0.04/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.15/0.34	% Computer   : n024.cluster.edu
0.15/0.34	% Model      : x86_64 x86_64
0.15/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.15/0.34	% Memory     : 8042.1875MB
0.15/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.15/0.34	% CPULimit   : 180
0.15/0.34	% DateTime   : Thu Aug 29 09:42:20 EDT 2019
0.15/0.34	% CPUTime    : 
0.21/0.45	% SZS status Unsatisfiable
0.21/0.45	
0.21/0.45	% SZS output start Proof
0.21/0.45	Take the following subset of the input axioms:
0.21/0.48	  fof(multiply, axiom, ![B, A]: inverse(double_divide(B, A))=multiply(A, B)).
0.21/0.48	  fof(prove_these_axioms_3, negated_conjecture, multiply(a3, multiply(b3, c3))!=multiply(multiply(a3, b3), c3)).
0.21/0.48	  fof(single_axiom, axiom, ![B, A, C]: double_divide(inverse(double_divide(double_divide(A, B), inverse(double_divide(A, inverse(C))))), B)=C).
0.21/0.48	
0.21/0.48	Now clausify the problem and encode Horn clauses using encoding 3 of
0.21/0.48	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.21/0.48	We repeatedly replace C & s=t => u=v by the two clauses:
0.21/0.48	  fresh(y, y, x1...xn) = u
0.21/0.48	  C => fresh(s, t, x1...xn) = v
0.21/0.48	where fresh is a fresh function symbol and x1..xn are the free
0.21/0.48	variables of u and v.
0.21/0.48	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.21/0.48	input problem has no model of domain size 1).
0.21/0.48	
0.21/0.48	The encoding turns the above axioms into the following unit equations and goals:
0.21/0.48	
0.21/0.48	Axiom 1 (single_axiom): double_divide(inverse(double_divide(double_divide(X, Y), inverse(double_divide(X, inverse(Z))))), Y) = Z.
0.21/0.49	Axiom 2 (multiply): inverse(double_divide(X, Y)) = multiply(Y, X).
0.21/0.49	
0.21/0.49	Lemma 3: multiply(X, multiply(multiply(inverse(Z), Y), double_divide(Y, X))) = inverse(Z).
0.21/0.49	Proof:
0.21/0.49	  multiply(X, multiply(multiply(inverse(Z), Y), double_divide(Y, X)))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  multiply(X, multiply(inverse(double_divide(Y, inverse(Z))), double_divide(Y, X)))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  multiply(X, inverse(double_divide(double_divide(Y, X), inverse(double_divide(Y, inverse(Z))))))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  inverse(double_divide(inverse(double_divide(double_divide(Y, X), inverse(double_divide(Y, inverse(Z))))), X))
0.21/0.49	= { by axiom 1 (single_axiom) }
0.21/0.49	  inverse(Z)
0.21/0.49	
0.21/0.49	Lemma 4: double_divide(multiply(multiply(inverse(X), Y), double_divide(Y, Z)), Z) = X.
0.21/0.49	Proof:
0.21/0.49	  double_divide(multiply(multiply(inverse(X), Y), double_divide(Y, Z)), Z)
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(multiply(inverse(double_divide(Y, inverse(X))), double_divide(Y, Z)), Z)
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(inverse(double_divide(double_divide(Y, Z), inverse(double_divide(Y, inverse(X))))), Z)
0.21/0.49	= { by axiom 1 (single_axiom) }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 5: double_divide(multiply(inverse(Y), Y), inverse(X)) = X.
0.21/0.49	Proof:
0.21/0.49	  double_divide(multiply(inverse(Y), Y), inverse(X))
0.21/0.49	= { by lemma 3 }
0.21/0.49	  double_divide(multiply(multiply(inverse(X), multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X)))), Y), inverse(X))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(multiply(inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))), Y), inverse(X))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(inverse(double_divide(Y, inverse(double_divide(multiply(multiply(inverse(Y), ?), double_divide(?, inverse(X))), inverse(X))))), inverse(X))
0.21/0.49	= { by lemma 4 }
0.21/0.49	  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))
0.21/0.49	= { by axiom 1 (single_axiom) }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 6: multiply(inverse(X), multiply(inverse(Y), Y)) = inverse(X).
0.21/0.49	Proof:
0.21/0.49	  multiply(inverse(X), multiply(inverse(Y), Y))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  inverse(double_divide(multiply(inverse(Y), Y), inverse(X)))
0.21/0.49	= { by lemma 5 }
0.21/0.49	  inverse(X)
0.21/0.49	
0.21/0.49	Lemma 7: double_divide(multiply(inverse(Z), Z), multiply(Y, X)) = double_divide(X, Y).
0.21/0.49	Proof:
0.21/0.49	  double_divide(multiply(inverse(Z), Z), multiply(Y, X))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(multiply(inverse(Z), Z), inverse(double_divide(X, Y)))
0.21/0.49	= { by lemma 5 }
0.21/0.49	  double_divide(X, Y)
0.21/0.49	
0.21/0.49	Lemma 8: double_divide(multiply(inverse(X), Y), inverse(Y)) = X.
