002ab5314402f354b6465f3", "root": 54, "statement": "forall ab ac L H J wb wc pb pc qb qc rb rc sb sc tb tc vb vc db dc eb ec N. Le(H,J) -> NatPowerTrace(ab,ac,L,H,pb,pc,qb,qc,N) -> NatPowerTrace(ab,ac,L,J,rb,rc,sb,sc,S N) -> NatWeightedSeriesTrace(H,wb,wc,pb,pc,qb,qc,tb,tc,vb,vc,S N) -> NatWeightedSeriesTrace(J,wb,wc,rb,rc,sb,sc,db,dc,eb,ec,S N) -> forall k ub uc fb fc i a. Le(k,S N) -> NatPowerRow(tb,tc,vb,vc,k,ub,uc) -> NatPowerRow(db,dc,eb,ec,k,fb,fc) -> Lt(i,H) -> BetaAt(ub,uc,i,a) -> BetaAt(fb,fc,i,a)", "target_ast_sha256": "f7c56ab23847715134689855bfe86d7d7162eddbd0c52f9112dda1a9a5a89e2b", "title": "successor composition transport", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0405", "ND0407", "ND0421" ] }, { "bundle_bytes": 2583524, "bundle_sha256": "739a6442d3cf4657fdb256c65867cd7b790f76676c457cd00f167d607a5b759c", "case": "CV080", "contains_checked_statements": [ "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079" ], "coverage": "same_actual_full_composition_has_proved_low_coefficient_pattern_geometric_low_prefix_value_factorial_normalization_and_rational_omitted_tail_bound_without_replacing_supplied_inputs_weights_or_assuming_code_equality_not_analytic_limit", "ha": "evidence/CV080-HA.json.gz", "internal_proof_nodes": [ 271, 412, 282, 422, 423, 424, 24, 30, 13, 47 ], "lean": "evidence/CV080-Lean.json.gz", "nodes": 426, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-same-table-composition-v3", "proof": "evidence/CV080-bundle.json.gz", "report_sha256": "af913805ff5af11d711eeb2e2781cf35d487bdce10a577d20e9f29d3c93e311f", "root": 425, "statement": "forall K. exists k N d ab ac wb wc z fn pn fpb fpc fqb fqc frb frc fsb fsc fs fv fub fuc hb hc r p3. (k=S K) /\\ ((N=S k) /\\ ((~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,N)) /\\ ((NatExpWeightTable(d,N,wb,wc)) /\\ ((BetaAt(wb,wc,0,z)) /\\ ((~z=0) /\\ ((Factorial(N,fn)) /\\ ((Pow(d,N,pn)) /\\ ((z=fn*pn) /\\ ((forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=z) /\\ ((NatPowerTrace(ab,ac,S k,S ((S N)*(S k)),fpb,fpc,fqb,fqc,S N)) /\\ ((NatWeightedSeriesTrace(S ((S N)*(S k)),wb,wc,fpb,fpc,fqb,fqc,frb,frc,fsb,fsc,S N)) /\\ ((NatPowerRow(frb,frc,fsb,fsc,S N,fub,fuc)) /\\ ((Pow(2,((S N)*(S k)),fv)) /\\ ((NatHornerValue(fub,fuc,2,S ((S N)*(S k)),fs)) /\\ ((Le(fs,9*(z*fv))) /\\ ((Le(N,((S N)*(S k)))) /\\ ((Le(N,S ((S N)*(S k)))) /\\ ((forall i a. Lt(i,N) -> BetaAt(fub,fuc,i,a) -> ((i=0 /\\ a=z) \\/ (exists j. i=S j /\\ a=2*z))) /\\ ((NatScaledHornerTrace(fub,fuc,3,N,hb,hc)) /\\ ((BetaAt(hb,hc,N,r)) /\\ ((Pow(3,k,p3)) /\\ ((~p3=0) /\\ ((r+z=(2*z)*p3) /\\ ((~(z*p3)=0) /\\ ((forall M. Le(M,((S N)*(S k))) -> exists r tb tc u p0 q0 h. (((S N)*(S k))=M+r) /\\ ((NatAffineSlice(fub,fuc,M,1,tb,tc,S r)) /\\ ((NatHornerValue(tb,tc,3,S r,u)) /\\ ((Pow(2,M,p0)) /\\ ((Pow(3,M,q0)) /\\ ((Pow(3,((S N)*(S k)),h)) /\\ ((~p0=0) /\\ ((~q0=0) /\\ ((~h=0) /\\ ((~(z*h)=0) /\\ ((Le(u*q0,(9*p0)*(z*h))) /\\ (IRatLe(u,0,z*h,9*p0,0,q0))))))))))))) /\\ (exists r tb tc u p0 q0 h. (((S N)*(S k))=N+r) /\\ ((NatAffineSlice(fub,fuc,N,1,tb,tc,S r)) /\\ ((NatHornerValue(tb,tc,3,S r,u)) /\\ ((Pow(2,N,p0)) /\\ ((Pow(3,N,q0)) /\\ ((Pow(3,((S N)*(S k)),h)) /\\ ((~p0=0) /\\ ((~q0=0) /\\ ((~h=0) /\\ ((~(z*h)=0) /\\ ((Le(u*q0,(9*p0)*(z*h))) /\\ (IRatLe(u,0,z*h,9*p0,0,q0)))))))))))))))))))))))))))))))))))))))", "target_ast_sha256": "e5ce4e46386b9c0aa0cde7db11202d410819547dc4b6b462458184f2871f9611", "title": "same table composition", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "ND0411", "ND0423", "ND0427", "PD0020", "PD0023", "ND0405", "ND0407", "ND0437", "ND0421", "ND0425", "ND0444" ] }, { "bundle_bytes": 1198253, "bundle_sha256": "fb723fc358c8247fa1a3bef7a64b5c8c4292429c439b19843823037a8c8e4873", "case": "CV081", "contains_checked_statements": [ "CV008", "CV009", "CV011", "CV032", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057" ], "coverage": "arbitrary_base_evaluation_of_supplied_whole_support_power_and_weighted_accumulation_traces_constructs_actual_row_values_products_sum_and_monomial_law_without_replacing_any_supplied_trace_not_analytic_limit", "ha": "evidence/CV081-HA.json.gz", "internal_proof_nodes": [ 207, 251, 259, 106, 246, 260, 261, 73, 169, 13 ], "lean": "evidence/CV081-Lean.json.gz", "nodes": 263, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supplied-composition-evaluation-v2", "proof": "evidence/CV081-bundle.json.gz", "report_sha256": "916ab53e37703ba9557c926b5c60cae600dd25db17d8db218daf33a4035110b5", "root": 262, "statement": "forall ab ac L t J K wb wc pb pc qb qc rb rc sb sc. Le(S(K*L),S J) -> NatPowerTrace(ab,ac,L,S J,pb,pc,qb,qc,K) -> NatWeightedSeriesTrace(S J,wb,wc,pb,pc,qb,qc,rb,rc,sb,sc,K) -> exists ahb ahc x p v vb vc zb zc s. (NatScaledHornerTrace(ab,ac,t,L,ahb,ahc)) /\\ ((BetaAt(ahb,ahc,L,x)) /\\ ((Pow(t,L,p)) /\\ ((Pow(t,J,v)) /\\ ((NatRowHornerPrefix(pb,pc,qb,qc,t,S J,vb,vc,K)) /\\ ((NatPointwiseProductPrefix(wb,wc,vb,vc,zb,zc,K)) /\\ ((Sum(zb,zc,K,s)) /\\ ((forall ub uc. NatPowerRow(rb,rc,sb,sc,K,ub,uc) -> NatHornerValue(ub,uc,t,S J,s)) /\\ (forall k y a b c. Lt(k,K) -> BetaAt(vb,vc,k,y) -> Pow(p,k,a) -> Pow(t,k,b) -> Pow(x,k,c) -> a*y=(v*b)*c))))))))", "target_ast_sha256": "afd8a2cb3249c998e8a875e5136ae8ea7ce51566569fff1a737c3bb58e6fb097", "title": "supplied composition evaluation", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0015", "ND0427", "PD0020", "ND0405", "ND0407", "ND0437", "ND0438", "ND0421", "ND0439" ] }, { "bundle_bytes": 106956, "bundle_sha256": "471fda92b4c23387af927e0452c262de5ebc4f043563b67c1eec37c75bc90b57", "case": "CV082", "contains_checked_statements": [], "coverage": "supplied_actual_tail_slice_and_independently_computed_prefix_tail_whole_values_obey_exact_scaled_split_using_proved_Horner_value_uniqueness_without_replacing_codes_values_or_cancelling_factors", "ha": "evidence/CV082-HA.json.gz", "internal_proof_nodes": [ 36, 35, 24, 34, 2, 11 ], "lean": "evidence/CV082-Lean.json.gz", "nodes": 38, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supplied-horner-splitting-v1", "proof": "evidence/CV082-bundle.json.gz", "report_sha256": "2867ba1c966fa493761e8ece62265243db23718046d9ef8d7f2ed7871c802652", "root": 37, "statement": "forall ab ac bb bc t M N x y z p. NatAffineSlice(ab,ac,M,1,bb,bc,N) -> NatHornerValue(ab,ac,t,M,x) -> NatHornerValue(bb,bc,t,N,y) -> NatHornerValue(ab,ac,t,M+N,z) -> Pow(t,N,p) -> z=p*x+y", "target_ast_sha256": "2060fe5fe40be2b8cca591f0f8c6e4703cbe9367a3616d60b77f5e6c457553fc", "title": "supplied horner splitting", "uses_definitions": [ "ND0411", "PD0020", "ND0437" ] }, { "bundle_bytes": 32215, "bundle_sha256": "c6ba62017eda3d139acc2d7694d8e37d2d02da904b4054901c0d2ac9103bde0a", "case": "CV083", "contains_checked_statements": [], "coverage": "natural_arithmetic_geometric_prefix_plus_actual_split_and_tail_bound_imply_exact_balance_and_both_denominator_cleared_bounds_without_division_cancellation_or_analytic_limit", "ha": "evidence/CV083-HA.json.gz", "internal_proof_nodes": [ 10, 11, 12 ], "lean": "evidence/CV083-Lean.json.gz", "nodes": 14, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-geometric-tail-sandwich-v1", "proof": "evidence/CV083-bundle.json.gz", "report_sha256": "ac71c14a33f33330e63a954bdd745c63d482fab8461f2ae4329d8cc7548175da", "root": 13, "statement": "forall r z p b y u h q a. r+z=(2*z)*p -> y=b*r+u -> h=p*b -> Le(u*q,(9*a)*(z*h)) -> (y+z*b=(2*z)*h+u) /\\ ((Le(y*q,(2*q+9*a)*(z*h))) /\\ (Le((2*p)*(z*h),y*p+z*h)))", "target_ast_sha256": "6ab738597b29754dfe8c2ecd8edd19ed02b665f859a9b263e2eb80903c75afbd", "title": "geometric tail sandwich", "uses_definitions": [ "PD0001" ] }, { "bundle_bytes": 354089, "bundle_sha256": "a503bbe03a5d157fc56e871be761c191ffd3f399c409968b955e31a4e5b6e300", "case": "CV084", "contains_checked_statements": [ "CV082", "CV083" ], "coverage": "supplied_actual_prefix_tail_and_full_Horner_values_with_actual_power_alignment_and_geometric_tail_premises_give_both_valid_rational_error_bounds_with_all_denominators_proved_nonzero_not_computation_existence_or_analytic_limit", "ha": "evidence/CV084-HA.json.gz", "internal_proof_nodes": [ 128, 132, 127, 111, 120, 126, 44, 12, 8, 6, 1, 10 ], "lean": "evidence/CV084-Lean.json.gz", "nodes": 134, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-supplied-geometric-error-v1", "proof": "evidence/CV084-bundle.json.gz", "report_sha256": "e9921d9144c8150e9c9de91991aafc0b3612d3f5a02053c5bf09d0ff5f5039e3", "root": 133, "statement": "forall ub uc tb tc k r z x y u p h q a. ~z=0 -> NatHornerValue(ub,uc,3,S k,x) -> x+z=(2*z)*p -> Pow(3,k,p) -> NatAffineSlice(ub,uc,S k,1,tb,tc,S r) -> NatHornerValue(tb,tc,3,S r,u) -> NatHornerValue(ub,uc,3,S ((S k)+r),y) -> Pow(3,((S k)+r),h) -> Pow(3,S k,q) -> Pow(2,S k,a) -> Le(u*q,(9*a)*(z*h)) -> exists b. (Pow(3,S r,b)) /\\ ((h=p*b) /\\ ((y=b*x+u) /\\ ((~p=0) /\\ ((~q=0) /\\ ((~h=0) /\\ ((~(z*h)=0) /\\ ((~((z*h)*p)=0) /\\ ((y+z*b=(2*z)*h+u) /\\ ((Le(y*q,(2*q+9*a)*(z*h))) /\\ ((Le((2*p)*(z*h),y*p+z*h)) /\\ ((IRatLe(y,0,z*h,2*q+9*a,0,q)) /\\ (IRatLe(2,0,1,y*p+z*h,0,(z*h)*p)))))))))))))", "target_ast_sha256": "4ee4edb3a27c63810499bc48918c92c504253b783efa6e306acf9c98ba79f558", "title": "supplied geometric error", "uses_definitions": [ "PD0001", "ND0411", "PD0020", "ND0437", "ND0444" ] }, { "bundle_bytes": 2727157, "bundle_sha256": "c2e9f924a90d6c5c2580db4e5dc7d18a89a02f6936f3c0dff0ebc0d566b2d833", "case": "CV085", "contains_checked_statements": [ "CV001", "CV002", "CV004", "CV008", "CV009", "CV011", "CV012", "CV019", "CV020", "CV021", "CV022", "CV024", "CV025", "CV027", "CV028", "CV029", "CV031", "CV032", "CV033", "CV037", "CV038", "CV039", "CV043", "CV044", "CV046", "CV047", "CV048", "CV049", "CV056", "CV057", "CV058", "CV059", "CV060", "CV061", "CV062", "CV063", "CV064", "CV065", "CV072", "CV073", "CV075", "CV077", "CV078", "CV079", "CV081", "CV082", "CV083", "CV084" ], "coverage": "unconditional_same_actual_atanh_input_factorial_weights_whole_support_composition_and_finite_weighted_sum_with_monomial_normalization_and_both_valid_rational_errors_no_geometric_or_tail_premises_no_analytic_limit", "ha": "evidence/CV085-HA.json.gz", "internal_proof_nodes": [ 432, 433 ], "lean": "evidence/CV085-Lean.json.gz", "nodes": 435, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-constructed-composition-error-v3", "proof": "evidence/CV085-bundle.json.gz", "report_sha256": "285fdc9a21e2f824f47392afb1d751d511d896084bae4ce490c664e32d339ba2", "root": 434, "statement": "forall K. exists k N d ab ac wb wc z fn pn ex ep ev evb evc ezb ezc es p3 a2. (k=S K) /\\ ((N=S k) /\\ ((~d=0) /\\ ((NatAtanhPrefix(d,ab,ac,N)) /\\ ((NatExpWeightTable(d,N,wb,wc)) /\\ ((BetaAt(wb,wc,0,z)) /\\ ((~z=0) /\\ ((Factorial(N,fn)) /\\ ((Pow(d,N,pn)) /\\ ((z=fn*pn) /\\ ((forall j w f p. Le(j,N) -> BetaAt(wb,wc,j,w) -> Factorial(j,f) -> Pow(d,j,p) -> (w*f)*p=z) /\\ ((NatHornerValue(ab,ac,3,N,ex)) /\\ ((Pow(3,N,ep)) /\\ ((Pow(3,((S N)*N),ev)) /\\ ((NatPointwiseProductPrefix(wb,wc,evb,evc,ezb,ezc,S N)) /\\ ((Sum(ezb,ezc,S N,es)) /\\ ((forall j y a b c. Lt(j,S N) -> BetaAt(evb,evc,j,y) -> Pow(ep,j,a) -> Pow(3,j,b) -> Pow(ex,j,c) -> a*y=(ev*b)*c) /\\ ((Pow(3,k,p3)) /\\ ((Pow(2,N,a2)) /\\ ((~ep=0) /\\ ((~ev=0) /\\ ((~p3=0) /\\ ((~(z*ev)=0) /\\ ((~((z*ev)*p3)=0) /\\ ((IRatLe(es,0,z*ev,2*ep+9*a2,0,ep)) /\\ (IRatLe(2,0,1,es*p3+z*ev,0,(z*ev)*p3))))))))))))))))))))))))))", "target_ast_sha256": "390079559204a44746d65722e27b386315e6b9830542df520c3ed5acd50b4d58", "title": "constructed composition error", "uses_definitions": [ "PD0001", "PD0002", "PD0013", "PD0015", "ND0423", "PD0020", "PD0023", "ND0437", "ND0439", "ND0425", "ND0444" ] }, { "bundle_bytes": 