0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n013.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 09:11:46 EDT 2019
0.12/0.34	% CPUTime    : 
0.19/0.38	% SZS status Unsatisfiable
0.19/0.38	
0.19/0.38	% SZS output start Proof
0.19/0.38	Take the following subset of the input axioms:
0.19/0.38	  fof(prove_wajsberg_lemma, negated_conjecture, not(x)!=implies(x, not(truth))).
0.19/0.38	  fof(wajsberg_1, axiom, ![X]: implies(truth, X)=X).
0.19/0.38	  fof(wajsberg_2, axiom, ![X, Y, Z]: truth=implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
0.19/0.38	  fof(wajsberg_3, axiom, ![X, Y]: implies(implies(Y, X), X)=implies(implies(X, Y), Y)).
0.19/0.38	  fof(wajsberg_4, axiom, ![X, Y]: truth=implies(implies(not(X), not(Y)), implies(Y, X))).
0.19/0.38	
0.19/0.38	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.38	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.38	  fresh(y, y, x1...xn) = u
0.19/0.38	  C => fresh(s, t, x1...xn) = v
0.19/0.38	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.38	variables of u and v.
0.19/0.38	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.38	input problem has no model of domain size 1).
0.19/0.38	
0.19/0.38	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.38	
0.19/0.38	Axiom 1 (wajsberg_4): truth = implies(implies(not(X), not(Y)), implies(Y, X)).
0.19/0.38	Axiom 2 (wajsberg_2): truth = implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))).
0.19/0.38	Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
0.19/0.38	Axiom 4 (wajsberg_1): implies(truth, X) = X.
0.19/0.38	
0.19/0.38	Lemma 5: implies(X, implies(implies(X, Y), Y)) = truth.
0.19/0.38	Proof:
0.19/0.38	  implies(X, implies(implies(X, Y), Y))
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(implies(truth, X), implies(implies(X, Y), Y))
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
0.19/0.38	= { by axiom 2 (wajsberg_2) }
0.19/0.38	  truth
0.19/0.38	
0.19/0.38	Lemma 6: implies(implies(implies(X, Y), Z), implies(Y, Z)) = truth.
0.19/0.38	Proof:
0.19/0.38	  implies(implies(implies(X, Y), Z), implies(Y, Z))
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(truth, implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by axiom 2 (wajsberg_2) }
0.19/0.38	  implies(implies(implies(X, truth), implies(implies(truth, Y), implies(X, Y))), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by lemma 5 }
0.19/0.38	  implies(implies(implies(X, implies(truth, implies(implies(truth, X), X))), implies(implies(truth, Y), implies(X, Y))), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(implies(implies(X, implies(implies(truth, X), X)), implies(implies(truth, Y), implies(X, Y))), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by axiom 3 (wajsberg_3) }
0.19/0.38	  implies(implies(implies(X, implies(implies(X, truth), truth)), implies(implies(truth, Y), implies(X, Y))), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by lemma 5 }
0.19/0.38	  implies(implies(truth, implies(implies(truth, Y), implies(X, Y))), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(implies(implies(truth, Y), implies(X, Y)), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(implies(Y, implies(X, Y)), implies(implies(implies(X, Y), Z), implies(Y, Z)))
0.19/0.38	= { by axiom 2 (wajsberg_2) }
0.19/0.38	  truth
0.19/0.38	
0.19/0.38	Lemma 7: implies(implies(not(X), not(truth)), X) = truth.
0.19/0.38	Proof:
0.19/0.38	  implies(implies(not(X), not(truth)), X)
0.19/0.38	= { by axiom 4 (wajsberg_1) }
0.19/0.38	  implies(implies(not(X), not(truth)), implies(truth, X))
0.19/0.38	= { by axiom 1 (wajsberg_4) }
0.19/0.39	  truth
0.19/0.39	
0.19/0.39	Goal 1 (prove_wajsberg_lemma): not(x) = implies(x, not(truth)).
0.19/0.39	Proof:
0.19/0.39	  not(x)
0.19/0.39	= { by axiom 4 (wajsberg_1) }
0.19/0.39	  implies(truth, not(x))
0.19/0.39	= { by axiom 2 (wajsberg_2) }
0.19/0.39	  implies(implies(implies(implies(not(x), not(truth)), x), implies(implies(x, not(truth)), implies(implies(not(x), not(truth)), not(truth)))), not(x))
0.19/0.39	= { by lemma 7 }
0.19/0.39	  implies(implies(truth, implies(implies(x, not(truth)), implies(implies(not(x), not(truth)), not(truth)))), not(x))
0.19/0.39	= { by axiom 4 (wajsberg_1) }
0.19/0.39	  implies(implies(implies(x, not(truth)), implies(implies(not(x), not(truth)), not(truth))), not(x))
0.19/0.39	= { by axiom 3 (wajsberg_3) }
0.19/0.39	  implies(implies(implies(x, not(truth)), implies(implies(not(truth), not(x)), not(x))), not(x))
0.19/0.39	= { by axiom 4 (wajsberg_1) }
0.19/0.39	  implies(implies(implies(x, not(truth)), implies(implies(truth, implies(not(truth), not(x))), not(x))), not(x))
0.19/0.39	= { by lemma 7 }
0.19/0.39	  implies(implies(implies(x, not(truth)), implies(implies(implies(implies(not(not(x)), not(truth)), not(x)), implies(not(truth), not(x))), not(x))), not(x))
0.19/0.39	= { by lemma 6 }
0.19/0.39	  implies(implies(implies(x, not(truth)), implies(truth, not(x))), not(x))
0.19/0.39	= { by axiom 4 (wajsberg_1) }
0.19/0.39	  implies(implies(implies(x, not(truth)), not(x)), not(x))
0.19/0.39	= { by axiom 3 (wajsberg_3) }
0.19/0.39	  implies(implies(not(x), implies(x, not(truth))), implies(x, not(truth)))
0.19/0.39	= { by axiom 4 (wajsberg_1) }
0.19/0.39	  implies(implies(truth, implies(not(x), implies(x, not(truth)))), implies(x, not(truth)))
0.19/0.39	= { by axiom 1 (wajsberg_4) }
0.19/0.39	  implies(implies(implies(implies(not(not(truth)), not(x)), implies(x, not(truth))), implies(not(x), implies(x, not(truth)))), implies(x, not(truth)))
0.19/0.39	= { by lemma 6 }
0.19/0.39	  implies(truth, implies(x, not(truth)))
0.19/0.39	= { by axiom 4 (wajsberg_1) }
0.19/0.39	  implies(x, not(truth))
0.19/0.39	% SZS output end Proof
0.19/0.39	
0.19/0.39	RESULT: Unsatisfiable (the axioms are contradictory).
0.19/0.39	EOF