TSTP Solution File: KLE010+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE010+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 : n013.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 05:35:26 EDT 2023
% Result : Theorem 0.20s 0.58s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : KLE010+4 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % 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 : n013.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 11:39:01 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.20/0.58 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.58
% 0.20/0.58 % SZS status Theorem
% 0.20/0.58
% 0.20/0.60 % SZS output start Proof
% 0.20/0.60 Take the following subset of the input axioms:
% 0.20/0.60 fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
% 0.20/0.60 fof(additive_commutativity, axiom, ![A3, B2]: addition(A3, B2)=addition(B2, A3)).
% 0.20/0.60 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.20/0.60 fof(goals, conjecture, ![X0, X1]: ((test(X1) & test(X0)) => (leq(one, addition(addition(addition(addition(multiplication(X1, X0), multiplication(c(X1), X0)), multiplication(X0, X1)), multiplication(c(X0), X1)), multiplication(c(X0), c(X1)))) & leq(addition(addition(addition(addition(multiplication(X1, X0), multiplication(c(X1), X0)), multiplication(X0, X1)), multiplication(c(X0), X1)), multiplication(c(X0), c(X1))), one)))).
% 0.20/0.60 fof(left_distributivity, axiom, ![A3, B2, C2]: multiplication(addition(A3, B2), C2)=addition(multiplication(A3, C2), multiplication(B2, C2))).
% 0.20/0.60 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.20/0.60 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 0.20/0.60 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.20/0.60 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 0.20/0.60 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1_2]: complement(X1_2, X0_2))).
% 0.20/0.60 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)))).
% 0.20/0.60 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.20/0.60 fof(test_deMorgan1, axiom, ![X0_2, X1_2]: ((test(X0_2) & test(X1_2)) => c(addition(X0_2, X1_2))=multiplication(c(X0_2), c(X1_2)))).
% 0.20/0.60
% 0.20/0.60 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.60 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.60 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.60 fresh(y, y, x1...xn) = u
% 0.20/0.60 C => fresh(s, t, x1...xn) = v
% 0.20/0.60 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.60 variables of u and v.
% 0.20/0.60 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.60 input problem has no model of domain size 1).
% 0.20/0.60
% 0.20/0.60 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.60
% 0.20/0.60 Axiom 1 (goals): test(x1) = true.
% 0.20/0.60 Axiom 2 (goals_1): test(x0) = true.
% 0.20/0.60 Axiom 3 (multiplicative_right_identity): multiplication(X, one) = X.
% 0.20/0.60 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.20/0.60 Axiom 5 (additive_idempotence): addition(X, X) = X.
% 0.20/0.60 Axiom 6 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.20/0.60 Axiom 7 (test_1): fresh16(X, X, Y) = true.
% 0.20/0.60 Axiom 8 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
% 0.20/0.60 Axiom 9 (test_1): fresh16(test(X), true, X) = complement(x1_2(X), X).
% 0.20/0.60 Axiom 10 (order): fresh15(X, X, Y, Z) = true.
% 0.20/0.60 Axiom 11 (test_2_1): fresh12(X, X, Y, Z) = one.
% 0.20/0.60 Axiom 12 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 0.20/0.60 Axiom 13 (test_3): fresh8(X, X, Y, Z) = true.
% 0.20/0.60 Axiom 14 (test_deMorgan1): fresh6(X, X, Y, Z) = multiplication(c(Y), c(Z)).
% 0.20/0.60 Axiom 15 (test_deMorgan1): fresh5(X, X, Y, Z) = c(addition(Y, Z)).
% 0.20/0.60 Axiom 16 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 0.20/0.60 Axiom 17 (test_deMorgan1): fresh6(test(X), true, Y, X) = fresh5(test(Y), true, Y, X).
% 0.20/0.60 Axiom 18 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 0.20/0.60 Axiom 19 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.20/0.60 Axiom 20 (order): fresh15(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.20/0.60 Axiom 21 (test_2_1): fresh12(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.20/0.60
% 0.20/0.60 Lemma 22: leq(X, X) = true.
% 0.20/0.60 Proof:
% 0.20/0.60 leq(X, X)
% 0.20/0.60 = { by axiom 20 (order) R->L }
% 0.20/0.60 fresh15(addition(X, X), X, X, X)
% 0.20/0.60 = { by axiom 5 (additive_idempotence) }
% 0.20/0.60 fresh15(X, X, X, X)
% 0.20/0.60 = { by axiom 10 (order) }
% 0.20/0.60 true
% 0.20/0.60
% 0.20/0.60 Lemma 23: addition(x0, c(x0)) = one.