97681, "bundle_sha256": "914f8df1b97d191c466fcb8decab245de1e5cb08bfbd9df904e403d18d09fb48", "case": "CV086", "contains_checked_statements": [], "coverage": "supplied_atanh_prefix_actual_base_three_Horner_value_and_power_obey_three_y_plus_three_d_at_most_d_times_p_by_zero_head_bounded_prefix_induction_no_division_or_analytic_limit", "ha": "evidence/CV086-HA.json.gz", "internal_proof_nodes": [ 42, 41, 34, 44, 13, 38, 7, 6, 8 ], "lean": "evidence/CV086-Lean.json.gz", "nodes": 46, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-ternary-atanh-bound-v1", "proof": "evidence/CV086-bundle.json.gz", "report_sha256": "e5984ff0102ed270f0525ebbac56fab8f22a9b579bfb7e74504cdc23b6c574d4", "root": 45, "statement": "forall d ab ac k y p. NatAtanhPrefix(d,ab,ac,S k) -> NatHornerValue(ab,ac,3,S k,y) -> Pow(3,S k,p) -> Le(3*y+3*d,d*p)", "target_ast_sha256": "7a0be8c3f072412b13318d7e4c22a5a86af4e3c3a1567ad788f60fc72b41a49a", "title": "ternary atanh bound", "uses_definitions": [ "PD0001", "PD0020", "ND0437", "ND0425" ] }, { "bundle_bytes": 216846, "bundle_sha256": "842b94791215ff765f03bf71b15cffbb29af38eece82ed279ea1f9636bb90d30", "case": "CV087", "contains_checked_statements": [ "CV068" ], "coverage": "construct_actual_shifted_finite_outer_exp_tail_and_sum_with_proved_ratio_half_and_sum_at_most_twice_first_omitted_term_under_nonzero_denominator_and_argument_at_most_one_not_infinite_remainder", "ha": "evidence/CV087-HA.json.gz", "internal_proof_nodes": [ 112, 118, 119, 115 ], "lean": "evidence/CV087-Lean.json.gz", "nodes": 121, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-scaled-exp-tail-v1", "proof": "evidence/CV087-bundle.json.gz", "report_sha256": "2acf27ab9131f8dd2e9e74da9c3e2cf19081260721f0e47b087a01cafedf6936", "root": 120, "statement": "forall d x tb tc m N a. ~d=0 -> Le(x,d) -> NatScaledExpRecurrence(d,x,tb,tc,(S m)+N) -> BetaAt(tb,tc,S m,a) -> exists ub uc s. NatAffineSlice(tb,tc,S m,1,ub,uc,S N) /\\ (NatRatioBoundPrefix(ub,uc,1,S N) /\\ (Sum(ub,uc,S N,s) /\\ Le(s,2*a)))", "target_ast_sha256": "d92d8780097ceaf3810aba46e31ffe2e3c8357929685f8314972342821425d8b", "title": "scaled exp tail", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0411", "ND0441", "ND0443" ] }, { "bundle_bytes": 283071, "bundle_sha256": "b69e30c4b5c8f199bb7dc45506f4d3476edf42e8acae440e9ee5c806e00fa303", "case": "CV088", "contains_checked_statements": [ "CV068", "CV087" ], "coverage": "construct_actual_cutoff_power_two_m_first_omitted_term_slice_ratio_and_sum_with_power_times_first_at_most_head_and_power_times_tail_at_most_twice_head_uniform_in_extra_finite_horizon_not_analytic_limit", "ha": "evidence/CV088-HA.json.gz", "internal_proof_nodes": [ 128, 73, 120, 133, 134, 2 ], "lean": "evidence/CV088-Lean.json.gz", "nodes": 136, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-scaled-exp-tail-decay-v1", "proof": "evidence/CV088-bundle.json.gz", "report_sha256": "d7c61b012275fb8f2625702f0a5edae24f8a4ed6ae846a5713ae42cd97c4c397", "root": 135, "statement": "forall d x tb tc m N a0. ~d=0 -> Le(x,d) -> NatScaledExpRecurrence(d,x,tb,tc,(S m)+N) -> BetaAt(tb,tc,0,a0) -> exists p ub uc a s. Pow(2,m,p) /\\ (BetaAt(tb,tc,S m,a) /\\ (NatAffineSlice(tb,tc,S m,1,ub,uc,S N) /\\ (NatRatioBoundPrefix(ub,uc,1,S N) /\\ (Sum(ub,uc,S N,s) /\\ (Le(p*a,a0) /\\ (Le(p*s,2*a0)))))))", "target_ast_sha256": "76d2878ab4276b62e8094e3b25956fc0b58fec9c0d936f9c3af4c253fe8cc5ba", "title": "scaled exp tail decay", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0411", "PD0020", "ND0441", "ND0443" ] }, { "bundle_bytes": 87358, "bundle_sha256": "24c5dab66abdf543bfa39db7e32f55de16c4e88cd4f17fc80bf203a809ddd6cc", "case": "CV089", "contains_checked_statements": [], "coverage": "supplied_base_three_Horner_value_and_actual_power_obey_y_plus_D_at_most_D_times_power_for_any_two_D_bounded_prefix_without_zero_head_including_empty_prefix_no_analytic_limit", "ha": "evidence/CV089-HA.json.gz", "internal_proof_nodes": [ 38, 12 ], "lean": "evidence/CV089-Lean.json.gz", "nodes": 40, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-ternary-prefix-bound-v1", "proof": "evidence/CV089-bundle.json.gz", "report_sha256": "115a2f2f88d642eb95c0b083e34085e2e2b4331954aa749b1add8f480457c6d0", "root": 39, "statement": "forall ab ac D N y p. NatBoundedPrefix(ab,ac,2*D,N) -> NatHornerValue(ab,ac,3,N,y) -> Pow(3,N,p) -> Le(y+D,D*p)", "target_ast_sha256": "5065e02e01a4babe946abf7b6ad03affafc1b509d98b79ed7d5f71995498c926", "title": "ternary prefix bound", "uses_definitions": [ "PD0001", "PD0020", "ND0437", "ND0440" ] }, { "bundle_bytes": 357579, "bundle_sha256": "0fdb76e370b33e761b9fd995aafbdc8a6d9bb0db5819bb8216a9d6d4ca7760c3", "case": "CV090", "contains_checked_statements": [ "CV032", "CV062", "CV089" ], "coverage": "supplied_actual_atanh_table_prefix_tail_whole_values_and_powers_constructed_with_exact_scaled_split_tail_bound_and_both_cross_multiplied_truncation_bounds_including_zero_lengths_no_division_or_analytic_limit", "ha": "evidence/CV090-HA.json.gz", "internal_proof_nodes": [ 131, 140 ], "lean": "evidence/CV090-Lean.json.gz", "nodes": 142, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-truncation-v1", "proof": "evidence/CV090-bundle.json.gz", "report_sha256": "b215ff648defc940d4612ed34dbe48f42b37dc804104c4b7dfea27ff492ff45f", "root": 141, "statement": "forall d ab ac M N. NatAtanhPrefix(d,ab,ac,M+N) -> exists bb bc x y z p q h. NatAffineSlice(ab,ac,M,1,bb,bc,N) /\\ (NatHornerValue(ab,ac,3,M,x) /\\ (NatHornerValue(bb,bc,3,N,y) /\\ (NatHornerValue(ab,ac,3,M+N,z) /\\ (Pow(3,M,p) /\\ (Pow(3,N,q) /\\ (Pow(3,M+N,h) /\\ (h=p*q /\\ (z=q*x+y /\\ (Le(y+d,d*q) /\\ (Le(h*x,p*z) /\\ (Le(p*z,h*x+d*h))))))))))))", "target_ast_sha256": "fe8e75a1b92914b876b3335a496085b7cea520d900aaedca8a40e6f31ac26fa6", "title": "atanh truncation", "uses_definitions": [ "PD0001", "ND0411", "PD0020", "ND0437", "ND0425" ] }, { "bundle_bytes": 386133, "bundle_sha256": "d46e09d94d9ac0ad74bd6dc8082e3e5d09cd31addc63b89fa81d81b4b6c92522", "case": "CV091", "contains_checked_statements": [ "CV032", "CV062", "CV089", "CV090" ], "coverage": "supplied_actual_atanh_table_prefix_and_whole_values_with_actual_powers_and_all_nonzero_denominators_give_valid_rational_monotonicity_and_error_at_most_three_over_three_to_cutoff_not_analytic_limit", "ha": "evidence/CV091-HA.json.gz", "internal_proof_nodes": [ 141, 144, 142, 11 ], "lean": "evidence/CV091-Lean.json.gz", "nodes": 146, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-rational-truncation-v1", "proof": "evidence/CV091-bundle.json.gz", "report_sha256": "b097bac9d3eef96d20930cd2deda826b8c26a12fbacb231f11176ceb589df156", "root": 145, "statement": "forall d ab ac M N. ~d=0 -> NatAtanhPrefix(d,ab,ac,M+N) -> exists x z p h. NatHornerValue(ab,ac,3,M,x) /\\ (NatHornerValue(ab,ac,3,M+N,z) /\\ (Pow(3,M,p) /\\ (Pow(3,M+N,h) /\\ (~(d*p)=0 /\\ (~(d*h)=0 /\\ (IRatLe(3*x,0,d*p,3*z,0,d*h) /\\ (IRatLe(3*z,0,d*h,3*x+3*d,0,d*p))))))))", "target_ast_sha256": "c1d96e8b045d76e79bd363c41f9562f92787aaef0867562246f3d89f06132418", "title": "atanh rational truncation", "uses_definitions": [ "PD0020", "ND0437", "ND0425", "ND0444" ] }, { "bundle_bytes": 72999, "bundle_sha256": "88406acbf3430be4143d97c26d2e1cb0a3f09d1a7b014add87323a1e08587074", "case": "CV092", "contains_checked_statements": [], "coverage": "two_independently_supplied_atanh_tables_and_horner_values_obey_exact_cross_scale_identity_including_zero_scales_base_and_empty_prefix_only_proved_nonzero_odd_index_cancelled_no_code_equality_or_analytic_limit", "ha": "evidence/CV092-HA.json.gz", "internal_proof_nodes": [ 35, 32, 10, 4 ], "lean": "evidence/CV092-Lean.json.gz", "nodes": 37, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-atanh-scale-independence-v1", "proof": "evidence/CV092-bundle.json.gz", "report_sha256": "f9a2de08583dec8ba02569bdb469ff0b491ca71342c4b46e8daaa92901edd888", "root": 36, "statement": "forall d e ab ac bb bc N t x y. NatAtanhPrefix(d,ab,ac,N) -> NatAtanhPrefix(e,bb,bc,N) -> NatHornerValue(ab,ac,t,N,x) -> NatHornerValue(bb,bc,t,N,y) -> e*x=d*y", "target_ast_sha256": "5b80b161ea46bbe0cbcdef8789de98218d6c1dfc077ff0023320f077fe3d8bc1", "title": "atanh scale independence", "uses_definitions": [ "ND0437", "ND0425" ] }, { "bundle_bytes": 430464, "bundle_sha256": "d687a27e5e4e4a010c27dcdc93fe18548e682cd1bfa3c03e7d32e04aa65f1acd", "case": "CV093", "contains_checked_statements": [ "CV032", "CV062", "CV089", "CV090", "CV092" ], "coverage": "independently_supplied_atanh_prefix_and_longer_table_scales_values_and_powers_give_valid_rational_monotonicity_and_uniform_error_three_over_three_to_cutoff_without_code_equality_or_analytic_limit", "ha": "evidence/CV093-HA.json.gz", "internal_proof_nodes": [ 141, 152, 142, 136, 144, 11, 60, 145, 153, 13, 2 ], "lean": "evidence/CV093-Lean.json.gz", "nodes": 155, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-independent-atanh-truncation-v1", "proof": "evidence/CV093-bundle.json.gz", "report_sha256": "0d1fde5b0fd84ef1f44bd07005862df74f17d44489aa63acc1063eafaa879a1e", "root": 154, "statement": "forall d e ab ac bb bc M N x z p h. ~d=0 -> ~e=0 -> NatAtanhPrefix(d,ab,ac,M) -> NatAtanhPrefix(e,bb,bc,M+N) -> NatHornerValue(ab,ac,3,M,x) -> NatHornerValue(bb,bc,3,M+N,z) -> Pow(3,M,p) -> Pow(3,M+N,h) -> IRatLe(3*x,0,d*p,3*z,0,e*h) /\\ IRatLe(3*z,0,e*h,3*x+3*d,0,d*p)", "target_ast_sha256": "c8bd73e82050b43d383204e7e5e1fc00bda755d88df07a5fb9b1c8f5700f3f6b", "title": "independent atanh truncation", "uses_definitions": [ "PD0020", "ND0437", "ND0425", "ND0444" ] }, { "bundle_bytes": 72987, "bundle_sha256": "44a7cc4e855ca9dc684c975187a4d1ab6ca93a7159fb656c8487dcc8daff9ea6", "case": "CV094", "contains_checked_statements": [], "coverage": "supplied_exponential_term_recurrences_with_common_nonzero_denominator_shared_actual_head_and_nonnegative_argument_gap_have_all_index_monotonicity_and_finite_difference_bound_not_derivative_or_infinite_sum", "ha": "evidence/CV094-HA.json.gz", "internal_proof_nodes": [ 37, 38, 30, 31, 9, 12, 20, 13 ], "lean": "evidence/CV094-Lean.json.gz", "nodes": 40, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-term-difference-v1", "proof": "evidence/CV094-bundle.json.gz", "report_sha256": "c277629d942f2043b68a9f3e29dd0659f079fabfd10cb5c2d8e77ec0bec873fb", "root": 39, "statement": "forall d x y g N ab ac bb bc a0. ~d=0 -> y=x+g -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(d,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,a0) -> forall k a b c. Le(S k,N) -> BetaAt(ab,ac,S k,a) -> BetaAt(bb,bc,S k,b) -> BetaAt(bb,bc,k,c) -> Le(a,b) /\\ Le(d*b,d*a+g*c)", "target_ast_sha256": "b58a497476351e3cc07a976d42021351568e27076943567eb7a2186d2dbc51d9", "title": "exp term difference", "uses_definitions": [ "PD0001", "PD0013", "ND0443" ] }, { "bundle_bytes": 59957, "bundle_sha256": "58ccc2beee0e324cc56f96da7460606f35d20ba422fa0f1e4270dd9c50719edf", "case": "CV095", "contains_checked_statements": [], "coverage": "explicit_pointwise_weighted_order_transports_to_three_supplied_actual_finite_sums_including_empty_prefix_and_zero_weights_via_conservative_alias_without_sum_accuracy_or_analytic_claim_in_definition", "ha": "evidence/CV095-HA.json.gz", "internal_proof_nodes": [ 24, 8 ], "lean": "evidence/CV095-Lean.json.gz", "nodes": 26, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-sum-linear-bound-v1", "proof": "evidence/CV095-bundle.json.gz", "report_sha256": "d44bd790e232b71dd9a1df45236ba5c186452021a2a62524b2b146cbdd23110e", "root": 25, "statement": "forall alpha beta gamma ab ac bb bc cb cc N u v w. NatLinearBound(alpha,beta,gamma,ab,ac,bb,bc,cb,cc,N) -> Sum(ab,ac,N,u) -> Sum(bb,bc,N,v) -> Sum(cb,cc,N,w) -> Le(gamma*w,alpha*u+beta*v)", "target_ast_sha256": "5c476b035cc43d4fd6b3e03efbe6c35fb50f132bb036051a3e1bb5e3c1ad422a", "title": "sum linear bound", "uses_definitions": [ "PD0001", "PD0015", "ND0445" ] }, { "bundle_bytes": 384106, "bundle_sha256": "ef000bac2887811708eb6851cb4a561d71102c44712840a7654723cc4543a689", "case": "CV096", "contains_checked_statements": [ "CV009", "CV094", "CV095" ], "coverage": "supplied_finite_exponential_recurrences_shared_head_and_sums_at_common_positive_denominator_obey_monotonicity_and_nine_times_argument_gap_bound_via_actual_tail_slices_and_proved_sum_majorant_not_analytic_limit", "ha": "evidence/CV096-HA.json.gz", "internal_proof_nodes": [ 142, 144, 44, 119, 143, 141, 13, 145, 9, 4 ], "lean": "evidence/CV096-Lean.json.gz", "nodes": 147, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-sum-continuity-v1", "proof": "evidence/CV096-bundle.json.gz", "report_sha256": "3839e668cd462080f6b5e3bacd04951e59cb8415dacf578f8142b2f5569deee8", "root": 146, "statement": "forall d x y g N ab ac bb bc a0 s t. ~d=0 -> y=x+g -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(d,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,a0) -> Le(y,2*d) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Le(s,t) /\\ Le(d*t,d*s+9*(g*a0))", "target_ast_sha256": "bcfe5d4711bda1d8f1b19f812c40c8a8cb6d8c2bf10fdb247fa69a32b08715d2", "title": "exp sum continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 215669, "bundle_sha256": "571d69eb382e46b1bd32c0d1a0b8ef59d2b3ca43a214e92a0e9ddcacd0191b0a", "case": "CV097", "contains_checked_statements": [], "coverage": "construct_actual_scaled_exponential_table_recurrence_at_common_multiplied_denominator_scaled_head_and_exact_scaled_sum_including_zero_scalars_and_degree_no_code_equality_or_rational_validity_assumption", "ha": "evidence/CV097-HA.json.gz", "internal_proof_nodes": [ 109, 105, 111, 112, 108, 9, 4 ], "lean": "evidence/CV097-Lean.json.gz", "nodes": 114, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-rescaling-v1", "proof": "evidence/CV097-bundle.json.gz", "report_sha256": "4074eb646b03a142f57753200d57b89ed724f022933c7e18e5947d1af268a34b", "root": 113, "statement": "forall d x f c N ab ac a0 s. NatScaledExpRecurrence(d,x,ab,ac,N) -> BetaAt(ab,ac,0,a0) -> Sum(ab,ac,S N,s) -> exists ub uc t. NatLinearTriple(c,0,1,ab,ac,ab,ac,ub,uc,S N) /\\ (NatScaledExpRecurrence(d*f,f*x,ub,uc,N) /\\ (BetaAt(ub,uc,0,c*a0) /\\ (Sum(ub,uc,S N,t) /\\ (t=c*s))))", "target_ast_sha256": "1366d756cc52f3aeda1363ed3188f534bc5489ed2977f09a101be2c92e875243", "title": "exp rescaling", "uses_definitions": [ "PD0013", "PD0015", "ND0409", "ND0443" ] }, { "bundle_bytes": 445508, "bundle_sha256": "e4d094f2bd99f93d7c2951a5a03e735e0d7ba193e0b6c117d1385ed11a50bc8d", "case": "CV098", "contains_checked_statements": [ "CV009", "CV094", "CV095", "CV096", "CV097" ], "coverage": "supplied_finite_exponential_recurrences_with_independent_nonzero_denominators_and_arbitrary_initial_scales_obey_cross_multiplied_continuity_by_constructing_actual_rescaled_tables_and_sums_not_code_equality_or_analytic_limit", "ha": "evidence/CV098-HA.json.gz", "internal_proof_nodes": [ 152, 154 ], "lean": "evidence/CV098-Lean.json.gz", "nodes": 156, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-cross-continuity-v1", "proof": "evidence/CV098-bundle.json.gz", "report_sha256": "7ff0ed166d2b4df05b875f1b11d5ef11fc4f090e33b5a0749e9e2a1964a126b6", "root": 155, "statement": "forall d e x y g N ab ac bb bc a0 b0 s t. ~d=0 -> ~e=0 -> d*y=e*x+g -> Le(y,2*e) -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> Le(b0*s,a0*t) /\\ Le((d*e)*(a0*t),(d*e)*(b0*s)+9*(g*(a0*b0)))", "target_ast_sha256": "cd70bc49ad3f8d993f61ab9ed4f3acc069f1cd0aac88b3f00bfd88e97d93da9c", "title": "exp cross continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443" ] }, { "bundle_bytes": 464104, "bundle_sha256": "a8da7ae8b4871484dc4eef2f3e675c1bfae67c2c29c1240fcbf03895414f3c6b", "case": "CV099", "contains_checked_statements": [ "CV009", "CV094", "CV095", "CV096", "CV097", "CV098" ], "coverage": "supplied_finite_exponential_sums_with_independent_positive_denominators_and_heads_have_valid_rational_monotonicity_and_nine_times_argument_difference_bound_after_explicit_nonzero_denominator_proofs_no_analytic_limit", "ha": "evidence/CV099-HA.json.gz", "internal_proof_nodes": [ 155, 157 ], "lean": "evidence/CV099-Lean.json.gz", "nodes": 159, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-rational-continuity-v1", "proof": "evidence/CV099-bundle.json.gz", "report_sha256": "68de49e18e8f1314a0befadd441515cfcd90087d648bf38c413b90061340d1c5", "root": 158, "statement": "forall d e x y g N ab ac bb bc a0 b0 s t. ~d=0 -> ~e=0 -> ~a0=0 -> ~b0=0 -> d*y=e*x+g -> Le(y,2*e) -> NatScaledExpRecurrence(d,x,ab,ac,N) -> NatScaledExpRecurrence(e,y,bb,bc,N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S N,s) -> Sum(bb,bc,S N,t) -> IRatLe(s,0,a0,t,0,b0) /\\ IRatLe(t,0,b0,(d*e)*s+(9*g)*a0,0,(d*e)*a0)", "target_ast_sha256": "82b80b04322470f648dfee85b1aab63cea1636dce56258cd64822934f90d7e32", "title": "exp rational continuity", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "ND0443", "ND0444" ] }, { "bundle_bytes": 385000, "bundle_sha256": "a8288777d6ca50ce713634fdda4f018996ec8ff25566bf936657bfb3630e1e5d", "case": "CV100", "contains_checked_statements": [ "CV009", "CV068", "CV087", "CV088" ], "coverage": "supplied_exponential_recurrence_head_prefix_and_longer_sum_and_actual_cutoff_power_obey_uniform_finite_truncation_bound_via_constructed_tail_and_actual_affine_sum_uniqueness_no_infinite_remainder_assumption", "ha": "evidence/CV100-HA.json.gz", "internal_proof_nodes": [ 135, 152, 151, 153 ], "lean": "evidence/CV100-Lean.json.gz", "nodes": 155, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-truncation-v1", "proof": "evidence/CV100-bundle.json.gz", "report_sha256": "a6bd0e579ffdc50954f7ac9208a05221de3f1cadfe22d5827d0126e796368405", "root": 154, "statement": "forall d x ab ac m N a0 s t p. ~d=0 -> Le(x,d) -> NatScaledExpRecurrence(d,x,ab,ac,(S m)+N) -> BetaAt(ab,ac,0,a0) -> Sum(ab,ac,S m,s) -> Sum(ab,ac,S((S m)+N),t) -> Pow(2,m,p) -> Le(s,t) /\\ Le(p*t,p*s+2*a0)", "target_ast_sha256": "38cc1a427fdb362d86fb580c3b3a939ae9e62900f35876de3390a804f26294b2", "title": "exp truncation", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 613675, "bundle_sha256": "ac305ff130d7a1936202c1022e0aa06acc322a556ae0d6684918bc30cb5533bd", "case": "CV101", "contains_checked_statements": [ "CV009", "CV068", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100" ], "coverage": "supplied_finite_exponential_sums_with_independent_arguments_denominators_heads_and_ordered_horizons_have_combined_argument_and_cutoff_error_via_constructed_middle_prefix_and_checked_polynomial_order_no_analytic_limit", "ha": "evidence/CV101-HA.json.gz", "internal_proof_nodes": [ 155, 185, 119, 186, 187, 2, 1, 13, 5, 9 ], "lean": "evidence/CV101-Lean.json.gz", "nodes": 189, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-two-error-bound-v2", "proof": "evidence/CV101-bundle.json.gz", "report_sha256": "15867ed6ec821300f7eb2961f2985f2229aea03456d816a0a95d20e46d6ec160", "root": 188, "statement": "forall d e x y g m N ab ac bb bc a0 b0 s t p. ~d=0 -> ~e=0 -> d*y=e*x+g -> Le(y,e) -> NatScaledExpRecurrence(d,x,ab,ac,m) -> NatScaledExpRecurrence(e,y,bb,bc,(S m)+N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S m,s) -> Sum(bb,bc,S((S m)+N),t) -> Pow(2,m,p) -> Le(b0*s,a0*t) /\\ Le((d*e)*(p*(a0*t)),(d*e)*(p*(b0*s))+(9*(g*p)+2*(d*e))*(a0*b0))", "target_ast_sha256": "99e2c67d567497309e07b57a9c96266fe4299a7a38f24bf605373710dc0b6857", "title": "exp two error bound", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443" ] }, { "bundle_bytes": 637312, "bundle_sha256": "dc1c490b0ea909db229a2530e7eb7c1ef55ea80df983a1c1351582e8c4802062", "case": "CV102", "contains_checked_statements": [ "CV009", "CV068", "CV087", "CV088", "CV094", "CV095", "CV096", "CV097", "CV098", "CV100", "CV101" ], "coverage": "independently_supplied_exponential_computations_at_ordered_arguments_and_distinct_horizons_have_valid_rational_bounds_with_separate_argument_and_dyadic_tail_errors_after_proving_all_denominators_and_cutoff_power_nonzero_no_analytic_limit", "ha": "evidence/CV102-HA.json.gz", "internal_proof_nodes": [ 188, 191, 27, 189, 11 ], "lean": "evidence/CV102-Lean.json.gz", "nodes": 193, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-exp-rational-two-error-v1", "proof": "evidence/CV102-bundle.json.gz", "report_sha256": "53a33287272403f8267c174df8f60ab30e95ec26836d72c6aa9dc3a1893bfc7b", "root": 192, "statement": "forall d e x y g m N ab ac bb bc a0 b0 s t p. ~d=0 -> ~e=0 -> ~a0=0 -> ~b0=0 -> d*y=e*x+g -> Le(y,e) -> NatScaledExpRecurrence(d,x,ab,ac,m) -> NatScaledExpRecurrence(e,y,bb,bc,(S m)+N) -> BetaAt(ab,ac,0,a0) -> BetaAt(bb,bc,0,b0) -> Sum(ab,ac,S m,s) -> Sum(bb,bc,S((S m)+N),t) -> Pow(2,m,p) -> IRatLe(s,0,a0,t,0,b0) /\\ IRatLe(t,0,b0,(d*e)*(p*s)+(9*(g*p)+2*(d*e))*a0,0,(d*e)*(p*a0))", "target_ast_sha256": "f3522601cd0c5260ecf6f5cde1c301a2d74e3f5b53f165eea0ef9f3bf32c837c", "title": "exp rational two error", "uses_definitions": [ "PD0001", "PD0013", "PD0015", "PD0020", "ND0443", "ND0444" ] }, { "bundle_bytes": 1406877, "bundle_sha256": "c7404c8766489feb5bb9994621535bb10d07907566e3f7a586ec383b98eaa225", "case": "IR016", "contains_checked_statements": [], "coverage": "IR016_natural_core_not_full_irrationality", "ha": "evidence/IR016-HA.json.gz", "internal_proof_nodes": [ 179, 180 ], "lean": "evidence/IR016-Lean.json.gz", "nodes": 182, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-ir016-v1", "proof": "evidence/IR016-bundle.json.gz", "report_sha256": "1e71af79d629fbb96f71286cd7a7d4dfaa71f9675891c5f5f367d9b9b0a390a9", "root": 181, "statement": "∀ x. ∀ y. x · x = 2 · y · y → x = 0 ∧ y = 0", "target_ast_sha256": "a1aaa15b01ead79382b92dff5be1ebd105745bc26b937451e73ecdec973209b7", "title": "ir016", "uses_definitions": [] }, { "bundle_bytes": 5777, "bundle_sha256": "e614473e462f9589988e7526d85de6427c03caca9edc9fc8ca5285514d1b1cc5", "case": "NG001", "contains_checked_statements": [], "coverage": "universal_natural_gap_not_full_campaign", "ha": "evidence/NG001-HA.json.gz", "internal_proof_nodes": [ 0, 1 ], "lean": "evidence/NG001-Lean.json.gz", "nodes": 3, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-natural-gap-v1", "proof": "evidence/NG001-bundle.json.gz", "report_sha256": "854da7402a369bf0e0e08713339fa93a4e8758af1d11d8197a5521a10a06a7b4", "root": 2, "statement": "∀ x. ∀ y. ¬x = y → ∃ z. x + S z = y ∨ y + S z = x", "target_ast_sha256": "f0598e325456d7992079c20535c5d15740fb9f4a35c1a2619d5983c7ac9ac64b", "title": "natural gap", "uses_definitions": [] }, { "bundle_bytes": 5866, "bundle_sha256": "04e8a51b9815eca002e0b490d9adfc470daa17f2733acf6a30f15489417dcc32", "case": "P04-C00", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C00-HA.json.gz", "internal_proof_nodes": [ 0, 1, 2, 3 ], "lean": "evidence/P04-C00-Lean.json.gz", "nodes": 5, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c00-v1", "proof": "evidence/P04-C00-bundle.json.gz", "report_sha256": "0524bf4afa01996cb3dbad20d33cf7fef3cc345b3e405f5f4ba1e538ba8737ad", "root": 4, "statement": "0 = 0 ∧ (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) = 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) ∧ 0 = 0) ∧ (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) = 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) ∧ (0 = 0 ∧ 2 · S (2 · (2 · S (2 · (2 · 1)))) = 2 · S (2 · (2 · S (2 · (2 · 1))))))", "target_ast_sha256": "e8fafbb03e9bf8c595485e474d9a258de8b542defdfbb20a3e3aa745dbc68cbb", "title": "p04 c00", "uses_definitions": [] }, { "bundle_bytes": 2141, "bundle_sha256": "164e24973a8368430969de032e87e44051c224bef163b1cddaf665684d900485", "case": "P04-C01", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C01-HA.json.gz", "internal_proof_nodes": [ 0, 1, 2 ], "lean": "evidence/P04-C01-Lean.json.gz", "nodes": 4, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c01-v1", "proof": "evidence/P04-C01-bundle.json.gz", "report_sha256": "4926e4248d24138b43cfccfcb72487571258f8fc84b24b7ed3a6a389b4cd03d2", "root": 