TSTP Solution File: LCL154-1 by Twee---2.4.2
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% File : Twee---2.4.2
% Problem : LCL154-1 : 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 : n020.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 08:17:40 EDT 2023
% Result : Unsatisfiable 0.21s 0.42s
% 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
% 0.00/0.12 % Problem : LCL154-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n020.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Thu Aug 24 22:22:45 EDT 2023
% 0.14/0.35 % CPUTime :
% 0.21/0.42 Command-line arguments: --flatten
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% 0.21/0.42 % SZS status Unsatisfiable
% 0.21/0.42
% 0.21/0.43 % SZS output start Proof
% 0.21/0.43 Axiom 1 (false_definition): not(truth) = falsehood.
% 0.21/0.43 Axiom 2 (and_star_commutativity): and_star(X, Y) = and_star(Y, X).
% 0.21/0.43 Axiom 3 (or_commutativity): or(X, Y) = or(Y, X).
% 0.21/0.43 Axiom 4 (wajsberg_1): implies(truth, X) = X.
% 0.21/0.43 Axiom 5 (or_definition): or(X, Y) = implies(not(X), Y).
% 0.21/0.43 Axiom 6 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 0.21/0.43 Axiom 7 (and_star_definition): and_star(X, Y) = not(or(not(X), not(Y))).
% 0.21/0.43 Axiom 8 (and_definition): and(X, Y) = not(or(not(X), not(Y))).
% 0.21/0.43 Axiom 9 (xor_definition): xor(X, Y) = or(and(X, not(Y)), and(not(X), Y)).
% 0.21/0.43 Axiom 10 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 0.21/0.43 Axiom 11 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 0.21/0.43
% 0.21/0.43 Lemma 12: implies(or(X, not(truth)), X) = truth.
% 0.21/0.43 Proof:
% 0.21/0.43 implies(or(X, not(truth)), X)
% 0.21/0.43 = { by axiom 4 (wajsberg_1) R->L }
% 0.21/0.43 implies(or(X, not(truth)), implies(truth, X))
% 0.21/0.43 = { by axiom 5 (or_definition) }
% 0.21/0.43 implies(implies(not(X), not(truth)), implies(truth, X))
% 0.21/0.43 = { by axiom 10 (wajsberg_4) }
% 0.21/0.43 truth
% 0.21/0.43
% 0.21/0.43 Lemma 13: implies(or(not(truth), X), X) = truth.
% 0.21/0.43 Proof:
% 0.21/0.43 implies(or(not(truth), X), X)
% 0.21/0.43 = { by axiom 3 (or_commutativity) R->L }
% 0.21/0.43 implies(or(X, not(truth)), X)
% 0.21/0.43 = { by lemma 12 }
% 0.21/0.43 truth
% 0.21/0.43
% 0.21/0.43 Lemma 14: implies(implies(X, truth), truth) = implies(X, X).
% 0.21/0.43 Proof:
% 0.21/0.43 implies(implies(X, truth), truth)
% 0.21/0.43 = { by axiom 6 (wajsberg_3) R->L }
% 0.21/0.43 implies(implies(truth, X), X)
% 0.21/0.43 = { by axiom 4 (wajsberg_1) }
% 0.21/0.43 implies(X, X)
% 0.21/0.43
% 0.21/0.43 Lemma 15: not(not(truth)) = truth.
% 0.21/0.43 Proof:
% 0.21/0.43 not(not(truth))
% 0.21/0.43 = { by axiom 4 (wajsberg_1) R->L }
% 0.21/0.43 implies(truth, not(not(truth)))
% 0.21/0.43 = { by lemma 13 R->L }
% 0.21/0.43 implies(implies(or(not(truth), truth), truth), not(not(truth)))
% 0.21/0.43 = { by axiom 5 (or_definition) }
% 0.21/0.43 implies(implies(implies(not(not(truth)), truth), truth), not(not(truth)))
% 0.21/0.43 = { by lemma 14 }
% 0.21/0.43 implies(implies(not(not(truth)), not(not(truth))), not(not(truth)))
% 0.21/0.43 = { by axiom 5 (or_definition) R->L }
% 0.21/0.43 implies(or(not(truth), not(not(truth))), not(not(truth)))
% 0.21/0.43 = { by lemma 13 }
% 0.21/0.43 truth
% 0.21/0.43
% 0.21/0.43 Lemma 16: or(X, not(truth)) = X.
% 0.21/0.43 Proof:
% 0.21/0.43 or(X, not(truth))
% 0.21/0.44 = { by axiom 3 (or_commutativity) R->L }
% 0.21/0.44 or(not(truth), X)
% 0.21/0.44 = { by axiom 5 (or_definition) }
% 0.21/0.44 implies(not(not(truth)), X)
% 0.21/0.44 = { by lemma 15 }
% 0.21/0.44 implies(truth, X)
% 0.21/0.44 = { by axiom 4 (wajsberg_1) }
% 0.21/0.44 X
% 0.21/0.44
% 0.21/0.44 Lemma 17: or(X, truth) = truth.
