0.00/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n006.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:10:39 EDT 2019
0.12/0.33	% CPUTime    : 
0.78/0.93	% SZS status Unsatisfiable
0.78/0.93	
0.78/0.93	% SZS output start Proof
0.78/0.93	Take the following subset of the input axioms:
0.78/0.93	  fof(prove_wajsberg_mv_4, negated_conjecture, implies(implies(implies(a, b), implies(b, a)), implies(b, a))!=truth).
0.78/0.93	  fof(wajsberg_1, axiom, ![X]: implies(truth, X)=X).
0.78/0.93	  fof(wajsberg_2, axiom, ![X, Y, Z]: truth=implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
0.78/0.93	  fof(wajsberg_3, axiom, ![X, Y]: implies(implies(Y, X), X)=implies(implies(X, Y), Y)).
0.78/0.93	  fof(wajsberg_4, axiom, ![X, Y]: truth=implies(implies(not(X), not(Y)), implies(Y, X))).
0.78/0.93	
0.78/0.93	Now clausify the problem and encode Horn clauses using encoding 3 of
0.78/0.93	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.78/0.93	We repeatedly replace C & s=t => u=v by the two clauses:
0.78/0.94	  fresh(y, y, x1...xn) = u
0.78/0.94	  C => fresh(s, t, x1...xn) = v
0.78/0.94	where fresh is a fresh function symbol and x1..xn are the free
0.78/0.94	variables of u and v.
0.78/0.94	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.78/0.94	input problem has no model of domain size 1).
0.78/0.94	
0.78/0.94	The encoding turns the above axioms into the following unit equations and goals:
0.78/0.94	
0.78/0.94	Axiom 1 (wajsberg_4): truth = implies(implies(not(X), not(Y)), implies(Y, X)).
0.78/0.94	Axiom 2 (wajsberg_2): truth = implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))).
0.78/0.94	Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
0.78/0.94	Axiom 4 (wajsberg_1): implies(truth, X) = X.
0.78/0.94	
0.78/0.94	Lemma 5: implies(X, implies(implies(X, Y), Y)) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(X, implies(implies(X, Y), Y))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(truth, X), implies(implies(X, Y), Y))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 6: implies(X, truth) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(X, truth)
0.78/0.94	= { by lemma 5 }
0.78/0.94	  implies(X, implies(truth, implies(implies(truth, X), X)))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(X, implies(implies(truth, X), X))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(X, implies(implies(X, truth), truth))
0.78/0.94	= { by lemma 5 }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 7: implies(X, implies(Y, X)) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(X, implies(Y, X))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(truth, X), implies(Y, X))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(implies(truth, X), implies(Y, X)))
0.78/0.94	= { by lemma 6 }
0.78/0.94	  implies(implies(Y, truth), implies(implies(truth, X), implies(Y, X)))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 8: implies(implies(implies(X, Y), Z), implies(Y, Z)) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(implies(implies(X, Y), Z), implies(Y, Z))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.78/0.94	= { by lemma 7 }
0.78/0.94	  implies(implies(Y, implies(X, Y)), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 9: implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))
0.78/0.94	= { by lemma 8 }
0.78/0.94	  implies(implies(implies(implies(implies(Z, Y), Y), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  implies(implies(implies(implies(X, implies(Y, Z)), implies(implies(implies(Y, Z), Z), implies(X, Z))), implies(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))))), implies(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(implies(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), implies(implies(X, implies(Y, Z)), implies(implies(implies(Y, Z), Z), implies(X, Z)))), implies(implies(X, implies(Y, Z)), implies(implies(implies(Y, Z), Z), implies(X, Z))))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  implies(implies(implies(implies(implies(implies(implies(Y, Z), Z), implies(X, Z)), implies(Y, implies(X, Z))), implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))), truth), implies(implies(X, implies(Y, Z)), implies(implies(implies(Y, Z), Z), implies(X, Z))))
0.78/0.94	= { by lemma 6 }
0.78/0.94	  implies(truth, implies(implies(X, implies(Y, Z)), implies(implies(implies(Y, Z), Z), implies(X, Z))))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  implies(truth, truth)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 10: implies(Y, implies(X, Z)) = implies(X, implies(Y, Z)).
