TSTP Solution File: LCL159-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL159-1 : 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 : n028.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 0.19s 0.51s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL159-1 : 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
% 0.13/0.34 % Computer : n028.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
% 0.19/0.34 % CPULimit : 300
% 0.19/0.34 % WCLimit : 300
% 0.19/0.34 % DateTime : Thu Aug 24 17:10:50 EDT 2023
% 0.19/0.34 % CPUTime :
% 0.19/0.51 Command-line arguments: --no-flatten-goal
% 0.19/0.51
% 0.19/0.51 % SZS status Unsatisfiable
% 0.19/0.51
% 0.19/0.53 % SZS output start Proof
% 0.19/0.53 Axiom 1 (false_definition): not(truth) = falsehood.
% 0.19/0.53 Axiom 2 (xor_commutativity): xor(X, Y) = xor(Y, X).
% 0.19/0.53 Axiom 3 (and_star_commutativity): and_star(X, Y) = and_star(Y, X).
% 0.19/0.54 Axiom 4 (or_commutativity): or(X, Y) = or(Y, X).
% 0.19/0.54 Axiom 5 (wajsberg_1): implies(truth, X) = X.
% 0.19/0.54 Axiom 6 (or_definition): or(X, Y) = implies(not(X), Y).
% 0.19/0.54 Axiom 7 (and_star_associativity): and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z)).
% 0.19/0.54 Axiom 8 (or_associativity): or(or(X, Y), Z) = or(X, or(Y, Z)).
% 0.19/0.54 Axiom 9 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 0.19/0.54 Axiom 10 (and_star_definition): and_star(X, Y) = not(or(not(X), not(Y))).
% 0.19/0.54 Axiom 11 (and_definition): and(X, Y) = not(or(not(X), not(Y))).
% 0.19/0.54 Axiom 12 (xor_definition): xor(X, Y) = or(and(X, not(Y)), and(not(X), Y)).
% 0.19/0.54 Axiom 13 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 0.19/0.54 Axiom 14 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 0.19/0.54
% 0.19/0.54 Lemma 15: implies(or(X, falsehood), X) = truth.
% 0.19/0.54 Proof:
% 0.19/0.54 implies(or(X, falsehood), X)
% 0.19/0.54 = { by axiom 5 (wajsberg_1) R->L }
% 0.19/0.54 implies(or(X, falsehood), implies(truth, X))
% 0.19/0.54 = { by axiom 1 (false_definition) R->L }
% 0.19/0.54 implies(or(X, not(truth)), implies(truth, X))
% 0.19/0.54 = { by axiom 6 (or_definition) }
% 0.19/0.54 implies(implies(not(X), not(truth)), implies(truth, X))
% 0.19/0.54 = { by axiom 13 (wajsberg_4) }
% 0.19/0.54 truth
% 0.19/0.54
% 0.19/0.54 Lemma 16: implies(or(falsehood, X), X) = truth.
% 0.19/0.54 Proof:
% 0.19/0.54 implies(or(falsehood, X), X)
% 0.19/0.54 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.54 implies(or(X, falsehood), X)
% 0.19/0.54 = { by lemma 15 }
% 0.19/0.54 truth
% 0.19/0.54
% 0.19/0.54 Lemma 17: implies(implies(X, truth), truth) = implies(X, X).
% 0.19/0.54 Proof:
% 0.19/0.54 implies(implies(X, truth), truth)
% 0.19/0.54 = { by axiom 9 (wajsberg_3) R->L }
% 0.19/0.54 implies(implies(truth, X), X)
% 0.19/0.54 = { by axiom 5 (wajsberg_1) }
% 0.19/0.54 implies(X, X)
% 0.19/0.54
% 0.19/0.54 Lemma 18: not(falsehood) = truth.
