TSTP Solution File: LCL145-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL145-1 : 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 : n005.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 08:17:38 EDT 2023
% Result : Unsatisfiable 0.20s 0.50s
% 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
% 0.11/0.12 % Problem : LCL145-1 : 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 : n005.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 : Thu Aug 24 20:07:08 EDT 2023
% 0.20/0.34 % CPUTime :
% 0.20/0.50 Command-line arguments: --no-flatten-goal
% 0.20/0.50
% 0.20/0.50 % SZS status Unsatisfiable
% 0.20/0.50
% 0.20/0.51 % SZS output start Proof
% 0.20/0.51 Take the following subset of the input axioms:
% 0.20/0.51 fof(big_V_definition, axiom, ![X, Y]: big_V(X, Y)=implies(implies(X, Y), Y)).
% 0.20/0.51 fof(big_hat_definition, axiom, ![X2, Y2]: big_hat(X2, Y2)=not(big_V(not(X2), not(Y2)))).
% 0.20/0.51 fof(prove_wajsberg_theorem, negated_conjecture, not(big_V(x, y))!=big_hat(not(x), not(y))).
% 0.20/0.51 fof(wajsberg_1, axiom, ![X2]: implies(truth, X2)=X2).
% 0.20/0.51 fof(wajsberg_2, axiom, ![Z, X2, Y2]: implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z)))=truth).
% 0.20/0.51 fof(wajsberg_3, axiom, ![X2, Y2]: implies(implies(X2, Y2), Y2)=implies(implies(Y2, X2), X2)).
% 0.20/0.51 fof(wajsberg_4, axiom, ![X2, Y2]: implies(implies(not(X2), not(Y2)), implies(Y2, X2))=truth).
% 0.20/0.51
% 0.20/0.51 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.20/0.51 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.20/0.51 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.20/0.51 fresh(y, y, x1...xn) = u
% 0.20/0.51 C => fresh(s, t, x1...xn) = v
% 0.20/0.51 where fresh is a fresh function symbol and x1..xn are the free
% 0.20/0.51 variables of u and v.
% 0.20/0.51 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.20/0.51 input problem has no model of domain size 1).
% 0.20/0.51
% 0.20/0.51 The encoding turns the above axioms into the following unit equations and goals:
% 0.20/0.51
% 0.20/0.51 Axiom 1 (wajsberg_1): implies(truth, X) = X.
% 0.20/0.51 Axiom 2 (big_V_definition): big_V(X, Y) = implies(implies(X, Y), Y).
% 0.20/0.51 Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 0.20/0.51 Axiom 4 (big_hat_definition): big_hat(X, Y) = not(big_V(not(X), not(Y))).
% 0.20/0.51 Axiom 5 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 0.20/0.51 Axiom 6 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 0.20/0.51
% 0.20/0.51 Lemma 7: big_V(Y, X) = big_V(X, Y).
% 0.20/0.51 Proof:
% 0.20/0.51 big_V(Y, X)
% 0.20/0.51 = { by axiom 2 (big_V_definition) }
% 0.20/0.51 implies(implies(Y, X), X)
% 0.20/0.51 = { by axiom 3 (wajsberg_3) R->L }
% 0.20/0.51 implies(implies(X, Y), Y)
% 0.20/0.52 = { by axiom 2 (big_V_definition) R->L }
% 0.20/0.52 big_V(X, Y)
% 0.20/0.52
% 0.20/0.52 Lemma 8: implies(implies(not(X), not(truth)), X) = truth.
% 0.20/0.52 Proof:
% 0.20/0.52 implies(implies(not(X), not(truth)), X)
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(implies(not(X), not(truth)), implies(truth, X))
% 0.20/0.52 = { by axiom 5 (wajsberg_4) }
% 0.20/0.52 truth
% 0.20/0.52
% 0.20/0.52 Lemma 9: implies(X, big_V(X, Y)) = truth.
% 0.20/0.52 Proof:
% 0.20/0.52 implies(X, big_V(X, Y))
% 0.20/0.52 = { by axiom 2 (big_V_definition) }
% 0.20/0.52 implies(X, implies(implies(X, Y), Y))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(X, implies(implies(X, Y), implies(truth, Y)))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
% 0.20/0.52 = { by axiom 6 (wajsberg_2) }
% 0.20/0.52 truth
% 0.20/0.52
% 0.20/0.52 Lemma 10: implies(X, big_V(Y, implies(Z, X))) = truth.
