TSTP Solution File: KLE027+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : KLE027+2 : 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 05:35:33 EDT 2023
% Result : Theorem 5.22s 1.06s
% Output : Proof 5.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.12 % Problem : KLE027+2 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 11:22:40 EDT 2023
% 0.12/0.34 % CPUTime :
% 5.22/1.06 Command-line arguments: --no-flatten-goal
% 5.22/1.06
% 5.22/1.06 % SZS status Theorem
% 5.22/1.06
% 5.22/1.06 % SZS output start Proof
% 5.22/1.06 Take the following subset of the input axioms:
% 5.22/1.07 fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
% 5.22/1.07 fof(additive_idempotence, axiom, ![A3]: addition(A3, A3)=A3).
% 5.22/1.07 fof(additive_identity, axiom, ![A3]: addition(A3, zero)=A3).
% 5.22/1.07 fof(goals, conjecture, ![X0, X1, X2, X3, X4]: ((test(X3) & test(X4)) => (leq(addition(multiplication(X3, addition(multiplication(X3, X0), multiplication(c(X3), X1))), multiplication(c(X3), X2)), addition(multiplication(X3, X0), multiplication(c(X3), X2))) & leq(addition(multiplication(X3, X0), multiplication(c(X3), X2)), addition(multiplication(X3, addition(multiplication(X3, X0), multiplication(c(X3), X1))), multiplication(c(X3), X2)))))).
% 5.22/1.07 fof(left_annihilation, axiom, ![A3]: multiplication(zero, A3)=zero).
% 5.22/1.07 fof(multiplicative_associativity, axiom, ![C, A3, B2]: multiplication(A3, multiplication(B2, C))=multiplication(multiplication(A3, B2), C)).
% 5.22/1.07 fof(multiplicative_right_identity, axiom, ![A3]: multiplication(A3, one)=A3).
% 5.22/1.07 fof(order, axiom, ![A2, B2]: (leq(A2, B2) <=> addition(A2, B2)=B2)).
% 5.22/1.07 fof(right_distributivity, axiom, ![A3, B2, C2]: multiplication(A3, addition(B2, C2))=addition(multiplication(A3, B2), multiplication(A3, C2))).
% 5.22/1.07 fof(test_1, axiom, ![X0_2]: (test(X0_2) <=> ?[X1_2]: complement(X1_2, X0_2))).
% 5.22/1.07 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)))).
% 5.22/1.07 fof(test_3, axiom, ![X0_2, X1_2]: (test(X0_2) => (c(X0_2)=X1_2 <=> complement(X0_2, X1_2)))).
% 5.22/1.07
% 5.22/1.07 Now clausify the problem and encode Horn clauses using encoding 3 of
% 5.22/1.07 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 5.22/1.07 We repeatedly replace C & s=t => u=v by the two clauses:
% 5.22/1.07 fresh(y, y, x1...xn) = u
% 5.22/1.07 C => fresh(s, t, x1...xn) = v
% 5.22/1.07 where fresh is a fresh function symbol and x1..xn are the free
% 5.22/1.07 variables of u and v.
% 5.22/1.07 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 5.22/1.07 input problem has no model of domain size 1).
% 5.22/1.07
% 5.22/1.07 The encoding turns the above axioms into the following unit equations and goals:
% 5.22/1.07
% 5.22/1.07 Axiom 1 (goals): test(x3) = true.
% 5.22/1.07 Axiom 2 (additive_idempotence): addition(X, X) = X.
% 5.22/1.07 Axiom 3 (additive_commutativity): addition(X, Y) = addition(Y, X).
% 5.22/1.07 Axiom 4 (additive_identity): addition(X, zero) = X.
% 5.22/1.07 Axiom 5 (multiplicative_right_identity): multiplication(X, one) = X.
% 5.22/1.07 Axiom 6 (left_annihilation): multiplication(zero, X) = zero.
% 5.22/1.07 Axiom 7 (test_1): fresh12(X, X, Y) = true.
% 5.22/1.07 Axiom 8 (test_3_1): fresh(X, X, Y, Z) = Z.
