0.07/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n027.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 10:04:09 EDT 2019
0.12/0.34	% 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:
0.12/0.36	  fof(multiply, axiom, ![A, B]: multiply(A, B)=inverse(double_divide(B, A))).
0.12/0.36	  fof(prove_these_axioms_1, negated_conjecture, multiply(inverse(b1), b1)!=multiply(inverse(a1), a1)).
0.12/0.36	  fof(single_axiom, axiom, ![A, B, C]: double_divide(inverse(double_divide(double_divide(A, B), inverse(double_divide(A, inverse(C))))), B)=C).
0.12/0.36	
0.12/0.36	Now clausify the problem and encode Horn clauses using encoding 3 of
0.12/0.36	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.12/0.36	We repeatedly replace C & s=t => u=v by the two clauses:
0.12/0.36	  fresh(y, y, x1...xn) = u
0.12/0.36	  C => fresh(s, t, x1...xn) = v
0.12/0.36	where fresh is a fresh function symbol and x1..xn are the free
0.12/0.36	variables of u and v.
0.12/0.36	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.12/0.36	input problem has no model of domain size 1).
0.12/0.36	
0.12/0.36	The encoding turns the above axioms into the following unit equations and goals:
0.12/0.36	
0.12/0.36	Axiom 1 (single_axiom): double_divide(inverse(double_divide(double_divide(X, Y), inverse(double_divide(X, inverse(Z))))), Y) = Z.
0.12/0.36	Axiom 2 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
0.12/0.36	
0.12/0.36	Lemma 3: multiply(X, multiply(multiply(inverse(Z), Y), double_divide(Y, X))) = inverse(Z).
0.12/0.36	Proof:
0.12/0.36	  multiply(X, multiply(multiply(inverse(Z), Y), double_divide(Y, X)))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  multiply(X, multiply(inverse(double_divide(Y, inverse(Z))), double_divide(Y, X)))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  multiply(X, inverse(double_divide(double_divide(Y, X), inverse(double_divide(Y, inverse(Z))))))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  inverse(double_divide(inverse(double_divide(double_divide(Y, X), inverse(double_divide(Y, inverse(Z))))), X))
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.36	  inverse(Z)
0.12/0.36	
0.12/0.36	Lemma 4: double_divide(multiply(inverse(Y), Y), inverse(X)) = X.
0.12/0.36	Proof:
0.12/0.36	  double_divide(multiply(inverse(Y), Y), inverse(X))
0.12/0.36	= { by lemma 3 }
0.12/0.36	  double_divide(multiply(multiply(inverse(X), multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X)))), Y), inverse(X))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  double_divide(multiply(inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))), Y), inverse(X))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  double_divide(inverse(double_divide(Y, inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))))), inverse(X))
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.36	  double_divide(inverse(double_divide(double_divide(inverse(double_divide(double_divide(?, inverse(X)), inverse(double_divide(?, inverse(Y))))), inverse(X)), inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))))), inverse(X))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  double_divide(inverse(double_divide(double_divide(multiply(inverse(double_divide(?, inverse(Y))), double_divide(?, inverse(X))), inverse(X)), inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))))), inverse(X))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  double_divide(inverse(double_divide(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X)), inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))))), inverse(X))
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.37	  X
0.12/0.37	
0.12/0.37	Lemma 5: multiply(inverse(X), multiply(inverse(Y), Y)) = inverse(X).
0.12/0.37	Proof:
0.12/0.37	  multiply(inverse(X), multiply(inverse(Y), Y))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  inverse(double_divide(multiply(inverse(Y), Y), inverse(X)))
0.12/0.37	= { by lemma 4 }
0.12/0.37	  inverse(X)
0.12/0.37	
0.12/0.37	Lemma 6: 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 5 }
0.12/0.37	  double_divide(multiply(multiply(inverse(X), multiply(inverse(?), ?)), Y), inverse(Y))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  double_divide(multiply(inverse(double_divide(multiply(inverse(?), ?), inverse(X))), Y), inverse(Y))
0.12/0.37	= { by axiom 1 (single_axiom) }
0.12/0.37	  double_divide(multiply(inverse(double_divide(multiply(inverse(?), ?), inverse(X))), double_divide(inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))), ?)), inverse(Y))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  double_divide(inverse(double_divide(double_divide(inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))), ?), inverse(double_divide(multiply(inverse(?), ?), inverse(X))))), inverse(Y))
0.12/0.37	= { by lemma 4 }
0.12/0.37	  double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), inverse(double_divide(inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))), ?))), inverse(double_divide(multiply(inverse(?), ?), inverse(X))))), inverse(Y))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), multiply(?, inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))))), inverse(double_divide(multiply(inverse(?), ?), inverse(X))))), inverse(Y))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), multiply(?, inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))))), inverse(double_divide(multiply(inverse(?), ?), inverse(X))))), multiply(?, multiply(multiply(inverse(Y), ?), double_divide(?, ?))))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), multiply(?, inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))))), inverse(double_divide(multiply(inverse(?), ?), inverse(X))))), multiply(?, multiply(inverse(double_divide(?, inverse(Y))), double_divide(?, ?))))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), multiply(?, inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))))), inverse(double_divide(multiply(inverse(?), ?), inverse(X))))), multiply(?, inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y)))))))
0.12/0.37	= { by axiom 1 (single_axiom) }
0.12/0.37	  X
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(inverse(double_divide(multiply(inverse(Y), X), inverse(X))), inverse(multiply(inverse(Y), X)))
0.12/0.37	= { by axiom 2 (multiply) }
0.12/0.37	  double_divide(multiply(inverse(X), multiply(inverse(Y), X)), inverse(multiply(inverse(Y), X)))
0.12/0.37	= { by lemma 6 }
0.12/0.37	  X
0.12/0.37	
0.12/0.37	Lemma 8: multiply(inverse(X), X) = multiply(inverse(?), ?).
0.12/0.37	Proof:
0.12/0.37	  multiply(inverse(X), X)
0.12/0.37	= { by lemma 7 }
0.12/0.37	  double_divide(inverse(Y), inverse(multiply(inverse(Y), multiply(inverse(X), X))))
0.12/0.37	= { by lemma 5 }
0.12/0.37	  double_divide(inverse(Y), inverse(inverse(Y)))
0.12/0.37	= { by lemma 5 }
0.12/0.37	  double_divide(inverse(Y), inverse(multiply(inverse(Y), multiply(inverse(?), ?))))
0.12/0.37	= { by lemma 7 }
0.12/0.37	  multiply(inverse(?), ?)
0.12/0.37	
0.12/0.37	Goal 1 (prove_these_axioms_1): multiply(inverse(b1), b1) = multiply(inverse(a1), a1).
0.12/0.37	Proof:
0.12/0.37	  multiply(inverse(b1), b1)
0.12/0.37	= { by lemma 8 }
0.12/0.37	  multiply(inverse(?), ?)
0.12/0.37	= { by lemma 8 }
0.12/0.37	  multiply(inverse(a1), a1)
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