TSTP Solution File: LCL109-4 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL109-4 : 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 : n009.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 51.06s 6.86s
% Output : Proof 51.53s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.07 % Problem : LCL109-4 : TPTP v8.1.2. Released v1.0.0.
% 0.04/0.08 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.07/0.26 % Computer : n009.cluster.edu
% 0.07/0.26 % Model : x86_64 x86_64
% 0.07/0.26 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.26 % Memory : 8042.1875MB
% 0.07/0.26 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.26 % CPULimit : 300
% 0.07/0.26 % WCLimit : 300
% 0.07/0.26 % DateTime : Fri Aug 25 06:57:50 EDT 2023
% 0.07/0.27 % CPUTime :
% 51.06/6.86 Command-line arguments: --ground-connectedness --complete-subsets
% 51.06/6.86
% 51.06/6.86 % SZS status Unsatisfiable
% 51.06/6.86
% 51.06/6.88 % SZS output start Proof
% 51.06/6.88 Take the following subset of the input axioms:
% 51.06/6.89 fof(big_V_definition, axiom, ![X, Y]: big_V(X, Y)=implies(implies(X, Y), Y)).
% 51.06/6.89 fof(big_hat_definition, axiom, ![X2, Y2]: big_hat(X2, Y2)=not(big_V(not(X2), not(Y2)))).
% 51.06/6.89 fof(prove_mv_4, negated_conjecture, big_V(implies(x, y), implies(y, x))!=truth).
% 51.06/6.89 fof(wajsberg_1, axiom, ![X2]: implies(truth, X2)=X2).
% 51.06/6.89 fof(wajsberg_2, axiom, ![Z, X2, Y2]: implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z)))=truth).
% 51.06/6.89 fof(wajsberg_3, axiom, ![X2, Y2]: implies(implies(X2, Y2), Y2)=implies(implies(Y2, X2), X2)).
% 51.06/6.89 fof(wajsberg_4, axiom, ![X2, Y2]: implies(implies(not(X2), not(Y2)), implies(Y2, X2))=truth).
% 51.06/6.89
% 51.06/6.89 Now clausify the problem and encode Horn clauses using encoding 3 of
% 51.06/6.89 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 51.06/6.89 We repeatedly replace C & s=t => u=v by the two clauses:
% 51.06/6.89 fresh(y, y, x1...xn) = u
% 51.06/6.89 C => fresh(s, t, x1...xn) = v
% 51.06/6.89 where fresh is a fresh function symbol and x1..xn are the free
% 51.06/6.89 variables of u and v.
% 51.06/6.89 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 51.06/6.89 input problem has no model of domain size 1).
% 51.06/6.89
% 51.06/6.89 The encoding turns the above axioms into the following unit equations and goals:
% 51.06/6.89
% 51.06/6.89 Axiom 1 (wajsberg_1): implies(truth, X) = X.
% 51.06/6.89 Axiom 2 (big_V_definition): big_V(X, Y) = implies(implies(X, Y), Y).
% 51.06/6.89 Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 51.06/6.89 Axiom 4 (big_hat_definition): big_hat(X, Y) = not(big_V(not(X), not(Y))).
% 51.06/6.89 Axiom 5 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 51.06/6.89 Axiom 6 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 51.06/6.89
% 51.06/6.89 Lemma 7: big_V(Y, X) = big_V(X, Y).
% 51.06/6.89 Proof:
% 51.06/6.89 big_V(Y, X)
% 51.06/6.89 = { by axiom 2 (big_V_definition) }
% 51.06/6.89 implies(implies(Y, X), X)
% 51.06/6.89 = { by axiom 3 (wajsberg_3) R->L }
% 51.06/6.89 implies(implies(X, Y), Y)
% 51.06/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.06/6.89 big_V(X, Y)
% 51.06/6.89
% 51.06/6.89 Lemma 8: implies(X, big_V(X, Y)) = truth.
% 51.06/6.89 Proof:
% 51.06/6.89 implies(X, big_V(X, Y))
% 51.06/6.89 = { by axiom 2 (big_V_definition) }
% 51.06/6.89 implies(X, implies(implies(X, Y), Y))
% 51.06/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.06/6.89 implies(X, implies(implies(X, Y), implies(truth, Y)))
% 51.06/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.06/6.89 implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
% 51.06/6.89 = { by axiom 6 (wajsberg_2) }
% 51.06/6.89 truth
% 51.06/6.89
% 51.06/6.89 Lemma 9: big_V(X, truth) = truth.
