TSTP Solution File: KLE025+2 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE025+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 : n006.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 0.19s 0.43s
% Output : Proof 0.19s
% 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 : KLE025+2 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n006.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Tue Aug 29 11:19:51 EDT 2023
% 0.13/0.33 % CPUTime :
% 0.19/0.43 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
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% 0.19/0.43 % SZS status Theorem
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% 0.19/0.45 % SZS output start Proof
% 0.19/0.45 Take the following subset of the input axioms:
% 0.19/0.45 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.19/0.45 fof(additive_identity, axiom, ![A2]: addition(A2, zero)=A2).
% 0.19/0.45 fof(goals, conjecture, ![X0, X1, X2]: ((test(X1) & test(X2)) => (multiplication(multiplication(X1, X0), c(X2))=zero => multiplication(X1, X0)=multiplication(multiplication(X1, X0), X2)))).
% 0.19/0.45 fof(multiplicative_associativity, axiom, ![C, A2, B2]: multiplication(A2, multiplication(B2, C))=multiplication(multiplication(A2, B2), C)).
% 0.19/0.45 fof(multiplicative_right_identity, axiom, ![A2]: multiplication(A2, one)=A2).
% 0.19/0.45 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 0.19/0.45 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)))).
% 0.19/0.45 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.19/0.45
% 0.19/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.45 fresh(y, y, x1...xn) = u
% 0.19/0.45 C => fresh(s, t, x1...xn) = v
% 0.19/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.45 variables of u and v.
% 0.19/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.45 input problem has no model of domain size 1).
% 0.19/0.45
% 0.19/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.45
% 0.19/0.45 Axiom 1 (goals_2): test(x2) = true.
% 0.19/0.45 Axiom 2 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.19/0.45 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.19/0.45 Axiom 4 (additive_identity): addition(X, zero) = X.
% 0.19/0.45 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 0.19/0.45 Axiom 6 (test_2_1): fresh12(X, X, Y, Z) = one.
% 0.19/0.45 Axiom 7 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 0.19/0.45 Axiom 8 (test_3): fresh8(X, X, Y, Z) = true.
% 0.19/0.45 Axiom 9 (goals): multiplication(multiplication(x1, x0), c(x2)) = zero.
% 0.19/0.45 Axiom 10 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 0.19/0.45 Axiom 11 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.19/0.45 Axiom 12 (test_2_1): fresh12(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.19/0.45
% 0.19/0.45 Goal 1 (goals_3): multiplication(x1, x0) = multiplication(multiplication(x1, x0), x2).
% 0.19/0.45 Proof:
% 0.19/0.45 multiplication(x1, x0)
% 0.19/0.45 = { by axiom 2 (multiplicative_right_identity) R->L }
% 0.19/0.45 multiplication(multiplication(x1, x0), one)
% 0.19/0.45 = { by axiom 6 (test_2_1) R->L }
% 0.19/0.45 multiplication(multiplication(x1, x0), fresh12(true, true, c(x2), x2))
% 0.19/0.45 = { by axiom 8 (test_3) R->L }
% 0.19/0.45 multiplication(multiplication(x1, x0), fresh12(fresh8(c(x2), c(x2), x2, c(x2)), true, c(x2), x2))
% 0.19/0.45 = { by axiom 10 (test_3) R->L }
% 0.19/0.45 multiplication(multiplication(x1, x0), fresh12(fresh9(test(x2), true, x2, c(x2)), true, c(x2), x2))
% 0.19/0.45 = { by axiom 1 (goals_2) }
% 0.19/0.45 multiplication(multiplication(x1, x0), fresh12(fresh9(true, true, x2, c(x2)), true, c(x2), x2))
% 0.19/0.45 = { by axiom 7 (test_3) }
% 0.19/0.45 multiplication(multiplication(x1, x0), fresh12(complement(x2, c(x2)), true, c(x2), x2))
% 0.19/0.45 = { by axiom 12 (test_2_1) }
% 0.19/0.45 multiplication(multiplication(x1, x0), addition(c(x2), x2))
% 0.19/0.45 = { by axiom 3 (additive_commutativity) }
% 0.19/0.45 multiplication(multiplication(x1, x0), addition(x2, c(x2)))
% 0.19/0.45 = { by axiom 5 (multiplicative_associativity) R->L }
% 0.19/0.45 multiplication(x1, multiplication(x0, addition(x2, c(x2))))
% 0.19/0.45 = { by axiom 11 (right_distributivity) }
% 0.19/0.45 multiplication(x1, addition(multiplication(x0, x2), multiplication(x0, c(x2))))
% 0.19/0.45 = { by axiom 11 (right_distributivity) }
% 0.19/0.45 addition(multiplication(x1, multiplication(x0, x2)), multiplication(x1, multiplication(x0, c(x2))))
% 0.19/0.45 = { by axiom 5 (multiplicative_associativity) }
% 0.19/0.45 addition(multiplication(x1, multiplication(x0, x2)), multiplication(multiplication(x1, x0), c(x2)))
% 0.19/0.45 = { by axiom 9 (goals) }
% 0.19/0.45 addition(multiplication(x1, multiplication(x0, x2)), zero)
% 0.19/0.45 = { by axiom 4 (additive_identity) }
% 0.19/0.45 multiplication(x1, multiplication(x0, x2))
% 0.19/0.45 = { by axiom 5 (multiplicative_associativity) }
% 0.19/0.45 multiplication(multiplication(x1, x0), x2)
% 0.19/0.45 % SZS output end Proof
% 0.19/0.45
% 0.19/0.45 RESULT: Theorem (the conjecture is true).
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