TSTP Solution File: LCL152-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL152-1 : 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 : n012.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 08:17:40 EDT 2023
% Result : Unsatisfiable 127.76s 16.70s
% Output : Proof 127.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : LCL152-1 : 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
% 0.13/0.35 % Computer : n012.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Thu Aug 24 20:38:56 EDT 2023
% 0.13/0.35 % CPUTime :
% 127.76/16.70 Command-line arguments: --no-flatten-goal
% 127.76/16.70
% 127.76/16.70 % SZS status Unsatisfiable
% 127.76/16.70
% 127.76/16.72 % SZS output start Proof
% 127.76/16.72 Take the following subset of the input axioms:
% 127.76/16.72 fof(big_V_definition, axiom, ![X, Y]: big_V(X, Y)=implies(implies(X, Y), Y)).
% 127.76/16.72 fof(big_hat_definition, axiom, ![X2, Y2]: big_hat(X2, Y2)=not(big_V(not(X2), not(Y2)))).
% 127.76/16.72 fof(prove_wajsberg_theorem, negated_conjecture, implies(big_hat(x, y), z)!=implies(implies(x, y), implies(x, z))).
% 127.76/16.72 fof(wajsberg_1, axiom, ![X2]: implies(truth, X2)=X2).
% 127.76/16.72 fof(wajsberg_2, axiom, ![Z, X2, Y2]: implies(implies(X2, Y2), implies(implies(Y2, Z), implies(X2, Z)))=truth).
% 127.76/16.72 fof(wajsberg_3, axiom, ![X2, Y2]: implies(implies(X2, Y2), Y2)=implies(implies(Y2, X2), X2)).
% 127.76/16.72 fof(wajsberg_4, axiom, ![X2, Y2]: implies(implies(not(X2), not(Y2)), implies(Y2, X2))=truth).
% 127.76/16.72
% 127.76/16.72 Now clausify the problem and encode Horn clauses using encoding 3 of
% 127.76/16.72 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 127.76/16.72 We repeatedly replace C & s=t => u=v by the two clauses:
% 127.76/16.72 fresh(y, y, x1...xn) = u
% 127.76/16.72 C => fresh(s, t, x1...xn) = v
% 127.76/16.72 where fresh is a fresh function symbol and x1..xn are the free
% 127.76/16.72 variables of u and v.
% 127.76/16.72 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 127.76/16.72 input problem has no model of domain size 1).
% 127.76/16.72
% 127.76/16.72 The encoding turns the above axioms into the following unit equations and goals:
% 127.76/16.72
% 127.76/16.72 Axiom 1 (wajsberg_1): implies(truth, X) = X.
% 127.76/16.72 Axiom 2 (big_V_definition): big_V(X, Y) = implies(implies(X, Y), Y).
% 127.76/16.72 Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 127.76/16.72 Axiom 4 (big_hat_definition): big_hat(X, Y) = not(big_V(not(X), not(Y))).
% 127.76/16.72 Axiom 5 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 127.76/16.72 Axiom 6 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 127.76/16.72
% 127.76/16.72 Lemma 7: big_V(Y, X) = big_V(X, Y).
% 127.76/16.72 Proof:
% 127.76/16.72 big_V(Y, X)
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(Y, X), X)
% 127.76/16.72 = { by axiom 3 (wajsberg_3) R->L }
% 127.76/16.72 implies(implies(X, Y), Y)
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 big_V(X, Y)
% 127.76/16.72
% 127.76/16.72 Lemma 8: implies(X, big_V(X, Y)) = truth.
% 127.76/16.72 Proof:
% 127.76/16.72 implies(X, big_V(X, Y))
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(X, implies(implies(X, Y), Y))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(X, implies(implies(X, Y), implies(truth, Y)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) }
% 127.76/16.72 truth
% 127.76/16.72
% 127.76/16.72 Lemma 9: big_V(X, truth) = truth.
% 127.76/16.72 Proof:
% 127.76/16.72 big_V(X, truth)
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 big_V(truth, X)
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(truth, big_V(truth, X))
% 127.76/16.72 = { by lemma 8 }
% 127.76/16.72 truth
% 127.76/16.72
% 127.76/16.72 Lemma 10: big_V(X, X) = X.
% 127.76/16.72 Proof:
% 127.76/16.72 big_V(X, X)
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 big_V(X, implies(truth, X))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 big_V(implies(truth, X), X)
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(implies(truth, X), X), X)
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 implies(big_V(truth, X), X)
% 127.76/16.72 = { by lemma 7 R->L }
% 127.76/16.72 implies(big_V(X, truth), X)
% 127.76/16.72 = { by lemma 9 }
% 127.76/16.72 implies(truth, X)
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 X
% 127.76/16.72
% 127.76/16.72 Lemma 11: not(not(X)) = big_hat(X, X).
