0.00/0.03 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.02/0.23 % Computer : n155.star.cs.uiowa.edu 0.02/0.23 % Model : x86_64 x86_64 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.23 % Memory : 32218.625MB 0.02/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.23 % CPULimit : 300 0.02/0.23 % DateTime : Sat Jul 14 05:18:41 CDT 2018 0.02/0.23 % CPUTime : 1.93/2.15 % SZS status Theorem 1.93/2.15 1.93/2.15 % SZS output start Proof 1.93/2.15 Take the following subset of the input axioms: 2.11/2.33 fof(additive_associativity, axiom, 2.11/2.33 ![A, B, C]: 2.11/2.33 addition(A, addition(B, C))=addition(addition(A, B), C)). 2.11/2.33 fof(additive_commutativity, axiom, 2.11/2.33 ![A, B]: addition(B, A)=addition(A, B)). 2.11/2.33 fof(distributivity1, axiom, 2.11/2.33 ![A, B, C]: 2.11/2.33 addition(multiplication(A, B), 2.11/2.33 multiplication(A, C))=multiplication(A, addition(B, C))). 2.11/2.33 fof(distributivity2, axiom, 2.11/2.33 ![A, B, C]: 2.11/2.33 multiplication(addition(A, B), C)=addition(multiplication(A, C), 2.11/2.33 multiplication(B, C))). 2.11/2.33 fof(goals, conjecture, 2.11/2.33 ![X0]: 2.11/2.33 (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), 2.11/2.33 strong_iteration(X0)) 2.11/2.33 & leq(strong_iteration(X0), 2.11/2.33 multiplication(strong_iteration(X0), strong_iteration(X0))))). 2.11/2.33 fof(idempotence, axiom, ![A]: addition(A, A)=A). 2.11/2.33 fof(infty_unfold1, axiom, 2.11/2.33 ![A]: 2.11/2.33 addition(multiplication(A, strong_iteration(A)), 2.11/2.33 one)=strong_iteration(A)). 2.11/2.33 fof(isolation, axiom, 2.11/2.33 ![A]: 2.11/2.33 addition(star(A), 2.11/2.33 multiplication(strong_iteration(A), zero))=strong_iteration(A)). 2.11/2.33 fof(left_annihilation, axiom, ![A]: multiplication(zero, A)=zero). 2.11/2.33 fof(multiplicative_associativity, axiom, 2.11/2.33 ![A, B, C]: 2.11/2.33 multiplication(A, 2.11/2.33 multiplication(B, C))=multiplication(multiplication(A, B), C)). 2.11/2.33 fof(multiplicative_left_identity, axiom, 2.11/2.33 ![A]: A=multiplication(one, A)). 2.11/2.33 fof(multiplicative_right_identity, axiom, 2.11/2.33 ![A]: multiplication(A, one)=A). 2.11/2.33 fof(order, axiom, ![A, B]: (leq(A, B) <=> B=addition(A, B))). 2.11/2.33 fof(star_induction1, axiom, 2.11/2.33 ![A, B, C]: 2.11/2.33 (leq(multiplication(star(A), B), C) 2.11/2.33 <= leq(addition(multiplication(A, C), B), C))). 2.11/2.33 fof(star_unfold1, axiom, 2.11/2.33 ![A]: star(A)=addition(one, multiplication(A, star(A)))). 2.11/2.33 fof(star_unfold2, axiom, 2.11/2.33 ![A]: addition(one, multiplication(star(A), A))=star(A)). 2.11/2.33 2.11/2.33 Now clausify the problem and encode Horn clauses using encoding 3 of 2.11/2.33 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 2.11/2.33 We repeatedly replace C & s=t => u=v by the two clauses: 2.11/2.33 $$fresh(y, y, x1...xn) = u 2.11/2.33 C => $$fresh(s, t, x1...xn) = v 2.11/2.33 where $$fresh is a fresh function symbol and x1..xn are the free 2.11/2.33 variables of u and v. 2.11/2.33 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 2.11/2.33 input problem has no model of domain size 1). 2.11/2.33 2.11/2.33 The encoding turns the above axioms into the following unit equations and goals: 2.11/2.33 2.11/2.33 Axiom 2 (order): $$fresh4(X, X, Y, Z) = $$true. 2.11/2.33 Axiom 3 (order_1): $$fresh(X, X, Y, Z) = Z. 2.11/2.33 Axiom 4 (star_induction1): $$fresh3(X, X, Y, Z, W) = $$true. 2.11/2.33 Axiom 6 (multiplicative_left_identity): X = multiplication(one, X). 2.11/2.33 Axiom 7 (distributivity1): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)). 2.11/2.33 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z). 2.11/2.33 Axiom 9 (order_1): $$fresh(leq(X, Y), $$true, X, Y) = addition(X, Y). 2.11/2.33 Axiom 10 (order): $$fresh4(X, addition(Y, X), Y, X) = leq(Y, X). 