0.08/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.35	% Computer   : n015.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 10:03:52 EDT 2019
0.13/0.36	% CPUTime    : 
0.21/0.44	% SZS status Unsatisfiable
0.21/0.44	
0.21/0.44	% SZS output start Proof
0.21/0.44	Take the following subset of the input axioms:
0.55/0.70	  fof(associativity, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
0.55/0.70	  fof(commutator, axiom, ![X, Y]: commutator(X, Y)=multiply(X, multiply(Y, multiply(inverse(X), inverse(Y))))).
0.55/0.70	  fof(left_identity, axiom, ![X]: X=multiply(identity, X)).
0.55/0.70	  fof(left_inverse, axiom, ![X]: multiply(inverse(X), X)=identity).
0.55/0.70	  fof(prove_commutator, negated_conjecture, identity!=commutator(commutator(a, b), b)).
0.55/0.70	  fof(right_identity, axiom, ![X]: multiply(X, identity)=X).
0.55/0.70	  fof(right_inverse, axiom, ![X]: identity=multiply(X, inverse(X))).
0.55/0.70	  fof(x_cubed_is_identity, hypothesis, ![X]: multiply(X, multiply(X, X))=identity).
0.55/0.70	
0.55/0.70	Now clausify the problem and encode Horn clauses using encoding 3 of
0.55/0.70	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.55/0.70	We repeatedly replace C & s=t => u=v by the two clauses:
0.55/0.70	  fresh(y, y, x1...xn) = u
0.55/0.70	  C => fresh(s, t, x1...xn) = v
0.55/0.70	where fresh is a fresh function symbol and x1..xn are the free
0.55/0.70	variables of u and v.
0.55/0.70	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.55/0.70	input problem has no model of domain size 1).
0.55/0.70	
0.55/0.70	The encoding turns the above axioms into the following unit equations and goals:
0.55/0.70	
0.55/0.71	Axiom 1 (left_inverse): multiply(inverse(X), X) = identity.
0.55/0.71	Axiom 2 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
0.55/0.71	Axiom 3 (left_identity): X = multiply(identity, X).
0.55/0.71	Axiom 4 (right_inverse): identity = multiply(X, inverse(X)).
0.55/0.71	Axiom 5 (x_cubed_is_identity): multiply(X, multiply(X, X)) = identity.
0.55/0.71	Axiom 6 (commutator): commutator(X, Y) = multiply(X, multiply(Y, multiply(inverse(X), inverse(Y)))).
0.55/0.71	Axiom 7 (right_identity): multiply(X, identity) = X.
0.55/0.71	
0.55/0.71	Lemma 8: multiply(inverse(X), multiply(X, Y)) = Y.
0.55/0.71	Proof:
0.55/0.71	  multiply(inverse(X), multiply(X, Y))
0.55/0.71	= { by axiom 2 (associativity) }
0.55/0.71	  multiply(multiply(inverse(X), X), Y)
0.55/0.71	= { by axiom 1 (left_inverse) }
0.55/0.71	  multiply(identity, Y)
0.55/0.71	= { by axiom 3 (left_identity) }
0.55/0.71	  Y
0.55/0.71	
0.55/0.71	Lemma 9: multiply(X, multiply(Y, inverse(multiply(X, Y)))) = identity.
0.55/0.71	Proof:
0.55/0.71	  multiply(X, multiply(Y, inverse(multiply(X, Y))))
0.55/0.71	= { by axiom 2 (associativity) }
0.55/0.71	  multiply(multiply(X, Y), inverse(multiply(X, Y)))
0.55/0.71	= { by axiom 4 (right_inverse) }
0.55/0.71	  identity
0.55/0.71	
0.55/0.71	Lemma 10: multiply(X, inverse(multiply(Y, X))) = inverse(Y).