% 127.76/16.72 Proof:
% 127.76/16.72 not(not(X))
% 127.76/16.72 = { by lemma 10 R->L }
% 127.76/16.72 not(big_V(not(X), not(X)))
% 127.76/16.72 = { by axiom 4 (big_hat_definition) R->L }
% 127.76/16.72 big_hat(X, X)
% 127.76/16.72
% 127.76/16.72 Lemma 12: implies(implies(not(X), not(truth)), X) = truth.
% 127.76/16.72 Proof:
% 127.76/16.72 implies(implies(not(X), not(truth)), X)
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(not(X), not(truth)), implies(truth, X))
% 127.76/16.72 = { by axiom 5 (wajsberg_4) }
% 127.76/16.72 truth
% 127.76/16.72
% 127.76/16.72 Lemma 13: implies(X, big_V(Y, implies(Z, X))) = truth.
% 127.76/16.72 Proof:
% 127.76/16.72 implies(X, big_V(Y, implies(Z, X)))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 implies(X, big_V(implies(Z, X), Y))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(truth, implies(X, big_V(implies(Z, X), Y)))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) R->L }
% 127.76/16.72 implies(implies(implies(Z, truth), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 127.76/16.72 = { by lemma 9 R->L }
% 127.76/16.72 implies(implies(implies(Z, big_V(Z, truth)), implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 127.76/16.72 = { by lemma 8 }
% 127.76/16.72 implies(implies(truth, implies(implies(truth, X), implies(Z, X))), implies(X, big_V(implies(Z, X), Y)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 implies(implies(implies(truth, X), implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 implies(implies(X, implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(X, implies(Z, X)), implies(truth, implies(X, big_V(implies(Z, X), Y))))
% 127.76/16.72 = { by lemma 8 R->L }
% 127.76/16.72 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))))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) }
% 127.76/16.72 truth
% 127.76/16.72
% 127.76/16.72 Lemma 14: big_V(X, not(truth)) = X.
% 127.76/16.72 Proof:
% 127.76/16.72 big_V(X, not(truth))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 big_V(not(truth), X)
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(not(truth), X), X)
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(not(truth), implies(truth, X)), X)
% 127.76/16.72 = { by lemma 12 R->L }
% 127.76/16.72 implies(implies(not(truth), implies(implies(implies(not(X), not(truth)), X), X)), X)
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 implies(implies(not(truth), big_V(implies(not(X), not(truth)), X)), X)
% 127.76/16.72 = { by lemma 7 R->L }
% 127.76/16.72 implies(implies(not(truth), big_V(X, implies(not(X), not(truth)))), X)
% 127.76/16.72 = { by lemma 13 }
% 127.76/16.72 implies(truth, X)
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 X
% 127.76/16.72
% 127.76/16.72 Lemma 15: not(not(X)) = big_hat(X, truth).
% 127.76/16.72 Proof:
% 127.76/16.72 not(not(X))
% 127.76/16.72 = { by lemma 14 R->L }
% 127.76/16.72 not(big_V(not(X), not(truth)))
% 127.76/16.72 = { by axiom 4 (big_hat_definition) R->L }
% 127.76/16.72 big_hat(X, truth)
% 127.76/16.72
% 127.76/16.72 Lemma 16: big_hat(X, truth) = big_hat(X, X).
% 127.76/16.72 Proof:
% 127.76/16.72 big_hat(X, truth)
% 127.76/16.72 = { by lemma 15 R->L }
% 127.76/16.72 not(not(X))
% 127.76/16.72 = { by lemma 11 }
% 127.76/16.72 big_hat(X, X)
% 127.76/16.72
% 127.76/16.72 Lemma 17: big_V(implies(X, Y), implies(not(Y), not(X))) = implies(X, Y).
% 127.76/16.72 Proof:
% 127.76/16.72 big_V(implies(X, Y), implies(not(Y), not(X)))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 big_V(implies(not(Y), not(X)), implies(X, Y))
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(implies(not(Y), not(X)), implies(X, Y)), implies(X, Y))
% 127.76/16.72 = { by axiom 5 (wajsberg_4) }
% 127.76/16.72 implies(truth, implies(X, Y))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 implies(X, Y)
% 127.76/16.72
% 127.76/16.72 Lemma 18: implies(X, not(truth)) = not(X).
