TSTP Solution File: KLE021+4 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : KLE021+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 : n001.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 05:35:30 EDT 2023
% Result : Theorem 0.20s 0.42s
% Output : Proof 0.20s
% 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 : KLE021+4 : 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
% 0.14/0.34 % Computer : n001.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 11:37:52 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.42 Command-line arguments: --flatten
% 0.20/0.42
% 0.20/0.42 % SZS status Theorem
% 0.20/0.42
% 0.20/0.42 % SZS output start Proof
% 0.20/0.42 Take the following subset of the input axioms:
% 0.20/0.42 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 0.20/0.42 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 0.20/0.42 fof(goals, conjecture, ![X0, X1]: (test(X1) => (leq(X0, addition(multiplication(X1, X0), multiplication(c(X1), X0))) & leq(addition(multiplication(X1, X0), multiplication(c(X1), X0)), X0)))).
% 0.20/0.42 fof(left_distributivity, axiom, ![C, B2, A3]: multiplication(addition(A3, B2), C)=addition(multiplication(A3, C), multiplication(B2, C))).
% 0.20/0.42 fof(multiplicative_left_identity, axiom, ![A3]: multiplication(one, A3)=A3).
% 0.20/0.42 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 0.20/0.42 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)))).
% 0.20/0.42 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 0.20/0.42
% 0.20/0.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.42 fresh(y, y, x1...xn) = u
% 0.20/0.42 C => fresh(s, t, x1...xn) = v
% 0.20/0.42 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.42 variables of u and v.
% 0.20/0.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.42 input problem has no model of domain size 1).
% 0.20/0.42
% 0.20/0.42 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.42
% 0.20/0.42 Axiom 1 (goals): test(x1) = true.
% 0.20/0.42 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 0.20/0.43 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 0.20/0.43 Axiom 4 (multiplicative_left_identity): multiplication(one, X) = X.
% 0.20/0.43 Axiom 5 (order): fresh15(X, X, Y, Z) = true.
% 0.20/0.43 Axiom 6 (test_2_1): fresh12(X, X, Y, Z) = one.
% 0.20/0.43 Axiom 7 (test_3): fresh9(X, X, Y, Z) = complement(Y, Z).
% 0.20/0.43 Axiom 8 (test_3): fresh8(X, X, Y, Z) = true.
% 0.20/0.43 Axiom 9 (test_3): fresh9(test(X), true, X, Y) = fresh8(c(X), Y, X, Y).
% 0.20/0.43 Axiom 10 (order): fresh15(addition(X, Y), Y, X, Y) = leq(X, Y).
% 0.20/0.43 Axiom 11 (test_2_1): fresh12(complement(X, Y), true, Y, X) = addition(Y, X).
% 0.20/0.43 Axiom 12 (left_distributivity): multiplication(addition(X, Y), Z) = addition(multiplication(X, Z), multiplication(Y, Z)).
% 0.20/0.43
% 0.20/0.43 Lemma 13: leq(X, X) = test(x1).
% 0.20/0.43 Proof:
% 0.20/0.43 leq(X, X)
% 0.20/0.43 = { by axiom 10 (order) R->L }
% 0.20/0.43 fresh15(addition(X, X), X, X, X)
% 0.20/0.43 = { by axiom 2 (additive_idempotence) }
% 0.20/0.43 fresh15(X, X, X, X)
% 0.20/0.43 = { by axiom 5 (order) }
% 0.20/0.43 true
% 0.20/0.43 = { by axiom 1 (goals) R->L }
% 0.20/0.43 test(x1)
% 0.20/0.43
% 0.20/0.43 Lemma 14: addition(multiplication(x1, X), multiplication(c(x1), X)) = X.
