0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.35	% Computer   : n003.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   : 180
0.13/0.35	% DateTime   : Thu Aug 29 11:45:47 EDT 2019
0.13/0.35	% CPUTime    : 
0.20/0.41	% SZS status Unsatisfiable
0.20/0.41	
0.20/0.41	% SZS output start Proof
0.20/0.41	Take the following subset of the input axioms:
0.20/0.41	  fof(and_definition, axiom, ![X, Y]: and(X, Y)=not(or(not(X), not(Y)))).
0.20/0.41	  fof(and_star_commutativity, axiom, ![X, Y]: and_star(X, Y)=and_star(Y, X)).
0.20/0.41	  fof(and_star_definition, axiom, ![X, Y]: not(or(not(X), not(Y)))=and_star(X, Y)).
0.20/0.41	  fof(false_definition, axiom, not(truth)=falsehood).
0.20/0.41	  fof(or_commutativity, axiom, ![X, Y]: or(X, Y)=or(Y, X)).
0.20/0.41	  fof(or_definition, axiom, ![X, Y]: or(X, Y)=implies(not(X), Y)).
0.20/0.41	  fof(prove_alternative_wajsberg_axiom, negated_conjecture, and_star(xor(and_star(xor(truth, y), x), truth), x)!=and_star(xor(and_star(xor(truth, x), y), truth), y)).
0.20/0.41	  fof(wajsberg_1, axiom, ![X]: implies(truth, X)=X).
0.20/0.41	  fof(wajsberg_2, axiom, ![X, Y, Z]: truth=implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))).
0.20/0.41	  fof(wajsberg_3, axiom, ![X, Y]: implies(implies(Y, X), X)=implies(implies(X, Y), Y)).
0.20/0.41	  fof(wajsberg_4, axiom, ![X, Y]: truth=implies(implies(not(X), not(Y)), implies(Y, X))).
0.20/0.41	  fof(xor_commutativity, axiom, ![X, Y]: xor(X, Y)=xor(Y, X)).
0.20/0.41	  fof(xor_definition, axiom, ![X, Y]: or(and(X, not(Y)), and(not(X), Y))=xor(X, Y)).
0.20/0.41	
0.20/0.41	Now clausify the problem and encode Horn clauses using encoding 3 of
0.20/0.41	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.20/0.41	We repeatedly replace C & s=t => u=v by the two clauses:
0.20/0.41	  fresh(y, y, x1...xn) = u
0.20/0.41	  C => fresh(s, t, x1...xn) = v
0.20/0.41	where fresh is a fresh function symbol and x1..xn are the free
0.20/0.41	variables of u and v.
0.20/0.41	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.20/0.41	input problem has no model of domain size 1).
0.20/0.41	
0.20/0.41	The encoding turns the above axioms into the following unit equations and goals:
0.20/0.41	
0.20/0.41	Axiom 1 (wajsberg_4): truth = implies(implies(not(X), not(Y)), implies(Y, X)).
0.20/0.41	Axiom 2 (wajsberg_2): truth = implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))).
0.20/0.41	Axiom 3 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
0.20/0.41	Axiom 4 (wajsberg_1): implies(truth, X) = X.
0.20/0.41	Axiom 5 (and_definition): and(X, Y) = not(or(not(X), not(Y))).
0.20/0.41	Axiom 6 (or_definition): or(X, Y) = implies(not(X), Y).
0.20/0.41	Axiom 7 (or_commutativity): or(X, Y) = or(Y, X).
0.20/0.41	Axiom 8 (xor_commutativity): xor(X, Y) = xor(Y, X).
0.20/0.41	Axiom 9 (xor_definition): or(and(X, not(Y)), and(not(X), Y)) = xor(X, Y).
0.20/0.41	Axiom 10 (and_star_definition): not(or(not(X), not(Y))) = and_star(X, Y).
0.20/0.41	Axiom 11 (false_definition): not(truth) = falsehood.
0.20/0.42	Axiom 12 (and_star_commutativity): and_star(X, Y) = and_star(Y, X).
0.20/0.42	
0.20/0.42	Lemma 13: and(X, Y) = and_star(X, Y).
0.20/0.42	Proof:
0.20/0.42	  and(X, Y)
0.20/0.42	= { by axiom 5 (and_definition) }
0.20/0.42	  not(or(not(X), not(Y)))
0.20/0.42	= { by axiom 10 (and_star_definition) }
0.20/0.42	  and_star(X, Y)
0.20/0.42	
0.20/0.42	Lemma 14: implies(X, implies(implies(X, Y), Y)) = truth.
