0.03/0.11 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.03/0.11 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.11/0.32 % Computer : n008.cluster.edu 0.11/0.32 % Model : x86_64 x86_64 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.11/0.32 % Memory : 8042.1875MB 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64 0.11/0.32 % CPULimit : 960 0.11/0.32 % WCLimit : 120 0.11/0.32 % DateTime : Thu Jul 2 07:58:44 EDT 2020 0.11/0.32 % CPUTime : 20.14/2.95 % SZS status Theorem 20.14/2.95 20.14/2.95 % SZS output start Proof 20.14/2.95 Take the following subset of the input axioms: 22.23/3.28 fof(additive_associativity, axiom, ![A, B, C]: addition(addition(A, B), C)=addition(A, addition(B, C))). 22.23/3.28 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)). 22.23/3.28 fof(distributivity1, axiom, ![A, B, C]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))). 22.23/3.28 fof(distributivity2, axiom, ![A, B, C]: multiplication(addition(A, B), C)=addition(multiplication(A, C), multiplication(B, C))). 22.23/3.28 fof(goals, conjecture, ![X0]: (leq(multiplication(strong_iteration(X0), strong_iteration(X0)), strong_iteration(X0)) & leq(strong_iteration(X0), multiplication(strong_iteration(X0), strong_iteration(X0))))). 22.23/3.28 fof(idempotence, axiom, ![A]: addition(A, A)=A). 22.23/3.28 fof(infty_unfold1, axiom, ![A]: addition(multiplication(A, strong_iteration(A)), one)=strong_iteration(A)). 22.23/3.28 fof(isolation, axiom, ![A]: strong_iteration(A)=addition(star(A), multiplication(strong_iteration(A), zero))). 22.23/3.28 fof(left_annihilation, axiom, ![A]: zero=multiplication(zero, A)). 22.23/3.28 fof(multiplicative_associativity, axiom, ![A, B, C]: multiplication(multiplication(A, B), C)=multiplication(A, multiplication(B, C))). 22.23/3.28 fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A). 22.23/3.28 fof(multiplicative_right_identity, axiom, ![A]: multiplication(A, one)=A). 22.23/3.28 fof(order, axiom, ![A, B]: (addition(A, B)=B <=> leq(A, B))). 22.23/3.28 fof(star_induction1, axiom, ![A, B, C]: (leq(multiplication(star(A), B), C) <= leq(addition(multiplication(A, C), B), C))). 22.23/3.28 fof(star_unfold1, axiom, ![A]: star(A)=addition(one, multiplication(A, star(A)))). 22.23/3.28 fof(star_unfold2, axiom, ![A]: star(A)=addition(one, multiplication(star(A), A))). 22.23/3.28 22.23/3.28 Now clausify the problem and encode Horn clauses using encoding 3 of 22.23/3.28 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 22.23/3.28 We repeatedly replace C & s=t => u=v by the two clauses: 22.23/3.28 fresh(y, y, x1...xn) = u 22.23/3.28 C => fresh(s, t, x1...xn) = v 22.23/3.28 where fresh is a fresh function symbol and x1..xn are the free 22.23/3.28 variables of u and v. 22.23/3.28 A predicate p(X) is encoded as p(X)=true (this is sound, because the 22.23/3.28 input problem has no model of domain size 1). 22.23/3.28 22.23/3.28 The encoding turns the above axioms into the following unit equations and goals: 22.23/3.28 22.23/3.28 Axiom 1 (order): fresh5(X, X, Y, Z) = true. 22.23/3.28 Axiom 2 (order_1): fresh(X, X, Y, Z) = Z. 22.23/3.28 Axiom 3 (star_induction1): fresh3(X, X, Y, Z, W) = true. 22.23/3.28 Axiom 4 (infty_unfold1): addition(multiplication(X, strong_iteration(X)), one) = strong_iteration(X). 22.23/3.28 Axiom 5 (left_annihilation): zero = multiplication(zero, X). 22.23/3.28 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X). 22.23/3.28 Axiom 7 (multiplicative_left_identity): multiplication(one, X) = X. 22.23/3.28 Axiom 8 (star_induction1): fresh3(leq(addition(multiplication(X, Y), Z), Y), true, X, Z, Y) = leq(multiplication(star(X), Z), Y). 22.23/3.28 Axiom 9 (additive_associativity): addition(addition(X, Y), Z) = addition(X, addition(Y, Z)). 