% 0.19/0.54 Proof:
% 0.19/0.54 not(falsehood)
% 0.19/0.54 = { by axiom 5 (wajsberg_1) R->L }
% 0.19/0.54 implies(truth, not(falsehood))
% 0.19/0.54 = { by lemma 16 R->L }
% 0.19/0.54 implies(implies(or(falsehood, truth), truth), not(falsehood))
% 0.19/0.54 = { by axiom 6 (or_definition) }
% 0.19/0.54 implies(implies(implies(not(falsehood), truth), truth), not(falsehood))
% 0.19/0.54 = { by lemma 17 }
% 0.19/0.54 implies(implies(not(falsehood), not(falsehood)), not(falsehood))
% 0.19/0.54 = { by axiom 6 (or_definition) R->L }
% 0.19/0.54 implies(or(falsehood, not(falsehood)), not(falsehood))
% 0.19/0.54 = { by lemma 16 }
% 0.19/0.54 truth
% 0.19/0.54
% 0.19/0.54 Lemma 19: or(X, falsehood) = X.
% 0.19/0.54 Proof:
% 0.19/0.54 or(X, falsehood)
% 0.19/0.54 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.54 or(falsehood, X)
% 0.19/0.54 = { by axiom 6 (or_definition) }
% 0.19/0.54 implies(not(falsehood), X)
% 0.19/0.54 = { by lemma 18 }
% 0.19/0.54 implies(truth, X)
% 0.19/0.54 = { by axiom 5 (wajsberg_1) }
% 0.19/0.54 X
% 0.19/0.54
% 0.19/0.54 Lemma 20: or(falsehood, X) = X.
% 0.19/0.54 Proof:
% 0.19/0.54 or(falsehood, X)
% 0.19/0.54 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.54 or(X, falsehood)
% 0.19/0.54 = { by lemma 19 }
% 0.19/0.54 X
% 0.19/0.54
% 0.19/0.54 Lemma 21: not(or(falsehood, not(X))) = and_star(X, truth).
% 0.19/0.54 Proof:
% 0.19/0.54 not(or(falsehood, not(X)))
% 0.19/0.54 = { by axiom 1 (false_definition) R->L }
% 0.19/0.54 not(or(not(truth), not(X)))
% 0.19/0.54 = { by axiom 10 (and_star_definition) R->L }
% 0.19/0.54 and_star(truth, X)
% 0.19/0.54 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.54 and_star(X, truth)
% 0.19/0.54
% 0.19/0.54 Lemma 22: not(not(X)) = and_star(X, truth).
% 0.19/0.54 Proof:
% 0.19/0.54 not(not(X))
% 0.19/0.54 = { by lemma 20 R->L }
% 0.19/0.54 not(or(falsehood, not(X)))
% 0.19/0.54 = { by lemma 21 }
% 0.19/0.54 and_star(X, truth)
% 0.19/0.54
% 0.19/0.54 Lemma 23: or(X, truth) = truth.
% 0.19/0.54 Proof:
% 0.19/0.54 or(X, truth)
% 0.19/0.54 = { by axiom 6 (or_definition) }
% 0.19/0.54 implies(not(X), truth)
% 0.19/0.54 = { by lemma 15 R->L }
% 0.19/0.54 implies(not(X), implies(or(not(X), falsehood), not(X)))
% 0.19/0.54 = { by lemma 19 }
% 0.19/0.54 implies(not(X), implies(not(X), not(X)))
% 0.19/0.54 = { by lemma 17 R->L }
% 0.19/0.54 implies(not(X), implies(implies(not(X), truth), truth))
% 0.19/0.54 = { by axiom 5 (wajsberg_1) R->L }
% 0.19/0.54 implies(not(X), implies(implies(not(X), truth), implies(truth, truth)))
% 0.19/0.54 = { by axiom 5 (wajsberg_1) R->L }
% 0.19/0.54 implies(implies(truth, not(X)), implies(implies(not(X), truth), implies(truth, truth)))
% 0.19/0.54 = { by axiom 14 (wajsberg_2) }
% 0.19/0.54 truth
% 0.19/0.54
% 0.19/0.54 Lemma 24: and(X, Y) = and_star(X, Y).
% 0.19/0.54 Proof:
% 0.19/0.54 and(X, Y)
% 0.19/0.54 = { by axiom 11 (and_definition) }
% 0.19/0.54 not(or(not(X), not(Y)))
% 0.19/0.54 = { by axiom 10 (and_star_definition) R->L }
% 0.19/0.54 and_star(X, Y)
% 0.19/0.54
% 0.19/0.54 Lemma 25: or(and_star(X, not(Y)), and_star(not(X), Y)) = xor(X, Y).