% 51.06/6.89 Proof:
% 51.06/6.89 big_V(X, truth)
% 51.06/6.89 = { by lemma 7 }
% 51.06/6.89 big_V(truth, X)
% 51.06/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.06/6.89 implies(truth, big_V(truth, X))
% 51.06/6.89 = { by lemma 8 }
% 51.06/6.89 truth
% 51.06/6.89
% 51.06/6.89 Lemma 10: big_V(X, X) = X.
% 51.06/6.89 Proof:
% 51.06/6.89 big_V(X, X)
% 51.06/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.06/6.89 big_V(X, implies(truth, X))
% 51.06/6.89 = { by lemma 7 }
% 51.06/6.89 big_V(implies(truth, X), X)
% 51.06/6.89 = { by axiom 2 (big_V_definition) }
% 51.06/6.89 implies(implies(implies(truth, X), X), X)
% 51.06/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.06/6.89 implies(big_V(truth, X), X)
% 51.06/6.89 = { by lemma 7 R->L }
% 51.06/6.89 implies(big_V(X, truth), X)
% 51.06/6.89 = { by lemma 9 }
% 51.06/6.89 implies(truth, X)
% 51.06/6.89 = { by axiom 1 (wajsberg_1) }
% 51.06/6.89 X
% 51.06/6.89
% 51.06/6.89 Lemma 11: implies(implies(not(X), not(truth)), X) = truth.
% 51.06/6.89 Proof:
% 51.06/6.89 implies(implies(not(X), not(truth)), X)
% 51.06/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.06/6.89 implies(implies(not(X), not(truth)), implies(truth, X))
% 51.06/6.89 = { by axiom 5 (wajsberg_4) }
% 51.06/6.89 truth
% 51.06/6.89
% 51.06/6.89 Lemma 12: implies(X, big_V(Y, implies(Z, X))) = truth.
% 51.06/6.89 Proof:
% 51.06/6.89 implies(X, big_V(Y, implies(Z, X)))
% 51.06/6.89 = { by lemma 7 }
% 51.06/6.89 implies(X, big_V(implies(Z, X), Y))
% 51.06/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(truth, implies(X, big_V(implies(Z, X), Y)))
% 51.53/6.89 = { by axiom 6 (wajsberg_2) R->L }
% 51.53/6.89 implies(implies(implies(Z, truth), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 51.53/6.89 = { by lemma 9 R->L }
% 51.53/6.89 implies(implies(implies(Z, big_V(Z, truth)), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 51.53/6.89 = { by lemma 8 }
% 51.53/6.89 implies(implies(truth, implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) }
% 51.53/6.89 implies(implies(implies(truth, X), implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) }
% 51.53/6.89 implies(implies(X, implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(implies(X, implies(Z, X)), implies(truth, implies(X, big_V(implies(Z, X), Y))))
% 51.53/6.89 = { by lemma 8 R->L }
% 51.53/6.89 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))))
% 51.53/6.89 = { by axiom 6 (wajsberg_2) }
% 51.53/6.89 truth
% 51.53/6.89
% 51.53/6.89 Lemma 13: big_V(X, not(truth)) = X.
% 51.53/6.89 Proof:
% 51.53/6.89 big_V(X, not(truth))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 big_V(not(truth), X)
% 51.53/6.89 = { by axiom 2 (big_V_definition) }
% 51.53/6.89 implies(implies(not(truth), X), X)
% 51.53/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(implies(not(truth), implies(truth, X)), X)
% 51.53/6.89 = { by lemma 11 R->L }
% 51.53/6.89 implies(implies(not(truth), implies(implies(implies(not(X), not(truth)), X), X)), X)
% 51.53/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.53/6.89 implies(implies(not(truth), big_V(implies(not(X), not(truth)), X)), X)
% 51.53/6.89 = { by lemma 7 R->L }
% 51.53/6.89 implies(implies(not(truth), big_V(X, implies(not(X), not(truth)))), X)
% 51.53/6.89 = { by lemma 12 }
% 51.53/6.89 implies(truth, X)
% 51.53/6.89 = { by axiom 1 (wajsberg_1) }
% 51.53/6.89 X
% 51.53/6.89
% 51.53/6.89 Lemma 14: not(not(X)) = big_hat(X, truth).
% 51.53/6.89 Proof:
% 51.53/6.89 not(not(X))
% 51.53/6.89 = { by lemma 13 R->L }
% 51.53/6.89 not(big_V(not(X), not(truth)))
% 51.53/6.89 = { by axiom 4 (big_hat_definition) R->L }
% 51.53/6.89 big_hat(X, truth)
% 51.53/6.89
% 51.53/6.89 Lemma 15: not(not(X)) = big_hat(X, X).
