TSTP Solution File: LCL163-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL163-1 : TPTP v8.1.2. Released v1.0.0.
% 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 : n025.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 08:17:42 EDT 2023
% Result : Unsatisfiable 0.20s 0.40s
% 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 : LCL163-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n025.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 06:52:37 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.40 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.40
% 0.20/0.40 % SZS status Unsatisfiable
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% 0.20/0.40 % SZS output start Proof
% 0.20/0.40 Axiom 1 (and_star_commutativity): and_star(X, Y) = and_star(Y, X).
% 0.20/0.40 Axiom 2 (axiom_3): xor(X, X) = falsehood.
% 0.20/0.40 Axiom 3 (xor_commutativity): xor(X, Y) = xor(Y, X).
% 0.20/0.40 Axiom 4 (axiom_2): xor(X, falsehood) = X.
% 0.20/0.40 Axiom 5 (axiom_1): not(X) = xor(X, truth).
% 0.20/0.40 Axiom 6 (axiom_7): xor(X, xor(truth, Y)) = xor(xor(X, truth), Y).
% 0.20/0.40 Axiom 7 (implies_definition): implies(X, Y) = xor(truth, and_star(X, xor(truth, Y))).
% 0.20/0.40 Axiom 8 (axiom_8): and_star(xor(and_star(xor(truth, X), Y), truth), Y) = and_star(xor(and_star(xor(truth, Y), X), truth), X).
% 0.20/0.40
% 0.20/0.40 Lemma 9: not(not(X)) = X.
% 0.20/0.41 Proof:
% 0.20/0.41 not(not(X))
% 0.20/0.41 = { by axiom 5 (axiom_1) }
% 0.20/0.41 xor(not(X), truth)
% 0.20/0.41 = { by axiom 5 (axiom_1) }
% 0.20/0.41 xor(xor(X, truth), truth)
% 0.20/0.41 = { by axiom 6 (axiom_7) R->L }
% 0.20/0.41 xor(X, xor(truth, truth))
% 0.20/0.41 = { by axiom 2 (axiom_3) }
% 0.20/0.41 xor(X, falsehood)
% 0.20/0.41 = { by axiom 4 (axiom_2) }
% 0.20/0.41 X
% 0.20/0.41
% 0.20/0.41 Lemma 10: xor(truth, X) = not(X).
% 0.20/0.41 Proof:
% 0.20/0.41 xor(truth, X)
% 0.20/0.41 = { by axiom 3 (xor_commutativity) R->L }
% 0.20/0.41 xor(X, truth)
% 0.20/0.41 = { by axiom 5 (axiom_1) R->L }
% 0.20/0.41 not(X)
% 0.20/0.41
% 0.20/0.41 Lemma 11: not(and_star(X, not(Y))) = implies(X, Y).
% 0.20/0.41 Proof:
% 0.20/0.41 not(and_star(X, not(Y)))
% 0.20/0.41 = { by axiom 1 (and_star_commutativity) R->L }
% 0.20/0.41 not(and_star(not(Y), X))
% 0.20/0.41 = { by lemma 10 R->L }
% 0.20/0.41 not(and_star(xor(truth, Y), X))
% 0.20/0.41 = { by lemma 10 R->L }
% 0.20/0.41 xor(truth, and_star(xor(truth, Y), X))
% 0.20/0.41 = { by axiom 1 (and_star_commutativity) R->L }
% 0.20/0.41 xor(truth, and_star(X, xor(truth, Y)))
% 0.20/0.41 = { by axiom 7 (implies_definition) R->L }
% 0.20/0.41 implies(X, Y)
% 0.20/0.41
% 0.20/0.41 Lemma 12: not(and_star(not(X), Y)) = implies(Y, X).
% 0.20/0.41 Proof:
% 0.20/0.41 not(and_star(not(X), Y))
% 0.20/0.41 = { by axiom 1 (and_star_commutativity) R->L }
% 0.20/0.41 not(and_star(Y, not(X)))
% 0.20/0.41 = { by lemma 11 }
% 0.20/0.41 implies(Y, X)
% 0.20/0.41
% 0.20/0.41 Goal 1 (prove_wajsberg_axiom): implies(implies(x, y), y) = implies(implies(y, x), x).
% 0.20/0.41 Proof:
% 0.20/0.41 implies(implies(x, y), y)
% 0.20/0.41 = { by lemma 9 R->L }
% 0.20/0.41 implies(implies(not(not(x)), y), y)
% 0.20/0.41 = { by lemma 10 R->L }
% 0.20/0.41 implies(implies(xor(truth, not(x)), y), y)
% 0.20/0.41 = { by lemma 11 R->L }
% 0.20/0.41 implies(not(and_star(xor(truth, not(x)), not(y))), y)
% 0.20/0.41 = { by lemma 12 R->L }
% 0.20/0.41 not(and_star(not(y), not(and_star(xor(truth, not(x)), not(y)))))
% 0.20/0.41 = { by axiom 1 (and_star_commutativity) R->L }
% 0.20/0.41 not(and_star(not(and_star(xor(truth, not(x)), not(y))), not(y)))
% 0.20/0.41 = { by axiom 5 (axiom_1) }
% 0.20/0.41 not(and_star(xor(and_star(xor(truth, not(x)), not(y)), truth), not(y)))
% 0.20/0.41 = { by axiom 8 (axiom_8) }
% 0.20/0.41 not(and_star(xor(and_star(xor(truth, not(y)), not(x)), truth), not(x)))
% 0.20/0.41 = { by axiom 5 (axiom_1) R->L }
% 0.20/0.41 not(and_star(not(and_star(xor(truth, not(y)), not(x))), not(x)))
% 0.20/0.41 = { by axiom 1 (and_star_commutativity) }
% 0.20/0.41 not(and_star(not(x), not(and_star(xor(truth, not(y)), not(x)))))
% 0.20/0.41 = { by lemma 11 }
% 0.20/0.41 implies(not(x), and_star(xor(truth, not(y)), not(x)))
% 0.20/0.41 = { by lemma 10 }
% 0.20/0.41 implies(not(x), and_star(not(not(y)), not(x)))
% 0.20/0.41 = { by lemma 9 }
% 0.20/0.41 implies(not(x), and_star(y, not(x)))
% 0.20/0.41 = { by lemma 11 R->L }
% 0.20/0.41 not(and_star(not(x), not(and_star(y, not(x)))))
% 0.20/0.41 = { by lemma 12 }
% 0.20/0.41 implies(not(and_star(y, not(x))), x)
% 0.20/0.41 = { by lemma 11 }
% 0.20/0.41 implies(implies(y, x), x)
% 0.20/0.41 % SZS output end Proof
% 0.20/0.41
% 0.20/0.41 RESULT: Unsatisfiable (the axioms are contradictory).
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