0.55/0.71	Proof:
0.55/0.71	  multiply(X, inverse(multiply(Y, X)))
0.55/0.71	= { by lemma 8 }
0.55/0.71	  multiply(inverse(Y), multiply(Y, multiply(X, inverse(multiply(Y, X)))))
0.55/0.71	= { by lemma 9 }
0.55/0.71	  multiply(inverse(Y), identity)
0.55/0.71	= { by axiom 7 (right_identity) }
0.55/0.71	  inverse(Y)
0.55/0.71	
0.55/0.71	Lemma 11: multiply(inverse(Y), inverse(X)) = inverse(multiply(X, Y)).
0.55/0.71	Proof:
0.55/0.71	  multiply(inverse(Y), inverse(X))
0.55/0.71	= { by lemma 10 }
0.55/0.71	  multiply(inverse(Y), multiply(Y, inverse(multiply(X, Y))))
0.55/0.71	= { by lemma 8 }
0.55/0.71	  inverse(multiply(X, Y))
0.55/0.71	
0.55/0.71	Lemma 12: multiply(X, multiply(inverse(X), Y)) = Y.
0.55/0.71	Proof:
0.55/0.71	  multiply(X, multiply(inverse(X), Y))
0.55/0.71	= { by axiom 2 (associativity) }
0.55/0.71	  multiply(multiply(X, inverse(X)), Y)
0.55/0.71	= { by axiom 4 (right_inverse) }
0.55/0.71	  multiply(identity, Y)
0.55/0.71	= { by axiom 3 (left_identity) }
0.55/0.71	  Y
0.55/0.71	
0.55/0.71	Lemma 13: inverse(inverse(X)) = X.
0.55/0.71	Proof:
0.55/0.71	  inverse(inverse(X))
0.55/0.71	= { by lemma 12 }
0.55/0.71	  multiply(X, multiply(inverse(X), inverse(inverse(X))))
0.55/0.71	= { by axiom 4 (right_inverse) }
0.55/0.71	  multiply(X, identity)
0.55/0.71	= { by axiom 7 (right_identity) }
0.55/0.71	  X
0.55/0.71	
0.55/0.71	Lemma 14: multiply(X, X) = inverse(X).
0.55/0.71	Proof:
0.55/0.71	  multiply(X, X)
0.55/0.71	= { by axiom 7 (right_identity) }
0.55/0.71	  multiply(X, multiply(X, identity))
0.55/0.71	= { by axiom 5 (x_cubed_is_identity) }
0.55/0.71	  multiply(X, multiply(X, multiply(inverse(X), multiply(inverse(X), inverse(X)))))
0.55/0.71	= { by lemma 12 }
0.55/0.71	  multiply(X, multiply(inverse(X), inverse(X)))
0.55/0.71	= { by lemma 12 }
0.55/0.71	  inverse(X)
0.55/0.71	
0.55/0.71	Lemma 15: commutator(X, multiply(Y, inverse(X))) = commutator(X, Y).
0.55/0.71	Proof:
0.55/0.71	  commutator(X, multiply(Y, inverse(X)))
0.55/0.71	= { by axiom 6 (commutator) }
0.55/0.71	  multiply(X, multiply(multiply(Y, inverse(X)), multiply(inverse(X), inverse(multiply(Y, inverse(X))))))
0.55/0.71	= { by lemma 10 }
0.55/0.71	  multiply(X, multiply(multiply(Y, inverse(X)), inverse(Y)))
0.55/0.71	= { by axiom 2 (associativity) }
0.55/0.71	  multiply(X, multiply(Y, multiply(inverse(X), inverse(Y))))
0.55/0.71	= { by axiom 6 (commutator) }
0.74/0.92	  commutator(X, Y)
0.74/0.92	
0.74/0.92	Goal 1 (prove_commutator): identity = commutator(commutator(a, b), b).
