0.03/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.13/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.35	% Computer   : n027.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 09:19:54 EDT 2019
0.13/0.35	% CPUTime    : 
0.13/0.38	% SZS status Unsatisfiable
0.13/0.38	
0.13/0.38	% SZS output start Proof
0.13/0.38	Take the following subset of the input axioms:
0.13/0.39	  fof(inverse, axiom, ![B, A]: divide(divide(B, B), A)=inverse(A)).
0.13/0.39	  fof(multiply, axiom, ![B, A, C]: multiply(A, B)=divide(A, divide(divide(C, C), B))).
0.13/0.39	  fof(prove_these_axioms_3, negated_conjecture, multiply(a3, multiply(b3, c3))!=multiply(multiply(a3, b3), c3)).
0.13/0.39	  fof(single_axiom, axiom, ![B, A, C]: divide(divide(A, divide(B, C)), divide(A, B))=C).
0.13/0.39	
0.13/0.39	Now clausify the problem and encode Horn clauses using encoding 3 of
0.13/0.39	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.13/0.39	We repeatedly replace C & s=t => u=v by the two clauses:
0.13/0.39	  fresh(y, y, x1...xn) = u
0.13/0.39	  C => fresh(s, t, x1...xn) = v
0.13/0.39	where fresh is a fresh function symbol and x1..xn are the free
0.13/0.39	variables of u and v.
0.13/0.39	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.13/0.39	input problem has no model of domain size 1).
0.13/0.39	
0.13/0.39	The encoding turns the above axioms into the following unit equations and goals:
0.13/0.39	
0.13/0.39	Axiom 1 (multiply): multiply(X, Y) = divide(X, divide(divide(Z, Z), Y)).
0.13/0.39	Axiom 2 (single_axiom): divide(divide(X, divide(Y, Z)), divide(X, Y)) = Z.
0.13/0.39	Axiom 3 (inverse): divide(divide(X, X), Y) = inverse(Y).
0.13/0.39	
0.13/0.39	Lemma 4: inverse(divide(divide(X, Y), X)) = Y.
0.13/0.39	Proof:
0.13/0.39	  inverse(divide(divide(X, Y), X))
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  divide(divide(divide(X, Y), divide(X, Y)), divide(divide(X, Y), X))
0.13/0.39	= { by axiom 2 (single_axiom) }
0.13/0.39	  Y
0.13/0.39	
0.13/0.39	Lemma 5: multiply(inverse(divide(Y, X)), Y) = X.
0.13/0.39	Proof:
0.13/0.39	  multiply(inverse(divide(Y, X)), Y)
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  divide(inverse(divide(Y, X)), divide(divide(?, ?), Y))
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  divide(divide(divide(?, ?), divide(Y, X)), divide(divide(?, ?), Y))
0.13/0.39	= { by axiom 2 (single_axiom) }
0.13/0.39	  X
0.13/0.39	
0.13/0.39	Lemma 6: multiply(Y, divide(X, Y)) = X.
0.13/0.39	Proof:
0.13/0.39	  multiply(Y, divide(X, Y))
0.13/0.39	= { by lemma 4 }
0.13/0.39	  multiply(inverse(divide(divide(X, Y), X)), divide(X, Y))
0.13/0.39	= { by lemma 5 }
0.13/0.39	  X
0.13/0.39	
0.13/0.39	Lemma 7: multiply(divide(X, Y), Z) = divide(X, divide(Y, Z)).
0.13/0.39	Proof:
0.13/0.39	  multiply(divide(X, Y), Z)
0.13/0.39	= { by axiom 2 (single_axiom) }
0.13/0.39	  multiply(divide(X, Y), divide(divide(X, divide(Y, Z)), divide(X, Y)))
0.13/0.39	= { by lemma 6 }
0.13/0.39	  divide(X, divide(Y, Z))
0.13/0.39	
0.13/0.39	Lemma 8: inverse(inverse(X)) = X.
0.13/0.39	Proof:
0.13/0.39	  inverse(inverse(X))
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  inverse(divide(divide(X, X), X))
0.13/0.39	= { by lemma 4 }
0.13/0.39	  X
0.13/0.39	
0.13/0.39	Lemma 9: divide(X, inverse(Y)) = multiply(X, Y).
0.13/0.39	Proof:
0.13/0.39	  divide(X, inverse(Y))
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  divide(X, divide(divide(?, ?), Y))
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  multiply(X, Y)
0.13/0.39	
0.13/0.39	Lemma 10: multiply(X, inverse(Y)) = divide(X, Y).
0.13/0.39	Proof:
0.13/0.39	  multiply(X, inverse(Y))
0.13/0.39	= { by lemma 9 }
0.13/0.39	  divide(X, inverse(inverse(Y)))
0.13/0.39	= { by lemma 8 }
0.13/0.39	  divide(X, Y)
0.13/0.39	
0.13/0.39	Lemma 11: multiply(inverse(multiply(X, Y)), X) = inverse(Y).
