TSTP Solution File: KLE093+1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE093+1 : 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 : n005.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 05:35:51 EDT 2023
% Result : Theorem 0.17s 0.39s
% Output : Proof 0.17s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.09/0.11 % Problem : KLE093+1 : TPTP v8.1.2. Released v4.0.0.
% 0.09/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n005.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 : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Tue Aug 29 11:54:52 EDT 2023
% 0.11/0.32 % CPUTime :
% 0.17/0.39 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.17/0.39
% 0.17/0.39 % SZS status Theorem
% 0.17/0.39
% 0.17/0.39 % SZS output start Proof
% 0.17/0.39 Take the following subset of the input axioms:
% 0.17/0.39 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.17/0.39 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.17/0.39 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.17/0.39 fof(domain1, axiom, ![X0]: addition(X0, multiplication(domain(X0), X0))=multiplication(domain(X0), X0)).
% 0.17/0.39 fof(domain3, axiom, ![X0_2]: addition(domain(X0_2), one)=one).
% 0.17/0.39 fof(domain5, axiom, ![X1, X0_2]: domain(addition(X0_2, X1))=addition(domain(X0_2), domain(X1))).
% 0.17/0.39 fof(goals, conjecture, ![X0_2]: domain(star(X0_2))=one).
% 0.17/0.39 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.17/0.39 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.17/0.39 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.17/0.39 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.17/0.39 fof(star_unfold_right, axiom, ![A3]: leq(addition(one, multiplication(A3, star(A3))), star(A3))).
% 0.17/0.39
% 0.17/0.39 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.17/0.39 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.17/0.39 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.17/0.39 fresh(y, y, x1...xn) = u
% 0.17/0.39 C => fresh(s, t, x1...xn) = v
% 0.17/0.39 where fresh is a fresh function symbol and x1..xn are the free
% 0.17/0.39 variables of u and v.
% 0.17/0.39 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.17/0.39 input problem has no model of domain size 1).
% 0.17/0.39
% 0.17/0.39 The encoding turns the above axioms into the following unit equations and goals:
% 0.17/0.39
% 0.17/0.39 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.17/0.39 Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.17/0.39 Axiom 3 (additive_idempotence): addition(X, X) = X.
% 0.17/0.39 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.17/0.39 Axiom 5 (domain3): addition(domain(X), one) = one.
% 0.17/0.39 Axiom 6 (domain5): domain(addition(X, Y)) = addition(domain(X), domain(Y)).
% 0.17/0.39 Axiom 7 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.17/0.39 Axiom 8 (order_1): fresh(X, X, Y, Z) = Z.
% 0.17/0.39 Axiom 9 (domain1): addition(X, multiplication(domain(X), X)) = multiplication(domain(X), X).
% 0.17/0.39 Axiom 10 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.17/0.39 Axiom 11 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.17/0.40 Axiom 12 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
% 0.17/0.40
% 0.17/0.40 Lemma 13: addition(one, domain(X)) = one.
% 0.17/0.40 Proof:
% 0.17/0.40 addition(one, domain(X))
% 0.17/0.40 = { by axiom 4 (additive_commutativity) R->L }
% 0.17/0.40 addition(domain(X), one)
% 0.17/0.40 = { by axiom 5 (domain3) }
% 0.17/0.40 one
% 0.17/0.40
% 0.17/0.40 Lemma 14: addition(one, multiplication(addition(X, one), star(X))) = star(X).
% 0.17/0.40 Proof:
% 0.17/0.40 addition(one, multiplication(addition(X, one), star(X)))
% 0.17/0.40 = { by axiom 4 (additive_commutativity) R->L }
% 0.17/0.40 addition(one, multiplication(addition(one, X), star(X)))
% 0.17/0.40 = { by axiom 10 (left_distributivity) }
% 0.17/0.40 addition(one, addition(multiplication(one, star(X)), multiplication(X, star(X))))
% 0.17/0.40 = { by axiom 2 (multiplicative_left_identity) }
% 0.17/0.40 addition(one, addition(star(X), multiplication(X, star(X))))
% 0.17/0.40 = { by axiom 4 (additive_commutativity) R->L }
% 0.17/0.40 addition(one, addition(multiplication(X, star(X)), star(X)))
% 0.17/0.40 = { by axiom 7 (additive_associativity) }
% 0.17/0.40 addition(addition(one, multiplication(X, star(X))), star(X))
% 0.17/0.40 = { by axiom 11 (order_1) R->L }
% 0.17/0.40 fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
% 0.17/0.40 = { by axiom 12 (star_unfold_right) }
% 0.17/0.40 fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
% 0.17/0.40 = { by axiom 8 (order_1) }
% 0.17/0.40 star(X)
% 0.17/0.40
% 0.17/0.40 Goal 1 (goals): domain(star(x0)) = one.
% 0.17/0.40 Proof:
% 0.17/0.40 domain(star(x0))
% 0.17/0.40 = { by lemma 14 R->L }
% 0.17/0.40 domain(addition(one, multiplication(addition(x0, one), star(x0))))
% 0.17/0.40 = { by axiom 3 (additive_idempotence) R->L }
% 0.17/0.40 domain(addition(addition(one, one), multiplication(addition(x0, one), star(x0))))
% 0.17/0.40 = { by axiom 7 (additive_associativity) R->L }
% 0.17/0.40 domain(addition(one, addition(one, multiplication(addition(x0, one), star(x0)))))
% 0.17/0.40 = { by lemma 14 }
% 0.17/0.40 domain(addition(one, star(x0)))
% 0.17/0.40 = { by axiom 6 (domain5) }
% 0.17/0.40 addition(domain(one), domain(star(x0)))
% 0.17/0.40 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.17/0.40 addition(multiplication(domain(one), one), domain(star(x0)))
% 0.17/0.40 = { by axiom 9 (domain1) R->L }
% 0.17/0.40 addition(addition(one, multiplication(domain(one), one)), domain(star(x0)))
% 0.17/0.40 = { by axiom 1 (multiplicative_right_identity) }
% 0.17/0.40 addition(addition(one, domain(one)), domain(star(x0)))
% 0.17/0.40 = { by lemma 13 }
% 0.17/0.40 addition(one, domain(star(x0)))
% 0.17/0.40 = { by lemma 13 }
% 0.17/0.40 one
% 0.17/0.40 % SZS output end Proof
% 0.17/0.40
% 0.17/0.40 RESULT: Theorem (the conjecture is true).
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