TSTP Solution File: KLE019+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE019+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 : n010.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 05:35:29 EDT 2023
% Result : Theorem 0.21s 0.59s
% Output : Proof 0.21s
% 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 : KLE019+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n010.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 29 12:03:34 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.59 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 0.21/0.59
% 0.21/0.59 % SZS status Theorem
% 0.21/0.59
% 0.21/0.60 % SZS output start Proof
% 0.21/0.60 Take the following subset of the input axioms:
% 0.21/0.60 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.21/0.60 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.21/0.60 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.21/0.60 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 0.21/0.60 fof(goals, conjecture, ![X0, X1, X2]: ((test(X0) & (test(X1) & test(X2))) => (leq(multiplication(X0, c(X1)), X2) <= leq(X0, addition(X1, X2))))).
% 0.21/0.60 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.21/0.60 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.21/0.60 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.21/0.60 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.21/0.60 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)))).
% 0.21/0.60 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.21/0.60
% 0.21/0.60 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.60 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.60 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.60 fresh(y, y, x1...xn) = u
% 0.21/0.60 C => fresh(s, t, x1...xn) = v
% 0.21/0.60 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.60 variables of u and v.
% 0.21/0.60 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.60 input problem has no model of domain size 1).
% 0.21/0.60
% 0.21/0.60 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.60
% 0.21/0.60 Axiom 1 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.21/0.60 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 0.21/0.60 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.21/0.60 Axiom 4 (additive_identity): addition(X, zero) = X.
% 0.21/0.60 Axiom 5 (goals_1): test(x1) = true.
% 0.21/0.60 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.21/0.60 Axiom 7 (goals): leq(x0, addition(x1, x2)) = true.
% 0.21/0.60 Axiom 8 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.21/0.60 Axiom 9 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.21/0.60 Axiom 10 (order): fresh11(X, X, Y, Z) = true.
% 0.21/0.60 Axiom 11 (test_2_1): fresh8(X, X, Y, Z) = one.
% 0.21/0.60 Axiom 12 (test_2_3): fresh6(X, X, Y, Z) = zero.
% 0.21/0.60 Axiom 13 (test_3): fresh5(X, X, Y, Z) = complement(Y, Z).
% 0.21/0.60 Axiom 14 (test_3): fresh4(X, X, Y, Z) = true.
% 0.21/0.60 Axiom 15 (order_1): fresh2(X, X, Y, Z) = Z.
% 0.21/0.60 Axiom 16 (order): fresh11(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.21/0.60 Axiom 17 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.21/0.60 Axiom 18 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 0.21/0.60 Axiom 19 (test_3): fresh5(test(X), true, X, Y) = fresh4(c(X), Y, X, Y).
% 0.21/0.60 Axiom 20 (order_1): fresh2(leq(X, Y), true, X, Y) = addition(X, Y).
% 0.21/0.60
% 0.21/0.60 Lemma 21: complement(x1, c(x1)) = true.
% 0.21/0.60 Proof:
% 0.21/0.60 complement(x1, c(x1))
% 0.21/0.60 = { by axiom 13 (test_3) R->L }
% 0.21/0.60 fresh5(true, true, x1, c(x1))
% 0.21/0.60 = { by axiom 5 (goals_1) R->L }
% 0.21/0.60 fresh5(test(x1), true, x1, c(x1))
% 0.21/0.60 = { by axiom 19 (test_3) }
% 0.21/0.60 fresh4(c(x1), c(x1), x1, c(x1))
% 0.21/0.60 = { by axiom 14 (test_3) }
% 0.21/0.60 true
% 0.21/0.60
% 0.21/0.60 Goal 1 (goals_4): leq(multiplication(x0, c(x1)), x2) = true.
% 0.21/0.60 Proof:
% 0.21/0.60 leq(multiplication(x0, c(x1)), x2)
% 0.21/0.60 = { by axiom 1 (multiplicative_right_identity) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), multiplication(x2, one))
% 0.21/0.60 = { by axiom 11 (test_2_1) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), multiplication(x2, fresh8(true, true, c(x1), x1)))
% 0.21/0.60 = { by lemma 21 R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), multiplication(x2, fresh8(complement(x1, c(x1)), true, c(x1), x1)))
% 0.21/0.60 = { by axiom 17 (test_2_1) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), multiplication(x2, addition(c(x1), x1)))
% 0.21/0.60 = { by axiom 3 (additive_commutativity) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), multiplication(x2, addition(x1, c(x1))))
% 0.21/0.60 = { by axiom 8 (right_distributivity) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(x2, c(x1))))
% 0.21/0.60 = { by axiom 4 (additive_identity) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(multiplication(x2, c(x1)), zero)))
% 0.21/0.60 = { by axiom 3 (additive_commutativity) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(zero, multiplication(x2, c(x1)))))
% 0.21/0.60 = { by axiom 12 (test_2_3) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(fresh6(true, true, c(x1), x1), multiplication(x2, c(x1)))))
% 0.21/0.60 = { by lemma 21 R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(fresh6(complement(x1, c(x1)), true, c(x1), x1), multiplication(x2, c(x1)))))
% 0.21/0.60 = { by axiom 18 (test_2_3) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(multiplication(x1, c(x1)), multiplication(x2, c(x1)))))
% 0.21/0.60 = { by axiom 9 (left_distributivity) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1))))
% 0.21/0.60 = { by axiom 15 (order_1) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(fresh2(true, true, x0, addition(x1, x2)), c(x1))))
% 0.21/0.60 = { by axiom 7 (goals) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(fresh2(leq(x0, addition(x1, x2)), true, x0, addition(x1, x2)), c(x1))))
% 0.21/0.60 = { by axiom 20 (order_1) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x0, addition(x1, x2)), c(x1))))
% 0.21/0.60 = { by axiom 9 (left_distributivity) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(multiplication(x0, c(x1)), multiplication(addition(x1, x2), c(x1)))))
% 0.21/0.60 = { by axiom 3 (additive_commutativity) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x2, x1), addition(multiplication(addition(x1, x2), c(x1)), multiplication(x0, c(x1)))))
% 0.21/0.60 = { by axiom 6 (additive_associativity) }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1))), multiplication(x0, c(x1))))
% 0.21/0.60 = { by axiom 3 (additive_commutativity) R->L }
% 0.21/0.60 leq(multiplication(x0, c(x1)), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))))
% 0.21/0.60 = { by axiom 16 (order) R->L }
% 0.21/0.60 fresh11(addition(multiplication(x0, c(x1)), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1))))), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))), multiplication(x0, c(x1)), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))))
% 0.21/0.60 = { by axiom 6 (additive_associativity) }
% 0.21/0.60 fresh11(addition(addition(multiplication(x0, c(x1)), multiplication(x0, c(x1))), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))), multiplication(x0, c(x1)), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))))
% 0.21/0.60 = { by axiom 2 (additive_idempotence) }
% 0.21/0.60 fresh11(addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))), multiplication(x0, c(x1)), addition(multiplication(x0, c(x1)), addition(multiplication(x2, x1), multiplication(addition(x1, x2), c(x1)))))
% 0.21/0.60 = { by axiom 10 (order) }
% 0.21/0.60 true
% 0.21/0.60 % SZS output end Proof
% 0.21/0.60
% 0.21/0.60 RESULT: Theorem (the conjecture is true).
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