3, "statement": "0 = 0 ∧ (2 · S (2 · S (2 · S (2 · 1))) = 2 · S (2 · S (2 · S (2 · 1))) ∧ 1 = 1) ∧ (0 = 0 ∧ 0 = 0 ∧ (0 = 0 ∧ 0 = 0))", "target_ast_sha256": "54843ab8cc85b0a20d691880db62805ab8fa519d9c6ca29dc906df1605e1274a", "title": "p04 c01", "uses_definitions": [] }, { "bundle_bytes": 11604, "bundle_sha256": "d0c7066cbbdf46a1c1e5cda65e8e6a6fcba3eab567a2b795f0345c61f9fa743f", "case": "P04-C02", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C02-HA.json.gz", "internal_proof_nodes": [ 0, 3, 11, 16 ], "lean": "evidence/P04-C02-Lean.json.gz", "nodes": 18, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c02-v1", "proof": "evidence/P04-C02-bundle.json.gz", "report_sha256": "5713f404a9876b0fe768c6c5dd946b51e92709a5f26571979d5e1e5a57802547", "root": 17, "statement": "0 = 0 ∧ (0 = 0 ∧ 0 = 0) ∧ (0 = 0 · 1 ∧ (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 1 ∧ 0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 1)))", "target_ast_sha256": "7e47828bf09996d16a8b6b4df1f9e4294d21df0390eaed4f546bc186ee88a535", "title": "p04 c02", "uses_definitions": [] }, { "bundle_bytes": 44798, "bundle_sha256": "5963edb25e695b84d72f8db7f91faf172057f436b14802157fcf4f4dab4df3f3", "case": "P04-C03", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C03-HA.json.gz", "internal_proof_nodes": [ 12, 19, 28, 34, 44, 45, 48 ], "lean": "evidence/P04-C03-Lean.json.gz", "nodes": 50, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c03-v1", "proof": "evidence/P04-C03-bundle.json.gz", "report_sha256": "a8610f43105dbf0cfa8d5fab42323fc88978bf8050488eee166a9e5815498d3e", "root": 49, "statement": "2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 1) ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 1)) ∧ 2 · S (2 · (2 · S (2 · (2 · 1)))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 1))) ∧ (0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0 + (2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + 0 · 1)) ∧ 2 · S (2 · S (2 · S (2 · 1))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + (0 · 0 + 2 · S (2 · S (2 · S (2 · 1))) · 1)) ∧ (0 = 0 · 0 ∧ 0 = 0 · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0))", "target_ast_sha256": "c24c821191d3d2a213cbcde4fb0a4a1f98276585434153becb4f9f83844abcd0", "title": "p04 c03", "uses_definitions": [] }, { "bundle_bytes": 596223, "bundle_sha256": "a67b191392d3d7fa47613cf06de7d18d25ab2c2056a195d70641bea9d4aaa774", "case": "P04-C04", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C04-HA.json.gz", "internal_proof_nodes": [ 95, 105, 205, 213, 390, 400 ], "lean": "evidence/P04-C04-Lean.json.gz", "nodes": 402, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c04-v1", "proof": "evidence/P04-C04-bundle.json.gz", "report_sha256": "d3fd39c537a801cbd4a2ccb31ff094deabd004347e73cc693fba89279fd11703", "root": 401, "statement": "2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1)))))))))))))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 0 · 0) ∧ (0 = 0 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) ∧ 2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · 1))))))))))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))) + (0 · 0 + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 0 · 0))) ∧ (0 = 0 · (2 · S (2 · (2 · S (2 · (2 · 1))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + (0 · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0)) ∧ (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · 1))))))))))))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · S (2 · (2 · S (2 · (2 · 1))))) + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))) + 0 · 0 + (2 · S (2 · (2 · S (2 · (2 · 1)))) · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 0 · 0)) ∧ 0 = 0 · (2 · S (2 · S (2 · S (2 · 1)))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · (2 · S (2 · (2 · S (2 · (2 · 1))))) + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) + (0 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))) + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + (0 · (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1))))))) + 2 · S (2 · S (2 · S (2 · 1))) · 0))))", "target_ast_sha256": "5a67781f28973fb59401b56a8a3f79fda8ee3aef85c4e6d982b25819cff0263d", "title": "p04 c04", "uses_definitions": [] }, { "bundle_bytes": 950498, "bundle_sha256": "8ba81fc9bda0364c60ee78c7198be6296945bd1a5db20503a3cf6059b725672e", "case": "P04-C05", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C05-HA.json.gz", "internal_proof_nodes": [ 2, 6, 11, 168, 174, 446, 454 ], "lean": "evidence/P04-C05-Lean.json.gz", "nodes": 456, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c05-v1", "proof": "evidence/P04-C05-bundle.json.gz", "report_sha256": "a1db1c1fbf9611e6e697da001b52ce428391902aaf8871baa3ac9a9fe307fa1f", "root": 455, "statement": "0 = 0 · 0 ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 ∧ 0 = 0 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0)) ∧ (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))))))))))))))))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))))) + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) ∧ 0 = 0 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · 1)))))))))))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))))) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0)) ∧ (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))))))))))))))))))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · 1)))))))))))))) + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))))) + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0)) ∧ 0 = 0 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · 1)))))))))))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · 1))))))))))))))) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))))) + (2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + 0 · 0))))", "target_ast_sha256": "d726769334d24cee7d918bed620ab71b89fabcd9e0a14bcd18f47d9399d8595e", "title": "p04 c05", "uses_definitions": [] }, { "bundle_bytes": 1702768, "bundle_sha256": "0ac9a051aaf3a996e2f7cf22692c5a31723132e719cca648d08f5a401f1933b3", "case": "P04-C06", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C06-HA.json.gz", "internal_proof_nodes": [ 461, 462, 465, 468, 475, 724, 731 ], "lean": "evidence/P04-C06-Lean.json.gz", "nodes": 733, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c06-v1", "proof": "evidence/P04-C06-bundle.json.gz", "report_sha256": "79daca42d7989ac3ac8e97a3254292de22bcae2f9aef1f6d11147af4ea85a342", "root": 732, "statement": "2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · 1)))))))))))))))))))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · 1)))))))))))))) + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · 1))))))))))))))) + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))))) + (0 · 0 + 2 · S (2 · S (2 · S (2 · 1))) · 0)) ∧ (0 = 0 · 0 ∧ 0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0) ∧ (0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) ∧ 0 = 0 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) ∧ (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1))))))))))))))))))))))))))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + (0 · 0 + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0)) ∧ 0 = 0 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))))))))))))))))))))) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0))))", "target_ast_sha256": "683e5eb3508f136eb0685dbe74f7f00ca7cc9ae3b8ba23f3361a5c7c6c5f5010", "title": "p04 c06", "uses_definitions": [] }, { "bundle_bytes": 1474604, "bundle_sha256": "57e97a07bf76bd3fcaad21d50e9dee22dc91ac94af063011cacd3a7539b89451", "case": "P04-C07", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C07-HA.json.gz", "internal_proof_nodes": [ 505, 519, 520, 522, 525, 528, 534 ], "lean": "evidence/P04-C07-Lean.json.gz", "nodes": 536, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c07-v1", "proof": "evidence/P04-C07-bundle.json.gz", "report_sha256": "750dd513ea6b87299fe3e73441289c0780f4c323e4f4a7a0359d6f0874a00591", "root": 535, "statement": "2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · 1)))))))))))))))))))))))))))))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + 0 · 0 + (2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + 0 · 0)) ∧ (0 = 0 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) + (0 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))))))))))))))))))))) + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + (0 · 0 + 2 · S (2 · S (2 · S (2 · 1))) · 0)) ∧ 0 = 0 · 0) ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 ∧ 0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) ∧ 0 = 0 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1)))))))))))))))))))))))))))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0))))", "target_ast_sha256": "e8276b96a062a8ba544cf2f5f1f411c49ffdd6952a615d5f06d4c323fd48cbf4", "title": "p04 c07", "uses_definitions": [] }, { "bundle_bytes": 2997726, "bundle_sha256": "0d4b5809a056f5dcc21f5bb7e2e6b390ee35d6e43c4e016076796ea450c1c04b", "case": "P04-C08", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C08-HA.json.gz", "internal_proof_nodes": [ 344, 355, 934, 935, 937, 940 ], "lean": "evidence/P04-C08-Lean.json.gz", "nodes": 942, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c08-v1", "proof": "evidence/P04-C08-bundle.json.gz", "report_sha256": "d757868dfc28ed369795ec7bba71d3940067747e501625eb5d2c5bc7aad9d5b2", "root": 941, "statement": "2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · 1))))))))))))))))))))))))))))))))))))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1)))))))))))))))))))))))))))))) + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0)) ∧ (0 = 0 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · 1))))))))))))))))))))))))))))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1))))))))))))))))))))))))))))))) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0 + (2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + 0 · 0)) ∧ 2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · 1)))))))))))))))))))))))))))))))))))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · 1))))))))))))))))))))))))))))))) + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1))))))))))))))))))))))))))))))) + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + (0 · 0 + 2 · S (2 · S (2 · S (2 · 1))) · 0))) ∧ (0 = 0 · 0 ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 ∧ 0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0)))", "target_ast_sha256": "4d5502076dff00227dc0f3c6c192d47ac1ca53b1043320c682270059c00803ef", "title": "p04 c08", "uses_definitions": [] }, { "bundle_bytes": 1540591, "bundle_sha256": "637cea85f00a69a75405dd187f22681e4e67a7350f1b11e1a77cb0df445d0865", "case": "P04-C09", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C09-HA.json.gz", "internal_proof_nodes": [ 8, 12, 21, 414, 423, 424, 425 ], "lean": "evidence/P04-C09-Lean.json.gz", "nodes": 427, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c09-v1", "proof": "evidence/P04-C09-bundle.json.gz", "report_sha256": "613d7be52e9476d88b7fca597e1f6d65af7ad3101c447267e12630db35c5738a", "root": 426, "statement": "0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0)) ∧ 0 = 0 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · 1)))))))))))))))))))))))))))))))))))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0))) ∧ (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · 1))))))))))))))))))))))))))))))))))))))))))))) = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · 1)))))))))))))))))))))))))))))))))))))) + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0 + (2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + 0 · 0)) ∧ 0 = 0 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · 1))))))))))))))))))))))))))))))))))))))) + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · 1)))))))))))))))))))))))))))))))))))))) + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + (0 · 0 + 2 · S (2 · S (2 · S (2 · 1))) · 0)) ∧ (0 = 0 · 0 ∧ 0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0))", "target_ast_sha256": "d7b8e60537999896b235e6d8e893aec799c0cfde827343add5ea0dfb53353611", "title": "p04 c09", "uses_definitions": [] }, { "bundle_bytes": 1996325, "bundle_sha256": "40c06aa37a7ab30b9755a1078abcecae7e26eceb1bb3c623ed199b9992c8145a", "case": "P04-C10", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C10-HA.json.gz", "internal_proof_nodes": [ 6, 11, 15, 20, 27, 449, 450 ], "lean": "evidence/P04-C10-Lean.json.gz", "nodes": 452, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c10-v1", "proof": "evidence/P04-C10-bundle.json.gz", "report_sha256": "0ac9fe7995311c8697f3e9b7609a637dcde57f6bf7144c867f8ec68ccae96c77", "root": 451, "statement": "0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) ∧ (0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) ∧ 0 = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + (0 · 0 + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0))) ∧ (0 = 0 · 0 + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0)) ∧ 0 = 0 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · 1)))))))))))))))))))))))))))))))))))))))))))))) + (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 0 + 0 · 0) + (2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0 + 0 · 0 + (2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + 0 · 0)) ∧ (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · 1)))))))))))))))))))))))))))))))))))))))))))))))))))) = 0 · 0 + 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · 1)))))))))))))))))))))))))))))))))))))))))))))) + (0 · 0 + 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) · 0) + (0 · 0 + 2 · S (2 · (2 · S (2 · (2 · 1)))) · 0 + (0 · 0 + 2 · S (2 · S (2 · S (2 · 1))) · 0)) ∧ 1 = 1))", "target_ast_sha256": "03b53e1693a14342d8e17250af6d1138635633cfa8f4a93403323b2d04f90610", "title": "p04 c10", "uses_definitions": [] }, { "bundle_bytes": 4948365, "bundle_sha256": "c38c21db40194a722ff5ebb5a12508206144f0622c4dd043dd19ced89757f09e", "case": "P04-C11", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C11-HA.json.gz", "internal_proof_nodes": [ 3, 85, 220, 457, 756, 1099, 1475 ], "lean": "evidence/P04-C11-Lean.json.gz", "nodes": 1477, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c11-v1", "proof": "evidence/P04-C11-bundle.json.gz", "report_sha256": "c22972b95f65ee533ab8e0ff1596c2141b14d2be62372f2df00acf9fe1e853e4", "root": 1476, "statement": "S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · 1 ∧ (S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1))))))))))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) ∧ S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))))))))))))))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · 1)))))))))))))) ∧ (S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1)))))))))))))))))))))))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · 1)))))))))))))))))))) ∧ S (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · 1))))))))))))))))))))))))))))))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · 1)))))))))))))))))))))))))) ∧ (S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · 1)))))))))))))))))))))))))))))))))))))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · S (2 · (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · 1))))))))))))))))))))))))))))))))) ∧ S (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · (2 · S (2 · (2 · S (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · S (2 · 1)))))))))))))))))))))))))))))))))))))))))))))) = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · S (2 · (2 · (2 · (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · S (2 · (2 · S (2 · S (2 · (2 · S (2 · (2 · (2 · S (2 · S (2 · (2 · S (2 · S (2 · S (2 · S (2 · S (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · (2 · S (2 · S (2 · S (2 · (2 · (2 · 1))))))))))))))))))))))))))))))))))))))))))", "target_ast_sha256": "8ae142a1eb8450325a6e51decbb05c458041dfdb5634d25386cbb702825cde05", "title": "p04 c11", "uses_definitions": [] }, { "bundle_bytes": 41188, "bundle_sha256": "9cac06c9d494e86d88aa7dff44e2fed35a5ef33f387f292aefa9c207a9f3bb57", "case": "P04-C12", "contains_checked_statements": [], "coverage": "complete_exact_subconjunction_child", "ha": "evidence/P04-C12-HA.json.gz", "internal_proof_nodes": [ 0, 3, 6, 13, 19, 37 ], "lean": "evidence/P04-C12-Lean.json.gz", "nodes": 39, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-c12-v1", "proof": "evidence/P04-C12-bundle.json.gz", "report_sha256": "d6a76f36a3f331d464753d299d2fcd6de07b851feb6f2a3c74fc35f9a78221c3", "root": 38, "statement": "1 = 1 ∧ (1 = 1 · 1 ∧ 2 · 1 = 2 · 1 · 1) ∧ (2 · S (2 · 1) = S (2 · 1) · (2 · 1) ∧ (2 · (2 · (2 · S (2 · 1))) = 2 · (2 · 1) · (2 · S (2 · 1)) ∧ 2 · (2 · (2 · S (2 · S (2 · S (2 · 1))))) = S (2 · (2 · 1)) · (2 · (2 · (2 · S (2 · 1))))))", "target_ast_sha256": "4a60e8f9088cac962514a8c81c382ce8c690d2c0342b7ef74876dd40d8a15002", "title": "p04 c12", "uses_definitions": [] }, { "bundle_bytes": 51651, "bundle_sha256": "45cecfc7deee6929545546cb47ce5878fe155cd0bff7da43a261d2c34d6f4dce", "case": "P04-soundness", "contains_checked_statements": [], "coverage": "supporting_conditional_trace_soundness", "ha": "evidence/P04-soundness-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/P04-soundness-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p04-soundness-v1", "proof": "evidence/P04-soundness-bundle.json.gz", "report_sha256": "ecbea908a68f0dd422ce56ec2214d822b0f85980938b108388a971a01f2fb9c7", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. ∀ x1. ∀ x2. ∀ x3. ∀ x4. ∀ x5. ∀ x6. ∀ x7. ∀ x8. ∀ x9. ∀ x10. ∀ x11. ∀ x12. ∀ x13. ∀ x14. ∀ x15. ∀ x16. ∀ x17. ∀ x18. ∀ x19. ∀ x20. ∀ x21. ∀ x22. ∀ x23. ∀ x24. ∀ x25. ∀ x26. ∀ x27. ∀ x28. ∀ x29. ∀ x30. ∀ x31. ∀ x32. ∀ x33. ∀ x34. ∀ x35. ∀ x36. ∀ x37. ∀ x38. ∀ x39. ∀ x40. ∀ x41. ∀ x42. ∀ x43. ∀ x44. ∀ x45. ∀ x46. ∀ x47. ∀ x48. ∀ x49. ∀ x50. ∀ x51. ∀ x52. ∀ x53. ∀ x54. ∀ x55. ∀ x56. ∀ x57. ∀ x58. ∀ x59. ∀ x60. ∀ x61. ∀ x62. ∀ x63. ∀ x64. ∀ x65. ∀ x66. ∀ x67. ∀ x68. ∀ x69. ∀ x70. ∀ x71. ∀ x72. ∀ x73. ∀ x74. ∀ x75. ∀ x76. ∀ x77. ∀ x78. ∀ x79. ∀ x80. ∀ x81. ∀ x82. ∀ x83. ∀ x84. ∀ x85. ∀ x86. ∀ x87. ∀ x88. ∀ x89. ∀ x90. ∀ x91. ∀ x92. ∀ x93. ∀ x94. x = 0 ∧ (y = 2 · S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) ∧ z = 0) ∧ (n = 2 · S (2 · S (2 · (2 · (2 · (2 · 1))))) ∧ (m = 0 ∧ k = 2 · S (2 · (2 · S (2 · (2 · 1)))))) ∧ (i = 0 ∧ (j = 2 · S (2 · S (2 · S (2 · 1))) ∧ u = 1) ∧ (v = 0 ∧ w = 0 ∧ (x0 = 0 ∧ x1 = 0))) ∧ (x2 = 0 ∧ (x3 = 0 ∧ x4 = 0) ∧ (x5 = x · u ∧ (x6 = x · v + y · u ∧ x7 = x · w + (y · v + z · u))) ∧ (x8 = x · x0 + y · w + (z · v + n · u) ∧ (x9 = x · x1 + y · x0 + (z · w + (n · v + m · u)) ∧ x10 = x · x2 + (y · x1 + z · x0) + (n · w + (m · v + k · u))) ∧ (x11 = x · x3 + (y · x2 + z · x1) + (n · x0 + m · w + (k · v + i · u)) ∧ x12 = x · x4 + y · x3 + (z · x2 + n · x1) + (m · x0 + k · w + (i · v + j · u)) ∧ (x13 = x · x5 ∧ x14 = x · x6 + y · x5)))) ∧ (x15 = x · x7 + (y · x6 + z · x5) ∧ (x16 = x · x8 + y · x7 + (z · x6 + n · x5) ∧ x17 = x · x9 + y · x8 + (z · x7 + (n · x6 + m · x5))) ∧ (x18 = x · x10 + (y · x9 + z · x8) + (n · x7 + (m · x6 + k · x5)) ∧ (x19 = x · x11 + (y · x10 + z · x9) + (n · x8 + m · x7 + (k · x6 + i · x5)) ∧ x20 = x · x12 + y · x11 + (z · x10 + n · x9) + (m · x8 + k · x7 + (i · x6 + j · x5)))) ∧ (x21 = x · x13 ∧ (x22 = x · x14 + y · x13 ∧ x23 = x · x15 + (y · x14 + z · x13)) ∧ (x24 = x · x16 + y · x15 + (z · x14 + n · x13) ∧ x25 = x · x17 + y · x16 + (z · x15 + (n · x14 + m · x13)) ∧ (x26 = x · x18 + (y · x17 + z · x16) + (n · x15 + (m · x14 + k · x13)) ∧ x27 = x · x19 + (y · x18 + z · x17) + (n · x16 + m · x15 + (k · x14 + i · x13))))) ∧ (x28 = x · x20 + y · x19 + (z · x18 + n · x17) + (m · x16 + k · x15 + (i · x14 + j · x13)) ∧ (x29 = x · x21 ∧ x30 = x · x22 + y · x21) ∧ (x31 = x · x23 + (y · x22 + z · x21) ∧ x32 = x · x24 + y · x23 + (z · x22 + n · x21) ∧ (x33 = x · x25 + y · x24 + (z · x23 + (n · x22 + m · x21)) ∧ x34 = x · x26 + (y · x25 + z · x24) + (n · x23 + (m · x22 + k · x21)))) ∧ (x35 = x · x27 + (y · x26 + z · x25) + (n · x24 + m · x23 + (k · x22 + i · x21)) ∧ (x36 = x · x28 + y · x27 + (z · x26 + n · x25) + (m · x24 + k · x23 + (i · x22 + j · x21)) ∧ x37 = x · x29) ∧ (x38 = x · x30 + y · x29 ∧ x39 = x · x31 + (y · x30 + z · x29) ∧ (x40 = x · x32 + y · x31 + (z · x30 + n · x29) ∧ x41 = x · x33 + y · x32 + (z · x31 + (n · x30 + m · x29))))))) ∧ (x42 = x · x34 + (y · x33 + z · x32) + (n · x31 + (m · x30 + k · x29)) ∧ (x43 = x · x35 + (y · x34 + z · x33) + (n · x32 + m · x31 + (k · x30 + i · x29)) ∧ x44 = x · x36 + y · x35 + (z · x34 + n · x33) + (m · x32 + k · x31 + (i · x30 + j · x29))) ∧ (x45 = x · x37 ∧ (x46 = x · x38 + y · x37 ∧ x47 = x · x39 + (y · x38 + z · x37))) ∧ (x48 = x · x40 + y · x39 + (z · x38 + n · x37) ∧ (x49 = x · x41 + y · x40 + (z · x39 + (n · x38 + m · x37)) ∧ x50 = x · x42 + (y · x41 + z · x40) + (n · x39 + (m · x38 + k · x37))) ∧ (x51 = x · x43 + (y · x42 + z · x41) + (n · x40 + m · x39 + (k · x38 + i · x37)) ∧ x52 = x · x44 + y · x43 + (z · x42 + n · x41) + (m · x40 + k · x39 + (i · x38 + j · x37)) ∧ (x53 = x · x45 ∧ x54 = x · x46 + y · x45))) ∧ (x55 = x · x47 + (y · x46 + z · x45) ∧ (x56 = x · x48 + y · x47 + (z · x46 + n · x45) ∧ x57 = x · x49 + y · x48 + (z · x47 + (n · x46 + m · x45))) ∧ (x58 = x · x50 + (y · x49 + z · x48) + (n · x47 + (m · x46 + k · x45)) ∧ (x59 = x · x51 + (y · x50 + z · x49) + (n · x48 + m · x47 + (k · x46 + i · x45)) ∧ x60 = x · x52 + y · x51 + (z · x50 + n · x49) + (m · x48 + k · x47 + (i · x46 + j · x45)))) ∧ (x61 = 1 ∧ (x62 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x61 ∧ x63 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x62) ∧ (x64 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x63 ∧ x65 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x64 ∧ (x66 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x65 ∧ x67 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x66)))) ∧ (x68 = S (2 · (2 · (2 · S (2 · (2 · S (2 · 1)))))) · x67 ∧ (x69 = 1 ∧ x70 = 1 · x69) ∧ (x71 = 2 · 1 · x70 ∧ (x72 = S (2 · 1) · x71 ∧ x73 = 2 · (2 · 1) · x72)) ∧ (x74 = S (2 · (2 · 1)) · x73 ∧ (x75 = 2 · S (2 · 1) · x74 ∧ x76 = S (2 · S (2 · 1)) · x75) ∧ (x93 = x68 · x76 ∧ x93 = S x94 ∧ (x77 · (x61 · x69) = x93 ∧ x78 · (x62 · x70) = x93))) ∧ (x79 · (x63 · x71) = x93 ∧ (x80 · (x64 · x72) = x93 ∧ x81 · (x65 · x73) = x93) ∧ (x82 · (x66 · x74) = x93 ∧ x83 · (x67 · x75) = x93 ∧ (x84 · (x68 · x76) = x93 ∧ x85 = x77 · u + x78 · x5 + (x79 · x13 + x80 · x21) + (x81 · x29 + x82 · x37 + (x83 · x45 + x84 · x53)))) ∧ (x86 = x77 · v + x78 · x6 + (x79 · x14 + x80 · x22) + (x81 · x30 + x82 · x38 + (x83 · x46 + x84 · x54)) ∧ (x87 = x77 · w + x78 · x7 + (x79 · x15 + x80 · x23) + (x81 · x31 + x82 · x39 + (x83 · x47 + x84 · x55)) ∧ x88 = x77 · x0 + x78 · x8 + (x79 · x16 + x80 · x24) + (x81 · x32 + x82 · x40 + (x83 · x48 + x84 · x56))) ∧ (x89 = x77 · x1 + x78 · x9 + (x79 · x17 + x80 · x25) + (x81 · x33 + x82 · x41 + (x83 · x49 + x84 · x57)) ∧ x90 = x77 · x2 + x78 · x10 + (x79 · x18 + x80 · x26) + (x81 · x34 + x82 · x42 + (x83 · x50 + x84 · x58)) ∧ (x91 = x77 · x3 + x78 · x11 + (x79 · x19 + x80 · x27) + (x81 · x35 + x82 · x43 + (x83 · x51 + x84 · x59)) ∧ x92 = x77 · x4 + x78 · x12 + (x79 · x20 + x80 · x28) + (x81 · x36 + x82 · x44 + (x83 · x52 + x84 · x60)))))))) → x85 = x93", "target_ast_sha256": "b8f9f93c128ddf06790122276e27c6e818daf7c31239959e4675c558002a189f", "title": "p04 