0.20/0.42	Proof:
0.20/0.42	  implies(X, implies(implies(X, Y), Y))
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(implies(truth, X), implies(implies(X, Y), Y))
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
0.20/0.42	= { by axiom 2 (wajsberg_2) }
0.20/0.42	  truth
0.20/0.42	
0.20/0.42	Lemma 15: implies(X, X) = truth.
0.20/0.42	Proof:
0.20/0.42	  implies(X, X)
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(implies(truth, X), X)
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(truth, implies(implies(truth, X), X))
0.20/0.42	= { by lemma 14 }
0.20/0.42	  truth
0.20/0.42	
0.20/0.42	Lemma 16: or(falsehood, X) = X.
0.20/0.42	Proof:
0.20/0.42	  or(falsehood, X)
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  or(X, falsehood)
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  or(falsehood, X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(not(falsehood), X)
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(implies(truth, not(falsehood)), X)
0.20/0.42	= { by lemma 15 }
0.20/0.42	  implies(implies(implies(not(falsehood), not(falsehood)), not(falsehood)), X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(implies(or(falsehood, not(falsehood)), not(falsehood)), X)
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  implies(implies(or(not(falsehood), falsehood), not(falsehood)), X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(implies(implies(not(not(falsehood)), falsehood), not(falsehood)), X)
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  implies(implies(implies(not(not(falsehood)), not(truth)), not(falsehood)), X)
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(implies(implies(not(not(falsehood)), not(truth)), implies(truth, not(falsehood))), X)
0.20/0.42	= { by axiom 1 (wajsberg_4) }
0.20/0.42	  implies(truth, X)
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  X
0.20/0.42	
0.20/0.42	Lemma 17: implies(implies(X, not(Y)), not(Y)) = implies(or(X, Y), X).
0.20/0.42	Proof:
0.20/0.42	  implies(implies(X, not(Y)), not(Y))
0.20/0.42	= { by axiom 3 (wajsberg_3) }
0.20/0.42	  implies(implies(not(Y), X), X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(or(Y, X), X)
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  implies(or(X, Y), X)
0.20/0.42	
0.20/0.42	Lemma 18: or(X, truth) = implies(falsehood, X).
0.20/0.42	Proof:
0.20/0.42	  or(X, truth)
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  or(truth, X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(not(truth), X)
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  implies(falsehood, X)
0.20/0.42	
0.20/0.42	Lemma 19: implies(X, truth) = truth.
0.20/0.42	Proof:
0.20/0.42	  implies(X, truth)
0.20/0.42	= { by lemma 15 }
0.20/0.42	  implies(X, implies(X, X))
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  implies(X, implies(implies(truth, X), X))
0.20/0.42	= { by axiom 3 (wajsberg_3) }
0.20/0.42	  implies(X, implies(implies(X, truth), truth))
0.20/0.42	= { by lemma 14 }
0.20/0.42	  truth
0.20/0.42	
0.20/0.42	Lemma 20: implies(falsehood, X) = truth.
0.20/0.42	Proof:
0.20/0.42	  implies(falsehood, X)
0.20/0.42	= { by lemma 18 }
0.20/0.42	  or(X, truth)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(not(X), truth)
0.20/0.42	= { by lemma 19 }
0.20/0.42	  truth
0.20/0.42	
0.20/0.42	Lemma 21: implies(X, falsehood) = not(X).
0.20/0.42	Proof:
0.20/0.42	  implies(X, falsehood)
0.20/0.42	= { by lemma 16 }
0.20/0.42	  implies(or(falsehood, X), falsehood)
0.20/0.42	= { by lemma 17 }
0.20/0.42	  implies(implies(falsehood, not(X)), not(X))
0.20/0.42	= { by lemma 20 }
0.20/0.42	  implies(truth, not(X))
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  not(X)
0.20/0.42	
0.20/0.42	Lemma 22: not(not(X)) = X.