% 0.20/0.52 Proof:
% 0.20/0.52 implies(X, big_V(Y, implies(Z, X)))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 implies(X, big_V(implies(Z, X), Y))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(truth, implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by axiom 6 (wajsberg_2) R->L }
% 0.20/0.52 implies(implies(implies(Z, truth), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by lemma 9 R->L }
% 0.20/0.52 implies(implies(implies(Z, implies(truth, big_V(truth, Z))), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) }
% 0.20/0.52 implies(implies(implies(Z, big_V(truth, Z)), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by lemma 7 R->L }
% 0.20/0.52 implies(implies(implies(Z, big_V(Z, truth)), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by lemma 9 }
% 0.20/0.52 implies(implies(truth, implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) }
% 0.20/0.52 implies(implies(implies(truth, X), implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) }
% 0.20/0.52 implies(implies(X, implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(implies(X, implies(Z, X)), implies(truth, implies(X, big_V(implies(Z, X), Y))))
% 0.20/0.52 = { by lemma 9 R->L }
% 0.20/0.52 implies(implies(X, implies(Z, X)), implies(implies(implies(Z, X), big_V(implies(Z, X), Y)), implies(X, big_V(implies(Z, X), Y))))
% 0.20/0.52 = { by axiom 6 (wajsberg_2) }
% 0.20/0.52 truth
% 0.20/0.52
% 0.20/0.52 Lemma 11: big_V(X, not(truth)) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 big_V(X, not(truth))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 big_V(not(truth), X)
% 0.20/0.52 = { by axiom 2 (big_V_definition) }
% 0.20/0.52 implies(implies(not(truth), X), X)
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(implies(not(truth), implies(truth, X)), X)
% 0.20/0.52 = { by lemma 8 R->L }
% 0.20/0.52 implies(implies(not(truth), implies(implies(implies(not(X), not(truth)), X), X)), X)
% 0.20/0.52 = { by axiom 2 (big_V_definition) R->L }
% 0.20/0.52 implies(implies(not(truth), big_V(implies(not(X), not(truth)), X)), X)
% 0.20/0.52 = { by lemma 7 R->L }
% 0.20/0.52 implies(implies(not(truth), big_V(X, implies(not(X), not(truth)))), X)
% 0.20/0.52 = { by lemma 10 }
% 0.20/0.52 implies(truth, X)
% 0.20/0.52 = { by axiom 1 (wajsberg_1) }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 12: not(not(X)) = big_hat(X, truth).
% 0.20/0.52 Proof:
% 0.20/0.52 not(not(X))
% 0.20/0.52 = { by lemma 11 R->L }
% 0.20/0.52 not(big_V(not(X), not(truth)))
% 0.20/0.52 = { by axiom 4 (big_hat_definition) R->L }
% 0.20/0.52 big_hat(X, truth)
% 0.20/0.52
% 0.20/0.52 Lemma 13: implies(X, not(truth)) = not(X).
% 0.20/0.52 Proof:
% 0.20/0.52 implies(X, not(truth))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(truth, implies(X, not(truth)))
% 0.20/0.52 = { by lemma 10 R->L }
% 0.20/0.52 implies(implies(not(X), big_V(implies(X, not(truth)), implies(not(not(truth)), not(X)))), implies(X, not(truth)))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 implies(implies(not(X), big_V(implies(not(not(truth)), not(X)), implies(X, not(truth)))), implies(X, not(truth)))
% 0.20/0.52 = { by axiom 2 (big_V_definition) }
% 0.20/0.52 implies(implies(not(X), implies(implies(implies(not(not(truth)), not(X)), implies(X, not(truth))), implies(X, not(truth)))), implies(X, not(truth)))
% 0.20/0.52 = { by axiom 5 (wajsberg_4) }
% 0.20/0.52 implies(implies(not(X), implies(truth, implies(X, not(truth)))), implies(X, not(truth)))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) }
% 0.20/0.52 implies(implies(not(X), implies(X, not(truth))), implies(X, not(truth)))
% 0.20/0.52 = { by axiom 2 (big_V_definition) R->L }
% 0.20/0.52 big_V(not(X), implies(X, not(truth)))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 big_V(implies(X, not(truth)), not(X))
% 0.20/0.52 = { by axiom 2 (big_V_definition) }
% 0.20/0.52 implies(implies(implies(X, not(truth)), not(X)), not(X))
% 0.20/0.52 = { by lemma 11 R->L }
% 0.20/0.52 implies(implies(implies(X, not(truth)), big_V(not(X), not(truth))), not(X))
% 0.20/0.52 = { by axiom 2 (big_V_definition) }
% 0.20/0.52 implies(implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth))), not(X))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) R->L }
% 0.20/0.52 implies(implies(truth, implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth)))), not(X))
% 0.20/0.52 = { by lemma 8 R->L }
% 0.20/0.52 implies(implies(implies(implies(not(X), not(truth)), X), implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth)))), not(X))
% 0.20/0.52 = { by axiom 6 (wajsberg_2) }
% 0.20/0.52 implies(truth, not(X))
% 0.20/0.52 = { by axiom 1 (wajsberg_1) }
% 0.20/0.52 not(X)
% 0.20/0.52
% 0.20/0.52 Lemma 14: big_hat(X, truth) = X.
