TSTP Solution File: LCL109-6 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL109-6 : 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 : n011.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 08:17:28 EDT 2023
% Result : Unsatisfiable 0.20s 0.48s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : LCL109-6 : TPTP v8.1.2. Released v1.0.0.
% 0.11/0.13 % 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 : n011.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
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 04:29:08 EDT 2023
% 0.13/0.34 % CPUTime :
% 0.20/0.48 Command-line arguments: --no-flatten-goal
% 0.20/0.48
% 0.20/0.48 % SZS status Unsatisfiable
% 0.20/0.48
% 0.20/0.50 % SZS output start Proof
% 0.20/0.50 Axiom 1 (false_definition): not(truth) = falsehood.
% 0.20/0.50 Axiom 2 (and_star_commutativity): and_star(X, Y) = and_star(Y, X).
% 0.20/0.50 Axiom 3 (axiom_5): and_star(X, falsehood) = falsehood.
% 0.20/0.50 Axiom 4 (axiom_4): and_star(X, truth) = X.
% 0.20/0.50 Axiom 5 (axiom_3): xor(X, X) = falsehood.
% 0.20/0.50 Axiom 6 (xor_commutativity): xor(X, Y) = xor(Y, X).
% 0.20/0.50 Axiom 7 (axiom_2): xor(X, falsehood) = X.
% 0.20/0.50 Axiom 8 (axiom_1): not(X) = xor(X, truth).
% 0.20/0.50 Axiom 9 (and_star_associativity): and_star(and_star(X, Y), Z) = and_star(X, and_star(Y, Z)).
% 0.20/0.50 Axiom 10 (axiom_6): and_star(xor(truth, X), X) = falsehood.
% 0.20/0.50 Axiom 11 (axiom_7): xor(X, xor(truth, Y)) = xor(xor(X, truth), Y).
% 0.20/0.50 Axiom 12 (implies_definition): implies(X, Y) = xor(truth, and_star(X, xor(truth, Y))).
% 0.20/0.50 Axiom 13 (axiom_8): and_star(xor(and_star(xor(truth, X), Y), truth), Y) = and_star(xor(and_star(xor(truth, Y), X), truth), X).
% 0.20/0.50
% 0.20/0.50 Lemma 14: not(falsehood) = truth.
% 0.20/0.50 Proof:
% 0.20/0.50 not(falsehood)
% 0.20/0.50 = { by axiom 8 (axiom_1) }
% 0.20/0.50 xor(falsehood, truth)
% 0.20/0.50 = { by axiom 6 (xor_commutativity) R->L }
% 0.20/0.50 xor(truth, falsehood)
% 0.20/0.50 = { by axiom 7 (axiom_2) }
% 0.20/0.50 truth
% 0.20/0.50
% 0.20/0.50 Lemma 15: xor(X, xor(truth, Y)) = xor(not(X), Y).
% 0.20/0.50 Proof:
% 0.20/0.50 xor(X, xor(truth, Y))
% 0.20/0.50 = { by axiom 11 (axiom_7) }
% 0.20/0.50 xor(xor(X, truth), Y)
% 0.20/0.50 = { by axiom 8 (axiom_1) R->L }
% 0.20/0.50 xor(not(X), Y)
% 0.20/0.50
% 0.20/0.50 Lemma 16: not(not(X)) = X.
% 0.20/0.50 Proof:
% 0.20/0.50 not(not(X))
% 0.20/0.50 = { by axiom 8 (axiom_1) }
% 0.20/0.50 xor(not(X), truth)
% 0.20/0.50 = { by lemma 15 R->L }
% 0.20/0.50 xor(X, xor(truth, truth))
% 0.20/0.50 = { by axiom 5 (axiom_3) }
% 0.20/0.50 xor(X, falsehood)
% 0.20/0.50 = { by axiom 7 (axiom_2) }
% 0.20/0.50 X
% 0.20/0.50
% 0.20/0.50 Lemma 17: xor(truth, X) = not(X).