% 0.20/0.60 Proof:
% 0.20/0.60 addition(x0, c(x0))
% 0.20/0.60 = { by axiom 6 (additive_commutativity) R->L }
% 0.20/0.60 addition(c(x0), x0)
% 0.20/0.60 = { by axiom 21 (test_2_1) R->L }
% 0.20/0.60 fresh12(complement(x0, c(x0)), true, c(x0), x0)
% 0.20/0.60 = { by axiom 12 (test_3) R->L }
% 0.20/0.60 fresh12(fresh9(true, true, x0, c(x0)), true, c(x0), x0)
% 0.20/0.60 = { by axiom 2 (goals_1) R->L }
% 0.20/0.60 fresh12(fresh9(test(x0), true, x0, c(x0)), true, c(x0), x0)
% 0.20/0.60 = { by axiom 16 (test_3) }
% 0.20/0.60 fresh12(fresh8(c(x0), c(x0), x0, c(x0)), true, c(x0), x0)
% 0.20/0.60 = { by axiom 13 (test_3) }
% 0.20/0.60 fresh12(true, true, c(x0), x0)
% 0.20/0.60 = { by axiom 11 (test_2_1) }
% 0.20/0.60 one
% 0.20/0.60
% 0.20/0.60 Lemma 24: addition(x0, x1_2(x0)) = one.
% 0.20/0.60 Proof:
% 0.20/0.60 addition(x0, x1_2(x0))
% 0.20/0.60 = { by axiom 21 (test_2_1) R->L }
% 0.20/0.60 fresh12(complement(x1_2(x0), x0), true, x0, x1_2(x0))
% 0.20/0.60 = { by axiom 9 (test_1) R->L }
% 0.20/0.60 fresh12(fresh16(test(x0), true, x0), true, x0, x1_2(x0))
% 0.20/0.60 = { by axiom 2 (goals_1) }
% 0.20/0.60 fresh12(fresh16(true, true, x0), true, x0, x1_2(x0))
% 0.20/0.60 = { by axiom 7 (test_1) }
% 0.20/0.60 fresh12(true, true, x0, x1_2(x0))
% 0.20/0.60 = { by axiom 11 (test_2_1) }
% 0.20/0.60 one
% 0.20/0.60
% 0.20/0.61 Lemma 25: addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))) = one.
% 0.20/0.61 Proof:
% 0.20/0.61 addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 8 (additive_associativity) R->L }
% 0.20/0.61 addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(x0, x1), multiplication(c(x0), x1))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 8 (additive_associativity) R->L }
% 0.20/0.61 addition(addition(multiplication(x1, x0), addition(multiplication(c(x1), x0), addition(multiplication(x0, x1), multiplication(c(x0), x1)))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 19 (left_distributivity) R->L }
% 0.20/0.61 addition(addition(multiplication(x1, x0), addition(multiplication(c(x1), x0), multiplication(addition(x0, c(x0)), x1))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 6 (additive_commutativity) }
% 0.20/0.61 addition(addition(addition(multiplication(c(x1), x0), multiplication(addition(x0, c(x0)), x1)), multiplication(x1, x0)), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 8 (additive_associativity) R->L }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), addition(multiplication(addition(x0, c(x0)), x1), multiplication(x1, x0))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by lemma 23 }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), addition(multiplication(one, x1), multiplication(x1, x0))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 4 (multiplicative_left_identity) }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), addition(x1, multiplication(x1, x0))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 3 (multiplicative_right_identity) R->L }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), addition(multiplication(x1, one), multiplication(x1, x0))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 18 (right_distributivity) R->L }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), multiplication(x1, addition(one, x0))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 6 (additive_commutativity) }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), multiplication(x1, addition(x0, one))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by lemma 24 R->L }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), multiplication(x1, addition(x0, addition(x0, x1_2(x0))))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 8 (additive_associativity) }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), multiplication(x1, addition(addition(x0, x0), x1_2(x0)))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 5 (additive_idempotence) }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), multiplication(x1, addition(x0, x1_2(x0)))), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by lemma 24 }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), multiplication(x1, one)), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 3 (multiplicative_right_identity) }
