TSTP Solution File: LCL146-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL146-1 : 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 : n021.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 08:17:39 EDT 2023
% Result : Unsatisfiable 0.21s 0.51s
% 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
% 0.00/0.14 % Problem : LCL146-1 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.36 % Computer : n021.cluster.edu
% 0.15/0.36 % Model : x86_64 x86_64
% 0.15/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36 % Memory : 8042.1875MB
% 0.15/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36 % CPULimit : 300
% 0.15/0.36 % WCLimit : 300
% 0.15/0.36 % DateTime : Fri Aug 25 01:10:11 EDT 2023
% 0.15/0.36 % CPUTime :
% 0.21/0.51 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.21/0.51
% 0.21/0.51 % SZS status Unsatisfiable
% 0.21/0.51
% 0.21/0.52 % SZS output start Proof
% 0.21/0.52 Take the following subset of the input axioms:
% 0.21/0.52 fof(big_V_definition, axiom, ![X, Y]: big_V(X, Y)=implies(implies(X, Y), Y)).
% 0.21/0.52 fof(big_hat_definition, axiom, ![X2, Y2]: big_hat(X2, Y2)=not(big_V(not(X2), not(Y2)))).
% 0.21/0.52 fof(prove_wajsberg_theorem, negated_conjecture, not(big_hat(x, y))!=big_V(not(x), not(y))).
% 0.21/0.52 fof(wajsberg_1, axiom, ![X2]: implies(truth, X2)=X2).
% 0.21/0.52 fof(wajsberg_2, axiom, ![Z, X2, Y2]: implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z)))=truth).
% 0.21/0.52 fof(wajsberg_3, axiom, ![X2, Y2]: implies(implies(X2, Y2), Y2)=implies(implies(Y2, X2), X2)).
% 0.21/0.52 fof(wajsberg_4, axiom, ![X2, Y2]: implies(implies(not(X2), not(Y2)), implies(Y2, X2))=truth).
% 0.21/0.52
% 0.21/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.52 fresh(y, y, x1...xn) = u
% 0.21/0.52 C => fresh(s, t, x1...xn) = v
% 0.21/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.52 variables of u and v.
% 0.21/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.52 input problem has no model of domain size 1).
% 0.21/0.52
% 0.21/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.52
% 0.21/0.52 Axiom 1 (wajsberg_1): implies(truth, X) = X.
% 0.21/0.52 Axiom 2 (big_V_definition): big_V(X, Y) = implies(implies(X, Y), Y).
% 0.21/0.52 Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 0.21/0.52 Axiom 4 (big_hat_definition): big_hat(X, Y) = not(big_V(not(X), not(Y))).
% 0.21/0.52 Axiom 5 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 0.21/0.52 Axiom 6 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 0.21/0.52
% 0.21/0.52 Lemma 7: big_V(Y, X) = big_V(X, Y).
% 0.21/0.52 Proof:
% 0.21/0.52 big_V(Y, X)
% 0.21/0.52 = { by axiom 2 (big_V_definition) }
% 0.21/0.52 implies(implies(Y, X), X)
% 0.21/0.52 = { by axiom 3 (wajsberg_3) R->L }
% 0.21/0.52 implies(implies(X, Y), Y)
% 0.21/0.52 = { by axiom 2 (big_V_definition) R->L }
% 0.21/0.52 big_V(X, Y)
% 0.21/0.52
% 0.21/0.52 Lemma 8: implies(X, big_V(X, Y)) = truth.
% 0.21/0.52 Proof:
% 0.21/0.52 implies(X, big_V(X, Y))
% 0.21/0.52 = { by axiom 2 (big_V_definition) }
% 0.21/0.52 implies(X, implies(implies(X, Y), Y))
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(X, implies(implies(X, Y), implies(truth, Y)))
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
% 0.21/0.52 = { by axiom 6 (wajsberg_2) }
% 0.21/0.52 truth
% 0.21/0.52
% 0.21/0.52 Lemma 9: big_V(X, truth) = truth.
% 0.21/0.52 Proof:
% 0.21/0.52 big_V(X, truth)
% 0.21/0.52 = { by lemma 7 }
% 0.21/0.52 big_V(truth, X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(truth, big_V(truth, X))
% 0.21/0.52 = { by lemma 8 }
% 0.21/0.52 truth
% 0.21/0.52
% 0.21/0.52 Lemma 10: implies(implies(not(X), not(truth)), X) = truth.
% 0.21/0.52 Proof:
% 0.21/0.52 implies(implies(not(X), not(truth)), X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(implies(not(X), not(truth)), implies(truth, X))
% 0.21/0.52 = { by axiom 5 (wajsberg_4) }
% 0.21/0.52 truth
% 0.21/0.52
% 0.21/0.52 Lemma 11: implies(not(not(X)), X) = truth.