0.78/0.94	Proof:
0.78/0.94	  implies(Y, implies(X, Z))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(Y, implies(X, Z)))
0.78/0.94	= { by lemma 9 }
0.78/0.94	  implies(implies(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))), implies(Y, implies(X, Z)))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(implies(Y, implies(X, Z)), implies(X, implies(Y, Z))), implies(X, implies(Y, Z)))
0.78/0.94	= { by lemma 9 }
0.78/0.94	  implies(truth, implies(X, implies(Y, Z)))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(X, implies(Y, Z))
0.78/0.94	
0.78/0.94	Lemma 11: implies(implies(implies(X, Y), Y), X) = implies(Y, X).
0.78/0.94	Proof:
0.78/0.94	  implies(implies(implies(X, Y), Y), X)
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(implies(Y, X), X), X)
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(X, implies(Y, X)), implies(Y, X))
0.78/0.94	= { by lemma 7 }
0.78/0.94	  implies(truth, implies(Y, X))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(Y, X)
0.78/0.94	
0.78/0.94	Lemma 12: implies(implies(not(X), not(truth)), X) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(implies(not(X), not(truth)), X)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(not(X), not(truth)), implies(truth, X))
0.78/0.94	= { by axiom 1 (wajsberg_4) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 13: implies(not(not(X)), X) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(not(not(X)), X)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(truth, not(not(X))), X)
0.78/0.94	= { by lemma 8 }
0.78/0.94	  implies(implies(implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth)))), not(not(X))), X)
0.78/0.94	= { by lemma 12 }
0.78/0.94	  implies(implies(implies(truth, implies(not(truth), not(not(truth)))), not(not(X))), X)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(implies(not(truth), not(not(truth))), not(not(X))), X)
0.78/0.94	= { by lemma 11 }
0.78/0.94	  implies(implies(implies(implies(implies(not(not(truth)), not(truth)), not(truth)), not(not(truth))), not(not(X))), X)
0.78/0.94	= { by lemma 12 }
0.78/0.94	  implies(implies(implies(truth, not(not(truth))), not(not(X))), X)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(not(not(truth)), not(not(X))), X)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(implies(not(not(truth)), not(not(X))), X))
0.78/0.94	= { by axiom 1 (wajsberg_4) }
0.78/0.94	  implies(implies(implies(not(not(truth)), not(not(X))), implies(not(X), not(truth))), implies(implies(not(not(truth)), not(not(X))), X))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(implies(not(not(truth)), not(not(X))), implies(not(X), not(truth))), implies(truth, implies(implies(not(not(truth)), not(not(X))), X)))
0.78/0.94	= { by lemma 12 }
0.78/0.94	  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)))
0.78/0.94	= { by axiom 2 (wajsberg_2) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 14: not(not(X)) = X.
0.78/0.94	Proof:
0.78/0.94	  not(not(X))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, not(not(X)))
0.78/0.94	= { by axiom 1 (wajsberg_4) }
0.78/0.94	  implies(implies(implies(not(not(not(X))), not(X)), implies(X, not(not(X)))), not(not(X)))
0.78/0.94	= { by lemma 13 }
0.78/0.94	  implies(implies(truth, implies(X, not(not(X)))), not(not(X)))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(implies(X, not(not(X))), not(not(X)))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(not(not(X)), X), X)
0.78/0.94	= { by lemma 13 }
0.78/0.94	  implies(truth, X)
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  X
0.78/0.94	
0.78/0.94	Lemma 15: implies(implies(X, not(Y)), implies(Y, not(X))) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(implies(X, not(Y)), implies(Y, not(X)))
0.78/0.94	= { by lemma 14 }
0.78/0.94	  implies(implies(not(not(X)), not(Y)), implies(Y, not(X)))
0.78/0.94	= { by axiom 1 (wajsberg_4) }
0.78/0.94	  truth
0.78/0.94	
0.78/0.94	Lemma 16: implies(X, not(Y)) = implies(Y, not(X)).