soundness", "uses_definitions": [] }, { "bundle_bytes": 18790, "bundle_sha256": "bfb83993603f8fc7413d0c87c264a030cb33d1859bacc853f278ad576545642c", "case": "P08-eprover", "contains_checked_statements": [], "coverage": "P08_supporting_leaf_not_full_pilot", "ha": "evidence/P08-eprover-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/P08-eprover-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p08-e-v1", "proof": "evidence/P08-eprover-bundle.json.gz", "report_sha256": "d4c486bd0dae0b14fb5d2611c38abdedb7b32788b514f56f26e699b91f089ee4", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. x · z = y → x · (n · z) = n · y", "target_ast_sha256": "5ab0ba252962549290db0651035439a1e1ab719b485f7c91893895b7a9ce8352", "title": "p08 e", "uses_definitions": [] }, { "bundle_bytes": 18790, "bundle_sha256": "bfb83993603f8fc7413d0c87c264a030cb33d1859bacc853f278ad576545642c", "case": "P08-vampire", "contains_checked_statements": [], "coverage": "P08_supporting_leaf_not_full_pilot", "ha": "evidence/P08-vampire-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/P08-vampire-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p08-v-v1", "proof": "evidence/P08-vampire-bundle.json.gz", "report_sha256": "d57db2aa9e9810cda524d6696277e96cc35c580038ab7e5b712ab53f42fb36a6", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. x · z = y → x · (n · z) = n · y", "target_ast_sha256": "5ab0ba252962549290db0651035439a1e1ab719b485f7c91893895b7a9ce8352", "title": "p08 v", "uses_definitions": [] }, { "bundle_bytes": 6015411, "bundle_sha256": "5e54a78d26b2ab9c9ea9159abbe793281c84bf9baab4312bf9bb0cbb59dae9f4", "case": "P10", "contains_checked_statements": [], "coverage": "full_fixed_P10_not_IR064", "ha": "evidence/P10-HA.json.gz", "internal_proof_nodes": [ 1484 ], "lean": "evidence/P10-Lean.json.gz", "nodes": 1486, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-p10-v1", "proof": "evidence/P10-bundle.json.gz", "report_sha256": "bd2a89f904da19b0c8f0991b361fe1374ee546cc67f0aa5e636cf1f94b147dea", "root": 1485, "statement": "∃ x. 2 · (2 · 2) · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · 2 · (2 · 2))) · (2 · (2 · 2) · (2 · 2 · (2 · 2)) · (2 · (2 · 2) · (2 · 2 · (2 · 2)))) · (2 · (2 · 2) · (2 · 2 · (2 · 2)) · (2 · (2 · 2) · (2 · 2 · (2 · 2))) · (2 · (2 · 2) · (2 · 2 · (2 · 2)) · (2 · (2 · 2) · (2 · 2 · (2 · 2))))) · (3 · 3 · (3 · (3 · 3)) · (3 · (3 · 3) · (3 · (3 · 3))) · (7 · 7 · (7 · (7 · 7)))) + x = 2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2)))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2))))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2)))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2))) · (2 · 2 · (2 · (2 · 2)) · (2 · (2 · 2) · (2 · (2 · 2))))))", "target_ast_sha256": "80dd8b78cf3a77c81e05f935df83684e6c758c2614521a877530097572cb61ee", "title": "p10", "uses_definitions": [] }, { "bundle_bytes": 3240416, "bundle_sha256": "4933e2fe5136b4c700fbe804b81d20ba2331ec53c1f0a18501bab03becf2675f", "case": "QF001", "contains_checked_statements": [ "IR016", "QN001", "SI001", "SN001", "SN003" ], "coverage": "quadratic_finite_trace_nonzero_not_trace_totality_or_real_IR046", "ha": "evidence/QF001-HA.json.gz", "internal_proof_nodes": [ 12, 13, 221, 222, 220 ], "lean": "evidence/QF001-Lean.json.gz", "nodes": 224, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-quadratic-product-trace-v1", "proof": "evidence/QF001-bundle.json.gz", "report_sha256": "26d7d75b2d04ad7bd43f4600d809570c94651be99800a71b719180e7a8668789", "root": 223, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. ∀ x1. ∀ x2. ∀ x3. ∀ x4. ∀ x5. ∀ x6. ∀ x7. ∀ x8. ∀ x9. 2 ≤ S (1 · v) ∧ (∃ x10. u = x10 · S (1 · v) + 1) ∧ (1 ≤ S (1 · x0) ∧ (∃ x10. w = x10 · S (1 · x0) + 0) ∧ (1 ≤ S (1 · x2) ∧ (∃ x10. x1 = x10 · S (1 · x2) + 0) ∧ (1 ≤ S (1 · x4) ∧ (∃ x10. x3 = x10 · S (1 · x4) + 0)))) ∧ (∀ x10. S x10 ≤ x5 → ∃ x11. ∃ x12. ∃ x13. ∃ x14. ∃ x15. ∃ x16. ∃ x17. ∃ x18. ∃ x19. ∃ x20. ∃ x21. ∃ x22. S x15 ≤ S (S x10 · y) ∧ (∃ x23. x = x23 · S (S x10 · y) + x15) ∧ (S x16 ≤ S (S x10 · n) ∧ (∃ x23. z = x23 · S (S x10 · n) + x16) ∧ (S x17 ≤ S (S x10 · k) ∧ (∃ x23. m = x23 · S (S x10 · k) + x17) ∧ (S x18 ≤ S (S x10 · j) ∧ (∃ x23. i = x23 · S (S x10 · j) + x18)))) ∧ (S x11 ≤ S (S x10 · v) ∧ (∃ x23. u = x23 · S (S x10 · v) + x11) ∧ (S x12 ≤ S (S x10 · x0) ∧ (∃ x23. w = x23 · S (S x10 · x0) + x12) ∧ (S x13 ≤ S (S x10 · x2) ∧ (∃ x23. x1 = x23 · S (S x10 · x2) + x13) ∧ (S x14 ≤ S (S x10 · x4) ∧ (∃ x23. x3 = x23 · S (S x10 · x4) + x14)))) ∧ (S x19 ≤ S (S S x10 · v) ∧ (∃ x23. u = x23 · S (S S x10 · v) + x19) ∧ (S x20 ≤ S (S S x10 · x0) ∧ (∃ x23. w = x23 · S (S S x10 · x0) + x20) ∧ (S x21 ≤ S (S S x10 · x2) ∧ (∃ x23. x1 = x23 · S (S S x10 · x2) + x21) ∧ (S x22 ≤ S (S S x10 · x4) ∧ (∃ x23. x3 = x23 · S (S S x10 · x4) + x22)))) ∧ (x19 + (x11 · x16 + x12 · x15 + 2 · (x13 · x18 + x14 · x17)) = x11 · x15 + x12 · x16 + 2 · (x13 · x17 + x14 · x18) + x20 ∧ x21 + (x11 · x18 + x12 · x17 + (x13 · x16 + x14 · x15)) = x11 · x17 + x12 · x18 + (x13 · x15 + x14 · x16) + x22)))) → (∀ x10. ∀ x11. ∀ x12. ∀ x13. ∀ x14. S x10 ≤ x5 → S x11 ≤ S (S x10 · y) ∧ (∃ x15. x = x15 · S (S x10 · y) + x11) ∧ (S x12 ≤ S (S x10 · n) ∧ (∃ x15. z = x15 · S (S x10 · n) + x12) ∧ (S x13 ≤ S (S x10 · k) ∧ (∃ x15. m = x15 · S (S x10 · k) + x13) ∧ (S x14 ≤ S (S x10 · j) ∧ (∃ x15. i = x15 · S (S x10 · j) + x14)))) → ¬(x11 = x12 ∧ x13 = x14)) → S x6 ≤ S (S x5 · v) ∧ (∃ x10. u = x10 · S (S x5 · v) + x6) ∧ (S x7 ≤ S (S x5 · x0) ∧ (∃ x10. w = x10 · S (S x5 · x0) + x7) ∧ (S x8 ≤ S (S x5 · x2) ∧ (∃ x10. x1 = x10 · S (S x5 · x2) + x8) ∧ (S x9 ≤ S (S x5 · x4) ∧ (∃ x10. x3 = x10 · S (S x5 · x4) + x9)))) → ¬(x6 = x7 ∧ x8 = x9)", "target_ast_sha256": "d27f07d45ef919c5b7597e15e774478ab7ae46fc82e18fd16fa82e860df90768", "title": "quadratic product trace", "uses_definitions": [] }, { "bundle_bytes": 207926, "bundle_sha256": "a27dba505ab5151a5842f58f5537219f9b5958bd3310103b98ac0dc6dff29b98", "case": "QF002", "contains_checked_statements": [], "coverage": "unconditional_integer_quadratic_product_trace_existence_not_real_IR046", "ha": "evidence/QF002-HA.json.gz", "internal_proof_nodes": [ 23, 36, 104, 106 ], "lean": "evidence/QF002-Lean.json.gz", "nodes": 108, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-quadratic-trace-exists-v2", "proof": "evidence/QF002-bundle.json.gz", "report_sha256": "006d1e3f21dcd7a0c5c8e7c235c6e5d08899dfa21baf0eb186365b39692c6f85", "root": 107, "statement": "forall ab ac bb bc cb cc db dc L. exists ub uc vb vc wb wc xb xc. IQuadProductTrace(ab,ac,bb,bc,cb,cc,db,dc,ub,uc,vb,vc,wb,wc,xb,xc,L)", "target_ast_sha256": "bce1bbd903c3dd33cd47e30166931e5de2a0dde0f5d058bab6051dcfa4e6400b", "title": "quadratic trace exists", "uses_definitions": [ "ND0392" ] }, { "bundle_bytes": 3376900, "bundle_sha256": "ed9f56f1fe3ea3f2e7df33ca725a9163b82883b9364433190cf3001be0abe1e4", "case": "QF003", "contains_checked_statements": [ "IR016", "QF001", "QF002", "QN001", "SI001", "SN001", "SN003" ], "coverage": "witnessed_nonzero_integer_quadratic_finite_product_not_real_IR046", "ha": "evidence/QF003-HA.json.gz", "internal_proof_nodes": [ 223, 270, 266 ], "lean": "evidence/QF003-Lean.json.gz", "nodes": 272, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-quadratic-product-exists-v1", "proof": "evidence/QF003-bundle.json.gz", "report_sha256": "ef364c33aca15a631a42f810cc7a4b3611eb4671f98311d12ae61dfd10a6c275", "root": 271, "statement": "forall ab ac bb bc cb cc db dc L. IQuadNonzeroFactors(ab,ac,bb,bc,cb,cc,db,dc,L) -> exists ub uc vb vc wb wc xb xc rp rn sp sn. IQuadProductTrace(ab,ac,bb,bc,cb,cc,db,dc,ub,uc,vb,vc,wb,wc,xb,xc,L) /\\ (IQuadAt(ub,uc,vb,vc,wb,wc,xb,xc,L,rp,rn,sp,sn) /\\ ~(rp=rn /\\ sp=sn))", "target_ast_sha256": "d8aa407a2364ad85cd2dec414b57462a4c78a2efe4b79b49e06a0b0bd78acb2d", "title": "quadratic product exists", "uses_definitions": [ "ND0390", "ND0392", "ND0393" ] }, { "bundle_bytes": 207847, "bundle_sha256": "405d151890344410934566e315ed6ec36bf471674f0c6e78f2122b0230678d01", "case": "QF004", "contains_checked_statements": [], "coverage": "canonical_integer_quadratic_trace_existence_not_finite_height_or_real_IR046", "ha": "evidence/QF004-HA.json.gz", "internal_proof_nodes": [ 23, 36, 104, 106 ], "lean": "evidence/QF004-Lean.json.gz", "nodes": 108, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-canonical-quadratic-trace-v1", "proof": "evidence/QF004-bundle.json.gz", "report_sha256": "8b332708d4d56e04db3796a1e2b1a0ffb2b6a08b874450bb00304e7fa336991e", "root": 107, "statement": "forall ab ac bb bc cb cc db dc L. exists ub uc vb vc wb wc xb xc. IQuadCanonicalTrace(ab,ac,bb,bc,cb,cc,db,dc,ub,uc,vb,vc,wb,wc,xb,xc,L)", "target_ast_sha256": "9c459742c8eb742cb4958f037c3b7492a8f773e609ef0f8e1d63f541c2e0a9b1", "title": "canonical quadratic trace", "uses_definitions": [ "ND0397" ] }, { "bundle_bytes": 3513591, "bundle_sha256": "c46933a920ba23989841a3f8e243c32c16a127b9a15cf699cd8d4b4ab448bad0", "case": "QF005", "contains_checked_statements": [ "IR016", "QF001", "QF004", "QH001", "QH002", "QN001", "SI001", "SN001", "SN003" ], "coverage": "constructed_bounded_nonzero_integer_quadratic_finite_product_not_real_IR046", "ha": "evidence/QF005-HA.json.gz", "internal_proof_nodes": [ 108, 277, 273, 294, 272, 295 ], "lean": "evidence/QF005-Lean.json.gz", "nodes": 297, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-bounded-quadratic-product-v1", "proof": "evidence/QF005-bundle.json.gz", "report_sha256": "9011ac4f16334663f692cebb86f0a10fac155b681629741afefe92e10e5989c8", "root": 296, "statement": "forall ab ac bb bc cb cc db dc L H. IQuadFactorHeightBound(ab,ac,bb,bc,cb,cc,db,dc,L,H) -> IQuadNonzeroFactors(ab,ac,bb,bc,cb,cc,db,dc,L) -> exists N ub uc vb vc wb wc xb xc rp rn sp sn. Pow(H,L,N) /\\ (IQuadCanonicalTrace(ab,ac,bb,bc,cb,cc,db,dc,ub,uc,vb,vc,wb,wc,xb,xc,L) /\\ (IQuadAt(ub,uc,vb,vc,wb,wc,xb,xc,L,rp,rn,sp,sn) /\\ (IQuadHeightBound(rp,rn,sp,sn,N) /\\ ~(rp=rn /\\ sp=sn))))", "target_ast_sha256": "dcb0b757df5bea77acb675edba870bb7cc99be0a5f700ee1782c1f55f324ec14", "title": "bounded quadratic product", "uses_definitions": [ "ND0390", "ND0393", "ND0397", "PD0020", "ND0398", "ND0399" ] }, { "bundle_bytes": 54153, "bundle_sha256": "d23dff4f88d369f48781cbe4293f474c6c16308a5679661b6cdd5295eb4c88ba", "case": "QH001", "contains_checked_statements": [], "coverage": "canonical_integer_quadratic_binary_height_not_finite_or_real_IR046", "ha": "evidence/QH001-HA.json.gz", "internal_proof_nodes": [ 8, 9, 10, 11 ], "lean": "evidence/QH001-Lean.json.gz", "nodes": 13, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-quadratic-height-v2", "proof": "evidence/QH001-bundle.json.gz", "report_sha256": "6c65f06642366525e03b76fa822f684f2699b9006c30408c8b410c87d65676c3", "root": 12, "statement": "forall ap an bp bn cp cn dp dn rp rn sp sn a b c. IQuadCanonicalProduct(ap,an,bp,bn,cp,cn,dp,dn,rp,rn,sp,sn) -> IQuadHeight(ap,an,bp,bn,a) -> IQuadHeight(cp,cn,dp,dn,b) -> IQuadHeight(rp,rn,sp,sn,c) -> exists gap. gap+c=a*b", "target_ast_sha256": "b9459095385fa95c3e8d4d7ff638db6464c2053389c1921031dabfe88560f7e6", "title": "quadratic height", "uses_definitions": [ "ND0394", "ND0395" ] }, { "bundle_bytes": 141562, "bundle_sha256": "080feebc1b7dcc21e09c4d259e5922341b0f4267f9df0e6b52fa2135c7f70ea2", "case": "QH002", "contains_checked_statements": [ "QH001" ], "coverage": "canonical_integer_quadratic_finite_height_not_real_IR046", "ha": "evidence/QH002-HA.json.gz", "internal_proof_nodes": [ 0, 9, 10, 15, 17, 18, 21, 25, 26, 39 ], "lean": "evidence/QH002-Lean.json.gz", "nodes": 41, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-quadratic-trace-height-v1", "proof": "evidence/QH002-bundle.json.gz", "report_sha256": "8de8285b1d3de12e53e49eaf5588e9d97a904d3500a04422f7ece99a267f03bb", "root": 40, "statement": "forall