% 5.22/1.07 Axiom 9 (test_2): fresh14(X, X, Y, Z) = true.
% 5.22/1.07 Axiom 10 (test_1): fresh12(test(X), true, X) = complement(x1_3(X), X).
% 5.22/1.07 Axiom 11 (order): fresh11(X, X, Y, Z) = true.
% 5.22/1.07 Axiom 12 (test_2): fresh9(X, X, Y, Z) = complement(Z, Y).
% 5.22/1.07 Axiom 13 (test_2_1): fresh8(X, X, Y, Z) = one.
% 5.22/1.07 Axiom 14 (test_2_2): fresh7(X, X, Y, Z) = zero.
% 5.22/1.07 Axiom 15 (test_2_3): fresh6(X, X, Y, Z) = zero.
% 5.22/1.07 Axiom 16 (test_3_1): fresh3(X, X, Y, Z) = c(Y).
% 5.22/1.07 Axiom 17 (multiplicative_associativity): multiplication(X, multiplication(Y, Z)) = multiplication(multiplication(X, Y), Z).
% 5.22/1.07 Axiom 18 (test_2): fresh13(X, X, Y, Z) = fresh14(addition(Y, Z), one, Y, Z).
% 5.22/1.07 Axiom 19 (order): fresh11(addition(X, Y), Y, X, Y) = leq(X, Y).
% 5.22/1.07 Axiom 20 (test_2): fresh13(multiplication(X, Y), zero, Y, X) = fresh9(multiplication(Y, X), zero, Y, X).
% 5.22/1.07 Axiom 21 (test_2_1): fresh8(complement(X, Y), true, Y, X) = addition(Y, X).
% 5.22/1.07 Axiom 22 (test_2_2): fresh7(complement(X, Y), true, Y, X) = multiplication(Y, X).
% 5.22/1.07 Axiom 23 (test_2_3): fresh6(complement(X, Y), true, Y, X) = multiplication(X, Y).
% 5.22/1.07 Axiom 24 (test_3_1): fresh3(complement(X, Y), true, X, Y) = fresh(test(X), true, X, Y).
% 5.22/1.07 Axiom 25 (right_distributivity): multiplication(X, addition(Y, Z)) = addition(multiplication(X, Y), multiplication(X, Z)).
% 5.22/1.07
% 5.22/1.07 Lemma 26: complement(x1_3(x3), x3) = true.
% 5.22/1.07 Proof:
% 5.22/1.07 complement(x1_3(x3), x3)
% 5.22/1.07 = { by axiom 10 (test_1) R->L }
% 5.22/1.07 fresh12(test(x3), true, x3)
% 5.22/1.07 = { by axiom 1 (goals) }
% 5.22/1.07 fresh12(true, true, x3)
% 5.22/1.07 = { by axiom 7 (test_1) }
% 5.22/1.07 true
% 5.22/1.07
% 5.22/1.07 Lemma 27: addition(x3, x1_3(x3)) = one.
% 5.22/1.07 Proof:
% 5.22/1.07 addition(x3, x1_3(x3))
% 5.22/1.07 = { by axiom 21 (test_2_1) R->L }
% 5.22/1.07 fresh8(complement(x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by lemma 26 }
% 5.22/1.07 fresh8(true, true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 13 (test_2_1) }
% 5.22/1.07 one
% 5.22/1.07
% 5.22/1.07 Lemma 28: multiplication(x3, x1_3(x3)) = zero.
% 5.22/1.07 Proof:
% 5.22/1.07 multiplication(x3, x1_3(x3))
% 5.22/1.07 = { by axiom 22 (test_2_2) R->L }
% 5.22/1.07 fresh7(complement(x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by lemma 26 }
% 5.22/1.07 fresh7(true, true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 14 (test_2_2) }
% 5.22/1.07 zero
% 5.22/1.07
% 5.22/1.07 Lemma 29: c(x3) = x1_3(x3).