% 0.19/0.54 Proof:
% 0.19/0.54 or(and_star(X, not(Y)), and_star(not(X), Y))
% 0.19/0.54 = { by lemma 24 R->L }
% 0.19/0.54 or(and_star(X, not(Y)), and(not(X), Y))
% 0.19/0.54 = { by lemma 24 R->L }
% 0.19/0.54 or(and(X, not(Y)), and(not(X), Y))
% 0.19/0.54 = { by axiom 12 (xor_definition) R->L }
% 0.19/0.54 xor(X, Y)
% 0.19/0.54
% 0.19/0.54 Lemma 26: and_star(X, truth) = X.
% 0.19/0.54 Proof:
% 0.19/0.54 and_star(X, truth)
% 0.19/0.54 = { by lemma 22 R->L }
% 0.19/0.54 not(not(X))
% 0.19/0.54 = { by axiom 5 (wajsberg_1) R->L }
% 0.19/0.54 implies(truth, not(not(X)))
% 0.19/0.54 = { by lemma 23 R->L }
% 0.19/0.54 implies(or(not(not(X)), truth), not(not(X)))
% 0.19/0.54 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.54 implies(or(truth, not(not(X))), not(not(X)))
% 0.19/0.54 = { by axiom 6 (or_definition) }
% 0.19/0.54 implies(implies(not(truth), not(not(X))), not(not(X)))
% 0.19/0.54 = { by axiom 1 (false_definition) }
% 0.19/0.54 implies(implies(falsehood, not(not(X))), not(not(X)))
% 0.19/0.54 = { by axiom 9 (wajsberg_3) }
% 0.19/0.54 implies(implies(not(not(X)), falsehood), falsehood)
% 0.19/0.54 = { by axiom 6 (or_definition) R->L }
% 0.19/0.54 implies(or(not(X), falsehood), falsehood)
% 0.19/0.54 = { by lemma 19 }
% 0.19/0.54 implies(not(X), falsehood)
% 0.19/0.54 = { by axiom 6 (or_definition) R->L }
% 0.19/0.54 or(X, falsehood)
% 0.19/0.54 = { by lemma 19 }
% 0.19/0.54 X
% 0.19/0.54
% 0.19/0.54 Lemma 27: xor(X, truth) = not(X).
% 0.19/0.54 Proof:
% 0.19/0.54 xor(X, truth)
% 0.19/0.55 = { by lemma 25 R->L }
% 0.19/0.55 or(and_star(X, not(truth)), and_star(not(X), truth))
% 0.19/0.55 = { by axiom 1 (false_definition) }
% 0.19/0.55 or(and_star(X, falsehood), and_star(not(X), truth))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) R->L }
% 0.19/0.55 or(and_star(falsehood, X), and_star(not(X), truth))
% 0.19/0.55 = { by axiom 10 (and_star_definition) }
% 0.19/0.55 or(not(or(not(falsehood), not(X))), and_star(not(X), truth))
% 0.19/0.55 = { by lemma 18 }
% 0.19/0.55 or(not(or(truth, not(X))), and_star(not(X), truth))
% 0.19/0.55 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.55 or(not(or(not(X), truth)), and_star(not(X), truth))
% 0.19/0.55 = { by lemma 23 }
% 0.19/0.55 or(not(truth), and_star(not(X), truth))
% 0.19/0.55 = { by axiom 1 (false_definition) }
% 0.19/0.55 or(falsehood, and_star(not(X), truth))
% 0.19/0.55 = { by lemma 20 }
% 0.19/0.55 and_star(not(X), truth)
% 0.19/0.55 = { by lemma 26 }
% 0.19/0.55 not(X)
% 0.19/0.55
% 0.19/0.55 Lemma 28: or(X, not(Y)) = implies(Y, X).
