0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n024.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 180
0.13/0.34	% DateTime   : Thu Aug 29 09:32:20 EDT 2019
0.13/0.34	% 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:
0.19/0.39	  fof(identity, axiom, ![A]: divide(A, A)=identity).
0.19/0.39	  fof(inverse, axiom, ![A, B]: inverse(A)=divide(divide(B, B), A)).
0.19/0.39	  fof(multiply, axiom, ![A, B, C]: multiply(A, B)=divide(A, divide(divide(C, C), B))).
0.19/0.39	  fof(prove_these_axioms_3, negated_conjecture, multiply(a3, multiply(b3, c3))!=multiply(multiply(a3, b3), c3)).
0.19/0.39	  fof(single_axiom, axiom, ![A, B, C]: divide(divide(A, divide(divide(A, B), C)), B)=C).
0.19/0.39	
0.19/0.39	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.39	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.39	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.39	  fresh(y, y, x1...xn) = u
0.19/0.39	  C => fresh(s, t, x1...xn) = v
0.19/0.39	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.39	variables of u and v.
0.19/0.39	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.39	input problem has no model of domain size 1).
0.19/0.39	
0.19/0.39	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.39	
0.19/0.39	Axiom 1 (multiply): multiply(X, Y) = divide(X, divide(divide(Z, Z), Y)).
0.19/0.39	Axiom 2 (inverse): inverse(X) = divide(divide(Y, Y), X).
0.19/0.39	Axiom 3 (single_axiom): divide(divide(X, divide(divide(X, Y), Z)), Y) = Z.
0.19/0.39	Axiom 4 (identity): divide(X, X) = identity.
0.19/0.39	
0.19/0.39	Lemma 5: divide(X, inverse(Y)) = multiply(X, Y).
0.19/0.39	Proof:
0.19/0.39	  divide(X, inverse(Y))
0.19/0.39	= { by axiom 2 (inverse) }
0.19/0.39	  divide(X, divide(divide(?, ?), Y))
0.19/0.39	= { by axiom 1 (multiply) }
0.19/0.39	  multiply(X, Y)
0.19/0.39	
0.19/0.39	Lemma 6: divide(identity, X) = inverse(X).
0.19/0.39	Proof:
0.19/0.39	  divide(identity, X)
0.19/0.39	= { by axiom 4 (identity) }
0.19/0.39	  divide(divide(?, ?), X)
0.19/0.39	= { by axiom 2 (inverse) }
0.19/0.39	  inverse(X)
0.19/0.39	
0.19/0.39	Lemma 7: inverse(inverse(X)) = multiply(identity, X).
0.19/0.39	Proof:
0.19/0.39	  inverse(inverse(X))
0.19/0.39	= { by axiom 2 (inverse) }
0.19/0.39	  inverse(divide(divide(?, ?), X))
0.19/0.39	= { by lemma 6 }
0.19/0.39	  divide(identity, divide(divide(?, ?), X))
0.19/0.39	= { by axiom 1 (multiply) }
0.19/0.39	  multiply(identity, X)
0.19/0.39	
0.19/0.39	Lemma 8: divide(multiply(Y, X), Y) = X.
0.19/0.39	Proof:
0.19/0.39	  divide(multiply(Y, X), Y)
0.19/0.39	= { by axiom 1 (multiply) }
0.19/0.39	  divide(divide(Y, divide(divide(Y, Y), X)), Y)
0.19/0.39	= { by axiom 3 (single_axiom) }
0.19/0.39	  X
0.19/0.39	
0.19/0.39	Lemma 9: multiply(identity, X) = X.
0.19/0.39	Proof:
0.19/0.39	  multiply(identity, X)
0.19/0.39	= { by lemma 7 }
0.19/0.39	  inverse(inverse(X))
0.19/0.39	= { by lemma 6 }
0.19/0.39	  divide(identity, inverse(X))
0.19/0.39	= { by axiom 4 (identity) }
0.19/0.39	  divide(divide(divide(divide(?, ?), X), divide(divide(?, ?), X)), inverse(X))
0.19/0.39	= { by axiom 1 (multiply) }
0.19/0.39	  divide(multiply(divide(divide(?, ?), X), X), inverse(X))
0.19/0.39	= { by axiom 2 (inverse) }
0.19/0.39	  divide(multiply(inverse(X), X), inverse(X))
0.19/0.39	= { by lemma 8 }
0.19/0.40	  X
0.19/0.40	
0.19/0.40	Lemma 10: multiply(inverse(divide(Z, X)), Z) = X.
