TSTP Solution File: KLE027+3 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE027+3 : 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 : n003.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 05:35:34 EDT 2023
% Result : Theorem 2.64s 0.73s
% Output : Proof 3.11s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : KLE027+3 : TPTP v8.1.2. Released v4.0.0.
% 0.04/0.13 % 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 : n003.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 11:24:26 EDT 2023
% 0.13/0.34 % CPUTime :
% 2.64/0.73 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 2.64/0.73
% 2.64/0.73 % SZS status Theorem
% 2.64/0.73
% 3.02/0.74 % SZS output start Proof
% 3.02/0.74 Take the following subset of the input axioms:
% 3.02/0.74 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 3.02/0.74 fof(goals, conjecture, ![X0, X1, X2, X3, X4]: ((test(X3) & test(X4)) => addition(multiplication(X3, addition(multiplication(X3, X0), multiplication(c(X3), X1))), multiplication(c(X3), X2))=addition(multiplication(X3, X0), multiplication(c(X3), X2)))).
% 3.02/0.74 fof(left_annihilation, axiom, ![A2]: multiplication(zero, A2)=zero).
% 3.02/0.74 fof(left_distributivity, axiom, ![C, A2, B2]: multiplication(addition(A2, B2), C)=addition(multiplication(A2, C), multiplication(B2, C))).
% 3.02/0.74 fof(multiplicative_associativity, axiom, ![A2, B2, C2]: multiplication(A2, multiplication(B2, C2))=multiplication(multiplication(A2, B2), C2)).
% 3.02/0.74 fof(multiplicative_left_identity, axiom, ![A2]: multiplication(one, A2)=A2).
% 3.02/0.74 fof(right_distributivity, axiom, ![A2, B2, C2]: multiplication(A2, addition(B2, C2))=addition(multiplication(A2, B2), multiplication(A2, C2))).
% 3.02/0.74 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)))).
% 3.02/0.74 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 3.02/0.74
% 3.02/0.74 Now clausify the problem and encode Horn clauses using encoding 3 of
% 3.02/0.74 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 3.02/0.74 We repeatedly replace C & s=t => u=v by the two clauses:
% 3.02/0.74 fresh(y, y, x1...xn) = u
% 3.02/0.74 C => fresh(s, t, x1...xn) = v
% 3.02/0.74 where fresh is a fresh function symbol and x1..xn are the free
% 3.02/0.74 variables of u and v.
% 3.02/0.74 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 3.02/0.74 input problem has no model of domain size 1).
% 3.02/0.74
% 3.02/0.74 The encoding turns the above axioms into the following unit equations and goals:
% 3.02/0.74
% 3.02/0.74 Axiom 1 (goals): test(x3) = true.
% 3.02/0.74 Axiom 2 (left_annihilation): multiplication(zero, X) = zero.
% 3.02/0.74 Axiom 3 (multiplicative_left_identity): multiplication(one, X) = X.
% 3.02/0.74 Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 3.02/0.74 Axiom 5 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 3.02/0.74 Axiom 6 (test_2_1): fresh12(X, X, Y, Z) = one.
% 3.02/0.74 Axiom 7 (test_2_2): fresh11(X, X, Y, Z) = zero.
% 3.02/0.74 Axiom 8 (test_2_3): fresh10(X, X, Y, Z) = zero.
% 3.02/0.74 Axiom 9 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 3.02/0.74 Axiom 10 (test_3): fresh8(X, X, Y, Z) = true.
% 3.02/0.74 Axiom 11 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 3.02/0.74 Axiom 12 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 3.02/0.74 Axiom 13 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 3.02/0.74 Axiom 14 (test_2_1): fresh12(complement(X, Y), true, Y, X) = addition(Y, X).
% 3.02/0.74 Axiom 15 (test_2_2): fresh11(complement(X, Y), true, Y, X) = multiplication(Y, X).
% 3.02/0.74 Axiom 16 (test_2_3): fresh10(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 3.02/0.74
% 3.02/0.74 Lemma 17: complement(x3, c(x3)) = true.
% 3.02/0.74 Proof:
% 3.02/0.74 complement(x3, c(x3))
% 3.02/0.74 = { by axiom 9 (test_3) R->L }
% 3.02/0.74 fresh9(true, true, x3, c(x3))
% 3.02/0.74 = { by axiom 1 (goals) R->L }
% 3.02/0.74 fresh9(test(x3), true, x3, c(x3))
% 3.02/0.74 = { by axiom 11 (test_3) }
% 3.02/0.74 fresh8(c(x3), c(x3), x3, c(x3))
% 3.02/0.74 = { by axiom 10 (test_3) }
% 3.11/0.74 true
% 3.11/0.74
% 3.11/0.74 Goal 1 (goals_2): addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))), multiplication(c(x3), x2)) = addition(multiplication(x3, x0), multiplication(c(x3), x2)).
% 3.11/0.74 Proof:
% 3.11/0.74 addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))), multiplication(c(x3), x2))
% 3.11/0.74 = { by axiom 4 (additive_commutativity) }
% 3.11/0.74 addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))
% 3.11/0.74 = { by axiom 4 (additive_commutativity) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(c(x3), x1), multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 12 (right_distributivity) }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(x3, multiplication(c(x3), x1)), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 5 (multiplicative_associativity) }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(multiplication(x3, c(x3)), x1), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 16 (test_2_3) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(fresh10(complement(x3, c(x3)), true, c(x3), x3), x1), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by lemma 17 }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(fresh10(true, true, c(x3), x3), x1), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 8 (test_2_3) }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(zero, x1), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 2 (left_annihilation) }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(zero, multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 2 (left_annihilation) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(zero, x0), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 7 (test_2_2) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(fresh11(true, true, c(x3), x3), x0), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by lemma 17 R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(fresh11(complement(x3, c(x3)), true, c(x3), x3), x0), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 15 (test_2_2) }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(multiplication(c(x3), x3), x0), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 5 (multiplicative_associativity) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), addition(multiplication(c(x3), multiplication(x3, x0)), multiplication(x3, multiplication(x3, x0))))
% 3.11/0.74 = { by axiom 13 (left_distributivity) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), multiplication(addition(c(x3), x3), multiplication(x3, x0)))
% 3.11/0.74 = { by axiom 14 (test_2_1) R->L }
% 3.11/0.74 addition(multiplication(c(x3), x2), multiplication(fresh12(complement(x3, c(x3)), true, c(x3), x3), multiplication(x3, x0)))
% 3.11/0.74 = { by lemma 17 }
% 3.11/0.75 addition(multiplication(c(x3), x2), multiplication(fresh12(true, true, c(x3), x3), multiplication(x3, x0)))
% 3.11/0.75 = { by axiom 6 (test_2_1) }
% 3.11/0.75 addition(multiplication(c(x3), x2), multiplication(one, multiplication(x3, x0)))
% 3.11/0.75 = { by axiom 3 (multiplicative_left_identity) }
% 3.11/0.75 addition(multiplication(c(x3), x2), multiplication(x3, x0))
% 3.11/0.75 = { by axiom 4 (additive_commutativity) R->L }
% 3.11/0.75 addition(multiplication(x3, x0), multiplication(c(x3), x2))
% 3.11/0.75 % SZS output end Proof
% 3.11/0.75
% 3.11/0.75 RESULT: Theorem (the conjecture is true).
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