% 127.76/16.72 Proof:
% 127.76/16.72 implies(X, not(truth))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(truth, implies(X, not(truth)))
% 127.76/16.72 = { by lemma 13 R->L }
% 127.76/16.72 implies(implies(not(X), big_V(implies(X, not(truth)), implies(not(not(truth)), not(X)))), implies(X, not(truth)))
% 127.76/16.72 = { by lemma 17 }
% 127.76/16.72 implies(implies(not(X), implies(X, not(truth))), implies(X, not(truth)))
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 big_V(not(X), implies(X, not(truth)))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 big_V(implies(X, not(truth)), not(X))
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(implies(X, not(truth)), not(X)), not(X))
% 127.76/16.72 = { by lemma 14 R->L }
% 127.76/16.72 implies(implies(implies(X, not(truth)), big_V(not(X), not(truth))), not(X))
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth))), not(X))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(truth, implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth)))), not(X))
% 127.76/16.72 = { by lemma 12 R->L }
% 127.76/16.72 implies(implies(implies(implies(not(X), not(truth)), X), implies(implies(X, not(truth)), implies(implies(not(X), not(truth)), not(truth)))), not(X))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) }
% 127.76/16.72 implies(truth, not(X))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 not(X)
% 127.76/16.72
% 127.76/16.72 Lemma 19: big_hat(X, truth) = X.
% 127.76/16.72 Proof:
% 127.76/16.72 big_hat(X, truth)
% 127.76/16.72 = { by lemma 15 R->L }
% 127.76/16.72 not(not(X))
% 127.76/16.72 = { by lemma 18 R->L }
% 127.76/16.72 not(implies(X, not(truth)))
% 127.76/16.72 = { by lemma 18 R->L }
% 127.76/16.72 implies(implies(X, not(truth)), not(truth))
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 big_V(X, not(truth))
% 127.76/16.72 = { by lemma 14 }
% 127.76/16.72 X
% 127.76/16.72
% 127.76/16.72 Lemma 20: big_hat(X, X) = X.
% 127.76/16.72 Proof:
% 127.76/16.72 big_hat(X, X)
% 127.76/16.72 = { by lemma 16 R->L }
% 127.76/16.72 big_hat(X, truth)
% 127.76/16.72 = { by lemma 19 }
% 127.76/16.72 X
% 127.76/16.72
% 127.76/16.72 Lemma 21: big_hat(Y, X) = big_hat(X, Y).
% 127.76/16.72 Proof:
% 127.76/16.72 big_hat(Y, X)
% 127.76/16.72 = { by axiom 4 (big_hat_definition) }
% 127.76/16.72 not(big_V(not(Y), not(X)))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 not(big_V(not(X), not(Y)))
% 127.76/16.72 = { by axiom 4 (big_hat_definition) R->L }
% 127.76/16.72 big_hat(X, Y)
% 127.76/16.72
% 127.76/16.72 Lemma 22: big_V(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))) = implies(Y, implies(X, Z)).
% 127.76/16.72 Proof:
% 127.76/16.72 big_V(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 2 (big_V_definition) }
% 127.76/16.72 implies(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(truth, implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) R->L }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(implies(implies(Y, Z), Z), implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Y, Z), implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by lemma 7 R->L }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Z, Y), implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Z, Y), implies(X, Z))), implies(truth, implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) R->L }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Z, Y), implies(X, Z))), implies(implies(implies(Y, big_V(Y, Z)), implies(implies(big_V(Y, Z), implies(X, Z)), implies(Y, implies(X, Z)))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by lemma 8 }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Z, Y), implies(X, Z))), implies(implies(truth, implies(implies(big_V(Y, Z), implies(X, Z)), implies(Y, implies(X, Z)))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Z, Y), implies(X, Z))), implies(implies(implies(big_V(Y, Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by lemma 7 R->L }
% 127.76/16.72 implies(implies(implies(implies(X, implies(Y, Z)), implies(big_V(Z, Y), implies(X, Z))), implies(implies(implies(big_V(Z, Y), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) }
% 127.76/16.72 implies(truth, implies(Y, implies(X, Z)))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) }
% 127.76/16.72 implies(Y, implies(X, Z))
% 127.76/16.72
% 127.76/16.72 Lemma 23: implies(X, implies(Y, Z)) = implies(Y, implies(X, Z)).
% 127.76/16.72 Proof:
% 127.76/16.72 implies(X, implies(Y, Z))
% 127.76/16.72 = { by lemma 22 R->L }
% 127.76/16.72 big_V(implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 127.76/16.72 = { by lemma 7 }
% 127.76/16.72 big_V(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 127.76/16.72 = { by lemma 22 }
% 127.76/16.72 implies(Y, implies(X, Z))
% 127.76/16.72
% 127.76/16.72 Lemma 24: implies(not(X), not(Y)) = implies(Y, X).