% 0.20/0.52 Proof:
% 0.20/0.52 big_hat(X, truth)
% 0.20/0.52 = { by lemma 12 R->L }
% 0.20/0.52 not(not(X))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 not(implies(X, not(truth)))
% 0.20/0.52 = { by lemma 13 R->L }
% 0.20/0.52 implies(implies(X, not(truth)), not(truth))
% 0.20/0.52 = { by axiom 2 (big_V_definition) R->L }
% 0.20/0.52 big_V(X, not(truth))
% 0.20/0.52 = { by lemma 11 }
% 0.20/0.52 X
% 0.20/0.52
% 0.20/0.52 Lemma 15: big_hat(Y, X) = big_hat(X, Y).
% 0.20/0.52 Proof:
% 0.20/0.52 big_hat(Y, X)
% 0.20/0.52 = { by axiom 4 (big_hat_definition) }
% 0.20/0.52 not(big_V(not(Y), not(X)))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 not(big_V(not(X), not(Y)))
% 0.20/0.52 = { by axiom 4 (big_hat_definition) R->L }
% 0.20/0.52 big_hat(X, Y)
% 0.20/0.52
% 0.20/0.52 Goal 1 (prove_wajsberg_theorem): not(big_V(x, y)) = big_hat(not(x), not(y)).
% 0.20/0.52 Proof:
% 0.20/0.52 not(big_V(x, y))
% 0.20/0.52 = { by lemma 14 R->L }
% 0.20/0.52 not(big_V(x, big_hat(y, truth)))
% 0.20/0.52 = { by lemma 12 R->L }
% 0.20/0.52 not(big_V(x, not(not(y))))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 not(big_V(not(not(y)), x))
% 0.20/0.52 = { by lemma 14 R->L }
% 0.20/0.52 not(big_V(not(not(y)), big_hat(x, truth)))
% 0.20/0.52 = { by lemma 15 }
% 0.20/0.52 not(big_V(not(not(y)), big_hat(truth, x)))
% 0.20/0.52 = { by lemma 7 }
% 0.20/0.52 not(big_V(big_hat(truth, x), not(not(y))))
% 0.20/0.52 = { by axiom 4 (big_hat_definition) }
% 0.20/0.52 not(big_V(not(big_V(not(truth), not(x))), not(not(y))))
% 0.20/0.52 = { by axiom 4 (big_hat_definition) R->L }
% 0.20/0.52 big_hat(big_V(not(truth), not(x)), not(y))
% 0.20/0.52 = { by lemma 15 R->L }
% 0.20/0.52 big_hat(not(y), big_V(not(truth), not(x)))
% 0.20/0.52 = { by lemma 7 R->L }
% 0.20/0.52 big_hat(not(y), big_V(not(x), not(truth)))
% 0.20/0.52 = { by lemma 11 }
% 0.20/0.52 big_hat(not(y), not(x))
% 0.20/0.52 = { by lemma 15 R->L }
% 0.20/0.52 big_hat(not(x), not(y))
% 0.20/0.52 % SZS output end Proof
% 0.20/0.52
% 0.20/0.52 RESULT: Unsatisfiable (the axioms are contradictory).
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