TSTP Solution File: KLE009+4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE009+4 : 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 : n008.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 05:35:25 EDT 2023
% Result : Theorem 0.21s 0.54s
% 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
% 0.00/0.13 % Problem : KLE009+4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n008.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.20/0.35 % WCLimit : 300
% 0.20/0.35 % DateTime : Tue Aug 29 12:35:32 EDT 2023
% 0.20/0.35 % CPUTime :
% 0.21/0.54 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.54
% 0.21/0.54 % SZS status Theorem
% 0.21/0.54
% 0.21/0.55 % SZS output start Proof
% 0.21/0.55 Take the following subset of the input axioms:
% 0.21/0.56 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.21/0.56 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.21/0.56 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.21/0.56 fof(goals, conjecture, ![X0, X1]: ((test(X1) & test(X0)) => (leq(one, addition(addition(addition(multiplication(X0, X1), multiplication(X0, c(X1))), multiplication(c(X0), X1)), multiplication(c(X0), c(X1)))) & leq(addition(addition(addition(multiplication(X0, X1), multiplication(X0, c(X1))), multiplication(c(X0), X1)), multiplication(c(X0), c(X1))), one)))).
% 0.21/0.56 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.21/0.56 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.21/0.56 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.21/0.56 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)))).
% 0.21/0.56 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.21/0.56
% 0.21/0.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.56 fresh(y, y, x1...xn) = u
% 0.21/0.56 C => fresh(s, t, x1...xn) = v
% 0.21/0.56 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.56 variables of u and v.
% 0.21/0.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.56 input problem has no model of domain size 1).
% 0.21/0.56
% 0.21/0.56 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.56
% 0.21/0.56 Axiom 1 (goals): test(x1) = true.
% 0.21/0.56 Axiom 2 (goals_1): test(x0) = true.
% 0.21/0.56 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.21/0.56 Axiom 4 (additive_idempotence): addition(X, X) = X.
% 0.21/0.56 Axiom 5 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.21/0.56 Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.21/0.56 Axiom 7 (order): fresh15(X, X, Y, Z) = true.
% 0.21/0.56 Axiom 8 (test_2_1): fresh12(X, X, Y, Z) = one.
% 0.21/0.56 Axiom 9 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 0.21/0.56 Axiom 10 (test_3): fresh8(X, X, Y, Z) = true.
% 0.21/0.56 Axiom 11 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 0.21/0.56 Axiom 12 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.21/0.56 Axiom 13 (order): fresh15(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.21/0.56 Axiom 14 (test_2_1): fresh12(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.21/0.56
% 0.21/0.56 Lemma 15: leq(X, X) = true.
% 0.21/0.56 Proof:
% 0.21/0.56 leq(X, X)
% 0.21/0.56 = { by axiom 13 (order) R->L }
% 0.21/0.56 fresh15(addition(X, X), X, X, X)
% 0.21/0.56 = { by axiom 4 (additive_idempotence) }
% 0.21/0.56 fresh15(X, X, X, X)
% 0.21/0.56 = { by axiom 7 (order) }
% 0.21/0.56 true
% 0.21/0.56
% 0.21/0.56 Lemma 16: addition(multiplication(X, c(x1)), multiplication(X, x1)) = X.
% 0.21/0.56 Proof:
% 0.21/0.56 addition(multiplication(X, c(x1)), multiplication(X, x1))
% 0.21/0.56 = { by axiom 12 (right_distributivity) R->L }
% 0.21/0.56 multiplication(X, addition(c(x1), x1))
% 0.21/0.56 = { by axiom 14 (test_2_1) R->L }
% 0.21/0.56 multiplication(X, fresh12(complement(x1, c(x1)), true, c(x1), x1))
% 0.21/0.56 = { by axiom 9 (test_3) R->L }
% 0.21/0.56 multiplication(X, fresh12(fresh9(true, true, x1, c(x1)), true, c(x1), x1))
% 0.21/0.56 = { by axiom 1 (goals) R->L }
% 0.21/0.56 multiplication(X, fresh12(fresh9(test(x1), true, x1, c(x1)), true, c(x1), x1))
% 0.21/0.56 = { by axiom 11 (test_3) }
% 0.21/0.56 multiplication(X, fresh12(fresh8(c(x1), c(x1), x1, c(x1)), true, c(x1), x1))
% 0.21/0.56 = { by axiom 10 (test_3) }
% 0.21/0.56 multiplication(X, fresh12(true, true, c(x1), x1))
% 0.21/0.56 = { by axiom 8 (test_2_1) }
% 0.21/0.56 multiplication(X, one)
% 0.21/0.56 = { by axiom 3 (multiplicative_right_identity) }
% 0.21/0.56 X
% 0.21/0.56
% 0.21/0.56 Lemma 17: addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))) = one.