% 0.21/0.44 Proof:
% 0.21/0.44 or(X, truth)
% 0.21/0.44 = { by axiom 5 (or_definition) }
% 0.21/0.44 implies(not(X), truth)
% 0.21/0.44 = { by lemma 12 R->L }
% 0.21/0.44 implies(not(X), implies(or(not(X), not(truth)), not(X)))
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 implies(not(X), implies(not(X), not(X)))
% 0.21/0.44 = { by lemma 14 R->L }
% 0.21/0.44 implies(not(X), implies(implies(not(X), truth), truth))
% 0.21/0.44 = { by axiom 4 (wajsberg_1) R->L }
% 0.21/0.44 implies(not(X), implies(implies(not(X), truth), implies(truth, truth)))
% 0.21/0.44 = { by axiom 4 (wajsberg_1) R->L }
% 0.21/0.44 implies(implies(truth, not(X)), implies(implies(not(X), truth), implies(truth, truth)))
% 0.21/0.44 = { by axiom 11 (wajsberg_2) }
% 0.21/0.44 truth
% 0.21/0.44
% 0.21/0.44 Lemma 18: and(X, Y) = and_star(X, Y).
% 0.21/0.44 Proof:
% 0.21/0.44 and(X, Y)
% 0.21/0.44 = { by axiom 8 (and_definition) }
% 0.21/0.44 not(or(not(X), not(Y)))
% 0.21/0.44 = { by axiom 7 (and_star_definition) R->L }
% 0.21/0.44 and_star(X, Y)
% 0.21/0.44
% 0.21/0.44 Goal 1 (prove_alternative_wajsberg_axiom): xor(x, falsehood) = x.
% 0.21/0.44 Proof:
% 0.21/0.44 xor(x, falsehood)
% 0.21/0.44 = { by axiom 1 (false_definition) R->L }
% 0.21/0.44 xor(x, not(truth))
% 0.21/0.44 = { by axiom 9 (xor_definition) }
% 0.21/0.44 or(and(x, not(not(truth))), and(not(x), not(truth)))
% 0.21/0.44 = { by lemma 18 }
% 0.21/0.44 or(and_star(x, not(not(truth))), and(not(x), not(truth)))
% 0.21/0.44 = { by lemma 18 }
% 0.21/0.44 or(and_star(x, not(not(truth))), and_star(not(x), not(truth)))
% 0.21/0.44 = { by axiom 2 (and_star_commutativity) R->L }
% 0.21/0.44 or(and_star(x, not(not(truth))), and_star(not(truth), not(x)))
% 0.21/0.44 = { by axiom 7 (and_star_definition) }
% 0.21/0.44 or(and_star(x, not(not(truth))), not(or(not(not(truth)), not(not(x)))))
% 0.21/0.44 = { by lemma 15 }
% 0.21/0.44 or(and_star(x, not(not(truth))), not(or(truth, not(not(x)))))
% 0.21/0.44 = { by axiom 3 (or_commutativity) R->L }
% 0.21/0.44 or(and_star(x, not(not(truth))), not(or(not(not(x)), truth)))
% 0.21/0.44 = { by lemma 17 }
% 0.21/0.44 or(and_star(x, not(not(truth))), not(truth))
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 and_star(x, not(not(truth)))
% 0.21/0.44 = { by lemma 15 }
% 0.21/0.44 and_star(x, truth)
% 0.21/0.44 = { by axiom 2 (and_star_commutativity) R->L }
% 0.21/0.44 and_star(truth, x)
% 0.21/0.44 = { by axiom 7 (and_star_definition) }
% 0.21/0.44 not(or(not(truth), not(x)))
% 0.21/0.44 = { by axiom 3 (or_commutativity) R->L }
% 0.21/0.44 not(or(not(x), not(truth)))
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 not(not(x))
% 0.21/0.44 = { by axiom 4 (wajsberg_1) R->L }
% 0.21/0.44 implies(truth, not(not(x)))
% 0.21/0.44 = { by lemma 17 R->L }
% 0.21/0.44 implies(or(not(not(x)), truth), not(not(x)))
% 0.21/0.44 = { by axiom 3 (or_commutativity) R->L }
% 0.21/0.44 implies(or(truth, not(not(x))), not(not(x)))
% 0.21/0.44 = { by axiom 5 (or_definition) }
% 0.21/0.44 implies(implies(not(truth), not(not(x))), not(not(x)))
% 0.21/0.44 = { by axiom 6 (wajsberg_3) }
% 0.21/0.44 implies(implies(not(not(x)), not(truth)), not(truth))
% 0.21/0.44 = { by axiom 5 (or_definition) R->L }
% 0.21/0.44 implies(or(not(x), not(truth)), not(truth))
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 implies(not(x), not(truth))
% 0.21/0.44 = { by axiom 5 (or_definition) R->L }
% 0.21/0.44 or(x, not(truth))
% 0.21/0.44 = { by lemma 16 }
% 0.21/0.44 x
% 0.21/0.44 % SZS output end Proof
% 0.21/0.44
% 0.21/0.44 RESULT: Unsatisfiable (the axioms are contradictory).
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