0.78/0.94	Proof:
0.78/0.94	  implies(X, not(Y))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(truth, implies(X, not(Y)))
0.78/0.94	= { by lemma 15 }
0.78/0.94	  implies(implies(implies(Y, not(X)), implies(X, not(Y))), implies(X, not(Y)))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(implies(implies(X, not(Y)), implies(Y, not(X))), implies(Y, not(X)))
0.78/0.94	= { by lemma 15 }
0.78/0.94	  implies(truth, implies(Y, not(X)))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(Y, not(X))
0.78/0.94	
0.78/0.94	Lemma 17: implies(not(X), not(Y)) = implies(Y, X).
0.78/0.94	Proof:
0.78/0.94	  implies(not(X), not(Y))
0.78/0.94	= { by lemma 16 }
0.78/0.94	  implies(Y, not(not(X)))
0.78/0.94	= { by lemma 14 }
0.78/0.94	  implies(Y, X)
0.78/0.94	
0.78/0.94	Lemma 18: implies(not(X), implies(Y, not(Z))) = implies(Y, implies(Z, X)).
0.78/0.94	Proof:
0.78/0.94	  implies(not(X), implies(Y, not(Z)))
0.78/0.94	= { by lemma 10 }
0.78/0.94	  implies(Y, implies(not(X), not(Z)))
0.78/0.94	= { by lemma 17 }
0.78/0.94	  implies(Y, implies(Z, X))
0.78/0.94	
0.78/0.94	Goal 1 (prove_wajsberg_mv_4): implies(implies(implies(a, b), implies(b, a)), implies(b, a)) = truth.
0.78/0.94	Proof:
0.78/0.94	  implies(implies(implies(a, b), implies(b, a)), implies(b, a))
0.78/0.94	= { by lemma 10 }
0.78/0.94	  implies(b, implies(implies(implies(a, b), implies(b, a)), a))
0.78/0.94	= { by lemma 11 }
0.78/0.94	  implies(b, implies(implies(implies(a, b), implies(implies(implies(a, b), b), a)), a))
0.78/0.94	= { by lemma 18 }
0.78/0.94	  implies(b, implies(implies(not(a), implies(implies(a, b), not(implies(implies(a, b), b)))), a))
0.78/0.94	= { by lemma 16 }
0.78/0.94	  implies(b, implies(implies(not(a), implies(implies(implies(a, b), b), not(implies(a, b)))), a))
0.78/0.94	= { by lemma 17 }
0.78/0.94	  implies(b, implies(implies(not(a), implies(implies(not(b), not(implies(a, b))), not(implies(a, b)))), a))
0.78/0.94	= { by axiom 3 (wajsberg_3) }
0.78/0.94	  implies(b, implies(implies(not(a), implies(implies(not(implies(a, b)), not(b)), not(b))), a))
0.78/0.94	= { by lemma 17 }
0.78/0.94	  implies(b, implies(implies(not(a), implies(implies(b, implies(a, b)), not(b))), a))
0.78/0.94	= { by lemma 16 }
0.78/0.94	  implies(b, implies(implies(not(a), implies(b, not(implies(b, implies(a, b))))), a))
0.78/0.94	= { by lemma 18 }
0.78/0.94	  implies(b, implies(implies(b, implies(implies(b, implies(a, b)), a)), a))
0.78/0.94	= { by lemma 7 }
0.78/0.94	  implies(b, implies(implies(b, implies(truth, a)), a))
0.78/0.94	= { by axiom 4 (wajsberg_1) }
0.78/0.94	  implies(b, implies(implies(b, a), a))
0.78/0.94	= { by lemma 5 }
0.78/0.94	  truth
0.78/0.94	% SZS output end Proof
0.78/0.94	
0.78/0.94	RESULT: Unsatisfiable (the axioms are contradictory).
0.79/0.94	EOF