% 0.19/0.55 Proof:
% 0.19/0.55 or(X, not(Y))
% 0.19/0.55 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.55 or(not(Y), X)
% 0.19/0.55 = { by lemma 20 R->L }
% 0.19/0.55 or(falsehood, or(not(Y), X))
% 0.19/0.55 = { by axiom 8 (or_associativity) R->L }
% 0.19/0.55 or(or(falsehood, not(Y)), X)
% 0.19/0.55 = { by axiom 6 (or_definition) }
% 0.19/0.55 implies(not(or(falsehood, not(Y))), X)
% 0.19/0.55 = { by lemma 21 }
% 0.19/0.55 implies(and_star(Y, truth), X)
% 0.19/0.55 = { by lemma 26 }
% 0.19/0.55 implies(Y, X)
% 0.19/0.55
% 0.19/0.55 Lemma 29: or(not(X), Y) = implies(X, Y).
% 0.19/0.55 Proof:
% 0.19/0.55 or(not(X), Y)
% 0.19/0.55 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.55 or(Y, not(X))
% 0.19/0.55 = { by lemma 28 }
% 0.19/0.55 implies(X, Y)
% 0.19/0.55
% 0.19/0.55 Lemma 30: not(implies(X, not(Y))) = and_star(X, Y).
% 0.19/0.55 Proof:
% 0.19/0.55 not(implies(X, not(Y)))
% 0.19/0.55 = { by lemma 28 R->L }
% 0.19/0.55 not(or(not(Y), not(X)))
% 0.19/0.55 = { by axiom 10 (and_star_definition) R->L }
% 0.19/0.55 and_star(Y, X)
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.55 and_star(X, Y)
% 0.19/0.55
% 0.19/0.55 Lemma 31: implies(X, not(Y)) = not(and_star(X, Y)).
% 0.19/0.55 Proof:
% 0.19/0.55 implies(X, not(Y))
% 0.19/0.55 = { by lemma 29 R->L }
% 0.19/0.55 or(not(X), not(Y))
% 0.19/0.55 = { by lemma 28 }
% 0.19/0.55 implies(Y, not(X))
% 0.19/0.55 = { by lemma 26 R->L }
% 0.19/0.55 and_star(implies(Y, not(X)), truth)
% 0.19/0.55 = { by lemma 21 R->L }
% 0.19/0.55 not(or(falsehood, not(implies(Y, not(X)))))
% 0.19/0.55 = { by lemma 30 }
% 0.19/0.55 not(or(falsehood, and_star(Y, X)))
% 0.19/0.55 = { by lemma 20 }
% 0.19/0.55 not(and_star(Y, X))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.55 not(and_star(X, Y))
% 0.19/0.55
% 0.19/0.55 Lemma 32: not(implies(X, Y)) = and_star(X, not(Y)).
% 0.19/0.55 Proof:
% 0.19/0.55 not(implies(X, Y))
% 0.19/0.55 = { by lemma 28 R->L }
% 0.19/0.55 not(or(Y, not(X)))
% 0.19/0.55 = { by axiom 6 (or_definition) }
% 0.19/0.55 not(implies(not(Y), not(X)))
% 0.19/0.55 = { by lemma 30 }
% 0.19/0.55 and_star(not(Y), X)
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.55 and_star(X, not(Y))
% 0.19/0.55
% 0.19/0.55 Lemma 33: implies(or(X, Y), and_star(X, Y)) = xor(X, not(Y)).
% 0.19/0.55 Proof:
% 0.19/0.55 implies(or(X, Y), and_star(X, Y))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) R->L }
% 0.19/0.55 implies(or(X, Y), and_star(Y, X))
% 0.19/0.55 = { by axiom 6 (or_definition) }
% 0.19/0.55 implies(implies(not(X), Y), and_star(Y, X))
% 0.19/0.55 = { by lemma 28 R->L }
% 0.19/0.55 or(and_star(Y, X), not(implies(not(X), Y)))
% 0.19/0.55 = { by lemma 32 }
% 0.19/0.55 or(and_star(Y, X), and_star(not(X), not(Y)))
% 0.19/0.55 = { by lemma 20 R->L }
% 0.19/0.55 or(and_star(Y, X), and_star(not(X), or(falsehood, not(Y))))
% 0.19/0.55 = { by lemma 26 R->L }
% 0.19/0.55 or(and_star(Y, and_star(X, truth)), and_star(not(X), or(falsehood, not(Y))))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.55 or(and_star(Y, and_star(truth, X)), and_star(not(X), or(falsehood, not(Y))))
% 0.19/0.55 = { by axiom 7 (and_star_associativity) R->L }
% 0.19/0.55 or(and_star(and_star(Y, truth), X), and_star(not(X), or(falsehood, not(Y))))
% 0.19/0.55 = { by lemma 21 R->L }
% 0.19/0.55 or(and_star(not(or(falsehood, not(Y))), X), and_star(not(X), or(falsehood, not(Y))))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) R->L }
% 0.19/0.55 or(and_star(X, not(or(falsehood, not(Y)))), and_star(not(X), or(falsehood, not(Y))))
% 0.19/0.55 = { by lemma 25 }
% 0.19/0.55 xor(X, or(falsehood, not(Y)))
% 0.19/0.55 = { by axiom 2 (xor_commutativity) }
% 0.19/0.55 xor(or(falsehood, not(Y)), X)
% 0.19/0.55 = { by lemma 20 }
% 0.19/0.55 xor(not(Y), X)
% 0.19/0.55 = { by axiom 2 (xor_commutativity) }
% 0.19/0.55 xor(X, not(Y))
% 0.19/0.55
% 0.19/0.55 Goal 1 (prove_alternative_wajsberg_axiom): xor(x, xor(truth, y)) = xor(xor(x, truth), y).