% 51.53/6.89 Proof:
% 51.53/6.89 not(not(X))
% 51.53/6.89 = { by lemma 10 R->L }
% 51.53/6.89 not(big_V(not(X), not(X)))
% 51.53/6.89 = { by axiom 4 (big_hat_definition) R->L }
% 51.53/6.89 big_hat(X, X)
% 51.53/6.89
% 51.53/6.89 Lemma 16: big_hat(X, truth) = big_hat(X, X).
% 51.53/6.89 Proof:
% 51.53/6.89 big_hat(X, truth)
% 51.53/6.89 = { by lemma 14 R->L }
% 51.53/6.89 not(not(X))
% 51.53/6.89 = { by lemma 15 }
% 51.53/6.89 big_hat(X, X)
% 51.53/6.89
% 51.53/6.89 Lemma 17: big_V(implies(X, Y), implies(not(Y), not(X))) = implies(X, Y).
% 51.53/6.89 Proof:
% 51.53/6.89 big_V(implies(X, Y), implies(not(Y), not(X)))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 big_V(implies(not(Y), not(X)), implies(X, Y))
% 51.53/6.89 = { by axiom 2 (big_V_definition) }
% 51.53/6.89 implies(implies(implies(not(Y), not(X)), implies(X, Y)), implies(X, Y))
% 51.53/6.89 = { by axiom 5 (wajsberg_4) }
% 51.53/6.89 implies(truth, implies(X, Y))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) }
% 51.53/6.89 implies(X, Y)
% 51.53/6.89
% 51.53/6.89 Lemma 18: implies(not(X), implies(X, Y)) = truth.
% 51.53/6.89 Proof:
% 51.53/6.89 implies(not(X), implies(X, Y))
% 51.53/6.89 = { by lemma 17 R->L }
% 51.53/6.89 implies(not(X), big_V(implies(X, Y), implies(not(Y), not(X))))
% 51.53/6.89 = { by lemma 12 }
% 51.53/6.89 truth
% 51.53/6.89
% 51.53/6.89 Lemma 19: implies(X, not(truth)) = not(X).
% 51.53/6.89 Proof:
% 51.53/6.89 implies(X, not(truth))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(truth, implies(X, not(truth)))
% 51.53/6.89 = { by lemma 18 R->L }
% 51.53/6.89 implies(implies(not(X), implies(X, not(truth))), implies(X, not(truth)))
% 51.53/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.53/6.89 big_V(not(X), implies(X, not(truth)))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 big_V(implies(X, not(truth)), not(X))
% 51.53/6.89 = { by axiom 2 (big_V_definition) }
% 51.53/6.89 implies(implies(implies(X, not(truth)), not(X)), not(X))
% 51.53/6.89 = { by lemma 13 R->L }
% 51.53/6.89 implies(implies(implies(X, not(truth)), big_V(not(X), not(truth))), not(X))
% 51.53/6.89 = { by axiom 2 (big_V_definition) }
% 51.53/6.89 implies(implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth))), not(X))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(implies(truth, implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth)))), not(X))
% 51.53/6.89 = { by lemma 11 R->L }
% 51.53/6.89 implies(implies(implies(implies(not(X), not(truth)), X), implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth)))), not(X))
% 51.53/6.89 = { by axiom 6 (wajsberg_2) }
% 51.53/6.89 implies(truth, not(X))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) }
% 51.53/6.89 not(X)
% 51.53/6.89
% 51.53/6.89 Lemma 20: big_hat(X, truth) = X.
% 51.53/6.89 Proof:
% 51.53/6.89 big_hat(X, truth)
% 51.53/6.89 = { by lemma 14 R->L }
% 51.53/6.89 not(not(X))
% 51.53/6.89 = { by lemma 19 R->L }
% 51.53/6.89 not(implies(X, not(truth)))
% 51.53/6.89 = { by lemma 19 R->L }
% 51.53/6.89 implies(implies(X, not(truth)), not(truth))
% 51.53/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.53/6.89 big_V(X, not(truth))
% 51.53/6.89 = { by lemma 13 }
% 51.53/6.89 X
% 51.53/6.89
% 51.53/6.89 Lemma 21: big_hat(X, X) = X.