ab ac bb bc cb cc db dc ub uc vb vc wb wc xb xc L H N rp rn sp sn. IQuadCanonicalTrace(ab,ac,bb,bc,cb,cc,db,dc,ub,uc,vb,vc,wb,wc,xb,xc,L) -> IQuadFactorHeightBound(ab,ac,bb,bc,cb,cc,db,dc,L,H) -> Pow(H,L,N) -> IQuadAt(ub,uc,vb,vc,wb,wc,xb,xc,L,rp,rn,sp,sn) -> IQuadHeightBound(rp,rn,sp,sn,N)", "target_ast_sha256": "e68c7893269115ba0b6b6505c3974c336b5fb98db8d47c2d3ef227003f8a388c", "title": "quadratic trace height", "uses_definitions": [ "ND0390", "ND0397", "PD0020", "ND0398", "ND0399" ] }, { "bundle_bytes": 3214045, "bundle_sha256": "faeb2544f48735855f80ba183dc596190cc4d788f804ed57af793bd6a7b89aa3", "case": "QN001", "contains_checked_statements": [ "IR016", "SI001", "SN001", "SN003" ], "coverage": "quadratic_integer_binary_nonzero_not_finite_IR046", "ha": "evidence/QN001-HA.json.gz", "internal_proof_nodes": [ 191, 214, 217, 215, 216, 218, 219 ], "lean": "evidence/QN001-Lean.json.gz", "nodes": 221, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-quadratic-nonzero-v1", "proof": "evidence/QN001-bundle.json.gz", "report_sha256": "0a548fe5be2d9083bf61eb5ce076ee2c4ebb2e3bf2f9a3281b43f7214a418375", "root": 220, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. u + (x · k + y · m + 2 · (z · j + n · i)) = x · m + y · k + 2 · (z · i + n · j) + v → w + (x · j + y · i + (z · k + n · m)) = x · i + y · j + (z · m + n · k) + x0 → ¬(x = y ∧ z = n) → ¬(m = k ∧ i = j) → ¬(u = v ∧ w = x0)", "target_ast_sha256": "f4d0bc4f03497b4b9d2b601380b0958486e631f8a2c5f4d0b647b43662f18ca9", "title": "quadratic nonzero", "uses_definitions": [] }, { "bundle_bytes": 3319, "bundle_sha256": "7417b07b645044b957b2977ee93d9769fb144f10622c64d29e3e22053fa0976f", "case": "RF001", "contains_checked_statements": [], "coverage": "full_named_rational_foundation_not_IR001", "ha": "evidence/RF001-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/RF001-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rf001-v1", "proof": "evidence/RF001-bundle.json.gz", "report_sha256": "1f1073dffac5de1bec24180109faee472ad4d4fd7a48bff122944ccdc72ca44e", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ¬z = 0 → ¬z = 0 ∧ (¬z = 0 ∧ x · z + y · z = y · z + x · z)", "target_ast_sha256": "e722f9ad9273a9bda607c55d52fa50d9db3ce71149d160f14a6a76ad24abbd63", "title": "rf001", "uses_definitions": [] }, { "bundle_bytes": 6478, "bundle_sha256": "2771ec79c87230edd6493a9bdae4fee95b4f6499905c51932b810e5f95b06a98", "case": "RF002", "contains_checked_statements": [], "coverage": "full_named_rational_foundation_not_IR001", "ha": "evidence/RF002-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/RF002-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rf002-v1", "proof": "evidence/RF002-bundle.json.gz", "report_sha256": "5af90c62b3522ea2e2475b2f7b01b8ad8180f8fbb8cc3eef43b1c3aefb05911d", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ¬z = 0 ∧ (¬k = 0 ∧ x · k + m · z = y · k + n · z) → ¬k = 0 ∧ (¬z = 0 ∧ n · z + y · k = m · z + x · k)", "target_ast_sha256": "f7d7e191f77999e19d76738748486cbece8288613c749b20716d0a473d6afaba", "title": "rf002", "uses_definitions": [] }, { "bundle_bytes": 2643423, "bundle_sha256": "44f7e7884fd34f582768032ea2b91b9a24e45744b83f23368eae49755d0aebbf", "case": "RF003", "contains_checked_statements": [], "coverage": "full_named_rational_foundation_not_IR001", "ha": "evidence/RF003-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/RF003-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rf003-typed-v2", "proof": "evidence/RF003-bundle.json.gz", "report_sha256": "3c28bc45cb18fb3f7a88a843eb1496e9c17534d855df080aef7a0196336a2340", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ¬z = 0 ∧ (¬k = 0 ∧ x · k + m · z = y · k + n · z) → ¬k = 0 ∧ (¬u = 0 ∧ n · u + j · k = m · u + i · k) → ¬z = 0 ∧ (¬u = 0 ∧ x · u + j · z = y · u + i · z)", "target_ast_sha256": "ece8e6152f2026a946640ae397e7bb249f4e0afa100380d3831522708328c8e8", "title": "rf003 typed", "uses_definitions": [] }, { "bundle_bytes": 662272, "bundle_sha256": "429fa6e2ea7a800c115327a3c465951c47da8e46ffcc71e40e3654e61c1303f0", "case": "RF004", "contains_checked_statements": [], "coverage": "full_named_rational_foundation_not_IR001", "ha": "evidence/RF004-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/RF004-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rf004-v1", "proof": "evidence/RF004-bundle.json.gz", "report_sha256": "f95c5942aeb6fa7aad063e01b987fab783b2e823beac64621c80639b19c8c1a7", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ¬z = 0 → ¬n = 0 → ¬z = 0 ∧ (¬z · n = 0 ∧ x · (z · n) + y · n · z = y · (z · n) + x · n · z)", "target_ast_sha256": "b07b8ed827c19e15e27f3dd746a89445ec5326208631ec81e298da144cd716fb", "title": "rf004", "uses_definitions": [] }, { "bundle_bytes": 513032, "bundle_sha256": "2a65c3ea35af78fb33f9939d215a4a11169112140ba30faa0726a22ec2d51db7", "case": "RF005", "contains_checked_statements": [], "coverage": "full_named_rational_foundation_not_IR001", "ha": "evidence/RF005-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/RF005-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rf005-v1", "proof": "evidence/RF005-bundle.json.gz", "report_sha256": "4d712d6efdacf3d676498c27d8c57f53c520b31785e4d1ae016aee0a8719ff40", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ¬z = 0 → ¬z = 0 ∧ (¬z = 0 ∧ x · z + (y + n) · z = y · z + (x + n) · z)", "target_ast_sha256": "75ff1c4c0f8885d395438dd9f33aaf6274d3fd1e4d34e982a9db50255d05105e", "title": "rf005", "uses_definitions": [] }, { "bundle_bytes": 1385, "bundle_sha256": "73a5ab398f643e71657f904a78d43e58a8c327ded87ae1eecb6376a5b3389aba", "case": "RF006", "contains_checked_statements": [], "coverage": "full_named_rational_foundation_not_IR001", "ha": "evidence/RF006-HA.json.gz", "internal_proof_nodes": [], "lean": "evidence/RF006-Lean.json.gz", "nodes": 1, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rf006-v1", "proof": "evidence/RF006-bundle.json.gz", "report_sha256": "de5efeaf6dcdf559314deac3dd29d50496e2a6197f531dd7813130611cf6b84e", "root": 0, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ¬z = 0 ∧ (¬k = 0 ∧ x · k + m · z = y · k + n · z) → ¬z = 0 ∧ (¬k = 0 ∧ y · k + n · z = x · k + m · z)", "target_ast_sha256": "c23712611e4fa9d7630c08e3f0543d4d8d6df6e7b03c01b0b5c02da8ea18c135", "title": "rf006", "uses_definitions": [] }, { "bundle_bytes": 168056, "bundle_sha256": "795cd64495f9ab37dc7b12add368225567bbb63bcca6e1f698ffbd08db431667", "case": "RN001", "contains_checked_statements": [], "coverage": "rational_norm_representative_transport_not_full_IR031", "ha": "evidence/RN001-HA.json.gz", "internal_proof_nodes": [ 2, 6, 8, 14, 21, 22, 27 ], "lean": "evidence/RN001-Lean.json.gz", "nodes": 29, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rational-norm-transport-v1", "proof": "evidence/RN001-bundle.json.gz", "report_sha256": "4b59b15ff66349238914c5fd959499914664623024adf792e4feea6c791b54a5", "root": 28, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. ∀ x1. ∀ x2. ¬m = 0 ∧ (¬v = 0 ∧ x · v + i · m = y · v + k · m) → ¬m = 0 ∧ (¬v = 0 ∧ z · v + u · m = n · v + j · m) → x · x + y · y = w + (x · y + y · x) → z · z + n · n = x0 + (z · n + n · z) → k · k + i · i = x1 + (k · i + i · k) → j · j + u · u = x2 + (j · u + u · j) → ¬m · m = 0 ∧ (¬v · v = 0 ∧ w · (v · v) + 2 · x2 · (m · m) = 2 · x0 · (v · v) + x1 · (m · m))", "target_ast_sha256": "db34efff2aecd8c2f5afdb13cad631533c9dd9cbe2de2a878318cd31ac2290da", "title": "rational norm transport", "uses_definitions": [] }, { "bundle_bytes": 1838656, "bundle_sha256": "70c033412b498cb8e180a800c45773610adb27aedad70716b6cc0e332edd9c68", "case": "RN002", "contains_checked_statements": [ "SN003" ], "coverage": "rational_norm_product_denominator_not_full_IR031", "ha": "evidence/RN002-HA.json.gz", "internal_proof_nodes": [ 36, 37, 38, 39, 40, 41, 42, 43, 44 ], "lean": "evidence/RN002-Lean.json.gz", "nodes": 46, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-rational-norm-product-v1", "proof": "evidence/RN002-bundle.json.gz", "report_sha256": "d348dafc788aacd7932ab27d2186f4315c9d6536e22547fe4eff12d3ae478dac", "root": 45, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. ∀ x1. ∀ x2. ∀ x3. ∀ x4. ∀ x5. ∀ x6. ∀ x7. ∀ x8. ¬x7 = 0 → ¬x8 = 0 → x · x + y · y = x1 + (x · y + y · x) → z · z + n · n = x2 + (z · n + n · z) → m · m + k · k = x3 + (m · k + k · m) → i · i + j · j = x4 + (i · j + j · i) → u · u + v · v = x5 + (u · v + v · u) → w · w + x0 · x0 = x6 + (w · x0 + x0 · w) → u + (x · k + y · m + 2 · (z · j + n · i)) = x · m + y · k + 2 · (z · i + n · j) + v → w + (x · j + y · i + (z · k + n · m)) = x · i + y · j + (z · m + n · k) + x0 → ¬x7 · x7 = 0 ∧ (¬x8 · x8 = 0 ∧ (¬x7 · x8 · (x7 · x8) = 0 ∧ (¬x7 · x7 · (x8 · x8) = 0 ∧ x5 · (x7 · x7 · (x8 · x8)) + (x1 · (2 · x4) + 2 · x2 · x3) · (x7 · x8 · (x7 · x8)) = 2 · x6 · (x7 · x7 · (x8 · x8)) + (x1 · x3 + 2 · x2 · (2 · x4)) · (x7 · x8 · (x7 · x8)))))", "target_ast_sha256": "ab88f7b597011f85f10d543e582abb37db52852d940b4b7a8c4c2cf7ae22c335", "title": "rational norm product", "uses_definitions": [] }, { "bundle_bytes": 153728, "bundle_sha256": "e0fd91506b33ed8a390173ed0599e8980e2c99ade59b24c37be92cf05bcfbc38", "case": "SI001", "contains_checked_statements": [], "coverage": "signed_integer_binary_nonzero_not_quadratic_finite_IR046", "ha": "evidence/SI001-HA.json.gz", "internal_proof_nodes": [ 26, 31, 32, 11, 12, 2 ], "lean": "evidence/SI001-Lean.json.gz", "nodes": 34, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-signed-product-nonzero-v1", "proof": "evidence/SI001-bundle.json.gz", "report_sha256": "095e0269a7ecfd0ce7ec2e4fda9a79b660779c37d65e408f9a87522e7e040950", "root": 33, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. m + (x · n + y · z) = x · z + y · n + k → ¬x = y → ¬z = n → ¬m = k", "target_ast_sha256": "ee1a6a3a9dfc5f7d45c67d4a040a8ff0ccf2d4422f2ddc42e1fad69334c2d581", "title": "signed product nonzero", "uses_definitions": [] }, { "bundle_bytes": 1439991, "bundle_sha256": "5685db9bdbe1ceb79e29a25dff767c519ccf424705a1e395dad5fcb3b01911de", "case": "SN001", "contains_checked_statements": [ "IR016" ], "coverage": "signed_norm_zero_core_not_IR032", "ha": "evidence/SN001-HA.json.gz", "internal_proof_nodes": [ 181, 202, 201, 203, 204 ], "lean": "evidence/SN001-Lean.json.gz", "nodes": 206, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-signed-norm-v1", "proof": "evidence/SN001-bundle.json.gz", "report_sha256": "55335796b5d134ac98f397e3dbd070e23829bd8747fef847c023b1612fcdfd72", "root": 205, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. x · x + y · y = m + (x · y + y · x) → z · z + n · n = k + (z · n + n · z) → m = 2 · k → x = y ∧ z = n", "target_ast_sha256": "c948aa0e7350e5c51c464721215573a489b0921c9d1a3263c9c5181f20164941", "title": "signed norm", "uses_definitions": [] }, { "bundle_bytes": 1446935, "bundle_sha256": "3b4257b6bd12ed44eb2de6342652a69480045fc09fe63dd62b9d94d0eabb2dc6", "case": "SN002", "contains_checked_statements": [ "IR016", "NG001", "SN001" ], "coverage": "signed_norm_integer_gap_not_full_IR032", "ha": "evidence/SN002-HA.json.gz", "internal_proof_nodes": [ 205, 208 ], "lean": "evidence/SN002-Lean.json.gz", "nodes": 210, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-norm-separation-v1", "proof": "evidence/SN002-bundle.json.gz", "report_sha256": "097840f02d46821cda83418b20728afdd5f87730f87c9660ec3eaa4fb617bcff", "root": 209, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. x · x + y · y = m + (x · y + y · x) → z · z + n · n = k + (z · n + n · z) → ¬(x = y ∧ z = n) → ∃ i. m + S i = 2 · k ∨ 2 · k + S i = m", "target_ast_sha256": "7a0c83cccc0c50cb487d5dda5fc177677b4aeb7758ac7179e2f46023dcccd4c5", "title": "norm separation", "uses_definitions": [] }, { "bundle_bytes": 