22.23/3.28 Axiom 10 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y). 22.23/3.28 Axiom 11 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y). 22.23/3.28 Axiom 12 (star_unfold1): star(X) = addition(one, multiplication(X, star(X))). 22.23/3.28 Axiom 13 (isolation): strong_iteration(X) = addition(star(X), multiplication(strong_iteration(X), zero)). 22.23/3.28 Axiom 14 (idempotence): addition(X, X) = X. 22.23/3.28 Axiom 15 (distributivity1): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)). 22.23/3.28 Axiom 16 (multiplicative_right_identity): multiplication(X, one) = X. 22.23/3.28 Axiom 17 (multiplicative_associativity): multiplication(multiplication(X, Y), Z) = multiplication(X, multiplication(Y, Z)). 22.23/3.28 Axiom 18 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)). 22.23/3.28 Axiom 19 (star_unfold2): star(X) = addition(one, multiplication(star(X), X)). 22.23/3.28 22.23/3.28 Lemma 20: addition(Y, multiplication(Y, X)) = multiplication(Y, addition(X, one)). 22.23/3.28 Proof: 22.23/3.28 addition(Y, multiplication(Y, X)) 22.23/3.28 = { by axiom 16 (multiplicative_right_identity) } 22.23/3.28 addition(multiplication(Y, one), multiplication(Y, X)) 22.23/3.28 = { by axiom 15 (distributivity1) } 22.23/3.28 multiplication(Y, addition(one, X)) 22.23/3.28 = { by axiom 6 (additive_commutativity) } 22.23/3.28 multiplication(Y, addition(X, one)) 22.23/3.28 22.23/3.28 Lemma 21: addition(X, addition(X, Y)) = addition(X, Y). 22.23/3.28 Proof: 22.23/3.28 addition(X, addition(X, Y)) 22.23/3.28 = { by axiom 9 (additive_associativity) } 22.23/3.28 addition(addition(X, X), Y) 22.23/3.28 = { by axiom 14 (idempotence) } 22.23/3.28 addition(X, Y) 22.23/3.28 22.23/3.28 Lemma 22: leq(X, addition(X, Y)) = true. 22.23/3.28 Proof: 22.23/3.28 leq(X, addition(X, Y)) 22.23/3.28 = { by axiom 11 (order) } 22.23/3.28 fresh5(addition(X, addition(X, Y)), addition(X, Y), X, addition(X, Y)) 22.23/3.28 = { by lemma 21 } 22.23/3.28 fresh5(addition(X, Y), addition(X, Y), X, addition(X, Y)) 22.23/3.28 = { by axiom 1 (order) } 22.23/3.28 true 22.23/3.28 22.23/3.28 Lemma 23: addition(one, multiplication(X, strong_iteration(X))) = strong_iteration(X). 22.23/3.28 Proof: 22.23/3.28 addition(one, multiplication(X, strong_iteration(X))) 22.23/3.28 = { by axiom 6 (additive_commutativity) } 22.23/3.28 addition(multiplication(X, strong_iteration(X)), one) 22.23/3.28 = { by axiom 4 (infty_unfold1) } 22.23/3.28 strong_iteration(X) 22.23/3.28 22.23/3.28 Lemma 24: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X). 22.23/3.28 Proof: 22.23/3.28 addition(X, multiplication(Y, X)) 22.23/3.28 = { by axiom 7 (multiplicative_left_identity) } 22.23/3.28 addition(multiplication(one, X), multiplication(Y, X)) 22.23/3.28 = { by axiom 18 (distributivity2) } 22.23/3.28 multiplication(addition(one, Y), X) 22.23/3.28 = { by axiom 6 (additive_commutativity) } 22.23/3.28 multiplication(addition(Y, one), X) 22.23/3.28 22.23/3.28 Lemma 25: addition(X, multiplication(Z, multiplication(Y, X))) = multiplication(addition(one, multiplication(Z, Y)), X). 22.23/3.28 Proof: 22.23/3.28 addition(X, multiplication(Z, multiplication(Y, X))) 22.23/3.28 = { by axiom 17 (multiplicative_associativity) } 22.23/3.28 addition(X, multiplication(multiplication(Z, Y), X)) 22.23/3.28 = { by lemma 24 } 22.23/3.28 multiplication(addition(multiplication(Z, Y), one), X) 22.23/3.28 = { by axiom 6 (additive_commutativity) } 22.23/3.28 multiplication(addition(one, multiplication(Z, Y)), X) 22.23/3.28 22.23/3.28 Lemma 26: addition(one, star(X)) = star(X). 