% 51.53/6.89 Proof:
% 51.53/6.89 big_hat(X, X)
% 51.53/6.89 = { by lemma 16 R->L }
% 51.53/6.89 big_hat(X, truth)
% 51.53/6.89 = { by lemma 20 }
% 51.53/6.89 X
% 51.53/6.89
% 51.53/6.89 Lemma 22: not(not(X)) = X.
% 51.53/6.89 Proof:
% 51.53/6.89 not(not(X))
% 51.53/6.89 = { by lemma 10 R->L }
% 51.53/6.89 not(big_V(not(X), not(X)))
% 51.53/6.89 = { by axiom 4 (big_hat_definition) R->L }
% 51.53/6.89 big_hat(X, X)
% 51.53/6.89 = { by lemma 21 }
% 51.53/6.89 X
% 51.53/6.89
% 51.53/6.89 Lemma 23: big_hat(Y, X) = big_hat(X, Y).
% 51.53/6.89 Proof:
% 51.53/6.89 big_hat(Y, X)
% 51.53/6.89 = { by axiom 4 (big_hat_definition) }
% 51.53/6.89 not(big_V(not(Y), not(X)))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 not(big_V(not(X), not(Y)))
% 51.53/6.89 = { by axiom 4 (big_hat_definition) R->L }
% 51.53/6.89 big_hat(X, Y)
% 51.53/6.89
% 51.53/6.89 Lemma 24: big_hat(truth, X) = X.
% 51.53/6.89 Proof:
% 51.53/6.89 big_hat(truth, X)
% 51.53/6.89 = { by lemma 23 }
% 51.53/6.89 big_hat(X, truth)
% 51.53/6.89 = { by lemma 20 }
% 51.53/6.89 X
% 51.53/6.89
% 51.53/6.89 Lemma 25: implies(not(X), not(Y)) = implies(Y, X).
% 51.53/6.89 Proof:
% 51.53/6.89 implies(not(X), not(Y))
% 51.53/6.89 = { by lemma 19 R->L }
% 51.53/6.89 implies(not(X), implies(Y, not(truth)))
% 51.53/6.89 = { by lemma 19 R->L }
% 51.53/6.89 implies(implies(X, not(truth)), implies(Y, not(truth)))
% 51.53/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(truth, implies(implies(X, not(truth)), implies(Y, not(truth))))
% 51.53/6.89 = { by axiom 6 (wajsberg_2) R->L }
% 51.53/6.89 implies(implies(implies(Y, X), implies(implies(X, not(truth)), implies(Y, not(truth)))), implies(implies(X, not(truth)), implies(Y, not(truth))))
% 51.53/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.53/6.89 big_V(implies(Y, X), implies(implies(X, not(truth)), implies(Y, not(truth))))
% 51.53/6.89 = { by lemma 19 }
% 51.53/6.89 big_V(implies(Y, X), implies(not(X), implies(Y, not(truth))))
% 51.53/6.89 = { by lemma 19 }
% 51.53/6.89 big_V(implies(Y, X), implies(not(X), not(Y)))
% 51.53/6.89 = { by lemma 17 }
% 51.53/6.89 implies(Y, X)
% 51.53/6.89
% 51.53/6.89 Lemma 26: not(big_V(X, not(Y))) = big_hat(Y, not(X)).
% 51.53/6.89 Proof:
% 51.53/6.89 not(big_V(X, not(Y)))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 not(big_V(not(Y), X))
% 51.53/6.89 = { by lemma 20 R->L }
% 51.53/6.89 not(big_V(not(Y), big_hat(X, truth)))
% 51.53/6.89 = { by lemma 23 }
% 51.53/6.89 not(big_V(not(Y), big_hat(truth, X)))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 not(big_V(big_hat(truth, X), not(Y)))
% 51.53/6.89 = { by axiom 4 (big_hat_definition) }
% 51.53/6.89 not(big_V(not(big_V(not(truth), not(X))), not(Y)))
% 51.53/6.89 = { by axiom 4 (big_hat_definition) R->L }
% 51.53/6.89 big_hat(big_V(not(truth), not(X)), Y)
% 51.53/6.89 = { by lemma 23 R->L }
% 51.53/6.89 big_hat(Y, big_V(not(truth), not(X)))
% 51.53/6.89 = { by lemma 7 R->L }
% 51.53/6.89 big_hat(Y, big_V(not(X), not(truth)))
% 51.53/6.89 = { by lemma 13 }
% 51.53/6.89 big_hat(Y, not(X))
% 51.53/6.89
% 51.53/6.89 Lemma 27: big_V(X, big_hat(X, Y)) = X.