% 127.76/16.72 Proof:
% 127.76/16.72 implies(not(X), not(Y))
% 127.76/16.72 = { by axiom 1 (wajsberg_1) R->L }
% 127.76/16.72 implies(truth, implies(not(X), not(Y)))
% 127.76/16.72 = { by axiom 6 (wajsberg_2) R->L }
% 127.76/16.72 implies(implies(implies(Y, X), implies(implies(X, not(truth)), implies(Y, not(truth)))), implies(not(X), not(Y)))
% 127.76/16.72 = { by lemma 18 }
% 127.76/16.72 implies(implies(implies(Y, X), implies(not(X), implies(Y, not(truth)))), implies(not(X), not(Y)))
% 127.76/16.72 = { by lemma 18 }
% 127.76/16.72 implies(implies(implies(Y, X), implies(not(X), not(Y))), implies(not(X), not(Y)))
% 127.76/16.72 = { by axiom 2 (big_V_definition) R->L }
% 127.76/16.72 big_V(implies(Y, X), implies(not(X), not(Y)))
% 127.76/16.72 = { by lemma 17 }
% 127.76/16.73 implies(Y, X)
% 127.76/16.73
% 127.76/16.73 Goal 1 (prove_wajsberg_theorem): implies(big_hat(x, y), z) = implies(implies(x, y), implies(x, z)).
% 127.76/16.73 Proof:
% 127.76/16.73 implies(big_hat(x, y), z)
% 127.76/16.73 = { by lemma 20 R->L }
% 127.76/16.73 implies(big_hat(big_hat(x, y), big_hat(x, y)), z)
% 127.76/16.73 = { by axiom 4 (big_hat_definition) }
% 127.76/16.73 implies(not(big_V(not(big_hat(x, y)), not(big_hat(x, y)))), z)
% 127.76/16.73 = { by lemma 10 }
% 127.76/16.73 implies(not(not(big_hat(x, y))), z)
% 127.76/16.73 = { by lemma 21 }
% 127.76/16.73 implies(not(not(big_hat(y, x))), z)
% 127.76/16.73 = { by axiom 4 (big_hat_definition) }
% 127.76/16.73 implies(not(not(not(big_V(not(y), not(x))))), z)
% 127.76/16.73 = { by lemma 11 }
% 127.76/16.73 implies(not(big_hat(big_V(not(y), not(x)), big_V(not(y), not(x)))), z)
% 127.76/16.73 = { by lemma 16 R->L }
% 127.76/16.73 implies(not(big_hat(big_V(not(y), not(x)), truth)), z)
% 127.76/16.73 = { by lemma 21 R->L }
% 127.76/16.73 implies(not(big_hat(truth, big_V(not(y), not(x)))), z)
% 127.76/16.73 = { by lemma 7 R->L }
% 127.76/16.73 implies(not(big_hat(truth, big_V(not(x), not(y)))), z)
% 127.76/16.73 = { by lemma 21 }
% 127.76/16.73 implies(not(big_hat(big_V(not(x), not(y)), truth)), z)
% 127.76/16.73 = { by lemma 19 }
% 127.76/16.73 implies(not(big_V(not(x), not(y))), z)
% 127.76/16.73 = { by lemma 7 R->L }
% 127.76/16.73 implies(not(big_V(not(y), not(x))), z)
% 127.76/16.73 = { by axiom 2 (big_V_definition) }
% 127.76/16.73 implies(not(implies(implies(not(y), not(x)), not(x))), z)
% 127.76/16.73 = { by lemma 24 }
% 127.76/16.73 implies(not(implies(implies(x, y), not(x))), z)
% 127.76/16.73 = { by lemma 24 R->L }
% 127.76/16.73 implies(not(implies(not(not(x)), not(implies(x, y)))), z)
% 127.76/16.73 = { by lemma 11 }
% 127.76/16.73 implies(not(implies(big_hat(x, x), not(implies(x, y)))), z)
% 127.76/16.73 = { by lemma 20 }
% 127.76/16.73 implies(not(implies(x, not(implies(x, y)))), z)
% 127.76/16.73 = { by lemma 20 R->L }
% 127.76/16.73 implies(not(implies(x, not(implies(x, y)))), big_hat(z, z))
% 127.76/16.73 = { by lemma 11 R->L }
% 127.76/16.73 implies(not(implies(x, not(implies(x, y)))), not(not(z)))
% 127.76/16.73 = { by lemma 24 }
% 127.76/16.73 implies(not(z), implies(x, not(implies(x, y))))
% 127.76/16.73 = { by lemma 23 }
% 127.76/16.73 implies(x, implies(not(z), not(implies(x, y))))
% 127.76/16.73 = { by lemma 24 }
% 127.76/16.73 implies(x, implies(implies(x, y), z))
% 127.76/16.73 = { by lemma 23 R->L }
% 127.76/16.73 implies(implies(x, y), implies(x, z))
% 127.76/16.73 % SZS output end Proof
% 127.76/16.73
% 127.76/16.73 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------