0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n004.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 09:50:08 EDT 2019
0.12/0.33	% CPUTime    : 
0.12/0.36	% SZS status Unsatisfiable
0.12/0.36	
0.12/0.36	% SZS output start Proof
0.12/0.36	Take the following subset of the input axioms:
0.12/0.37	  fof(multiply, axiom, ![A, B]: multiply(A, B)=inverse(double_divide(B, A))).
0.12/0.37	  fof(prove_these_axioms_4, negated_conjecture, multiply(a, b)!=multiply(b, a)).
0.12/0.37	  fof(single_axiom, axiom, ![A, B, C]: double_divide(A, inverse(double_divide(inverse(double_divide(double_divide(A, B), inverse(C))), B)))=C).
0.12/0.37	
0.12/0.37	Now clausify the problem and encode Horn clauses using encoding 3 of
0.12/0.37	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.12/0.37	We repeatedly replace C & s=t => u=v by the two clauses:
0.12/0.37	  fresh(y, y, x1...xn) = u
0.12/0.37	  C => fresh(s, t, x1...xn) = v
0.12/0.37	where fresh is a fresh function symbol and x1..xn are the free
0.12/0.37	variables of u and v.
0.12/0.37	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.12/0.37	input problem has no model of domain size 1).
0.12/0.37	
0.12/0.37	The encoding turns the above axioms into the following unit equations and goals:
0.12/0.37	
0.12/0.37	Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
0.12/0.37	Axiom 2 (single_axiom): double_divide(X, inverse(double_divide(inverse(double_divide(double_divide(X, Y), inverse(Z))), Y))) = Z.
0.12/0.37	
0.12/0.37	Lemma 3: double_divide(X, multiply(Y, multiply(inverse(Z), double_divide(X, Y)))) = Z.
0.12/0.37	Proof:
0.12/0.37	  double_divide(X, multiply(Y, multiply(inverse(Z), double_divide(X, Y))))
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  double_divide(X, multiply(Y, inverse(double_divide(double_divide(X, Y), inverse(Z)))))
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  double_divide(X, inverse(double_divide(inverse(double_divide(double_divide(X, Y), inverse(Z))), Y)))
0.12/0.37	= { by axiom 2 (single_axiom) }
0.12/0.37	  Z
0.12/0.37	
0.12/0.37	Lemma 4: multiply(multiply(X, multiply(inverse(Y), double_divide(Z, X))), Z) = inverse(Y).
0.12/0.37	Proof:
0.12/0.37	  multiply(multiply(X, multiply(inverse(Y), double_divide(Z, X))), Z)
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  inverse(double_divide(Z, multiply(X, multiply(inverse(Y), double_divide(Z, X)))))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  inverse(Y)
0.12/0.37	
0.12/0.37	Lemma 5: double_divide(multiply(inverse(X), Y), inverse(Y)) = X.
0.12/0.37	Proof:
0.12/0.37	  double_divide(multiply(inverse(X), Y), inverse(Y))
0.12/0.37	= { by lemma 4 }
0.12/0.37	  double_divide(multiply(inverse(X), Y), multiply(multiply(?, multiply(inverse(Y), double_divide(multiply(inverse(X), Y), ?))), multiply(inverse(X), Y)))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  double_divide(multiply(inverse(X), Y), multiply(multiply(?, multiply(inverse(Y), double_divide(multiply(inverse(X), Y), ?))), multiply(inverse(X), double_divide(multiply(inverse(X), Y), multiply(?, multiply(inverse(Y), double_divide(multiply(inverse(X), Y), ?)))))))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  X
0.12/0.37	
0.12/0.37	Lemma 6: multiply(inverse(X), multiply(inverse(Y), X)) = inverse(Y).
0.12/0.37	Proof:
0.12/0.37	  multiply(inverse(X), multiply(inverse(Y), X))
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  inverse(double_divide(multiply(inverse(Y), X), inverse(X)))
0.12/0.37	= { by lemma 5 }
0.12/0.37	  inverse(Y)
0.12/0.37	
0.12/0.37	Lemma 7: double_divide(inverse(Y), inverse(multiply(inverse(Y), X))) = X.
0.12/0.37	Proof:
0.12/0.37	  double_divide(inverse(Y), inverse(multiply(inverse(Y), X)))
0.12/0.37	= { by lemma 6 }
0.12/0.37	  double_divide(multiply(inverse(X), multiply(inverse(Y), X)), inverse(multiply(inverse(Y), X)))
0.12/0.37	= { by lemma 5 }
0.12/0.37	  X
0.12/0.37	
0.12/0.37	Lemma 8: multiply(multiply(inverse(X), Y), X) = Y.
0.12/0.37	Proof:
0.12/0.37	  multiply(multiply(inverse(X), Y), X)
0.12/0.37	= { by lemma 7 }
0.12/0.37	  double_divide(inverse(X), inverse(multiply(inverse(X), multiply(multiply(inverse(X), Y), X))))
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  double_divide(inverse(X), inverse(multiply(inverse(X), multiply(inverse(double_divide(Y, inverse(X))), X))))
0.12/0.37	= { by lemma 6 }
0.12/0.37	  double_divide(inverse(X), inverse(inverse(double_divide(Y, inverse(X)))))
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  double_divide(inverse(X), inverse(multiply(inverse(X), Y)))
0.12/0.37	= { by lemma 7 }
0.12/0.37	  Y
0.12/0.37	
0.12/0.37	Lemma 9: multiply(multiply(Y, X), Z) = multiply(multiply(Y, Z), X).
0.12/0.37	Proof:
0.12/0.37	  multiply(multiply(Y, X), Z)
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  multiply(inverse(double_divide(X, Y)), Z)
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  inverse(double_divide(Z, inverse(double_divide(X, Y))))
0.12/0.37	= { by lemma 4 }
0.12/0.37	  multiply(multiply(Y, multiply(inverse(double_divide(Z, inverse(double_divide(X, Y)))), double_divide(X, Y))), X)
0.12/0.37	= { by axiom 1 (multiply) }
0.12/0.37	  multiply(multiply(Y, multiply(multiply(inverse(double_divide(X, Y)), Z), double_divide(X, Y))), X)
0.12/0.37	= { by lemma 8 }
0.12/0.37	  multiply(multiply(Y, Z), X)
0.12/0.37	
0.12/0.37	Lemma 10: multiply(multiply(inverse(X), X), Y) = Y.
0.12/0.37	Proof:
0.12/0.37	  multiply(multiply(inverse(X), X), Y)
0.12/0.37	= { by lemma 9 }
0.12/0.37	  multiply(multiply(inverse(X), Y), X)
0.12/0.37	= { by lemma 8 }
0.12/0.37	  Y
0.12/0.37	
0.12/0.37	Goal 1 (prove_these_axioms_4): multiply(a, b) = multiply(b, a).
0.12/0.37	Proof:
0.12/0.37	  multiply(a, b)
0.12/0.37	= { by lemma 10 }
0.12/0.37	  multiply(multiply(multiply(inverse(?), ?), a), b)
0.12/0.37	= { by lemma 9 }
0.12/0.37	  multiply(multiply(multiply(inverse(?), ?), b), a)
0.12/0.37	= { by lemma 10 }
0.12/0.37	  multiply(b, a)
0.12/0.37	% SZS output end Proof
0.12/0.37	
0.12/0.37	RESULT: Unsatisfiable (the axioms are contradictory).
0.12/0.37	EOF