TSTP Solution File: KLE139+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE139+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 05:36:00 EDT 2023
% Result : Theorem 4.73s 1.01s
% Output : Proof 5.30s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : KLE139+2 : TPTP v8.1.2. Released v4.0.0.
% 0.03/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n009.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 11:44:05 EDT 2023
% 0.13/0.34 % CPUTime :
% 4.73/1.01 Command-line arguments: --no-flatten-goal
% 4.73/1.01
% 4.73/1.01 % SZS status Theorem
% 4.73/1.01
% 5.30/1.02 % SZS output start Proof
% 5.30/1.02 Take the following subset of the input axioms:
% 5.30/1.02 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 5.30/1.02 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 5.30/1.02 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 5.30/1.02 fof(distributivity1, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 5.30/1.02 fof(distributivity2, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 5.30/1.02 fof(goals, conjecture, ![X0]: (leq(strong_iteration(X0), addition(multiplication(strong_iteration(X0), X0), one)) & leq(addition(multiplication(strong_iteration(X0), X0), one), strong_iteration(X0)))).
% 5.30/1.02 fof(idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 5.30/1.02 fof(isolation, axiom, ![A3]: strong_iteration(A3)=addition(star(A3), multiplication(strong_iteration(A3), zero))).
% 5.30/1.02 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 5.30/1.02 fof(multiplicative_associativity, axiom, ![A3, B2, C2]: multiplication(A3, multiplication(B2, C2))=multiplication(multiplication(A3, B2), C2)).
% 5.30/1.02 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 5.30/1.02 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 5.30/1.02 fof(star_unfold2, axiom, ![A3]: addition(one, multiplication(star(A3), A3))=star(A3)).
% 5.30/1.02
% 5.30/1.02 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.30/1.02 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.30/1.02 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.30/1.02 fresh(y, y, x1...xn) = u
% 5.30/1.02 C => fresh(s, t, x1...xn) = v
% 5.30/1.02 where fresh is a fresh function symbol and x1..xn are the free
% 5.30/1.02 variables of u and v.
% 5.30/1.02 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.30/1.02 input problem has no model of domain size 1).
% 5.30/1.02
% 5.30/1.02 The encoding turns the above axioms into the following unit equations and goals:
% 5.30/1.02
% 5.30/1.02 Axiom 1 (idempotence): addition(X, X) = X.
% 5.30/1.02 Axiom 2 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 5.30/1.02 Axiom 3 (additive_identity): addition(X, zero) = X.
% 5.30/1.02 Axiom 4 (multiplicative_right_identity): multiplication(X, one) = X.
% 5.30/1.02 Axiom 5 (left_annihilation): multiplication(zero, X) = zero.
% 5.30/1.02 Axiom 6 (order): fresh5(X, X, Y, Z) = true.
% 5.30/1.02 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 5.30/1.02 Axiom 8 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 5.30/1.02 Axiom 9 (star_unfold2): addition(one, multiplication(star(X), X)) = star(X).
% 5.30/1.02 Axiom 10 (order): fresh5(addition(X, Y), Y, X, Y) = leq(X, Y).
% 5.30/1.02 Axiom 11 (isolation): strong_iteration(X) = addition(star(X), multiplication(strong_iteration(X), zero)).
% 5.30/1.02 Axiom 12 (distributivity1): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 5.30/1.02 Axiom 13 (distributivity2): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 5.30/1.02
% 5.30/1.02 Lemma 14: leq(X, X) = true.
% 5.30/1.02 Proof:
% 5.30/1.02 leq(X, X)
% 5.30/1.02 = { by axiom 10 (order) R->L }
% 5.30/1.02 fresh5(addition(X, X), X, X, X)
% 5.30/1.02 = { by axiom 1 (idempotence) }
% 5.30/1.02 fresh5(X, X, X, X)
% 5.30/1.02 = { by axiom 6 (order) }
% 5.30/1.02 true
% 5.30/1.02
% 5.30/1.02 Lemma 15: addition(one, multiplication(strong_iteration(X), X)) = strong_iteration(X).