% 51.53/6.89 Proof:
% 51.53/6.89 big_V(X, big_hat(X, Y))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 big_V(big_hat(X, Y), X)
% 51.53/6.89 = { by axiom 2 (big_V_definition) }
% 51.53/6.89 implies(implies(big_hat(X, Y), X), X)
% 51.53/6.89 = { by lemma 20 R->L }
% 51.53/6.89 implies(implies(big_hat(X, Y), big_hat(X, truth)), X)
% 51.53/6.89 = { by lemma 14 R->L }
% 51.53/6.89 implies(implies(big_hat(X, Y), not(not(X))), X)
% 51.53/6.89 = { by axiom 4 (big_hat_definition) }
% 51.53/6.89 implies(implies(not(big_V(not(X), not(Y))), not(not(X))), X)
% 51.53/6.89 = { by lemma 19 R->L }
% 51.53/6.89 implies(implies(not(big_V(not(X), not(Y))), implies(not(X), not(truth))), X)
% 51.53/6.89 = { by lemma 19 R->L }
% 51.53/6.89 implies(implies(implies(big_V(not(X), not(Y)), not(truth)), implies(not(X), not(truth))), X)
% 51.53/6.89 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.89 implies(implies(truth, implies(implies(big_V(not(X), not(Y)), not(truth)), implies(not(X), not(truth)))), X)
% 51.53/6.89 = { by lemma 8 R->L }
% 51.53/6.89 implies(implies(implies(not(X), big_V(not(X), not(Y))), implies(implies(big_V(not(X), not(Y)), not(truth)), implies(not(X), not(truth)))), X)
% 51.53/6.89 = { by axiom 6 (wajsberg_2) }
% 51.53/6.89 implies(truth, X)
% 51.53/6.89 = { by axiom 1 (wajsberg_1) }
% 51.53/6.89 X
% 51.53/6.89
% 51.53/6.89 Lemma 28: implies(implies(X, Y), not(X)) = big_V(not(X), not(Y)).
% 51.53/6.89 Proof:
% 51.53/6.89 implies(implies(X, Y), not(X))
% 51.53/6.89 = { by lemma 25 R->L }
% 51.53/6.89 implies(implies(not(Y), not(X)), not(X))
% 51.53/6.89 = { by axiom 2 (big_V_definition) R->L }
% 51.53/6.89 big_V(not(Y), not(X))
% 51.53/6.89 = { by lemma 7 R->L }
% 51.53/6.89 big_V(not(X), not(Y))
% 51.53/6.89
% 51.53/6.89 Lemma 29: big_hat(truth, big_V(not(X), not(Y))) = not(big_hat(X, Y)).
% 51.53/6.89 Proof:
% 51.53/6.89 big_hat(truth, big_V(not(X), not(Y)))
% 51.53/6.89 = { by lemma 7 }
% 51.53/6.89 big_hat(truth, big_V(not(Y), not(X)))
% 51.53/6.89 = { by lemma 23 }
% 51.53/6.89 big_hat(big_V(not(Y), not(X)), truth)
% 51.53/6.89 = { by lemma 16 }
% 51.53/6.89 big_hat(big_V(not(Y), not(X)), big_V(not(Y), not(X)))
% 51.53/6.89 = { by lemma 15 R->L }
% 51.53/6.89 not(not(big_V(not(Y), not(X))))
% 51.53/6.90 = { by axiom 4 (big_hat_definition) R->L }
% 51.53/6.90 not(big_hat(Y, X))
% 51.53/6.90 = { by lemma 23 R->L }
% 51.53/6.90 not(big_hat(X, Y))
% 51.53/6.90
% 51.53/6.90 Lemma 30: implies(implies(X, Y), not(X)) = not(big_hat(X, Y)).
% 51.53/6.90 Proof:
% 51.53/6.90 implies(implies(X, Y), not(X))
% 51.53/6.90 = { by lemma 24 R->L }
% 51.53/6.90 big_hat(truth, implies(implies(X, Y), not(X)))
% 51.53/6.90 = { by lemma 28 }
% 51.53/6.90 big_hat(truth, big_V(not(X), not(Y)))
% 51.53/6.90 = { by lemma 29 }
% 51.53/6.90 not(big_hat(X, Y))
% 51.53/6.90
% 51.53/6.90 Lemma 31: big_V(implies(X, Y), not(X)) = implies(X, Y).
