0.10/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.10/0.11	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.11/0.32	% Computer   : n009.cluster.edu
0.11/0.32	% Model      : x86_64 x86_64
0.11/0.32	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.32	% Memory     : 8042.1875MB
0.11/0.32	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.32	% CPULimit   : 180
0.11/0.32	% DateTime   : Thu Aug 29 09:58:27 EDT 2019
0.11/0.32	% CPUTime    : 
0.11/0.34	% SZS status Unsatisfiable
0.11/0.34	
0.11/0.34	% SZS output start Proof
0.11/0.34	Take the following subset of the input axioms:
0.11/0.34	  fof(multiply, axiom, ![A, B]: divide(A, inverse(B))=multiply(A, B)).
0.11/0.34	  fof(prove_these_axioms_4, negated_conjecture, multiply(b, a)!=multiply(a, b)).
0.11/0.34	  fof(single_axiom, axiom, ![A, B, C]: divide(divide(divide(A, inverse(B)), C), divide(A, C))=B).
0.11/0.34	
0.11/0.34	Now clausify the problem and encode Horn clauses using encoding 3 of
0.11/0.34	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.11/0.34	We repeatedly replace C & s=t => u=v by the two clauses:
0.11/0.34	  fresh(y, y, x1...xn) = u
0.11/0.34	  C => fresh(s, t, x1...xn) = v
0.11/0.34	where fresh is a fresh function symbol and x1..xn are the free
0.11/0.34	variables of u and v.
0.11/0.34	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.11/0.34	input problem has no model of domain size 1).
0.11/0.34	
0.11/0.34	The encoding turns the above axioms into the following unit equations and goals:
0.11/0.34	
0.11/0.34	Axiom 1 (single_axiom): divide(divide(divide(X, inverse(Y)), Z), divide(X, Z)) = Y.
0.11/0.34	Axiom 2 (multiply): divide(X, inverse(Y)) = multiply(X, Y).
0.11/0.34	
0.11/0.34	Lemma 3: divide(divide(multiply(Y, X), Z), divide(Y, Z)) = X.
0.11/0.34	Proof:
0.11/0.34	  divide(divide(multiply(Y, X), Z), divide(Y, Z))
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(divide(Y, inverse(X)), Z), divide(Y, Z))
0.11/0.34	= { by axiom 1 (single_axiom) }
0.11/0.34	  X
0.11/0.34	
0.11/0.34	Lemma 4: divide(multiply(multiply(Y, X), Z), multiply(Y, Z)) = X.
0.11/0.34	Proof:
0.11/0.34	  divide(multiply(multiply(Y, X), Z), multiply(Y, Z))
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(multiply(divide(Y, inverse(X)), Z), multiply(Y, Z))
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(divide(Y, inverse(X)), inverse(Z)), multiply(Y, Z))
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(divide(Y, inverse(X)), inverse(Z)), divide(Y, inverse(Z)))
0.11/0.34	= { by axiom 1 (single_axiom) }
0.11/0.34	  X
0.11/0.34	
0.11/0.34	Lemma 5: divide(multiply(Y, X), Y) = X.
0.11/0.34	Proof:
0.11/0.34	  divide(multiply(Y, X), Y)
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(Y, inverse(X)), Y)
0.11/0.34	= { by axiom 1 (single_axiom) }
0.11/0.34	  divide(divide(divide(divide(divide(multiply(multiply(?, inverse(X)), ?), inverse(Y)), multiply(?, ?)), divide(multiply(multiply(?, inverse(X)), ?), multiply(?, ?))), inverse(X)), Y)
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(divide(divide(multiply(multiply(multiply(?, inverse(X)), ?), Y), multiply(?, ?)), divide(multiply(multiply(?, inverse(X)), ?), multiply(?, ?))), inverse(X)), Y)
0.11/0.34	= { by lemma 4 }
0.11/0.34	  divide(divide(divide(divide(multiply(multiply(multiply(?, inverse(X)), ?), Y), multiply(?, ?)), inverse(X)), inverse(X)), Y)
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(multiply(divide(multiply(multiply(multiply(?, inverse(X)), ?), Y), multiply(?, ?)), X), inverse(X)), Y)
0.11/0.34	= { by axiom 2 (multiply) }
0.11/0.34	  divide(divide(multiply(divide(divide(multiply(multiply(?, inverse(X)), ?), inverse(Y)), multiply(?, ?)), X), inverse(X)), Y)
0.11/0.34	= { by lemma 4 }
0.11/0.34	  divide(divide(multiply(divide(divide(multiply(multiply(?, inverse(X)), ?), inverse(Y)), multiply(?, ?)), X), divide(multiply(multiply(?, inverse(X)), ?), multiply(?, ?))), Y)
0.11/0.34	= { by axiom 1 (single_axiom) }
0.11/0.34	  divide(divide(multiply(divide(divide(multiply(multiply(?, inverse(X)), ?), inverse(Y)), multiply(?, ?)), X), divide(multiply(multiply(?, inverse(X)), ?), multiply(?, ?))), divide(divide(divide(multiply(multiply(?, inverse(X)), ?), inverse(Y)), multiply(?, ?)), divide(multiply(multiply(?, inverse(X)), ?), multiply(?, ?))))