% 5.30/1.02 Proof:
% 5.30/1.02 addition(one, multiplication(strong_iteration(X), X))
% 5.30/1.02 = { by axiom 11 (isolation) }
% 5.30/1.02 addition(one, multiplication(addition(star(X), multiplication(strong_iteration(X), zero)), X))
% 5.30/1.02 = { by axiom 2 (additive_commutativity) R->L }
% 5.30/1.02 addition(one, multiplication(addition(multiplication(strong_iteration(X), zero), star(X)), X))
% 5.30/1.02 = { by axiom 13 (distributivity2) }
% 5.30/1.02 addition(one, addition(multiplication(multiplication(strong_iteration(X), zero), X), multiplication(star(X), X)))
% 5.30/1.02 = { by axiom 2 (additive_commutativity) R->L }
% 5.30/1.02 addition(one, addition(multiplication(star(X), X), multiplication(multiplication(strong_iteration(X), zero), X)))
% 5.30/1.02 = { by axiom 7 (additive_associativity) }
% 5.30/1.02 addition(addition(one, multiplication(star(X), X)), multiplication(multiplication(strong_iteration(X), zero), X))
% 5.30/1.02 = { by axiom 9 (star_unfold2) }
% 5.30/1.02 addition(star(X), multiplication(multiplication(strong_iteration(X), zero), X))
% 5.30/1.02 = { by axiom 8 (multiplicative_associativity) R->L }
% 5.30/1.02 addition(star(X), multiplication(strong_iteration(X), multiplication(zero, X)))
% 5.30/1.02 = { by axiom 3 (additive_identity) R->L }
% 5.30/1.02 addition(star(X), multiplication(strong_iteration(X), addition(multiplication(zero, X), zero)))
% 5.30/1.02 = { by axiom 12 (distributivity1) }
% 5.30/1.02 addition(star(X), addition(multiplication(strong_iteration(X), multiplication(zero, X)), multiplication(strong_iteration(X), zero)))
% 5.30/1.02 = { by axiom 2 (additive_commutativity) R->L }
% 5.30/1.02 addition(star(X), addition(multiplication(strong_iteration(X), zero), multiplication(strong_iteration(X), multiplication(zero, X))))
% 5.30/1.02 = { by axiom 7 (additive_associativity) }
% 5.30/1.02 addition(addition(star(X), multiplication(strong_iteration(X), zero)), multiplication(strong_iteration(X), multiplication(zero, X)))
% 5.30/1.02 = { by axiom 11 (isolation) R->L }
% 5.30/1.02 addition(strong_iteration(X), multiplication(strong_iteration(X), multiplication(zero, X)))
% 5.30/1.02 = { by axiom 4 (multiplicative_right_identity) R->L }
% 5.30/1.02 addition(multiplication(strong_iteration(X), one), multiplication(strong_iteration(X), multiplication(zero, X)))
% 5.30/1.02 = { by axiom 12 (distributivity1) R->L }
% 5.30/1.02 multiplication(strong_iteration(X), addition(one, multiplication(zero, X)))
% 5.30/1.02 = { by axiom 2 (additive_commutativity) }
% 5.30/1.02 multiplication(strong_iteration(X), addition(multiplication(zero, X), one))
% 5.30/1.02 = { by axiom 5 (left_annihilation) }
% 5.30/1.02 multiplication(strong_iteration(X), addition(zero, one))
% 5.30/1.02 = { by axiom 2 (additive_commutativity) R->L }
% 5.30/1.02 multiplication(strong_iteration(X), addition(one, zero))
% 5.30/1.02 = { by axiom 3 (additive_identity) }
% 5.30/1.02 multiplication(strong_iteration(X), one)
% 5.30/1.02 = { by axiom 4 (multiplicative_right_identity) }
% 5.30/1.02 strong_iteration(X)
% 5.30/1.02
% 5.30/1.02 Goal 1 (goals): tuple(leq(addition(multiplication(strong_iteration(x0), x0), one), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(multiplication(strong_iteration(x0_2), x0_2), one))) = tuple(true, true).
% 5.30/1.02 Proof:
% 5.30/1.02 tuple(leq(addition(multiplication(strong_iteration(x0), x0), one), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(multiplication(strong_iteration(x0_2), x0_2), one)))
% 5.30/1.02 = { by axiom 2 (additive_commutativity) }
% 5.30/1.02 tuple(leq(addition(one, multiplication(strong_iteration(x0), x0)), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(multiplication(strong_iteration(x0_2), x0_2), one)))
% 5.30/1.03 = { by axiom 2 (additive_commutativity) }
% 5.30/1.03 tuple(leq(addition(one, multiplication(strong_iteration(x0), x0)), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 5.30/1.03 = { by lemma 15 }
% 5.30/1.03 tuple(leq(strong_iteration(x0), strong_iteration(x0)), leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 5.30/1.03 = { by lemma 14 }
% 5.30/1.03 tuple(true, leq(strong_iteration(x0_2), addition(one, multiplication(strong_iteration(x0_2), x0_2))))
% 5.30/1.03 = { by lemma 15 }
% 5.30/1.03 tuple(true, leq(strong_iteration(x0_2), strong_iteration(x0_2)))
% 5.30/1.03 = { by lemma 14 }
% 5.30/1.03 tuple(true, true)
% 5.30/1.03 % SZS output end Proof
% 5.30/1.03
% 5.30/1.03 RESULT: Theorem (the conjecture is true).
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