0.74/0.92	Proof:
0.74/0.92	  identity
0.74/0.92	= { by lemma 9 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(a, b), b))))
0.74/0.92	= { by lemma 12 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(a, multiply(inverse(a), multiply(inverse(inverse(a)), b))), b))))
0.74/0.92	= { by axiom 6 (commutator) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(a, multiply(multiply(inverse(a), multiply(inverse(inverse(a)), b)), multiply(inverse(a), inverse(multiply(inverse(a), multiply(inverse(inverse(a)), b)))))), b))))
0.74/0.92	= { by lemma 11 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(a, multiply(multiply(inverse(a), multiply(inverse(inverse(a)), b)), multiply(inverse(a), multiply(inverse(multiply(inverse(inverse(a)), b)), inverse(inverse(a)))))), b))))
0.74/0.92	= { by lemma 8 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(a, multiply(multiply(inverse(a), multiply(inverse(inverse(a)), b)), multiply(inverse(multiply(inverse(inverse(a)), b)), multiply(multiply(inverse(inverse(a)), b), multiply(inverse(a), multiply(inverse(multiply(inverse(inverse(a)), b)), inverse(inverse(a)))))))), b))))
0.74/0.92	= { by axiom 6 (commutator) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(a, multiply(multiply(inverse(a), multiply(inverse(inverse(a)), b)), multiply(inverse(multiply(inverse(inverse(a)), b)), commutator(multiply(inverse(inverse(a)), b), inverse(a))))), b))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(a, multiply(inverse(a), multiply(multiply(inverse(inverse(a)), b), multiply(inverse(multiply(inverse(inverse(a)), b)), commutator(multiply(inverse(inverse(a)), b), inverse(a)))))), b))))
0.74/0.92	= { by lemma 12 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(inverse(inverse(a)), b), multiply(inverse(multiply(inverse(inverse(a)), b)), commutator(multiply(inverse(inverse(a)), b), inverse(a)))), b))))
0.74/0.92	= { by lemma 12 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(inverse(inverse(a)), b), inverse(a)), b))))
0.74/0.92	= { by lemma 13 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), b))))
0.74/0.92	= { by lemma 8 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), multiply(inverse(a), multiply(a, b))))))
0.74/0.92	= { by lemma 13 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), inverse(inverse(multiply(inverse(a), multiply(a, b))))))))
0.74/0.92	= { by lemma 11 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), inverse(multiply(inverse(multiply(a, b)), inverse(inverse(a))))))))
0.74/0.92	= { by lemma 8 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), inverse(multiply(inverse(multiply(multiply(a, b), inverse(a))), multiply(multiply(multiply(a, b), inverse(a)), multiply(inverse(multiply(a, b)), inverse(inverse(a))))))))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), inverse(multiply(inverse(multiply(multiply(a, b), inverse(a))), multiply(multiply(a, b), multiply(inverse(a), multiply(inverse(multiply(a, b)), inverse(inverse(a)))))))))))
0.74/0.92	= { by axiom 6 (commutator) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(commutator(multiply(a, b), inverse(a)), inverse(multiply(inverse(multiply(multiply(a, b), inverse(a))), commutator(multiply(a, b), inverse(a))))))))
0.74/0.92	= { by lemma 10 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(inverse(inverse(multiply(multiply(a, b), inverse(a)))))))
0.74/0.92	= { by lemma 13 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(a, b), inverse(a)))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(a, multiply(b, inverse(a))))))
0.74/0.92	= { by lemma 13 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(a, inverse(inverse(multiply(b, inverse(a))))))))
0.74/0.92	= { by lemma 14 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(a, inverse(multiply(multiply(b, inverse(a)), multiply(b, inverse(a))))))))
0.74/0.92	= { by lemma 10 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(a, multiply(inverse(a), inverse(multiply(multiply(multiply(b, inverse(a)), multiply(b, inverse(a))), inverse(a))))))))
0.74/0.92	= { by lemma 12 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(inverse(multiply(multiply(multiply(b, inverse(a)), multiply(b, inverse(a))), inverse(a))))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(inverse(multiply(multiply(b, inverse(a)), multiply(multiply(b, inverse(a)), inverse(a)))))))