% 0.20/0.50 Proof:
% 0.20/0.50 xor(truth, X)
% 0.20/0.50 = { by axiom 6 (xor_commutativity) R->L }
% 0.20/0.50 xor(X, truth)
% 0.20/0.50 = { by axiom 8 (axiom_1) R->L }
% 0.20/0.50 not(X)
% 0.20/0.50
% 0.20/0.50 Lemma 18: implies(truth, X) = X.
% 0.20/0.50 Proof:
% 0.20/0.50 implies(truth, X)
% 0.20/0.50 = { by axiom 12 (implies_definition) }
% 0.20/0.50 xor(truth, and_star(truth, xor(truth, X)))
% 0.20/0.50 = { by axiom 2 (and_star_commutativity) R->L }
% 0.20/0.50 xor(truth, and_star(xor(truth, X), truth))
% 0.20/0.50 = { by axiom 4 (axiom_4) }
% 0.20/0.50 xor(truth, xor(truth, X))
% 0.20/0.50 = { by lemma 17 }
% 0.20/0.50 not(xor(truth, X))
% 0.20/0.50 = { by lemma 17 }
% 0.20/0.50 not(not(X))
% 0.20/0.50 = { by lemma 16 }
% 0.20/0.50 X
% 0.20/0.50
% 0.20/0.50 Lemma 19: and_star(X, not(X)) = falsehood.
% 0.20/0.50 Proof:
% 0.20/0.50 and_star(X, not(X))
% 0.20/0.50 = { by axiom 8 (axiom_1) }
% 0.20/0.50 and_star(X, xor(X, truth))
% 0.20/0.50 = { by axiom 6 (xor_commutativity) R->L }
% 0.20/0.50 and_star(X, xor(truth, X))
% 0.20/0.50 = { by axiom 2 (and_star_commutativity) R->L }
% 0.20/0.50 and_star(xor(truth, X), X)
% 0.20/0.50 = { by axiom 10 (axiom_6) }
% 0.20/0.50 falsehood
% 0.20/0.50
% 0.20/0.50 Lemma 20: not(and_star(X, not(Y))) = implies(X, Y).
% 0.20/0.50 Proof:
% 0.20/0.50 not(and_star(X, not(Y)))
% 0.20/0.50 = { by lemma 17 R->L }
% 0.20/0.50 not(and_star(X, xor(truth, Y)))
% 0.20/0.50 = { by lemma 17 R->L }
% 0.20/0.50 xor(truth, and_star(X, xor(truth, Y)))
% 0.20/0.50 = { by axiom 12 (implies_definition) R->L }
% 0.20/0.50 implies(X, Y)
% 0.20/0.50
% 0.20/0.50 Lemma 21: and_star(X, not(Y)) = not(implies(X, Y)).
% 0.20/0.50 Proof:
% 0.20/0.50 and_star(X, not(Y))
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 not(not(and_star(X, not(Y))))
% 0.20/0.50 = { by lemma 20 }
% 0.20/0.50 not(implies(X, Y))
% 0.20/0.50
% 0.20/0.50 Lemma 22: and_star(not(X), Y) = not(implies(Y, X)).
% 0.20/0.50 Proof:
% 0.20/0.50 and_star(not(X), Y)
% 0.20/0.50 = { by axiom 2 (and_star_commutativity) R->L }
% 0.20/0.50 and_star(Y, not(X))
% 0.20/0.50 = { by lemma 21 }
% 0.20/0.50 not(implies(Y, X))
% 0.20/0.50
% 0.20/0.50 Lemma 23: implies(X, not(Y)) = not(and_star(X, Y)).