1794186, "bundle_sha256": "93465c3527692a4b2614c0b54d30457b7f70682203477afd9afe9775d3977ebf", "case": "SN003", "contains_checked_statements": [], "coverage": "signed_norm_multiplicativity_not_rational_IR031", "ha": "evidence/SN003-HA.json.gz", "internal_proof_nodes": [ 19, 23, 26, 31, 32, 13, 33, 34, 35 ], "lean": "evidence/SN003-Lean.json.gz", "nodes": 37, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-norm-product-v1", "proof": "evidence/SN003-bundle.json.gz", "report_sha256": "13fc6ea3d3c390b315e972cee397adcc873796e242c1f65459866e64be477567", "root": 36, "statement": "∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. ∀ x1. ∀ x2. ∀ x3. ∀ x4. ∀ x5. ∀ x6. x · x + y · y = x1 + (x · y + y · x) → z · z + n · n = x2 + (z · n + n · z) → m · m + k · k = x3 + (m · k + k · m) → i · i + j · j = x4 + (i · j + j · i) → u · u + v · v = x5 + (u · v + v · u) → w · w + x0 · x0 = x6 + (w · x0 + x0 · w) → u + (x · k + y · m + 2 · (z · j + n · i)) = x · m + y · k + 2 · (z · i + n · j) + v → w + (x · j + y · i + (z · k + n · m)) = x · i + y · j + (z · m + n · k) + x0 → x5 + 2 · (x1 · x4 + x2 · x3) = x1 · x3 + 2 · 2 · (x2 · x4) + 2 · x6", "target_ast_sha256": "e123c3a589c565bfb3da565b09d0d7fd4e0ca431f884f790f02899bc10711bfa", "title": "norm product", "uses_definitions": [] }, { "bundle_bytes": 29866, "bundle_sha256": "c67e379ff1997694a46682d276a9f81661370a42a311190580828229450435de", "case": "SQ001", "contains_checked_statements": [], "coverage": "unconditional_floor_square_root_totality_and_uniqueness_from_exact_replayed_Bertrand_ancestor_bodies_and_original_conservative_FloorSqrt_definition_not_dyadic_accuracy_or_real_interpretation", "ha": "evidence/SQ001-HA.json.gz", "internal_proof_nodes": [ 28, 29, 10, 15 ], "lean": "evidence/SQ001-Lean.json.gz", "nodes": 31, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-floor-sqrt-v1", "proof": "evidence/SQ001-bundle.json.gz", "report_sha256": "40c38cc4bd68078914582831715bed7c3c935cb09b67fac3eff73cf2a41aab47", "root": 30, "statement": "forall n. exists s. FloorSqrt(n,s) /\\ forall t. FloorSqrt(n,t) -> t=s", "target_ast_sha256": "da6b93cc7d0f9f4458be509f5f2405cac0063f79ab70acfef60768c09a0f2d5c", "title": "floor sqrt", "uses_definitions": [ "PD0051" ] }, { "bundle_bytes": 18278, "bundle_sha256": "ee5be134efec33276a298e8ac6119db809740d9fe595e3586e9c51d175e60f96", "case": "SQ002", "contains_checked_statements": [], "coverage": "actual_floor_root_lies_between_roots_of_supplied_nonstrict_lower_and_strict_upper_square_bounds_via_replayed_natural_decidable_order_no_real_number_or_accuracy_axiom", "ha": "evidence/SQ002-HA.json.gz", "internal_proof_nodes": [ 14, 10, 9, 11 ], "lean": "evidence/SQ002-Lean.json.gz", "nodes": 16, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-floor-sqrt-comparison-v1", "proof": "evidence/SQ002-bundle.json.gz", "report_sha256": "efdf6befcde250e60a342e473b3d8edc55c331266ec714c4723f4ad550cc6b26", "root": 15, "statement": "forall n s a b. FloorSqrt(n,s) -> Le(a*a,n) -> Lt(n,b*b) -> Le(a,s) /\\ Lt(s,b)", "target_ast_sha256": "87748e18b800d65594ab1288e0586b089302a29354fd0c34e978ab8a54a76e5a", "title": "floor sqrt comparison", "uses_definitions": [ "PD0001", "PD0002", "PD0051" ] }, { "bundle_bytes": 43642, "bundle_sha256": "05d951bee0ed77b01b03ff51e51da3ea4e8349a397c563cbe6b119670a83271f", "case": "SQ003", "contains_checked_statements": [ "SQ001", "SQ002" ], "coverage": "construct_unique_actual_floor_root_of_twice_any_positive_scale_squared_and_prove_it_lies_between_the_scale_and_twice_scale_with_nonzero_root_no_analytic_interpretation", "ha": "evidence/SQ003-HA.json.gz", "internal_proof_nodes": [ 30, 32, 39, 40 ], "lean": "evidence/SQ003-Lean.json.gz", "nodes": 42, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-scaled-sqrt2-v2", "proof": "evidence/SQ003-bundle.json.gz", "report_sha256": "9720483066b3fd9f154aec44b7f76b07b2402e2c6393b7d34addf6f7fee5016b", "root": 41, "statement": "forall d. ~d=0 -> exists s. (FloorSqrt(2*(d*d),s)) /\\ (Le(d,s)) /\\ (Lt(s,2*d)) /\\ (~s=0) /\\ (forall t. FloorSqrt(2*(d*d),t) -> t=s)", "target_ast_sha256": "4e1139169e23ee835cb5f0eb9371fa4b6a641079db34b76bcb25f692bd88efb8", "title": "scaled sqrt2", "uses_definitions": [ "PD0001", "PD0002", "PD0051" ] }, { "bundle_bytes": 252950, "bundle_sha256": "f73908713a8b4912da9bc340d3a4e9e51f8ef156993f9527040eaaad0cf34ebf", "case": "SQ004", "contains_checked_statements": [ "SQ001", "SQ002", "SQ003" ], "coverage": "unconditional_total_unique_positive_dyadic_sqrt2_computation_using_actual_power_and_floor_root_graph_with_separate_proved_integer_bounds_no_accuracy_clause_or_analytic_interpretation", "ha": "evidence/SQ004-HA.json.gz", "internal_proof_nodes": [ 41, 128, 136, 137, 9 ], "lean": "evidence/SQ004-Lean.json.gz", "nodes": 139, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-dyadic-sqrt2-v1", "proof": "evidence/SQ004-bundle.json.gz", "report_sha256": "83905582c574545942d6fbb4b775c635b987cb4071b5dcbe8fd132f00872359e", "root": 138, "statement": "forall k. exists d s. (DyadicSqrt2(k,d,s)) /\\ ((~d=0) /\\ ((~s=0) /\\ ((Le(d,s)) /\\ ((Lt(s,2*d)) /\\ (forall q t. DyadicSqrt2(k,q,t) -> q=d /\\ t=s)))))", "target_ast_sha256": "949e42462649aba0033ab2dacf74c2aba14cced0c759d3efedb524aa34ee12ce", "title": "dyadic sqrt2", "uses_definitions": [ "PD0001", "PD0002", "ND0446" ] }, { "bundle_bytes": 29300, "bundle_sha256": "281d07e71b9dffd4902db8ba8c2a02b0ae4a9073befdb0d3ed21490253c78a82", "case": "SQ005", "contains_checked_statements": [ "SQ002" ], "coverage": "supplied_floor_root_at_scaled_square_input_is_between_positive_integer_multiples_of_the_original_floor_root_and_its_successor_no_real_limit_assumption", "ha": "evidence/SQ005-HA.json.gz", "internal_proof_nodes": [ 17, 14, 22, 21, 18, 6 ], "lean": "evidence/SQ005-Lean.json.gz", "nodes": 24, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-floor-sqrt-scaling-v1", "proof": "evidence/SQ005-bundle.json.gz", "report_sha256": "d19aa957a4a9cfc3de4025a75ad2b87c12d6a34f0884aebc3c1cc10653b6e059", "root": 23, "statement": "forall n s e t. ~e=0 -> FloorSqrt(n,s) -> FloorSqrt(n*(e*e),t) -> Le(s*e,t) /\\ Lt(t,(S s)*e)", "target_ast_sha256": "af5a7f392980a78a1da6c3dd0057aaee1ddbfe5ffb78712c7317553f73567df8", "title": "floor sqrt scaling", "uses_definitions": [ "PD0001", "PD0002", "PD0051" ] }, { "bundle_bytes": 271968, "bundle_sha256": "de94cca849970d3b9ae6794bc012b3ac94b78e32d9475c5069ca02df86f5ba33", "case": "SQ006", "contains_checked_statements": [ "SQ002", "SQ005" ], "coverage": "preserve_supplied_dyadic_sqrt2_computations_at_ordered_precisions_construct_actual_intervening_power_prove_denominator_product_and_integer_nesting_without_real_limit", "ha": "evidence/SQ006-HA.json.gz", "internal_proof_nodes": [ 23, 119, 128, 129, 26, 18, 6 ], "lean": "evidence/SQ006-Lean.json.gz", "nodes": 131, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-dyadic-sqrt2-refinement-v1", "proof": "evidence/SQ006-bundle.json.gz", "report_sha256": "dad554f143ebb431c847331aa2086967f88ba96e583f0c365e2cd83f34f92f02", "root": 130, "statement": "forall k l d s q t. DyadicSqrt2(k,d,s) -> DyadicSqrt2(k+l,q,t) -> exists e. (Pow(2,l,e)) /\\ ((~e=0) /\\ ((q=d*e) /\\ ((Le(s*e,t)) /\\ (Lt(t,(S s)*e)))))", "target_ast_sha256": "4c4a92be36b18b982f9f2f1196e3d2d996ddb4e94fa07479a82bc2eb01d548df", "title": "dyadic sqrt2 refinement", "uses_definitions": [ "PD0001", "PD0002", "PD0020", "ND0446" ] }, { "bundle_bytes": 281610, "bundle_sha256": "662b208170553daf44e743701543229b6fc24f7f1542c2fd68db47c81d1001f3", "case": "SQ007", "contains_checked_statements": [ "SQ002", "SQ005", "SQ006" ], "coverage": "actual_dyadic_sqrt2_intervals_at_ordered_precisions_have_valid_rational_nondecreasing_lower_and_nonincreasing_upper_endpoints_with_proved_denominator_positivity_no_limit", "ha": "evidence/SQ007-HA.json.gz", "internal_proof_nodes": [ 130, 132, 129, 26 ], "lean": "evidence/SQ007-Lean.json.gz", "nodes": 134, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-dyadic-sqrt2-intervals-v1", "proof": "evidence/SQ007-bundle.json.gz", "report_sha256": "88acf005b34630d1d5f1d49cf5db2023fb5788d5e169703edb12b4778033a9f9", "root": 133, "statement": "forall k l d s q t. DyadicSqrt2(k,d,s) -> DyadicSqrt2(k+l,q,t) -> IRatLe(s,0,d,t,0,q) /\\ IRatLe(S t,0,q,S s,0,d)", "target_ast_sha256": "13d665c846b9974b8406b7dbafd3d385a28847ea32ce289f7bf5caf79f2d8a32", "title": "dyadic sqrt2 intervals", "uses_definitions": [ "ND0446", "ND0444" ] }, { "bundle_bytes": 288954, "bundle_sha256": "fa202c285e41ec0f1b6619dc5605fd748dba041a74efb3ee72cd9abbb5861f32", "case": "SQ008", "contains_checked_statements": [ "SQ002", "SQ005", "SQ006", "SQ007" ], "coverage": "actual_dyadic_sqrt2_computations_at_ordered_precisions_have_explicit_positive_denominators_and_cross_multiplied_error_at_most_one_coarse_denominator_reciprocal_no_analytic_limit", "ha": "evidence/SQ008-HA.json.gz", "internal_proof_nodes": [ 133, 134, 5, 14, 9, 35 ], "lean": "evidence/SQ008-Lean.json.gz", "nodes": 136, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-dyadic-sqrt2-modulus-v2", "proof": "evidence/SQ008-bundle.json.gz", "report_sha256": "8c76390b377d64eff1a3aa17ab42abd5b09ff15f3e85723884c81355c3724c51", "root": 135, "statement": "forall k l d s q t. DyadicSqrt2(k,d,s) -> DyadicSqrt2(k+l,q,t) -> (~d=0) /\\ ((~q=0) /\\ ((Le(s*q,t*d)) /\\ (Le(t*d,s*q+q))))", "target_ast_sha256": "a431f58073293b4d6efa8d8122c194a4bcbaa3f8ad022a73b9f8d2a0b8182910", "title": "dyadic sqrt2 modulus", "uses_definitions": [ "PD0001", "ND0446" ] }, { "bundle_bytes": 91493, "bundle_sha256": "1189e78e28fb6697459f8f7ff4696adaa4d8f739643e53c11fa82c491066e274", "case": "SQ009", "contains_checked_statements": [ "SQ002" ], "coverage": "supplied_actual_dyadic_sqrt2_computation_has_positive_denominator_and_root_root_range_and_explicit_square_residual_at_most_four_times_denominator_no_limit_or_square_root_identity_assumption", "ha": "evidence/SQ009-HA.json.gz", "internal_proof_nodes": [ 32, 25, 33, 49, 9, 17, 11, 50, 51, 48 ], "lean": "evidence/SQ009-Lean.json.gz", "nodes": 53, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-dyadic-sqrt2-square-error-v1", "proof": "evidence/SQ009-bundle.json.gz", "report_sha256": "baa3ca085a50f25900deb6edd081517ce387b16357c7e3383b44d9371a62eb62", "root": 52, "statement": "forall k d s. DyadicSqrt2(k,d,s) -> (~d=0) /\\ ((~s=0) /\\ ((Le(d,s)) /\\ ((Lt(s,2*d)) /\\ ((Le(s*s,2*(d*d))) /\\ (Le(2*(d*d),s*s+2*(2*d)))))))", "target_ast_sha256": "c5d6445cccb9460719eff249437f8849968eaa22bcc68dc2d5615589e7873a0c", "title": "dyadic sqrt2 square error", "uses_definitions": [ "PD0001", "PD0002", "ND0446" ] }, { "bundle_bytes": 318039, "bundle_sha256": "16ba3030487cf8f07c10bd9a3cc909569392a2b677c8bbd6322d8bcd4e403ab4", "case": "SQ010", "contains_checked_statements": [ "SQ002", "SQ005", "SQ006", "SQ007", "SQ008", "SQ009" ], "coverage": "actual_dyadic_sqrt2_reciprocal_approximants_have_proved_positive_denominators_reversed_order_and_explicit_finite_cross_multiplied_error_at_most_2_to_minus_coarse_precision_no_real_reciprocal_assumption", "ha": "evidence/SQ010-HA.json.gz", "internal_proof_nodes": [ 135, 145, 147 ], "lean": "evidence/SQ010-Lean.json.gz", "nodes": 149, "original_observation": "research/arithmetic-library/sqrt2-power/observations/shared-wave-dyadic-sqrt2-reciprocal-v1", "proof": "evidence/SQ010-bundle.json.gz", "report_sha256": "b07cc58cb1903875925d6302101ad800f00d27e86ee98d4f7eddf349d1d9c9f6", "root": 148, "statement": "forall k l d s q t. DyadicSqrt2(k,d,s) -> DyadicSqrt2(k+l,q,t) -> (~d=0) /\\ ((~s=0) /\\ ((~t=0) /\\ ((Le(q*s,d*t)) /\\ (Le(d*(d*t),d*(q*s)+s*t)))))", "target_ast_sha256": "4e5333c99ca1ed711e3c4059b5352c1b5545b4d3d6ce9e26db04482a0f15cdcd", "title": "dyadic sqrt2 reciprocal", "uses_definitions": [ "PD0001", "ND0446" ] } ], "schema": "sqrt2-research-checkpoint-corpus-v1", "unique_statements": 245 }