0.11/0.34	= { by lemma 3 }
0.11/0.35	  X
0.11/0.35	
0.11/0.35	Goal 1 (prove_these_axioms_4): multiply(b, a) = multiply(a, b).
0.11/0.35	Proof:
0.11/0.35	  multiply(b, a)
0.11/0.35	= { by lemma 5 }
0.11/0.35	  multiply(divide(multiply(a, b), a), a)
0.11/0.35	= { by axiom 2 (multiply) }
0.11/0.35	  divide(divide(multiply(a, b), a), inverse(a))
0.11/0.35	= { by lemma 3 }
0.11/0.35	  divide(divide(divide(divide(multiply(inverse(a), multiply(a, b)), inverse(multiply(inverse(a), multiply(a, b)))), divide(inverse(a), inverse(multiply(inverse(a), multiply(a, b))))), a), inverse(a))
0.11/0.35	= { by lemma 5 }
0.11/0.35	  divide(divide(divide(divide(multiply(inverse(a), multiply(a, b)), inverse(multiply(inverse(a), multiply(a, b)))), divide(inverse(a), inverse(multiply(inverse(a), multiply(a, b))))), divide(multiply(multiply(inverse(a), multiply(inverse(a), multiply(a, b))), a), multiply(inverse(a), multiply(inverse(a), multiply(a, b))))), inverse(a))
0.11/0.35	= { by axiom 2 (multiply) }
0.11/0.35	  divide(divide(divide(divide(multiply(inverse(a), multiply(a, b)), inverse(multiply(inverse(a), multiply(a, b)))), divide(inverse(a), inverse(multiply(inverse(a), multiply(a, b))))), divide(divide(multiply(inverse(a), multiply(inverse(a), multiply(a, b))), inverse(a)), multiply(inverse(a), multiply(inverse(a), multiply(a, b))))), inverse(a))
0.11/0.35	= { by lemma 5 }
0.11/0.35	  divide(divide(divide(divide(multiply(inverse(a), multiply(a, b)), inverse(multiply(inverse(a), multiply(a, b)))), divide(inverse(a), inverse(multiply(inverse(a), multiply(a, b))))), divide(multiply(inverse(a), multiply(a, b)), multiply(inverse(a), multiply(inverse(a), multiply(a, b))))), inverse(a))
0.11/0.35	= { by axiom 2 (multiply) }
0.11/0.35	  divide(divide(divide(divide(multiply(inverse(a), multiply(a, b)), inverse(multiply(inverse(a), multiply(a, b)))), divide(inverse(a), inverse(multiply(inverse(a), multiply(a, b))))), divide(multiply(inverse(a), multiply(a, b)), divide(inverse(a), inverse(multiply(inverse(a), multiply(a, b)))))), inverse(a))
0.11/0.35	= { by axiom 1 (single_axiom) }
0.11/0.35	  divide(multiply(inverse(a), multiply(a, b)), inverse(a))
0.11/0.35	= { by lemma 5 }
0.11/0.35	  multiply(a, b)
0.11/0.35	% SZS output end Proof
0.11/0.35	
0.11/0.35	RESULT: Unsatisfiable (the axioms are contradictory).
0.11/0.35	EOF