2.11/2.33 Axiom 11 (multiplicative_right_identity): multiplication(X, one) = X. 2.11/2.33 Axiom 14 (idempotence): addition(X, X) = X. 2.11/2.33 Axiom 15 (isolation): addition(star(X), multiplication(strong_iteration(X), zero)) = strong_iteration(X). 2.11/2.33 Axiom 16 (infty_unfold1): addition(multiplication(X, strong_iteration(X)), one) = strong_iteration(X). 2.11/2.33 Axiom 17 (additive_commutativity): addition(X, Y) = addition(Y, X). 2.11/2.33 Axiom 18 (star_unfold2): addition(one, multiplication(star(X), X)) = star(X). 2.11/2.33 Axiom 20 (star_unfold1): star(X) = addition(one, multiplication(X, star(X))). 2.11/2.33 Axiom 21 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)). 2.11/2.33 Axiom 22 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z). 2.11/2.33 Axiom 23 (left_annihilation): multiplication(zero, X) = zero. 2.11/2.33 Axiom 24 (star_induction1): $$fresh3(leq(addition(multiplication(X, Y), Z), Y), $$true, X, Z, Y) = leq(multiplication(star(X), Z), Y). 2.11/2.33 2.11/2.33 Lemma 25: addition(X, addition(X, Y)) = addition(X, Y). 2.11/2.33 Proof: 2.11/2.33 addition(X, addition(X, Y)) 2.11/2.33 = { by axiom 8 (additive_associativity) } 2.11/2.33 addition(addition(X, X), Y) 2.11/2.33 = { by axiom 14 (idempotence) } 2.11/2.33 addition(X, Y) 2.11/2.33 2.11/2.33 Lemma 26: leq(X, addition(X, Y)) = $$true. 2.11/2.33 Proof: 2.11/2.33 leq(X, addition(X, Y)) 2.11/2.33 = { by axiom 10 (order) } 2.11/2.33 $$fresh4(addition(X, Y), addition(X, addition(X, Y)), X, addition(X, Y)) 2.11/2.33 = { by lemma 25 } 2.11/2.33 $$fresh4(addition(X, Y), addition(X, Y), X, addition(X, Y)) 2.11/2.33 = { by axiom 2 (order) } 2.11/2.33 $$true 2.11/2.33 2.11/2.33 Lemma 27: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)). 2.11/2.33 Proof: 2.11/2.33 addition(X, multiplication(X, Y)) 2.11/2.33 = { by axiom 11 (multiplicative_right_identity) } 2.11/2.33 addition(multiplication(X, one), multiplication(X, Y)) 2.11/2.33 = { by axiom 7 (distributivity1) } 2.11/2.33 multiplication(X, addition(one, Y)) 2.11/2.33 = { by axiom 17 (additive_commutativity) } 2.11/2.33 multiplication(X, addition(Y, one)) 2.11/2.33 2.11/2.33 Lemma 28: addition(Y, multiplication(X, Y)) = multiplication(addition(X, one), Y). 2.11/2.33 Proof: 2.11/2.33 addition(Y, multiplication(X, Y)) 2.11/2.33 = { by axiom 6 (multiplicative_left_identity) } 2.11/2.33 addition(multiplication(one, Y), multiplication(X, Y)) 2.11/2.33 = { by axiom 21 (distributivity2) } 2.11/2.33 multiplication(addition(one, X), Y) 2.11/2.33 = { by axiom 17 (additive_commutativity) } 2.11/2.33 multiplication(addition(X, one), Y) 2.11/2.33 2.11/2.33 Lemma 29: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X). 2.11/2.33 Proof: 2.11/2.33 addition(one, multiplication(X, strong_iteration(X))) 2.11/2.33 = { by axiom 17 (additive_commutativity) } 2.11/2.33 addition(multiplication(X, strong_iteration(X)), one) 2.11/2.33 = { by axiom 16 (infty_unfold1) } 2.11/2.33 strong_iteration(X) 2.11/2.33 2.11/2.33 Lemma 30: addition(Z, multiplication(X, multiplication(Y, Z))) = multiplication(addition(one, multiplication(X, Y)), Z). 2.11/2.33 Proof: 2.11/2.33 addition(Z, multiplication(X, multiplication(Y, Z))) 2.11/2.33 = { by axiom 22 (multiplicative_associativity) } 2.11/2.33 addition(Z, multiplication(multiplication(X, Y), Z)) 2.11/2.33 = { by lemma 28 } 2.11/2.33 multiplication(addition(multiplication(X, Y), one), Z) 2.11/2.33 = { by axiom 17 (additive_commutativity) } 2.31/2.51 multiplication(addition(one, multiplication(X, Y)), Z) 2.31/2.51 2.31/2.51 Goal 1 (goals): tuple(leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) = tuple($$true, $$true). 