% 0.20/0.50 Proof:
% 0.20/0.50 implies(X, not(Y))
% 0.20/0.50 = { by axiom 12 (implies_definition) }
% 0.20/0.50 xor(truth, and_star(X, xor(truth, not(Y))))
% 0.20/0.50 = { by axiom 6 (xor_commutativity) R->L }
% 0.20/0.50 xor(truth, and_star(X, xor(not(Y), truth)))
% 0.20/0.50 = { by lemma 15 R->L }
% 0.20/0.50 xor(truth, and_star(X, xor(Y, xor(truth, truth))))
% 0.20/0.50 = { by axiom 8 (axiom_1) R->L }
% 0.20/0.50 xor(truth, and_star(X, xor(Y, not(truth))))
% 0.20/0.50 = { by lemma 17 }
% 0.20/0.50 not(and_star(X, xor(Y, not(truth))))
% 0.20/0.50 = { by axiom 1 (false_definition) }
% 0.20/0.50 not(and_star(X, xor(Y, falsehood)))
% 0.20/0.50 = { by axiom 7 (axiom_2) }
% 0.20/0.50 not(and_star(X, Y))
% 0.20/0.50
% 0.20/0.50 Lemma 24: implies(X, implies(Y, X)) = truth.
% 0.20/0.50 Proof:
% 0.20/0.50 implies(X, implies(Y, X))
% 0.20/0.50 = { by lemma 16 R->L }
% 0.20/0.50 not(not(implies(X, implies(Y, X))))
% 0.20/0.50 = { by lemma 21 R->L }
% 0.20/0.50 not(and_star(X, not(implies(Y, X))))
% 0.20/0.50 = { by lemma 22 R->L }
% 0.20/0.50 not(and_star(X, and_star(not(X), Y)))
% 0.20/0.50 = { by axiom 9 (and_star_associativity) R->L }
% 0.20/0.50 not(and_star(and_star(X, not(X)), Y))
% 0.20/0.50 = { by lemma 19 }
% 0.20/0.50 not(and_star(falsehood, Y))
% 0.20/0.50 = { by axiom 2 (and_star_commutativity) R->L }
% 0.20/0.50 not(and_star(Y, falsehood))
% 0.20/0.51 = { by axiom 3 (axiom_5) }
% 0.20/0.51 not(falsehood)
% 0.20/0.51 = { by lemma 14 }
% 0.20/0.51 truth
% 0.20/0.51
% 0.20/0.51 Lemma 25: implies(and_star(X, Y), Z) = implies(X, implies(Y, Z)).
% 0.20/0.51 Proof:
% 0.20/0.51 implies(and_star(X, Y), Z)
% 0.20/0.51 = { by lemma 16 R->L }
% 0.20/0.51 not(not(implies(and_star(X, Y), Z)))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 not(and_star(and_star(X, Y), not(Z)))
% 0.20/0.51 = { by axiom 9 (and_star_associativity) }
% 0.20/0.51 not(and_star(X, and_star(Y, not(Z))))
% 0.20/0.51 = { by lemma 21 }
% 0.20/0.51 not(and_star(X, not(implies(Y, Z))))
% 0.20/0.51 = { by lemma 21 }
% 0.20/0.51 not(not(implies(X, implies(Y, Z))))
% 0.20/0.51 = { by lemma 16 }
% 0.20/0.51 implies(X, implies(Y, Z))
% 0.20/0.51
% 0.20/0.51 Lemma 26: and_star(X, not(and_star(xor(truth, Y), X))) = and_star(Y, not(and_star(xor(truth, X), Y))).
% 0.20/0.51 Proof:
% 0.20/0.51 and_star(X, not(and_star(xor(truth, Y), X)))
% 0.20/0.51 = { by axiom 2 (and_star_commutativity) R->L }
% 0.20/0.51 and_star(not(and_star(xor(truth, Y), X)), X)
% 0.20/0.51 = { by axiom 8 (axiom_1) }
% 0.20/0.51 and_star(xor(and_star(xor(truth, Y), X), truth), X)
% 0.20/0.51 = { by axiom 13 (axiom_8) }
% 0.20/0.51 and_star(xor(and_star(xor(truth, X), Y), truth), Y)
% 0.20/0.51 = { by axiom 8 (axiom_1) R->L }
% 0.20/0.51 and_star(not(and_star(xor(truth, X), Y)), Y)
% 0.20/0.51 = { by axiom 2 (and_star_commutativity) }
% 0.20/0.51 and_star(Y, not(and_star(xor(truth, X), Y)))
% 0.20/0.51
% 0.20/0.51 Lemma 27: implies(implies(not(X), Y), X) = implies(Y, and_star(X, Y)).