22.23/3.28 Proof: 22.23/3.28 addition(one, star(X)) 22.23/3.28 = { by axiom 12 (star_unfold1) } 22.23/3.28 addition(one, addition(one, multiplication(X, star(X)))) 22.23/3.28 = { by lemma 21 } 22.23/3.28 addition(one, multiplication(X, star(X))) 22.23/3.28 = { by axiom 12 (star_unfold1) } 22.23/3.28 star(X) 22.23/3.28 22.23/3.28 Lemma 27: addition(Z, addition(X, Y)) = addition(X, addition(Z, Y)). 22.23/3.28 Proof: 22.23/3.28 addition(Z, addition(X, Y)) 22.23/3.28 = { by axiom 6 (additive_commutativity) } 22.23/3.28 addition(Z, addition(Y, X)) 22.23/3.28 = { by axiom 9 (additive_associativity) } 22.23/3.28 addition(addition(Z, Y), X) 22.23/3.28 = { by axiom 6 (additive_commutativity) } 22.23/3.29 addition(X, addition(Z, Y)) 22.23/3.29 22.23/3.29 Lemma 28: addition(one, multiplication(star(X), strong_iteration(X))) = multiplication(star(X), strong_iteration(X)). 22.23/3.29 Proof: 22.23/3.29 addition(one, multiplication(star(X), strong_iteration(X))) 22.23/3.29 = { by axiom 12 (star_unfold1) } 22.23/3.29 addition(one, multiplication(addition(one, multiplication(X, star(X))), strong_iteration(X))) 22.23/3.29 = { by lemma 26 } 22.23/3.29 addition(one, multiplication(addition(one, multiplication(X, addition(one, star(X)))), strong_iteration(X))) 22.23/3.29 = { by axiom 6 (additive_commutativity) } 22.23/3.29 addition(one, multiplication(addition(one, multiplication(X, addition(star(X), one))), strong_iteration(X))) 22.23/3.29 = { by lemma 20 } 22.23/3.29 addition(one, multiplication(addition(one, addition(X, multiplication(X, star(X)))), strong_iteration(X))) 22.23/3.29 = { by axiom 6 (additive_commutativity) } 22.23/3.29 addition(one, multiplication(addition(one, addition(multiplication(X, star(X)), X)), strong_iteration(X))) 22.23/3.29 = { by axiom 9 (additive_associativity) } 22.23/3.29 addition(one, multiplication(addition(addition(one, multiplication(X, star(X))), X), strong_iteration(X))) 22.23/3.29 = { by axiom 12 (star_unfold1) } 22.23/3.29 addition(one, multiplication(addition(star(X), X), strong_iteration(X))) 22.23/3.29 = { by axiom 18 (distributivity2) } 22.23/3.29 addition(one, addition(multiplication(star(X), strong_iteration(X)), multiplication(X, strong_iteration(X)))) 22.23/3.29 = { by axiom 6 (additive_commutativity) } 22.23/3.29 addition(one, addition(multiplication(X, strong_iteration(X)), multiplication(star(X), strong_iteration(X)))) 22.23/3.29 = { by lemma 27 } 22.23/3.29 addition(multiplication(X, strong_iteration(X)), addition(one, multiplication(star(X), strong_iteration(X)))) 22.23/3.29 = { by axiom 9 (additive_associativity) } 22.23/3.29 addition(addition(multiplication(X, strong_iteration(X)), one), multiplication(star(X), strong_iteration(X))) 22.23/3.29 = { by axiom 4 (infty_unfold1) } 22.23/3.29 addition(strong_iteration(X), multiplication(star(X), strong_iteration(X))) 22.23/3.29 = { by lemma 24 } 22.23/3.29 multiplication(addition(star(X), one), strong_iteration(X)) 22.23/3.29 = { by axiom 6 (additive_commutativity) } 22.23/3.29 multiplication(addition(one, star(X)), strong_iteration(X)) 22.23/3.29 = { by lemma 26 } 24.49/3.59 multiplication(star(X), strong_iteration(X)) 24.49/3.59 24.49/3.59 Goal 1 (goals): tuple(leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0))), leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0))) = tuple(true, true). 