0.13/0.39	Proof:
0.13/0.39	  multiply(inverse(multiply(X, Y)), X)
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  multiply(inverse(divide(X, divide(divide(?, ?), Y))), X)
0.13/0.39	= { by lemma 5 }
0.13/0.39	  divide(divide(?, ?), Y)
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  inverse(Y)
0.13/0.39	
0.13/0.39	Lemma 12: multiply(X, Y) = multiply(Y, X).
0.13/0.39	Proof:
0.13/0.39	  multiply(X, Y)
0.13/0.39	= { by lemma 8 }
0.13/0.39	  inverse(inverse(multiply(X, Y)))
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  inverse(divide(divide(?, ?), multiply(X, Y)))
0.13/0.39	= { by axiom 3 (inverse) }
0.13/0.39	  divide(divide(Y, Y), divide(divide(?, ?), multiply(X, Y)))
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  multiply(divide(Y, Y), multiply(X, Y))
0.13/0.39	= { by lemma 7 }
0.13/0.39	  divide(Y, divide(Y, multiply(X, Y)))
0.13/0.39	= { by lemma 8 }
0.13/0.39	  divide(Y, divide(inverse(inverse(Y)), multiply(X, Y)))
0.13/0.39	= { by lemma 10 }
0.13/0.39	  divide(Y, multiply(inverse(inverse(Y)), inverse(multiply(X, Y))))
0.13/0.39	= { by lemma 11 }
0.13/0.39	  divide(Y, multiply(inverse(multiply(inverse(multiply(X, Y)), X)), inverse(multiply(X, Y))))
0.13/0.39	= { by lemma 11 }
0.13/0.39	  divide(Y, inverse(X))
0.13/0.39	= { by lemma 9 }
0.13/0.39	  multiply(Y, X)
0.13/0.39	
0.13/0.39	Lemma 13: divide(divide(X, Y), Z) = divide(X, multiply(Y, Z)).
0.13/0.39	Proof:
0.13/0.39	  divide(divide(X, Y), Z)
0.13/0.39	= { by lemma 10 }
0.13/0.39	  multiply(divide(X, Y), inverse(Z))
0.13/0.39	= { by lemma 7 }
0.13/0.39	  divide(X, divide(Y, inverse(Z)))
0.13/0.39	= { by lemma 9 }
0.13/0.39	  divide(X, multiply(Y, Z))
0.13/0.39	
0.13/0.39	Goal 1 (prove_these_axioms_3): multiply(a3, multiply(b3, c3)) = multiply(multiply(a3, b3), c3).
0.13/0.39	Proof:
0.13/0.39	  multiply(a3, multiply(b3, c3))
0.13/0.39	= { by lemma 12 }
0.13/0.39	  multiply(a3, multiply(c3, b3))
0.13/0.39	= { by lemma 12 }
0.13/0.39	  multiply(multiply(c3, b3), a3)
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  divide(multiply(c3, b3), divide(divide(?, ?), a3))
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  divide(divide(c3, divide(divide(?, ?), b3)), divide(divide(?, ?), a3))
0.13/0.39	= { by lemma 13 }
0.13/0.39	  divide(c3, multiply(divide(divide(?, ?), b3), divide(divide(?, ?), a3)))
0.13/0.39	= { by lemma 7 }
0.13/0.39	  divide(c3, divide(divide(?, ?), divide(b3, divide(divide(?, ?), a3))))
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  multiply(c3, divide(b3, divide(divide(?, ?), a3)))
0.13/0.39	= { by lemma 9 }
0.13/0.39	  divide(c3, inverse(divide(b3, divide(divide(?, ?), a3))))
0.13/0.39	= { by lemma 11 }
0.13/0.39	  divide(c3, multiply(inverse(multiply(divide(divide(?, ?), a3), divide(b3, divide(divide(?, ?), a3)))), divide(divide(?, ?), a3)))
0.13/0.39	= { by lemma 6 }
0.13/0.39	  divide(c3, multiply(inverse(b3), divide(divide(?, ?), a3)))
0.13/0.39	= { by lemma 12 }
0.13/0.39	  divide(c3, multiply(divide(divide(?, ?), a3), inverse(b3)))
0.13/0.39	= { by lemma 10 }
0.13/0.39	  divide(c3, divide(divide(divide(?, ?), a3), b3))
0.13/0.39	= { by lemma 13 }
0.13/0.39	  divide(c3, divide(divide(?, ?), multiply(a3, b3)))
0.13/0.39	= { by axiom 1 (multiply) }
0.13/0.39	  multiply(c3, multiply(a3, b3))
0.13/0.39	= { by lemma 12 }
0.13/0.39	  multiply(multiply(a3, b3), c3)
0.13/0.39	% SZS output end Proof
0.13/0.39	
0.13/0.39	RESULT: Unsatisfiable (the axioms are contradictory).
0.13/0.39	EOF