% 0.21/0.52 Proof:
% 0.21/0.52 implies(not(not(X)), X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(implies(truth, not(not(X))), X)
% 0.21/0.52 = { by lemma 9 R->L }
% 0.21/0.52 implies(implies(big_V(not(not(truth)), truth), not(not(X))), X)
% 0.21/0.52 = { by lemma 10 R->L }
% 0.21/0.52 implies(implies(big_V(not(not(truth)), implies(implies(not(not(truth)), not(truth)), not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 2 (big_V_definition) R->L }
% 0.21/0.52 implies(implies(big_V(not(not(truth)), big_V(not(not(truth)), not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 2 (big_V_definition) }
% 0.21/0.52 implies(implies(implies(implies(not(not(truth)), big_V(not(not(truth)), not(truth))), big_V(not(not(truth)), not(truth))), not(not(X))), X)
% 0.21/0.52 = { by lemma 8 }
% 0.21/0.52 implies(implies(implies(truth, big_V(not(not(truth)), not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) }
% 0.21/0.52 implies(implies(big_V(not(not(truth)), not(truth)), not(not(X))), X)
% 0.21/0.52 = { by lemma 7 }
% 0.21/0.52 implies(implies(big_V(not(truth), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 2 (big_V_definition) }
% 0.21/0.52 implies(implies(implies(implies(not(truth), not(not(truth))), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(implies(implies(implies(truth, implies(not(truth), not(not(truth)))), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by lemma 10 R->L }
% 0.21/0.52 implies(implies(implies(implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth)))), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.52 implies(implies(implies(implies(truth, implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 6 (wajsberg_2) R->L }
% 0.21/0.52 implies(implies(implies(implies(implies(implies(not(not(not(truth))), truth), implies(implies(truth, not(truth)), implies(not(not(not(truth))), not(truth)))), implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by axiom 1 (wajsberg_1) }
% 0.21/0.52 implies(implies(implies(implies(implies(implies(not(not(not(truth))), truth), implies(not(truth), implies(not(not(not(truth))), not(truth)))), implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(truth))), not(not(X))), X)
% 0.21/0.52 = { by lemma 9 R->L }
% 0.21/0.53 implies(implies(implies(implies(implies(implies(not(not(not(truth))), big_V(not(not(not(truth))), truth)), implies(not(truth), implies(not(not(not(truth))), not(truth)))), implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(truth))), not(not(X))), X)
% 0.21/0.53 = { by lemma 8 }
% 0.21/0.53 implies(implies(implies(implies(implies(truth, implies(not(truth), implies(not(not(not(truth))), not(truth)))), implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(truth))), not(not(X))), X)
% 0.21/0.53 = { by axiom 1 (wajsberg_1) }
% 0.21/0.53 implies(implies(implies(implies(implies(not(truth), implies(not(not(not(truth))), not(truth))), implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(truth))), not(not(X))), X)
% 0.21/0.53 = { by axiom 6 (wajsberg_2) }
% 0.21/0.53 implies(implies(implies(truth, not(not(truth))), not(not(X))), X)
% 0.21/0.53 = { by axiom 1 (wajsberg_1) }
% 0.21/0.53 implies(implies(not(not(truth)), not(not(X))), X)
% 0.21/0.53 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.53 implies(truth, implies(implies(not(not(truth)), not(not(X))), X))
% 0.21/0.53 = { by lemma 10 R->L }
% 0.21/0.53 implies(implies(implies(not(X), not(truth)), X), implies(implies(not(not(truth)), not(not(X))), X))
% 0.21/0.53 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.53 implies(truth, implies(implies(implies(not(X), not(truth)), X), implies(implies(not(not(truth)), not(not(X))), X)))
% 0.21/0.53 = { by axiom 5 (wajsberg_4) R->L }
% 0.21/0.53 implies(implies(implies(not(not(truth)), not(not(X))), implies(not(X), not(truth))), implies(implies(implies(not(X), not(truth)), X), implies(implies(not(not(truth)), not(not(X))), X)))
% 0.21/0.53 = { by axiom 6 (wajsberg_2) }
% 0.21/0.53 truth
% 0.21/0.53
% 0.21/0.53 Goal 1 (prove_wajsberg_theorem): not(big_hat(x, y)) = big_V(not(x), not(y)).
% 0.21/0.53 Proof:
% 0.21/0.53 not(big_hat(x, y))
% 0.21/0.53 = { by axiom 4 (big_hat_definition) }
% 0.21/0.53 not(not(big_V(not(x), not(y))))
% 0.21/0.53 = { by axiom 1 (wajsberg_1) R->L }
% 0.21/0.53 implies(truth, not(not(big_V(not(x), not(y)))))
% 0.21/0.53 = { by axiom 5 (wajsberg_4) R->L }
% 0.21/0.53 implies(implies(implies(not(not(not(big_V(not(x), not(y))))), not(big_V(not(x), not(y)))), implies(big_V(not(x), not(y)), not(not(big_V(not(x), not(y)))))), not(not(big_V(not(x), not(y)))))
% 0.21/0.53 = { by lemma 11 }
% 0.21/0.53 implies(implies(truth, implies(big_V(not(x), not(y)), not(not(big_V(not(x), not(y)))))), not(not(big_V(not(x), not(y)))))
% 0.21/0.53 = { by axiom 1 (wajsberg_1) }
% 0.21/0.53 implies(implies(big_V(not(x), not(y)), not(not(big_V(not(x), not(y))))), not(not(big_V(not(x), not(y)))))
% 0.21/0.53 = { by axiom 2 (big_V_definition) R->L }
% 0.21/0.53 big_V(big_V(not(x), not(y)), not(not(big_V(not(x), not(y)))))
% 0.21/0.53 = { by lemma 7 }
% 0.21/0.53 big_V(not(not(big_V(not(x), not(y)))), big_V(not(x), not(y)))
% 0.21/0.53 = { by axiom 2 (big_V_definition) }
% 0.21/0.53 implies(implies(not(not(big_V(not(x), not(y)))), big_V(not(x), not(y))), big_V(not(x), not(y)))
% 0.21/0.53 = { by lemma 11 }
% 0.21/0.53 implies(truth, big_V(not(x), not(y)))
% 0.21/0.53 = { by axiom 1 (wajsberg_1) }
% 0.21/0.53 big_V(not(x), not(y))
% 0.21/0.53 % SZS output end Proof
% 0.21/0.53
% 0.21/0.53 RESULT: Unsatisfiable (the axioms are contradictory).
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