24.49/3.59 Proof: 24.49/3.59 tuple(leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), strong_iteration(sK1_goals_X0))), leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by axiom 4 (infty_unfold1) } 24.49/3.59 tuple(leq(strong_iteration(sK1_goals_X0), multiplication(strong_iteration(sK1_goals_X0), addition(multiplication(sK1_goals_X0, strong_iteration(sK1_goals_X0)), one))), leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by lemma 20 } 24.49/3.59 tuple(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))))), leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by lemma 22 } 24.49/3.59 tuple(true, leq(multiplication(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by lemma 23 } 24.49/3.59 tuple(true, leq(multiplication(addition(one, multiplication(sK2_goals_X0, strong_iteration(sK2_goals_X0))), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by lemma 25 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 10 (order_1) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 13 (isolation) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 23 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 21 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 23 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 18 (distributivity2) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 17 (multiplicative_associativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 5 (left_annihilation) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 20 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 9 (additive_associativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 9 (additive_associativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 13 (isolation) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 28 } 24.49/3.59 tuple(true, leq(fresh(leq(multiplication(sK2_goals_X0, addition(strong_iteration(sK2_goals_X0), addition(one, 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))) 24.49/3.59 = { by lemma 27 } 24.49/3.59 tuple(true, leq(fresh(leq(multiplication(sK2_goals_X0, addition(one, 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))) 24.49/3.59 = { by lemma 24 } 24.49/3.59 tuple(true, leq(fresh(leq(multiplication(sK2_goals_X0, addition(one, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, leq(fresh(leq(multiplication(sK2_goals_X0, addition(one, 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))) 24.49/3.59 = { by lemma 26 } 24.49/3.59 tuple(true, leq(fresh(leq(multiplication(sK2_goals_X0, addition(one, 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))) 24.49/3.59 = { by lemma 28 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 17 (multiplicative_associativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 19 (star_unfold2) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 6 (additive_commutativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 20 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 25 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 12 (star_unfold1) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 17 (multiplicative_associativity) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 8 (star_induction1) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 14 (idempotence) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 4 (infty_unfold1) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by lemma 22 } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 3 (star_induction1) } 24.49/3.59 tuple(true, 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))) 24.49/3.59 = { by axiom 2 (order_1) } 24.49/3.59 tuple(true, leq(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by axiom 11 (order) } 24.49/3.59 tuple(true, fresh5(addition(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0)), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by axiom 14 (idempotence) } 24.49/3.59 tuple(true, fresh5(strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0), strong_iteration(sK2_goals_X0))) 24.49/3.59 = { by axiom 1 (order) } 24.49/3.59 tuple(true, true) 24.49/3.59 % SZS output end Proof 24.49/3.59 24.49/3.59 RESULT: Theorem (the conjecture is true). 24.49/3.60 EOF