% 0.21/0.56 Proof:
% 0.21/0.56 addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))
% 0.21/0.56 = { by axiom 5 (additive_commutativity) R->L }
% 0.21/0.56 addition(multiplication(c(x0), c(x1)), addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)))
% 0.21/0.56 = { by axiom 6 (additive_associativity) R->L }
% 0.21/0.56 addition(multiplication(c(x0), c(x1)), addition(multiplication(x0, x1), addition(multiplication(x0, c(x1)), multiplication(c(x0), x1))))
% 0.21/0.56 = { by axiom 5 (additive_commutativity) }
% 0.21/0.56 addition(multiplication(c(x0), c(x1)), addition(multiplication(x0, x1), addition(multiplication(c(x0), x1), multiplication(x0, c(x1)))))
% 0.21/0.56 = { by axiom 5 (additive_commutativity) }
% 0.21/0.56 addition(multiplication(c(x0), c(x1)), addition(addition(multiplication(c(x0), x1), multiplication(x0, c(x1))), multiplication(x0, x1)))
% 0.21/0.56 = { by axiom 6 (additive_associativity) R->L }
% 0.21/0.56 addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), addition(multiplication(x0, c(x1)), multiplication(x0, x1))))
% 0.21/0.56 = { by lemma 16 }
% 0.21/0.56 addition(multiplication(c(x0), c(x1)), addition(multiplication(c(x0), x1), x0))
% 0.21/0.56 = { by axiom 6 (additive_associativity) }
% 0.21/0.56 addition(addition(multiplication(c(x0), c(x1)), multiplication(c(x0), x1)), x0)
% 0.21/0.56 = { by lemma 16 }
% 0.21/0.56 addition(c(x0), x0)
% 0.21/0.56 = { by axiom 14 (test_2_1) R->L }
% 0.21/0.56 fresh12(complement(x0, c(x0)), true, c(x0), x0)
% 0.21/0.56 = { by axiom 9 (test_3) R->L }
% 0.21/0.56 fresh12(fresh9(true, true, x0, c(x0)), true, c(x0), x0)
% 0.21/0.56 = { by axiom 2 (goals_1) R->L }
% 0.21/0.56 fresh12(fresh9(test(x0), true, x0, c(x0)), true, c(x0), x0)
% 0.21/0.56 = { by axiom 11 (test_3) }
% 0.21/0.56 fresh12(fresh8(c(x0), c(x0), x0, c(x0)), true, c(x0), x0)
% 0.21/0.56 = { by axiom 10 (test_3) }
% 0.21/0.56 fresh12(true, true, c(x0), x0)
% 0.21/0.56 = { by axiom 8 (test_2_1) }
% 0.21/0.57 one
% 0.21/0.57
% 0.21/0.57 Goal 1 (goals_2): tuple(leq(addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))))) = tuple(true, true).
% 0.21/0.57 Proof:
% 0.21/0.57 tuple(leq(addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))))
% 0.21/0.57 = { by lemma 17 }
% 0.21/0.57 tuple(leq(one, one), leq(one, addition(addition(addition(multiplication(x0, x1), multiplication(x0, c(x1))), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))))
% 0.21/0.57 = { by lemma 17 }
% 0.21/0.57 tuple(leq(one, one), leq(one, one))
% 0.21/0.57 = { by lemma 15 }
% 0.21/0.57 tuple(true, leq(one, one))
% 0.21/0.57 = { by lemma 15 }
% 0.21/0.57 tuple(true, true)
% 0.21/0.57 % SZS output end Proof
% 0.21/0.57
% 0.21/0.57 RESULT: Theorem (the conjecture is true).
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