0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n008.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   : 180
0.13/0.34	% DateTime   : Thu Aug 29 10:12:53 EDT 2019
0.13/0.34	% CPUTime    : 
0.13/0.36	% SZS status Unsatisfiable
0.13/0.36	
0.13/0.37	% SZS output start Proof
0.13/0.37	Take the following subset of the input axioms:
0.13/0.37	  fof(prove_wajsberg_lemma, negated_conjecture, not(not(x))!=x).
0.13/0.37	  fof(wajsberg_1, axiom, ![X]: implies(truth, X)=X).
0.13/0.37	  fof(wajsberg_2, axiom, ![X, Y, Z]: truth=implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
0.13/0.37	  fof(wajsberg_3, axiom, ![X, Y]: implies(implies(Y, X), X)=implies(implies(X, Y), Y)).
0.13/0.37	  fof(wajsberg_4, axiom, ![X, Y]: truth=implies(implies(not(X), not(Y)), implies(Y, X))).
0.13/0.37	
0.13/0.37	Now clausify the problem and encode Horn clauses using encoding 3 of
0.13/0.37	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.13/0.37	We repeatedly replace C & s=t => u=v by the two clauses:
0.13/0.37	  fresh(y, y, x1...xn) = u
0.13/0.37	  C => fresh(s, t, x1...xn) = v
0.13/0.37	where fresh is a fresh function symbol and x1..xn are the free
0.13/0.37	variables of u and v.
0.13/0.37	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.13/0.37	input problem has no model of domain size 1).
0.13/0.37	
0.13/0.37	The encoding turns the above axioms into the following unit equations and goals:
0.13/0.37	
0.13/0.37	Axiom 1 (wajsberg_4): truth = implies(implies(not(X), not(Y)), implies(Y, X)).
0.13/0.37	Axiom 2 (wajsberg_2): truth = implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))).
0.13/0.37	Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
0.13/0.37	Axiom 4 (wajsberg_1): implies(truth, X) = X.
0.13/0.37	
0.13/0.37	Lemma 5: implies(X, implies(implies(X, Y), Y)) = truth.
0.13/0.37	Proof:
0.13/0.37	  implies(X, implies(implies(X, Y), Y))
0.13/0.37	= { by axiom 4 (wajsberg_1) }
0.13/0.37	  implies(implies(truth, X), implies(implies(X, Y), Y))
0.13/0.37	= { by axiom 4 (wajsberg_1) }
0.13/0.37	  implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
0.13/0.37	= { by axiom 2 (wajsberg_2) }
0.13/0.37	  truth
0.13/0.37	
0.13/0.37	Lemma 6: implies(X, implies(Y, X)) = truth.
0.13/0.37	Proof:
0.13/0.37	  implies(X, implies(Y, X))
0.13/0.37	= { by axiom 4 (wajsberg_1) }
0.13/0.37	  implies(implies(truth, X), implies(Y, X))
0.13/0.37	= { by axiom 4 (wajsberg_1) }
0.13/0.37	  implies(truth, implies(implies(truth, X), implies(Y, X)))
0.13/0.37	= { by lemma 5 }
0.13/0.37	  implies(implies(Y, implies(implies(Y, truth), truth)), implies(implies(truth, X), implies(Y, X)))
0.13/0.37	= { by axiom 3 (wajsberg_3) }
0.13/0.37	  implies(implies(Y, implies(implies(truth, Y), Y)), implies(implies(truth, X), implies(Y, X)))
0.13/0.37	= { by axiom 4 (wajsberg_1) }
0.13/0.37	  implies(implies(Y, implies(truth, implies(implies(truth, Y), Y))), implies(implies(truth, X), implies(Y, X)))
0.13/0.37	= { by lemma 5 }
0.13/0.37	  implies(implies(Y, truth), implies(implies(truth, X), implies(Y, X)))
0.13/0.37	= { by axiom 2 (wajsberg_2) }
0.13/0.37	  truth
0.13/0.37	
0.13/0.37	Lemma 7: implies(implies(not(X), not(truth)), X) = truth.