0.19/0.40	Proof:
0.19/0.40	  multiply(inverse(divide(Z, X)), Z)
0.19/0.40	= { by lemma 9 }
0.19/0.40	  multiply(inverse(divide(multiply(identity, Z), X)), Z)
0.19/0.40	= { by lemma 7 }
0.19/0.40	  multiply(inverse(divide(inverse(inverse(Z)), X)), Z)
0.19/0.40	= { by axiom 2 (inverse) }
0.19/0.40	  multiply(inverse(divide(inverse(divide(divide(?, ?), Z)), X)), Z)
0.19/0.40	= { by axiom 1 (multiply) }
0.19/0.40	  divide(inverse(divide(inverse(divide(divide(?, ?), Z)), X)), divide(divide(?, ?), Z))
0.19/0.40	= { by axiom 2 (inverse) }
0.19/0.40	  divide(inverse(divide(divide(divide(?, ?), divide(divide(?, ?), Z)), X)), divide(divide(?, ?), Z))
0.19/0.40	= { by axiom 2 (inverse) }
0.19/0.40	  divide(divide(divide(?, ?), divide(divide(divide(?, ?), divide(divide(?, ?), Z)), X)), divide(divide(?, ?), Z))
0.19/0.40	= { by axiom 3 (single_axiom) }
0.19/0.40	  X
0.19/0.40	
0.19/0.40	Lemma 11: multiply(X, inverse(Y)) = divide(X, Y).
0.19/0.40	Proof:
0.19/0.40	  multiply(X, inverse(Y))
0.19/0.40	= { by axiom 1 (multiply) }
0.19/0.40	  divide(X, divide(divide(?, ?), inverse(Y)))
0.19/0.40	= { by lemma 5 }
0.19/0.40	  divide(X, multiply(divide(?, ?), Y))
0.19/0.40	= { by axiom 4 (identity) }
0.19/0.40	  divide(X, multiply(identity, Y))
0.19/0.40	= { by lemma 9 }
0.19/0.40	  divide(X, Y)
0.19/0.40	
0.19/0.40	Lemma 12: divide(divide(Y, X), Y) = inverse(X).
0.19/0.40	Proof:
0.19/0.40	  divide(divide(Y, X), Y)
0.19/0.40	= { by lemma 11 }
0.19/0.40	  divide(multiply(Y, inverse(X)), Y)
0.19/0.40	= { by lemma 8 }
0.19/0.40	  inverse(X)
0.19/0.40	
0.19/0.40	Lemma 13: multiply(X, Y) = multiply(Y, X).
0.19/0.40	Proof:
0.19/0.40	  multiply(X, Y)
0.19/0.40	= { by lemma 8 }
0.19/0.40	  divide(multiply(inverse(divide(multiply(X, Y), Y)), multiply(X, Y)), inverse(divide(multiply(X, Y), Y)))
0.19/0.40	= { by lemma 10 }
0.19/0.40	  divide(Y, inverse(divide(multiply(X, Y), Y)))
0.19/0.40	= { by lemma 5 }
0.19/0.40	  multiply(Y, divide(multiply(X, Y), Y))
0.19/0.40	= { by lemma 5 }
0.19/0.40	  multiply(Y, divide(divide(X, inverse(Y)), Y))
0.19/0.40	= { by lemma 12 }
0.19/0.40	  multiply(Y, divide(divide(X, divide(divide(X, Y), X)), Y))
0.19/0.40	= { by axiom 3 (single_axiom) }
0.19/0.40	  multiply(Y, X)
0.19/0.40	
0.19/0.40	Lemma 14: multiply(W, divide(Y, divide(multiply(Y, W), X))) = X.
0.19/0.40	Proof:
0.19/0.40	  multiply(W, divide(Y, divide(multiply(Y, W), X)))
0.19/0.40	= { by lemma 13 }
0.19/0.40	  multiply(divide(Y, divide(multiply(Y, W), X)), W)
0.19/0.40	= { by axiom 1 (multiply) }
0.19/0.40	  multiply(divide(Y, divide(divide(Y, divide(divide(?, ?), W)), X)), W)
0.19/0.40	= { by axiom 1 (multiply) }
0.19/0.40	  divide(divide(Y, divide(divide(Y, divide(divide(?, ?), W)), X)), divide(divide(?, ?), W))
0.19/0.40	= { by axiom 3 (single_axiom) }
0.19/0.40	  X
0.19/0.40	
0.19/0.40	Lemma 15: divide(X, divide(multiply(X, Y), Z)) = divide(Z, Y).
0.19/0.40	Proof:
0.19/0.40	  divide(X, divide(multiply(X, Y), Z))
0.19/0.40	= { by lemma 8 }
0.19/0.40	  divide(multiply(Y, divide(X, divide(multiply(X, Y), Z))), Y)
0.19/0.40	= { by lemma 14 }
0.19/0.40	  divide(Z, Y)
0.19/0.40	
0.19/0.40	Lemma 16: multiply(X, divide(Z, W)) = divide(X, divide(W, Z)).