% 51.53/6.90 Proof:
% 51.53/6.90 big_V(implies(X, Y), not(X))
% 51.53/6.90 = { by lemma 7 }
% 51.53/6.90 big_V(not(X), implies(X, Y))
% 51.53/6.90 = { by axiom 2 (big_V_definition) }
% 51.53/6.90 implies(implies(not(X), implies(X, Y)), implies(X, Y))
% 51.53/6.90 = { by lemma 18 }
% 51.53/6.90 implies(truth, implies(X, Y))
% 51.53/6.90 = { by axiom 1 (wajsberg_1) }
% 51.53/6.90 implies(X, Y)
% 51.53/6.90
% 51.53/6.90 Goal 1 (prove_mv_4): big_V(implies(x, y), implies(y, x)) = truth.
% 51.53/6.90 Proof:
% 51.53/6.90 big_V(implies(x, y), implies(y, x))
% 51.53/6.90 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.90 implies(truth, big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 9 R->L }
% 51.53/6.90 implies(big_V(implies(y, x), truth), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 7 }
% 51.53/6.90 implies(big_V(truth, implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 2 (big_V_definition) }
% 51.53/6.90 implies(implies(implies(truth, implies(y, x)), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 1 (wajsberg_1) }
% 51.53/6.90 implies(implies(implies(y, x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 27 R->L }
% 51.53/6.90 implies(implies(implies(big_V(y, big_hat(y, x)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 23 R->L }
% 51.53/6.90 implies(implies(implies(big_V(y, big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 2 (big_V_definition) }
% 51.53/6.90 implies(implies(implies(implies(implies(y, big_hat(x, y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 25 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(not(big_hat(x, y)), not(y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 30 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(implies(implies(x, y), not(x)), not(y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 28 }
% 51.53/6.90 implies(implies(implies(implies(implies(big_V(not(x), not(y)), not(y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 2 (big_V_definition) }
% 51.53/6.90 implies(implies(implies(implies(implies(implies(implies(not(x), not(y)), not(y)), not(y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 25 }
% 51.53/6.90 implies(implies(implies(implies(implies(implies(implies(y, x), not(y)), not(y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 2 (big_V_definition) R->L }
% 51.53/6.90 implies(implies(implies(implies(big_V(implies(y, x), not(y)), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 31 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), x), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 27 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_V(x, big_hat(x, not(implies(x, y))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 26 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_V(x, not(big_V(implies(x, y), not(x))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 31 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_V(x, not(implies(x, y)))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 21 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_hat(big_V(x, not(implies(x, y))), big_V(x, not(implies(x, y))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 15 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), not(not(big_V(x, not(implies(x, y)))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 26 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), not(big_hat(implies(x, y), not(x)))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 29 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_hat(truth, big_V(not(implies(x, y)), not(not(x))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 24 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_V(not(implies(x, y)), not(not(x)))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 7 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), big_V(not(not(x)), not(implies(x, y)))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 2 (big_V_definition) }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(not(not(x)), not(implies(x, y))), not(implies(x, y)))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 25 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(implies(x, y), not(x)), not(implies(x, y)))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 25 R->L }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(not(not(implies(x, y))), not(implies(implies(x, y), not(x))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 22 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), not(implies(implies(x, y), not(x))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 30 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), not(not(big_hat(x, y))))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by lemma 22 }
% 51.53/6.90 implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), big_hat(x, y))), implies(y, x)), big_V(implies(x, y), implies(y, x)))
% 51.53/6.90 = { by axiom 1 (wajsberg_1) R->L }
% 51.53/6.90 implies(truth, implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), big_hat(x, y))), implies(y, x)), big_V(implies(x, y), implies(y, x))))
% 51.53/6.90 = { by axiom 6 (wajsberg_2) R->L }
% 51.53/6.90 implies(implies(implies(implies(x, y), implies(y, x)), implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), big_hat(x, y)))), implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), big_hat(x, y))), implies(y, x)), big_V(implies(x, y), implies(y, x))))
% 51.53/6.90 = { by axiom 2 (big_V_definition) }
% 51.53/6.90 implies(implies(implies(implies(x, y), implies(y, x)), implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), big_hat(x, y)))), implies(implies(implies(implies(implies(y, x), big_hat(x, y)), implies(implies(x, y), big_hat(x, y))), implies(y, x)), implies(implies(implies(x, y), implies(y, x)), implies(y, x))))
% 51.53/6.90 = { by axiom 6 (wajsberg_2) }
% 51.53/6.90 truth
% 51.53/6.90 % SZS output end Proof
% 51.53/6.90
% 51.53/6.90 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------