% 0.19/0.55 Proof:
% 0.19/0.55 xor(x, xor(truth, y))
% 0.19/0.55 = { by axiom 2 (xor_commutativity) }
% 0.19/0.55 xor(x, xor(y, truth))
% 0.19/0.55 = { by lemma 27 }
% 0.19/0.55 xor(x, not(y))
% 0.19/0.55 = { by lemma 33 R->L }
% 0.19/0.55 implies(or(x, y), and_star(x, y))
% 0.19/0.55 = { by lemma 26 R->L }
% 0.19/0.55 and_star(implies(or(x, y), and_star(x, y)), truth)
% 0.19/0.55 = { by lemma 21 R->L }
% 0.19/0.55 not(or(falsehood, not(implies(or(x, y), and_star(x, y)))))
% 0.19/0.55 = { by lemma 32 }
% 0.19/0.55 not(or(falsehood, and_star(or(x, y), not(and_star(x, y)))))
% 0.19/0.55 = { by lemma 20 }
% 0.19/0.55 not(and_star(or(x, y), not(and_star(x, y))))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.55 not(and_star(not(and_star(x, y)), or(x, y)))
% 0.19/0.55 = { by lemma 31 R->L }
% 0.19/0.55 not(and_star(implies(x, not(y)), or(x, y)))
% 0.19/0.55 = { by lemma 29 R->L }
% 0.19/0.55 not(and_star(or(not(x), not(y)), or(x, y)))
% 0.19/0.55 = { by lemma 31 R->L }
% 0.19/0.55 implies(or(not(x), not(y)), not(or(x, y)))
% 0.19/0.55 = { by axiom 4 (or_commutativity) R->L }
% 0.19/0.55 implies(or(not(x), not(y)), not(or(y, x)))
% 0.19/0.55 = { by axiom 6 (or_definition) }
% 0.19/0.55 implies(or(not(x), not(y)), not(implies(not(y), x)))
% 0.19/0.55 = { by lemma 32 }
% 0.19/0.55 implies(or(not(x), not(y)), and_star(not(y), not(x)))
% 0.19/0.55 = { by axiom 3 (and_star_commutativity) }
% 0.19/0.55 implies(or(not(x), not(y)), and_star(not(x), not(y)))
% 0.19/0.55 = { by lemma 33 }
% 0.19/0.55 xor(not(x), not(not(y)))
% 0.19/0.55 = { by lemma 22 }
% 0.19/0.55 xor(not(x), and_star(y, truth))
% 0.19/0.55 = { by lemma 26 }
% 0.19/0.55 xor(not(x), y)
% 0.19/0.55 = { by axiom 2 (xor_commutativity) }
% 0.19/0.55 xor(y, not(x))
% 0.19/0.55 = { by lemma 27 R->L }
% 0.19/0.55 xor(y, xor(x, truth))
% 0.19/0.55 = { by axiom 2 (xor_commutativity) R->L }
% 0.19/0.55 xor(xor(x, truth), y)
% 0.19/0.55 % SZS output end Proof
% 0.19/0.55
% 0.19/0.55 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------