2.31/2.51 Proof: 2.31/2.51 tuple(leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 29 } 2.31/2.51 tuple(leq(multiplication(addition(one, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 30 } 2.31/2.51 tuple(leq(addition(strong_iteration(sK2_goals_X0), multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)))), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq(addition(multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 9 (order_1) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 15 (isolation) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(star(sK2_goals_X0), multiplication(strong_iteration(sK2_goals_X0), zero)), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(multiplication(strong_iteration(sK2_goals_X0), zero), star(sK2_goals_X0)), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 29 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(multiplication(strong_iteration(sK2_goals_X0), zero), star(sK2_goals_X0)), addition(one, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0))))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 25 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(multiplication(strong_iteration(sK2_goals_X0), zero), star(sK2_goals_X0)), addition(one, addition(one, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)))))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 29 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(multiplication(strong_iteration(sK2_goals_X0), zero), star(sK2_goals_X0)), addition(one, strong_iteration(sK2_goals_X0)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(multiplication(strong_iteration(sK2_goals_X0), zero), star(sK2_goals_X0)), addition(strong_iteration(sK2_goals_X0), one))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 21 (distributivity2) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(multiplication(multiplication(strong_iteration(sK2_goals_X0), zero), addition(strong_iteration(sK2_goals_X0), one)), multiplication(star(sK2_goals_X0), addition(strong_iteration(sK2_goals_X0), one)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 22 (multiplicative_associativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(multiplication(strong_iteration(sK2_goals_X0), multiplication(zero, addition(strong_iteration(sK2_goals_X0), one))), multiplication(star(sK2_goals_X0), addition(strong_iteration(sK2_goals_X0), one)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 23 (left_annihilation) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(multiplication(strong_iteration(sK2_goals_X0), zero), multiplication(star(sK2_goals_X0), addition(strong_iteration(sK2_goals_X0), one)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(multiplication(star(sK2_goals_X0), addition(strong_iteration(sK2_goals_X0), one)), multiplication(strong_iteration(sK2_goals_X0), zero))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 27 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(addition(star(sK2_goals_X0), multiplication(star(sK2_goals_X0), strong_iteration(sK2_goals_X0))), multiplication(strong_iteration(sK2_goals_X0), zero))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 8 (additive_associativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(star(sK2_goals_X0), addition(multiplication(star(sK2_goals_X0), strong_iteration(sK2_goals_X0)), multiplication(strong_iteration(sK2_goals_X0), zero)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(star(sK2_goals_X0), addition(multiplication(strong_iteration(sK2_goals_X0), zero), multiplication(star(sK2_goals_X0), strong_iteration(sK2_goals_X0))))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 8 (additive_associativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(addition(star(sK2_goals_X0), multiplication(strong_iteration(sK2_goals_X0), zero)), multiplication(star(sK2_goals_X0), strong_iteration(sK2_goals_X0)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 15 (isolation) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, addition(strong_iteration(sK2_goals_X0), multiplication(star(sK2_goals_X0), strong_iteration(sK2_goals_X0)))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 28 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(star(sK2_goals_X0), one), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(one, star(sK2_goals_X0)), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 