0.74/0.92	= { by lemma 8 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(inverse(multiply(multiply(b, inverse(a)), multiply(inverse(a), multiply(a, multiply(multiply(b, inverse(a)), inverse(a)))))))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(a, multiply(multiply(b, inverse(a)), inverse(a))))))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(inverse(multiply(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a), multiply(multiply(b, inverse(a)), inverse(a)))))))
0.74/0.92	= { by lemma 11 }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(inverse(multiply(multiply(b, inverse(a)), inverse(a))), inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a))))))
0.74/0.92	= { by axiom 3 (left_identity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(identity, multiply(inverse(multiply(multiply(b, inverse(a)), inverse(a))), inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a)))))))
0.74/0.92	= { by axiom 5 (x_cubed_is_identity) }
0.74/0.92	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(b, inverse(a)), inverse(a)))), multiply(inverse(multiply(multiply(b, inverse(a)), inverse(a))), inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a)))))))
0.74/0.92	= { by axiom 2 (associativity) }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(b, inverse(a)), inverse(a))), multiply(inverse(multiply(multiply(b, inverse(a)), inverse(a))), inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a))))))))
0.74/0.93	= { by axiom 2 (associativity) }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(inverse(multiply(multiply(b, inverse(a)), inverse(a))), inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a)))))))))
0.74/0.93	= { by lemma 12 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(multiply(b, inverse(a)), inverse(a)), inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), a)))))))
0.74/0.93	= { by lemma 11 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(inverse(a), inverse(multiply(multiply(b, inverse(a)), inverse(a)))))))))
0.74/0.93	= { by lemma 8 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(inverse(a), multiply(a, multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(inverse(a), inverse(multiply(multiply(b, inverse(a)), inverse(a)))))))))))
0.74/0.93	= { by axiom 6 (commutator) }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(inverse(a), commutator(a, multiply(multiply(b, inverse(a)), inverse(a))))))))
0.74/0.93	= { by lemma 15 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(multiply(b, inverse(a)), inverse(a)), multiply(inverse(a), commutator(a, multiply(b, inverse(a))))))))
0.74/0.93	= { by axiom 2 (associativity) }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(b, inverse(a)), multiply(inverse(a), multiply(inverse(a), commutator(a, multiply(b, inverse(a)))))))))
0.74/0.93	= { by axiom 2 (associativity) }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(b, inverse(a)), multiply(multiply(inverse(a), inverse(a)), commutator(a, multiply(b, inverse(a))))))))
0.74/0.93	= { by lemma 11 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(b, inverse(a)), multiply(inverse(multiply(a, a)), commutator(a, multiply(b, inverse(a))))))))
0.74/0.93	= { by lemma 14 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(b, inverse(a)), multiply(inverse(inverse(a)), commutator(a, multiply(b, inverse(a))))))))
0.74/0.93	= { by lemma 13 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(b, inverse(a)), multiply(a, commutator(a, multiply(b, inverse(a))))))))
0.74/0.93	= { by lemma 15 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(multiply(b, inverse(a)), multiply(a, commutator(a, b))))))
0.74/0.93	= { by axiom 2 (associativity) }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(b, multiply(inverse(a), multiply(a, commutator(a, b)))))))
0.74/0.93	= { by lemma 8 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, inverse(multiply(b, commutator(a, b)))))
0.74/0.93	= { by lemma 11 }
0.74/0.93	  multiply(commutator(a, b), multiply(b, multiply(inverse(commutator(a, b)), inverse(b))))
0.74/0.93	= { by lemma 8 }
0.74/0.93	  multiply(commutator(a, b), multiply(inverse(commutator(a, b)), multiply(commutator(a, b), multiply(b, multiply(inverse(commutator(a, b)), inverse(b))))))
0.74/0.93	= { by axiom 6 (commutator) }
0.74/0.93	  multiply(commutator(a, b), multiply(inverse(commutator(a, b)), commutator(commutator(a, b), b)))
0.74/0.93	= { by lemma 12 }
0.74/0.93	  commutator(commutator(a, b), b)
0.74/0.93	% SZS output end Proof
0.74/0.93	
0.74/0.93	RESULT: Unsatisfiable (the axioms are contradictory).
0.74/0.93	EOF