% 5.22/1.07 Proof:
% 5.22/1.07 c(x3)
% 5.22/1.07 = { by axiom 16 (test_3_1) R->L }
% 5.22/1.07 fresh3(true, true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 9 (test_2) R->L }
% 5.22/1.07 fresh3(fresh14(one, one, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by lemma 27 R->L }
% 5.22/1.07 fresh3(fresh14(addition(x3, x1_3(x3)), one, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 3 (additive_commutativity) R->L }
% 5.22/1.07 fresh3(fresh14(addition(x1_3(x3), x3), one, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 18 (test_2) R->L }
% 5.22/1.07 fresh3(fresh13(zero, zero, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by lemma 28 R->L }
% 5.22/1.07 fresh3(fresh13(multiplication(x3, x1_3(x3)), zero, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 20 (test_2) }
% 5.22/1.07 fresh3(fresh9(multiplication(x1_3(x3), x3), zero, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 23 (test_2_3) R->L }
% 5.22/1.07 fresh3(fresh9(fresh6(complement(x1_3(x3), x3), true, x3, x1_3(x3)), zero, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by lemma 26 }
% 5.22/1.07 fresh3(fresh9(fresh6(true, true, x3, x1_3(x3)), zero, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 15 (test_2_3) }
% 5.22/1.07 fresh3(fresh9(zero, zero, x1_3(x3), x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 12 (test_2) }
% 5.22/1.07 fresh3(complement(x3, x1_3(x3)), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 24 (test_3_1) }
% 5.22/1.07 fresh(test(x3), true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 1 (goals) }
% 5.22/1.07 fresh(true, true, x3, x1_3(x3))
% 5.22/1.07 = { by axiom 8 (test_3_1) }
% 5.22/1.07 x1_3(x3)
% 5.22/1.07
% 5.22/1.07 Lemma 30: addition(zero, X) = X.
% 5.22/1.07 Proof:
% 5.22/1.07 addition(zero, X)
% 5.22/1.07 = { by axiom 3 (additive_commutativity) R->L }
% 5.22/1.07 addition(X, zero)
% 5.22/1.07 = { by axiom 4 (additive_identity) }
% 5.22/1.07 X
% 5.22/1.07
% 5.22/1.07 Lemma 31: leq(X, X) = true.
% 5.22/1.07 Proof:
% 5.22/1.07 leq(X, X)
% 5.22/1.07 = { by axiom 19 (order) R->L }
% 5.22/1.07 fresh11(addition(X, X), X, X, X)
% 5.22/1.07 = { by axiom 2 (additive_idempotence) }
% 5.22/1.07 fresh11(X, X, X, X)
% 5.22/1.07 = { by axiom 11 (order) }
% 5.22/1.07 true
% 5.22/1.07
% 5.22/1.07 Lemma 32: multiplication(x3, multiplication(x3, X)) = multiplication(x3, X).
% 5.22/1.07 Proof:
% 5.22/1.07 multiplication(x3, multiplication(x3, X))
% 5.22/1.07 = { by axiom 17 (multiplicative_associativity) }
% 5.22/1.07 multiplication(multiplication(x3, x3), X)
% 5.22/1.07 = { by lemma 30 R->L }
% 5.22/1.07 multiplication(addition(zero, multiplication(x3, x3)), X)
% 5.22/1.07 = { by lemma 28 R->L }
% 5.22/1.07 multiplication(addition(multiplication(x3, x1_3(x3)), multiplication(x3, x3)), X)
% 5.22/1.07 = { by axiom 25 (right_distributivity) R->L }
% 5.22/1.07 multiplication(multiplication(x3, addition(x1_3(x3), x3)), X)
% 5.22/1.07 = { by axiom 3 (additive_commutativity) }
% 5.22/1.07 multiplication(multiplication(x3, addition(x3, x1_3(x3))), X)
% 5.22/1.07 = { by lemma 27 }
% 5.22/1.07 multiplication(multiplication(x3, one), X)
% 5.22/1.07 = { by axiom 5 (multiplicative_right_identity) }
% 5.22/1.07 multiplication(x3, X)
% 5.22/1.07
% 5.22/1.07 Lemma 33: multiplication(x3, addition(X, multiplication(x1_3(x3), Y))) = multiplication(x3, X).