18 (star_unfold2) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(one, addition(one, multiplication(star(sK2_goals_X0), sK2_goals_X0))), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 25 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(addition(one, multiplication(star(sK2_goals_X0), sK2_goals_X0)), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 18 (star_unfold2) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(sK2_goals_X0, multiplication(star(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 22 (multiplicative_associativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(multiplication(sK2_goals_X0, star(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 18 (star_unfold2) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(multiplication(sK2_goals_X0, addition(one, multiplication(star(sK2_goals_X0), sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 17 (additive_commutativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(multiplication(sK2_goals_X0, addition(multiplication(star(sK2_goals_X0), sK2_goals_X0), one)), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 27 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(addition(sK2_goals_X0, multiplication(sK2_goals_X0, multiplication(star(sK2_goals_X0), sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 30 } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(multiplication(addition(one, multiplication(sK2_goals_X0, star(sK2_goals_X0))), sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 20 (star_unfold1) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(multiplication(star(sK2_goals_X0), sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 22 (multiplicative_associativity) } 2.31/2.51 tuple(leq($$fresh(leq(multiplication(star(sK2_goals_X0), multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 24 (star_induction1) } 2.31/2.51 tuple(leq($$fresh($$fresh3(leq(addition(multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), $$true, sK2_goals_X0, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 14 (idempotence) } 2.31/2.51 tuple(leq($$fresh($$fresh3(leq(multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, sK2_goals_X0, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 16 (infty_unfold1) } 2.31/2.51 tuple(leq($$fresh($$fresh3(leq(multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), addition(multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), one)), $$true, sK2_goals_X0, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by lemma 26 } 2.31/2.51 tuple(leq($$fresh($$fresh3($$true, $$true, sK2_goals_X0, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 4 (star_induction1) } 2.31/2.51 tuple(leq($$fresh($$true, $$true, multiplication(sK2_goals_X0, multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 3 (order_1) } 2.31/2.51 tuple(leq(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 10 (order) } 2.31/2.51 tuple($$fresh4(strong_iteration(sK2_goals_X0), addition(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 14 (idempotence) } 2.31/2.51 tuple($$fresh4(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 2 (order) } 2.31/2.51 tuple($$true, leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0)))) 2.31/2.51 = { by axiom 16 (infty_unfold1) } 2.31/2.51 tuple($$true, leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), addition(multiplication(sK1_goals_X0, strong_iteration(sK1_goals_X0)), one)))) 2.31/2.51 = { by lemma 27 } 2.31/2.51 tuple($$true, leq(strong_iteration(sK1_goals_X0), addition(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), multiplication(sK1_goals_X0, strong_iteration(sK1_goals_X0)))))) 2.31/2.51 = { by lemma 26 } 2.31/2.51 tuple($$true, $$true) 2.31/2.51 % SZS output end Proof 2.31/2.51 2.31/2.51 RESULT: Theorem (the conjecture is true). 2.31/2.52 EOF