% 0.20/0.51 Proof:
% 0.20/0.51 implies(implies(not(X), Y), X)
% 0.20/0.51 = { by lemma 16 R->L }
% 0.20/0.51 not(not(implies(implies(not(X), Y), X)))
% 0.20/0.51 = { by lemma 22 R->L }
% 0.20/0.51 not(and_star(not(X), implies(not(X), Y)))
% 0.20/0.51 = { by lemma 23 R->L }
% 0.20/0.51 implies(not(X), not(implies(not(X), Y)))
% 0.20/0.51 = { by lemma 22 R->L }
% 0.20/0.51 implies(not(X), and_star(not(Y), not(X)))
% 0.20/0.51 = { by lemma 17 R->L }
% 0.20/0.51 implies(not(X), and_star(xor(truth, Y), not(X)))
% 0.20/0.51 = { by lemma 20 R->L }
% 0.20/0.51 not(and_star(not(X), not(and_star(xor(truth, Y), not(X)))))
% 0.20/0.51 = { by lemma 26 R->L }
% 0.20/0.51 not(and_star(Y, not(and_star(xor(truth, not(X)), Y))))
% 0.20/0.51 = { by lemma 20 }
% 0.20/0.51 implies(Y, and_star(xor(truth, not(X)), Y))
% 0.20/0.51 = { by lemma 17 }
% 0.20/0.51 implies(Y, and_star(not(not(X)), Y))
% 0.20/0.51 = { by lemma 22 }
% 0.20/0.51 implies(Y, not(implies(Y, not(X))))
% 0.20/0.51 = { by lemma 23 }
% 0.20/0.51 not(and_star(Y, implies(Y, not(X))))
% 0.20/0.51 = { by lemma 23 }
% 0.20/0.51 not(and_star(Y, not(and_star(Y, X))))
% 0.20/0.51 = { by lemma 20 }
% 0.20/0.51 implies(Y, and_star(Y, X))
% 0.20/0.51 = { by axiom 2 (and_star_commutativity) }
% 0.20/0.51 implies(Y, and_star(X, Y))
% 0.20/0.51
% 0.20/0.51 Goal 1 (prove_wajsberg_mv_4): implies(implies(implies(a, b), implies(b, a)), implies(b, a)) = truth.
% 0.20/0.51 Proof:
% 0.20/0.51 implies(implies(implies(a, b), implies(b, a)), implies(b, a))
% 0.20/0.51 = { by lemma 25 R->L }
% 0.20/0.51 implies(and_star(implies(implies(a, b), implies(b, a)), b), a)
% 0.20/0.51 = { by axiom 2 (and_star_commutativity) }
% 0.20/0.51 implies(and_star(b, implies(implies(a, b), implies(b, a))), a)
% 0.20/0.51 = { by lemma 25 }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, a)), a))
% 0.20/0.51 = { by lemma 18 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, implies(truth, a))), a))
% 0.20/0.51 = { by lemma 24 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, implies(implies(a, implies(b, a)), a))), a))
% 0.20/0.51 = { by lemma 20 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), not(and_star(b, not(implies(implies(a, implies(b, a)), a))))), a))
% 0.20/0.51 = { by lemma 22 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), not(and_star(b, and_star(not(a), implies(a, implies(b, a)))))), a))
% 0.20/0.51 = { by lemma 23 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, not(and_star(not(a), implies(a, implies(b, a)))))), a))
% 0.20/0.51 = { by lemma 23 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, implies(not(a), not(implies(a, implies(b, a)))))), a))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, implies(not(a), and_star(a, not(implies(b, a)))))), a))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(b, implies(not(a), and_star(a, and_star(b, not(a)))))), a))
% 0.20/0.51 = { by lemma 25 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(and_star(b, not(a)), and_star(a, and_star(b, not(a))))), a))
% 0.20/0.51 = { by lemma 27 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(implies(not(a), and_star(b, not(a))), a)), a))