0.19/0.40	Proof:
0.19/0.40	  multiply(X, divide(Z, W))
0.19/0.40	= { by lemma 9 }
0.19/0.40	  multiply(X, divide(multiply(identity, Z), W))
0.19/0.40	= { by axiom 4 (identity) }
0.19/0.40	  multiply(X, divide(multiply(divide(?, ?), Z), W))
0.19/0.40	= { by axiom 1 (multiply) }
0.19/0.40	  divide(X, divide(divide(?, ?), divide(multiply(divide(?, ?), Z), W)))
0.19/0.40	= { by lemma 15 }
0.19/0.41	  divide(X, divide(W, Z))
0.19/0.41	
0.19/0.41	Goal 1 (prove_these_axioms_3): multiply(a3, multiply(b3, c3)) = multiply(multiply(a3, b3), c3).
0.19/0.41	Proof:
0.19/0.41	  multiply(a3, multiply(b3, c3))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  multiply(a3, multiply(c3, b3))
0.19/0.41	= { by lemma 14 }
0.19/0.41	  multiply(b3, divide(c3, divide(multiply(c3, b3), multiply(a3, multiply(c3, b3)))))
0.19/0.41	= { by lemma 8 }
0.19/0.41	  multiply(b3, divide(c3, divide(divide(multiply(a3, multiply(c3, b3)), a3), multiply(a3, multiply(c3, b3)))))
0.19/0.41	= { by lemma 12 }
0.19/0.41	  multiply(b3, divide(c3, inverse(a3)))
0.19/0.41	= { by lemma 16 }
0.19/0.41	  divide(b3, divide(inverse(a3), c3))
0.19/0.41	= { by lemma 15 }
0.19/0.41	  divide(b3, divide(?, divide(multiply(?, c3), inverse(a3))))
0.19/0.41	= { by lemma 15 }
0.19/0.41	  divide(c3, divide(multiply(c3, divide(?, divide(multiply(?, c3), inverse(a3)))), b3))
0.19/0.41	= { by lemma 14 }
0.19/0.41	  divide(c3, divide(inverse(a3), b3))
0.19/0.41	= { by lemma 15 }
0.19/0.41	  divide(c3, divide(divide(?, ?), divide(multiply(divide(?, ?), b3), inverse(a3))))
0.19/0.41	= { by lemma 5 }
0.19/0.41	  divide(c3, divide(divide(?, ?), multiply(multiply(divide(?, ?), b3), a3)))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  divide(c3, divide(divide(?, ?), multiply(a3, multiply(divide(?, ?), b3))))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  divide(c3, divide(divide(?, ?), multiply(a3, multiply(b3, divide(?, ?)))))
0.19/0.41	= { by axiom 1 (multiply) }
0.19/0.41	  multiply(c3, multiply(a3, multiply(b3, divide(?, ?))))
0.19/0.41	= { by lemma 16 }
0.19/0.41	  multiply(c3, multiply(a3, divide(b3, divide(?, ?))))
0.19/0.41	= { by lemma 16 }
0.19/0.41	  multiply(c3, divide(a3, divide(divide(?, ?), b3)))
0.19/0.41	= { by lemma 16 }
0.19/0.41	  divide(c3, divide(divide(divide(?, ?), b3), a3))
0.19/0.41	= { by lemma 15 }
0.19/0.41	  divide(c3, divide(divide(a3, divide(multiply(a3, b3), divide(?, ?))), a3))
0.19/0.41	= { by lemma 12 }
0.19/0.41	  divide(c3, inverse(divide(multiply(a3, b3), divide(?, ?))))
0.19/0.41	= { by lemma 10 }
0.19/0.41	  divide(c3, multiply(inverse(divide(multiply(inverse(divide(multiply(a3, b3), divide(?, ?))), multiply(a3, b3)), inverse(divide(multiply(a3, b3), divide(?, ?))))), multiply(inverse(divide(multiply(a3, b3), divide(?, ?))), multiply(a3, b3))))
0.19/0.41	= { by lemma 8 }
0.19/0.41	  divide(c3, multiply(inverse(multiply(a3, b3)), multiply(inverse(divide(multiply(a3, b3), divide(?, ?))), multiply(a3, b3))))
0.19/0.41	= { by lemma 10 }
0.19/0.41	  divide(c3, multiply(inverse(multiply(a3, b3)), divide(?, ?)))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  divide(c3, multiply(divide(?, ?), inverse(multiply(a3, b3))))
0.19/0.41	= { by lemma 11 }
0.19/0.41	  divide(c3, divide(divide(?, ?), multiply(a3, b3)))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  divide(c3, divide(divide(?, ?), multiply(b3, a3)))
0.19/0.41	= { by axiom 1 (multiply) }
0.19/0.41	  multiply(c3, multiply(b3, a3))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  multiply(c3, multiply(a3, b3))
0.19/0.41	= { by lemma 13 }
0.19/0.41	  multiply(multiply(a3, b3), c3)
0.19/0.41	% SZS output end Proof
0.19/0.41	
0.19/0.41	RESULT: Unsatisfiable (the axioms are contradictory).
0.19/0.41	EOF