0.20/0.42	Proof:
0.20/0.42	  not(not(X))
0.20/0.42	= { by lemma 21 }
0.20/0.42	  not(implies(X, falsehood))
0.20/0.42	= { by lemma 21 }
0.20/0.42	  implies(implies(X, falsehood), falsehood)
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  implies(implies(X, not(truth)), falsehood)
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  implies(implies(X, not(truth)), not(truth))
0.20/0.42	= { by axiom 3 (wajsberg_3) }
0.20/0.42	  implies(implies(not(truth), X), X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(or(truth, X), X)
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  implies(or(X, truth), X)
0.20/0.42	= { by lemma 18 }
0.20/0.42	  implies(implies(falsehood, X), X)
0.20/0.42	= { by lemma 20 }
0.20/0.42	  implies(truth, X)
0.20/0.42	= { by axiom 4 (wajsberg_1) }
0.20/0.42	  X
0.20/0.42	
0.20/0.42	Lemma 23: or(X, not(Y)) = implies(Y, X).
0.20/0.42	Proof:
0.20/0.42	  or(X, not(Y))
0.20/0.42	= { by axiom 7 (or_commutativity) }
0.20/0.42	  or(not(Y), X)
0.20/0.42	= { by axiom 6 (or_definition) }
0.20/0.42	  implies(not(not(Y)), X)
0.20/0.42	= { by lemma 22 }
0.20/0.42	  implies(Y, X)
0.20/0.42	
0.20/0.42	Lemma 24: and_star(X, not(Y)) = not(implies(X, Y)).
0.20/0.42	Proof:
0.20/0.42	  and_star(X, not(Y))
0.20/0.42	= { by axiom 12 (and_star_commutativity) }
0.20/0.42	  and_star(not(Y), X)
0.20/0.42	= { by lemma 13 }
0.20/0.42	  and(not(Y), X)
0.20/0.42	= { by axiom 5 (and_definition) }
0.20/0.42	  not(or(not(not(Y)), not(X)))
0.20/0.42	= { by lemma 22 }
0.20/0.42	  not(or(Y, not(X)))
0.20/0.42	= { by lemma 23 }
0.20/0.42	  not(implies(X, Y))
0.20/0.42	
0.20/0.42	Lemma 25: xor(truth, X) = not(X).
0.20/0.42	Proof:
0.20/0.42	  xor(truth, X)
0.20/0.42	= { by axiom 8 (xor_commutativity) }
0.20/0.42	  xor(X, truth)
0.20/0.42	= { by axiom 9 (xor_definition) }
0.20/0.42	  or(and(X, not(truth)), and(not(X), truth))
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  or(and(X, falsehood), and(not(X), truth))
0.20/0.42	= { by lemma 13 }
0.20/0.42	  or(and_star(X, falsehood), and(not(X), truth))
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  or(and_star(X, not(truth)), and(not(X), truth))
0.20/0.42	= { by lemma 24 }
0.20/0.42	  or(not(implies(X, truth)), and(not(X), truth))
0.20/0.42	= { by lemma 19 }
0.20/0.42	  or(not(truth), and(not(X), truth))
0.20/0.42	= { by axiom 11 (false_definition) }
0.20/0.42	  or(falsehood, and(not(X), truth))
0.20/0.42	= { by lemma 13 }
0.20/0.42	  or(falsehood, and_star(not(X), truth))
0.20/0.42	= { by axiom 12 (and_star_commutativity) }
0.20/0.42	  or(falsehood, and_star(truth, not(X)))
0.20/0.42	= { by lemma 24 }
0.20/0.43	  or(falsehood, not(implies(truth, X)))
0.20/0.43	= { by lemma 16 }
0.20/0.43	  not(implies(truth, X))
0.20/0.43	= { by axiom 4 (wajsberg_1) }
0.20/0.43	  not(X)
0.20/0.43	
0.20/0.43	Lemma 26: or(not(X), Y) = implies(X, Y).
0.20/0.43	Proof:
0.20/0.43	  or(not(X), Y)
0.20/0.43	= { by axiom 7 (or_commutativity) }
0.20/0.43	  or(Y, not(X))
0.20/0.43	= { by lemma 23 }
0.20/0.43	  implies(X, Y)
0.20/0.43	
0.20/0.43	Lemma 27: not(and_star(X, Y)) = implies(X, not(Y)).
0.20/0.43	Proof:
0.20/0.43	  not(and_star(X, Y))
0.20/0.43	= { by lemma 13 }
0.20/0.43	  not(and(X, Y))
0.20/0.43	= { by axiom 5 (and_definition) }
0.20/0.43	  not(not(or(not(X), not(Y))))
0.20/0.43	= { by lemma 22 }
0.20/0.43	  or(not(X), not(Y))
0.20/0.43	= { by lemma 26 }
0.20/0.43	  implies(X, not(Y))
0.20/0.43	
0.20/0.43	Lemma 28: implies(X, not(Y)) = implies(Y, not(X)).