% 0.20/0.51 = { by lemma 27 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(implies(implies(not(b), not(a)), b), a)), a))
% 0.20/0.51 = { by lemma 23 }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(implies(not(and_star(not(b), a)), b), a)), a))
% 0.20/0.51 = { by lemma 22 }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(implies(not(not(implies(a, b))), b), a)), a))
% 0.20/0.51 = { by lemma 16 }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(implies(implies(a, b), b), a)), a))
% 0.20/0.51 = { by lemma 16 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(not(not(implies(implies(a, b), b))), a)), a))
% 0.20/0.51 = { by lemma 22 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(not(and_star(not(b), implies(a, b))), a)), a))
% 0.20/0.51 = { by lemma 17 R->L }
% 0.20/0.51 implies(b, implies(implies(implies(a, b), implies(not(and_star(xor(truth, b), implies(a, b))), a)), a))
% 0.20/0.51 = { by lemma 25 R->L }
% 0.20/0.51 implies(b, implies(implies(and_star(implies(a, b), not(and_star(xor(truth, b), implies(a, b)))), a), a))
% 0.20/0.51 = { by lemma 26 }
% 0.20/0.51 implies(b, implies(implies(and_star(b, not(and_star(xor(truth, implies(a, b)), b))), a), a))
% 0.20/0.51 = { by lemma 25 }
% 0.20/0.51 implies(b, implies(implies(b, implies(not(and_star(xor(truth, implies(a, b)), b)), a)), a))
% 0.20/0.51 = { by lemma 17 }
% 0.20/0.51 implies(b, implies(implies(b, implies(not(and_star(not(implies(a, b)), b)), a)), a))
% 0.20/0.51 = { by lemma 22 }
% 0.20/0.51 implies(b, implies(implies(b, implies(not(not(implies(b, implies(a, b)))), a)), a))
% 0.20/0.51 = { by lemma 16 }
% 0.20/0.51 implies(b, implies(implies(b, implies(implies(b, implies(a, b)), a)), a))
% 0.20/0.51 = { by lemma 24 }
% 0.20/0.51 implies(b, implies(implies(b, implies(truth, a)), a))
% 0.20/0.51 = { by lemma 18 }
% 0.20/0.51 implies(b, implies(implies(b, a), a))
% 0.20/0.51 = { by lemma 20 R->L }
% 0.20/0.51 not(and_star(b, not(implies(implies(b, a), a))))
% 0.20/0.51 = { by lemma 22 R->L }
% 0.20/0.51 not(and_star(b, and_star(not(a), implies(b, a))))
% 0.20/0.51 = { by lemma 23 R->L }
% 0.20/0.51 implies(b, not(and_star(not(a), implies(b, a))))
% 0.20/0.51 = { by lemma 23 R->L }
% 0.20/0.51 implies(b, implies(not(a), not(implies(b, a))))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 implies(b, implies(not(a), and_star(b, not(a))))
% 0.20/0.51 = { by lemma 16 R->L }
% 0.20/0.51 not(not(implies(b, implies(not(a), and_star(b, not(a))))))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 not(and_star(b, not(implies(not(a), and_star(b, not(a))))))
% 0.20/0.51 = { by lemma 21 R->L }
% 0.20/0.51 not(and_star(b, and_star(not(a), not(and_star(b, not(a))))))
% 0.20/0.51 = { by axiom 9 (and_star_associativity) R->L }
% 0.20/0.51 not(and_star(and_star(b, not(a)), not(and_star(b, not(a)))))
% 0.20/0.51 = { by lemma 19 }
% 0.20/0.51 not(falsehood)
% 0.20/0.51 = { by lemma 14 }
% 0.20/0.51 truth
% 0.20/0.51 % SZS output end Proof
% 0.20/0.51
% 0.20/0.51 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------