% 0.20/0.43 Proof:
% 0.20/0.43 addition(multiplication(x1, X), multiplication(c(x1), X))
% 0.20/0.43 = { by axiom 12 (left_distributivity) R->L }
% 0.20/0.43 multiplication(addition(x1, c(x1)), X)
% 0.20/0.43 = { by axiom 3 (additive_commutativity) }
% 0.20/0.43 multiplication(addition(c(x1), x1), X)
% 0.20/0.43 = { by axiom 11 (test_2_1) R->L }
% 0.20/0.43 multiplication(fresh12(complement(x1, c(x1)), true, c(x1), x1), X)
% 0.20/0.43 = { by axiom 1 (goals) R->L }
% 0.20/0.43 multiplication(fresh12(complement(x1, c(x1)), test(x1), c(x1), x1), X)
% 0.20/0.43 = { by axiom 7 (test_3) R->L }
% 0.20/0.43 multiplication(fresh12(fresh9(test(x1), test(x1), x1, c(x1)), test(x1), c(x1), x1), X)
% 0.20/0.43 = { by axiom 1 (goals) }
% 0.20/0.43 multiplication(fresh12(fresh9(test(x1), true, x1, c(x1)), test(x1), c(x1), x1), X)
% 0.20/0.43 = { by axiom 9 (test_3) }
% 0.20/0.43 multiplication(fresh12(fresh8(c(x1), c(x1), x1, c(x1)), test(x1), c(x1), x1), X)
% 0.20/0.43 = { by axiom 8 (test_3) }
% 0.20/0.43 multiplication(fresh12(true, test(x1), c(x1), x1), X)
% 0.20/0.43 = { by axiom 1 (goals) R->L }
% 0.20/0.43 multiplication(fresh12(test(x1), test(x1), c(x1), x1), X)
% 0.20/0.43 = { by axiom 6 (test_2_1) }
% 0.20/0.43 multiplication(one, X)
% 0.20/0.43 = { by axiom 4 (multiplicative_left_identity) }
% 0.20/0.43 X
% 0.20/0.43
% 0.20/0.43 Goal 1 (goals_1): tuple(leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), x0), leq(x0, addition(multiplication(x1, x0), multiplication(c(x1), x0)))) = tuple(true, true).
% 0.20/0.43 Proof:
% 0.20/0.43 tuple(leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), x0), leq(x0, addition(multiplication(x1, x0), multiplication(c(x1), x0))))
% 0.20/0.43 = { by axiom 3 (additive_commutativity) }
% 0.20/0.43 tuple(leq(addition(multiplication(c(x1), x0), multiplication(x1, x0)), x0), leq(x0, addition(multiplication(x1, x0), multiplication(c(x1), x0))))
% 0.20/0.43 = { by axiom 3 (additive_commutativity) }
% 0.20/0.43 tuple(leq(addition(multiplication(c(x1), x0), multiplication(x1, x0)), x0), leq(x0, addition(multiplication(c(x1), x0), multiplication(x1, x0))))
% 0.20/0.43 = { by lemma 14 R->L }
% 0.20/0.43 tuple(leq(addition(multiplication(c(x1), x0), multiplication(x1, x0)), addition(multiplication(x1, x0), multiplication(c(x1), x0))), leq(x0, addition(multiplication(c(x1), x0), multiplication(x1, x0))))
% 0.20/0.43 = { by lemma 14 R->L }
% 0.20/0.43 tuple(leq(addition(multiplication(c(x1), x0), multiplication(x1, x0)), addition(multiplication(x1, x0), multiplication(c(x1), x0))), leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(c(x1), x0), multiplication(x1, x0))))
% 0.20/0.43 = { by axiom 3 (additive_commutativity) R->L }
% 0.20/0.43 tuple(leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(x1, x0), multiplication(c(x1), x0))), leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(c(x1), x0), multiplication(x1, x0))))
% 0.20/0.43 = { by axiom 3 (additive_commutativity) R->L }
% 0.20/0.43 tuple(leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(x1, x0), multiplication(c(x1), x0))), leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(x1, x0), multiplication(c(x1), x0))))
% 0.20/0.43 = { by lemma 13 }
% 0.20/0.43 tuple(test(x1), leq(addition(multiplication(x1, x0), multiplication(c(x1), x0)), addition(multiplication(x1, x0), multiplication(c(x1), x0))))
% 0.20/0.43 = { by lemma 13 }
% 0.20/0.43 tuple(test(x1), test(x1))
% 0.20/0.43 = { by axiom 1 (goals) }
% 0.20/0.43 tuple(true, test(x1))
% 0.20/0.43 = { by axiom 1 (goals) }
% 0.20/0.43 tuple(true, true)
% 0.20/0.43 % SZS output end Proof
% 0.20/0.43
% 0.20/0.43 RESULT: Theorem (the conjecture is true).
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