0.20/0.43	Proof:
0.20/0.43	  implies(X, not(Y))
0.20/0.43	= { by lemma 27 }
0.20/0.43	  not(and_star(X, Y))
0.20/0.43	= { by axiom 12 (and_star_commutativity) }
0.20/0.43	  not(and_star(Y, X))
0.20/0.43	= { by lemma 27 }
0.20/0.43	  implies(Y, not(X))
0.20/0.43	
0.20/0.43	Lemma 29: not(implies(X, not(Y))) = and_star(X, Y).
0.20/0.43	Proof:
0.20/0.43	  not(implies(X, not(Y)))
0.20/0.43	= { by lemma 28 }
0.20/0.43	  not(implies(Y, not(X)))
0.20/0.43	= { by lemma 23 }
0.20/0.43	  not(or(not(X), not(Y)))
0.20/0.43	= { by axiom 5 (and_definition) }
0.20/0.43	  and(X, Y)
0.20/0.43	= { by lemma 13 }
0.20/0.43	  and_star(X, Y)
0.20/0.43	
0.20/0.43	Goal 1 (prove_alternative_wajsberg_axiom): and_star(xor(and_star(xor(truth, y), x), truth), x) = and_star(xor(and_star(xor(truth, x), y), truth), y).
0.20/0.43	Proof:
0.20/0.43	  and_star(xor(and_star(xor(truth, y), x), truth), x)
0.20/0.43	= { by axiom 8 (xor_commutativity) }
0.20/0.43	  and_star(xor(truth, and_star(xor(truth, y), x)), x)
0.20/0.43	= { by axiom 12 (and_star_commutativity) }
0.20/0.43	  and_star(x, xor(truth, and_star(xor(truth, y), x)))
0.20/0.43	= { by axiom 12 (and_star_commutativity) }
0.20/0.43	  and_star(x, xor(truth, and_star(x, xor(truth, y))))
0.20/0.43	= { by lemma 25 }
0.20/0.43	  and_star(x, not(and_star(x, xor(truth, y))))
0.20/0.43	= { by lemma 25 }
0.20/0.43	  and_star(x, not(and_star(x, not(y))))
0.20/0.43	= { by lemma 24 }
0.20/0.43	  not(implies(x, and_star(x, not(y))))
0.20/0.43	= { by lemma 24 }
0.20/0.43	  not(implies(x, not(implies(x, y))))
0.20/0.43	= { by lemma 28 }
0.20/0.43	  not(implies(implies(x, y), not(x)))
0.20/0.43	= { by lemma 26 }
0.20/0.43	  not(implies(or(not(x), y), not(x)))
0.20/0.43	= { by lemma 17 }
0.20/0.43	  not(implies(implies(not(x), not(y)), not(y)))
0.20/0.43	= { by axiom 6 (or_definition) }
0.20/0.43	  not(implies(or(x, not(y)), not(y)))
0.20/0.43	= { by lemma 23 }
0.20/0.43	  not(implies(implies(y, x), not(y)))
0.20/0.43	= { by lemma 28 }
0.20/0.43	  not(implies(y, not(implies(y, x))))
0.20/0.43	= { by lemma 24 }
0.20/0.43	  not(implies(y, and_star(y, not(x))))
0.20/0.43	= { by lemma 24 }
0.20/0.43	  and_star(y, not(and_star(y, not(x))))
0.20/0.43	= { by lemma 25 }
0.20/0.43	  and_star(y, not(and_star(y, xor(truth, x))))
0.20/0.43	= { by lemma 25 }
0.20/0.43	  and_star(y, xor(truth, and_star(y, xor(truth, x))))
0.20/0.43	= { by axiom 12 (and_star_commutativity) }
0.20/0.43	  and_star(y, xor(truth, and_star(xor(truth, x), y)))
0.20/0.43	= { by axiom 12 (and_star_commutativity) }
0.20/0.43	  and_star(xor(truth, and_star(xor(truth, x), y)), y)
0.20/0.43	= { by axiom 8 (xor_commutativity) }
0.20/0.43	  and_star(xor(and_star(xor(truth, x), y), truth), y)
0.20/0.43	% SZS output end Proof
0.20/0.43	
0.20/0.43	RESULT: Unsatisfiable (the axioms are contradictory).
0.20/0.43	EOF