TSTP Solution File: KLE024+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE024+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 : n029.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 05:35:32 EDT 2023
% Result : Theorem 0.18s 0.48s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : KLE024+2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n029.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 12:22:26 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.18/0.48 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.18/0.48
% 0.18/0.48 % SZS status Theorem
% 0.18/0.48
% 0.18/0.49 % SZS output start Proof
% 0.18/0.49 Take the following subset of the input axioms:
% 0.18/0.49 fof(additive_identity, axiom, ![A]: addition(A, zero)=A).
% 0.18/0.49 fof(goals, conjecture, ![X0, X1, X2]: ((test(X1) & test(X2)) => (addition(multiplication(X0, c(X2)), multiplication(c(X1), X0))=multiplication(c(X1), X0) => multiplication(multiplication(X1, X0), c(X2))=zero))).
% 0.18/0.49 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 0.18/0.49 fof(multiplicative_associativity, axiom, ![B, C, A2]: multiplication(A2, multiplication(B, C))=multiplication(multiplication(A2, B), C)).
% 0.18/0.49 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.18/0.49 fof(test_2, axiom, ![X0_2, X1_2]: (complement(X1_2, X0_2) <=> (multiplication(X0_2, X1_2)=zero & (multiplication(X1_2, X0_2)=zero & addition(X0_2, X1_2)=one)))).
% 0.18/0.49 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.18/0.49
% 0.18/0.49 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.18/0.49 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.18/0.49 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.18/0.49 fresh(y, y, x1...xn) = u
% 0.18/0.49 C => fresh(s, t, x1...xn) = v
% 0.18/0.49 where fresh is a fresh function symbol and x1..xn are the free
% 0.18/0.49 variables of u and v.
% 0.18/0.49 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.18/0.49 input problem has no model of domain size 1).
% 0.18/0.49
% 0.18/0.49 The encoding turns the above axioms into the following unit equations and goals:
% 0.18/0.49
% 0.18/0.49 Axiom 1 (goals_1): test(x1) = true.
% 0.18/0.49 Axiom 2 (left_annihilation): multiplication(zero, X) = zero.
% 0.18/0.49 Axiom 3 (additive_identity): addition(X, zero) = X.
% 0.18/0.49 Axiom 4 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.18/0.49 Axiom 5 (test_2_3): fresh10(X, X, Y, Z) = zero.
% 0.18/0.49 Axiom 6 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 0.18/0.49 Axiom 7 (test_3): fresh8(X, X, Y, Z) = true.
% 0.18/0.49 Axiom 8 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 0.18/0.49 Axiom 9 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.18/0.49 Axiom 10 (test_2_3): fresh10(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 0.18/0.49 Axiom 11 (goals): addition(multiplication(x0, c(x2)), multiplication(c(x1), x0)) = multiplication(c(x1), x0).
% 0.18/0.49
% 0.18/0.49 Lemma 12: multiplication(x1, multiplication(c(x1), X)) = zero.
% 0.18/0.49 Proof:
% 0.18/0.49 multiplication(x1, multiplication(c(x1), X))
% 0.18/0.49 = { by axiom 4 (multiplicative_associativity) }
% 0.18/0.49 multiplication(multiplication(x1, c(x1)), X)
% 0.18/0.49 = { by axiom 10 (test_2_3) R->L }
% 0.18/0.49 multiplication(fresh10(complement(x1, c(x1)), true, c(x1), x1), X)
% 0.18/0.49 = { by axiom 6 (test_3) R->L }
% 0.18/0.49 multiplication(fresh10(fresh9(true, true, x1, c(x1)), true, c(x1), x1), X)
% 0.18/0.49 = { by axiom 1 (goals_1) R->L }
% 0.18/0.49 multiplication(fresh10(fresh9(test(x1), true, x1, c(x1)), true, c(x1), x1), X)
% 0.18/0.49 = { by axiom 8 (test_3) }
% 0.18/0.49 multiplication(fresh10(fresh8(c(x1), c(x1), x1, c(x1)), true, c(x1), x1), X)
% 0.18/0.49 = { by axiom 7 (test_3) }
% 0.18/0.49 multiplication(fresh10(true, true, c(x1), x1), X)
% 0.18/0.49 = { by axiom 5 (test_2_3) }
% 0.18/0.49 multiplication(zero, X)
% 0.18/0.49 = { by axiom 2 (left_annihilation) }
% 0.18/0.49 zero
% 0.18/0.49
% 0.18/0.49 Goal 1 (goals_3): multiplication(multiplication(x1, x0), c(x2)) = zero.
% 0.18/0.49 Proof:
% 0.18/0.49 multiplication(multiplication(x1, x0), c(x2))
% 0.18/0.49 = { by axiom 3 (additive_identity) R->L }
% 0.18/0.49 addition(multiplication(multiplication(x1, x0), c(x2)), zero)
% 0.18/0.49 = { by lemma 12 R->L }
% 0.18/0.49 addition(multiplication(multiplication(x1, x0), c(x2)), multiplication(x1, multiplication(c(x1), x0)))
% 0.18/0.49 = { by axiom 4 (multiplicative_associativity) R->L }
% 0.18/0.49 addition(multiplication(x1, multiplication(x0, c(x2))), multiplication(x1, multiplication(c(x1), x0)))
% 0.18/0.49 = { by axiom 9 (right_distributivity) R->L }
% 0.18/0.49 multiplication(x1, addition(multiplication(x0, c(x2)), multiplication(c(x1), x0)))
% 0.18/0.49 = { by axiom 11 (goals) }
% 0.18/0.49 multiplication(x1, multiplication(c(x1), x0))
% 0.18/0.49 = { by lemma 12 }
% 0.18/0.49 zero
% 0.18/0.49 % SZS output end Proof
% 0.18/0.49
% 0.18/0.49 RESULT: Theorem (the conjecture is true).
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