% 0.20/0.61 addition(addition(multiplication(c(x1), x0), x1), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 6 (additive_commutativity) }
% 0.20/0.61 addition(addition(x1, multiplication(c(x1), x0)), multiplication(c(x0), c(x1)))
% 0.20/0.61 = { by axiom 8 (additive_associativity) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), multiplication(c(x0), c(x1))))
% 0.20/0.61 = { by axiom 14 (test_deMorgan1) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh6(true, true, x0, x1)))
% 0.20/0.61 = { by axiom 1 (goals) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh6(test(x1), true, x0, x1)))
% 0.20/0.61 = { by axiom 17 (test_deMorgan1) }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh5(test(x0), true, x0, x1)))
% 0.20/0.61 = { by axiom 2 (goals_1) }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh5(true, true, x0, x1)))
% 0.20/0.61 = { by axiom 15 (test_deMorgan1) }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), c(addition(x0, x1))))
% 0.20/0.61 = { by axiom 6 (additive_commutativity) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), c(addition(x1, x0))))
% 0.20/0.61 = { by axiom 15 (test_deMorgan1) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh5(true, true, x1, x0)))
% 0.20/0.61 = { by axiom 1 (goals) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh5(test(x1), true, x1, x0)))
% 0.20/0.61 = { by axiom 17 (test_deMorgan1) R->L }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh6(test(x0), true, x1, x0)))
% 0.20/0.61 = { by axiom 2 (goals_1) }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), fresh6(true, true, x1, x0)))
% 0.20/0.61 = { by axiom 14 (test_deMorgan1) }
% 0.20/0.61 addition(x1, addition(multiplication(c(x1), x0), multiplication(c(x1), c(x0))))
% 0.20/0.61 = { by axiom 18 (right_distributivity) R->L }
% 0.20/0.61 addition(x1, multiplication(c(x1), addition(x0, c(x0))))
% 0.20/0.61 = { by lemma 23 }
% 0.20/0.61 addition(x1, multiplication(c(x1), one))
% 0.20/0.61 = { by axiom 3 (multiplicative_right_identity) }
% 0.20/0.61 addition(x1, c(x1))
% 0.20/0.61 = { by axiom 6 (additive_commutativity) R->L }
% 0.20/0.61 addition(c(x1), x1)
% 0.20/0.61 = { by axiom 21 (test_2_1) R->L }
% 0.20/0.61 fresh12(complement(x1, c(x1)), true, c(x1), x1)
% 0.20/0.61 = { by axiom 12 (test_3) R->L }
% 0.20/0.61 fresh12(fresh9(true, true, x1, c(x1)), true, c(x1), x1)
% 0.20/0.61 = { by axiom 1 (goals) R->L }
% 0.20/0.61 fresh12(fresh9(test(x1), true, x1, c(x1)), true, c(x1), x1)
% 0.20/0.61 = { by axiom 16 (test_3) }
% 0.20/0.61 fresh12(fresh8(c(x1), c(x1), x1, c(x1)), true, c(x1), x1)
% 0.20/0.61 = { by axiom 13 (test_3) }
% 0.20/0.61 fresh12(true, true, c(x1), x1)
% 0.20/0.61 = { by axiom 11 (test_2_1) }
% 0.20/0.61 one
% 0.20/0.61
% 0.20/0.61 Goal 1 (goals_2): tuple(leq(addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))))) = tuple(true, true).
% 0.20/0.61 Proof:
% 0.20/0.61 tuple(leq(addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1))), one), leq(one, addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))))
% 0.20/0.61 = { by lemma 25 }
% 0.20/0.61 tuple(leq(one, one), leq(one, addition(addition(addition(addition(multiplication(x1, x0), multiplication(c(x1), x0)), multiplication(x0, x1)), multiplication(c(x0), x1)), multiplication(c(x0), c(x1)))))
% 0.20/0.61 = { by lemma 25 }
% 0.20/0.61 tuple(leq(one, one), leq(one, one))
% 0.20/0.61 = { by lemma 22 }
% 0.20/0.61 tuple(true, leq(one, one))
% 0.20/0.61 = { by lemma 22 }
% 0.20/0.61 tuple(true, true)
% 0.20/0.61 % SZS output end Proof
% 0.20/0.61
% 0.20/0.61 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------