0.13/0.37	Proof:
0.13/0.37	  implies(implies(not(X), not(truth)), X)
0.13/0.37	= { by axiom 4 (wajsberg_1) }
0.13/0.37	  implies(implies(not(X), not(truth)), implies(truth, X))
0.13/0.37	= { by axiom 1 (wajsberg_4) }
0.21/0.37	  truth
0.21/0.37	
0.21/0.37	Lemma 8: implies(not(not(X)), X) = truth.
0.21/0.37	Proof:
0.21/0.37	  implies(not(not(X)), X)
0.21/0.37	= { by axiom 4 (wajsberg_1) }
0.21/0.37	  implies(implies(truth, not(not(X))), X)
0.21/0.37	= { by axiom 2 (wajsberg_2) }
0.21/0.37	  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(X))), X)
0.21/0.37	= { by lemma 6 }
0.21/0.37	  implies(implies(implies(truth, implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth))))), not(not(X))), X)
0.21/0.37	= { by axiom 4 (wajsberg_1) }
0.21/0.37	  implies(implies(implies(implies(implies(not(not(not(truth))), not(truth)), not(not(truth))), implies(not(truth), not(not(truth)))), not(not(X))), X)
0.21/0.37	= { by lemma 7 }
0.21/0.37	  implies(implies(implies(truth, implies(not(truth), not(not(truth)))), not(not(X))), X)
0.21/0.37	= { by lemma 6 }
0.21/0.37	  implies(implies(implies(implies(not(not(truth)), implies(not(truth), not(not(truth)))), implies(not(truth), not(not(truth)))), not(not(X))), X)
0.21/0.37	= { by axiom 3 (wajsberg_3) }
0.21/0.37	  implies(implies(implies(implies(implies(not(truth), not(not(truth))), not(not(truth))), not(not(truth))), not(not(X))), X)
0.21/0.37	= { by axiom 3 (wajsberg_3) }
0.21/0.37	  implies(implies(implies(implies(implies(not(not(truth)), not(truth)), not(truth)), not(not(truth))), not(not(X))), X)
0.21/0.37	= { by lemma 7 }
0.21/0.37	  implies(implies(implies(truth, not(not(truth))), not(not(X))), X)
0.21/0.37	= { by axiom 4 (wajsberg_1) }
0.21/0.37	  implies(implies(not(not(truth)), not(not(X))), X)
0.21/0.37	= { by axiom 4 (wajsberg_1) }
0.21/0.37	  implies(truth, implies(implies(not(not(truth)), not(not(X))), X))
0.21/0.37	= { by axiom 1 (wajsberg_4) }
0.21/0.37	  implies(implies(implies(not(not(truth)), not(not(X))), implies(not(X), not(truth))), implies(implies(not(not(truth)), not(not(X))), X))
0.21/0.37	= { by axiom 4 (wajsberg_1) }
0.21/0.37	  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)))
0.21/0.37	= { by lemma 7 }
0.21/0.37	  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)))
0.21/0.37	= { by axiom 2 (wajsberg_2) }
0.21/0.38	  truth
0.21/0.38	
0.21/0.38	Goal 1 (prove_wajsberg_lemma): not(not(x)) = x.
0.21/0.38	Proof:
0.21/0.38	  not(not(x))
0.21/0.38	= { by axiom 4 (wajsberg_1) }
0.21/0.38	  implies(truth, not(not(x)))
0.21/0.38	= { by axiom 1 (wajsberg_4) }
0.21/0.38	  implies(implies(implies(not(not(not(x))), not(x)), implies(x, not(not(x)))), not(not(x)))
0.21/0.38	= { by lemma 8 }
0.21/0.38	  implies(implies(truth, implies(x, not(not(x)))), not(not(x)))
0.21/0.38	= { by axiom 4 (wajsberg_1) }
0.21/0.38	  implies(implies(x, not(not(x))), not(not(x)))
0.21/0.38	= { by axiom 3 (wajsberg_3) }
0.21/0.38	  implies(implies(not(not(x)), x), x)
0.21/0.38	= { by lemma 8 }
0.21/0.38	  implies(truth, x)
0.21/0.38	= { by axiom 4 (wajsberg_1) }
0.21/0.38	  x
0.21/0.38	% SZS output end Proof
0.21/0.38	
0.21/0.38	RESULT: Unsatisfiable (the axioms are contradictory).
0.21/0.38	EOF