% 5.22/1.07 Proof:
% 5.22/1.07 multiplication(x3, addition(X, multiplication(x1_3(x3), Y)))
% 5.22/1.07 = { by axiom 3 (additive_commutativity) R->L }
% 5.22/1.07 multiplication(x3, addition(multiplication(x1_3(x3), Y), X))
% 5.22/1.07 = { by axiom 25 (right_distributivity) }
% 5.22/1.07 addition(multiplication(x3, multiplication(x1_3(x3), Y)), multiplication(x3, X))
% 5.22/1.07 = { by axiom 17 (multiplicative_associativity) }
% 5.22/1.07 addition(multiplication(multiplication(x3, x1_3(x3)), Y), multiplication(x3, X))
% 5.22/1.07 = { by lemma 28 }
% 5.22/1.07 addition(multiplication(zero, Y), multiplication(x3, X))
% 5.22/1.07 = { by axiom 6 (left_annihilation) }
% 5.22/1.07 addition(zero, multiplication(x3, X))
% 5.22/1.07 = { by lemma 30 }
% 5.22/1.07 multiplication(x3, X)
% 5.22/1.07
% 5.22/1.07 Goal 1 (goals_2): tuple(leq(addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1_2))), multiplication(c(x3), x2_2)), addition(multiplication(x3, x0), multiplication(c(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))), multiplication(c(x3), x2)))) = tuple(true, true).
% 5.22/1.07 Proof:
% 5.22/1.07 tuple(leq(addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1_2))), multiplication(c(x3), x2_2)), addition(multiplication(x3, x0), multiplication(c(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))), multiplication(c(x3), x2))))
% 5.22/1.07 = { by axiom 3 (additive_commutativity) }
% 5.22/1.07 tuple(leq(addition(multiplication(c(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(c(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))), multiplication(c(x3), x2))))
% 5.22/1.07 = { by axiom 3 (additive_commutativity) }
% 5.22/1.07 tuple(leq(addition(multiplication(c(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(c(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))))
% 5.22/1.07 = { by lemma 29 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(c(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))))
% 5.22/1.07 = { by lemma 29 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(c(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))))
% 5.22/1.07 = { by lemma 29 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(c(x3), x2)), addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))))
% 5.22/1.07 = { by lemma 29 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(c(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))))
% 5.22/1.07 = { by lemma 29 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(c(x3), x1))))))
% 5.22/1.07 = { by lemma 29 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1_2)))), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1))))))
% 5.22/1.07 = { by lemma 33 }
% 5.22/1.07 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, multiplication(x3, x0))), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, addition(multiplication(x3, x0), multiplication(x1_3(x3), x1))))))
% 5.22/1.08 = { by lemma 33 }
% 5.22/1.08 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, multiplication(x3, x0))), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, multiplication(x3, x0)))))
% 5.22/1.08 = { by lemma 32 }
% 5.22/1.08 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, x0)), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, multiplication(x3, x0)))))
% 5.22/1.08 = { by lemma 32 }
% 5.22/1.08 tuple(leq(addition(multiplication(x1_3(x3), x2_2), multiplication(x3, x0)), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, x0))))
% 5.22/1.08 = { by axiom 3 (additive_commutativity) }
% 5.22/1.08 tuple(leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2)), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2_2))), leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, x0))))
% 5.22/1.08 = { by lemma 31 }
% 5.22/1.08 tuple(true, leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x1_3(x3), x2), multiplication(x3, x0))))
% 5.22/1.08 = { by axiom 3 (additive_commutativity) }
% 5.22/1.08 tuple(true, leq(addition(multiplication(x3, x0), multiplication(x1_3(x3), x2)), addition(multiplication(x3, x0), multiplication(x1_3(x3), x2))))
% 5.22/1.08 = { by lemma 31 }
% 5.22/1.08 tuple(true, true)
% 5.22/1.08 % SZS output end Proof
% 5.22/1.08
% 5.22/1.08 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------