0.21/0.49	Proof:
0.21/0.49	  double_divide(multiply(inverse(X), Y), inverse(Y))
0.21/0.49	= { by lemma 6 }
0.21/0.49	  double_divide(multiply(multiply(inverse(X), multiply(inverse(?), ?)), Y), inverse(Y))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(multiply(inverse(double_divide(multiply(inverse(?), ?), inverse(X))), Y), inverse(Y))
0.21/0.49	= { by axiom 1 (single_axiom) }
0.21/0.49	  double_divide(multiply(inverse(double_divide(multiply(inverse(?), ?), inverse(X))), double_divide(inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))), ?)), inverse(Y))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  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))
0.21/0.49	= { by lemma 7 }
0.21/0.49	  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))
0.21/0.49	= { by lemma 3 }
0.21/0.49	  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(?, ?))))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  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(?, ?))))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  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)))))))
0.21/0.49	= { by axiom 1 (single_axiom) }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 9: multiply(inverse(X), multiply(inverse(Y), X)) = inverse(Y).
0.21/0.49	Proof:
0.21/0.49	  multiply(inverse(X), multiply(inverse(Y), X))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  inverse(double_divide(multiply(inverse(Y), X), inverse(X)))
0.21/0.49	= { by lemma 8 }
0.21/0.49	  inverse(Y)
0.21/0.49	
0.21/0.49	Lemma 10: multiply(inverse(multiply(inverse(Y), X)), inverse(Y)) = inverse(X).
0.21/0.49	Proof:
0.21/0.49	  multiply(inverse(multiply(inverse(Y), X)), inverse(Y))
0.21/0.49	= { by lemma 9 }
0.21/0.49	  multiply(inverse(multiply(inverse(Y), X)), multiply(inverse(X), multiply(inverse(Y), X)))
0.21/0.49	= { by lemma 9 }
0.21/0.49	  inverse(X)
0.21/0.49	
0.21/0.49	Lemma 11: double_divide(inverse(Y), inverse(multiply(inverse(Y), X))) = X.
0.21/0.49	Proof:
0.21/0.49	  double_divide(inverse(Y), inverse(multiply(inverse(Y), X)))
0.21/0.49	= { by lemma 9 }
0.21/0.49	  double_divide(multiply(inverse(X), multiply(inverse(Y), X)), inverse(multiply(inverse(Y), X)))
0.21/0.49	= { by lemma 8 }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 12: multiply(inverse(X), X) = multiply(inverse(?), ?).
0.21/0.49	Proof:
0.21/0.49	  multiply(inverse(X), X)
0.21/0.49	= { by lemma 11 }
0.21/0.49	  double_divide(inverse(Y), inverse(multiply(inverse(Y), multiply(inverse(X), X))))
0.21/0.49	= { by lemma 6 }
0.21/0.49	  double_divide(inverse(Y), inverse(inverse(Y)))
0.21/0.49	= { by lemma 6 }
0.21/0.49	  double_divide(inverse(Y), inverse(multiply(inverse(Y), multiply(inverse(?), ?))))
0.21/0.49	= { by lemma 11 }
0.21/0.49	  multiply(inverse(?), ?)
0.21/0.49	
0.21/0.49	Lemma 13: double_divide(inverse(X), inverse(multiply(inverse(?), ?))) = X.
0.21/0.49	Proof:
0.21/0.49	  double_divide(inverse(X), inverse(multiply(inverse(?), ?)))
0.21/0.49	= { by lemma 6 }
0.21/0.49	  double_divide(multiply(inverse(X), multiply(inverse(?), ?)), inverse(multiply(inverse(?), ?)))
0.21/0.49	= { by lemma 8 }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 14: double_divide(Y, multiply(inverse(X), inverse(Y))) = X.
0.21/0.49	Proof:
0.21/0.49	  double_divide(Y, multiply(inverse(X), inverse(Y)))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  double_divide(Y, inverse(double_divide(inverse(Y), inverse(X))))
0.21/0.49	= { by lemma 13 }
0.21/0.49	  double_divide(inverse(double_divide(Y, inverse(double_divide(inverse(Y), inverse(X))))), inverse(multiply(inverse(?), ?)))
0.21/0.49	= { by lemma 13 }
0.21/0.49	  double_divide(inverse(double_divide(double_divide(inverse(Y), inverse(multiply(inverse(?), ?))), inverse(double_divide(inverse(Y), inverse(X))))), inverse(multiply(inverse(?), ?)))
0.21/0.49	= { by axiom 1 (single_axiom) }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 15: double_divide(X, inverse(Y)) = multiply(inverse(X), Y).
0.21/0.49	Proof:
0.21/0.49	  double_divide(X, inverse(Y))
0.21/0.49	= { by lemma 10 }
0.21/0.49	  double_divide(X, multiply(inverse(multiply(inverse(X), Y)), inverse(X)))
0.21/0.49	= { by lemma 14 }
0.21/0.49	  multiply(inverse(X), Y)
0.21/0.49	
0.21/0.49	Lemma 16: inverse(inverse(X)) = X.
0.21/0.49	Proof:
0.21/0.49	  inverse(inverse(X))
0.21/0.49	= { by lemma 6 }
0.21/0.49	  multiply(inverse(inverse(X)), multiply(inverse(?), ?))
0.21/0.49	= { by lemma 15 }
0.21/0.49	  double_divide(inverse(X), inverse(multiply(inverse(?), ?)))
0.21/0.49	= { by lemma 13 }
0.21/0.49	  X
0.21/0.49	
0.21/0.49	Lemma 17: inverse(multiply(Y, X)) = double_divide(X, Y).
0.21/0.49	Proof:
0.21/0.49	  inverse(multiply(Y, X))
0.21/0.49	= { by axiom 2 (multiply) }
0.21/0.49	  inverse(inverse(double_divide(X, Y)))
0.21/0.49	= { by lemma 16 }
0.21/0.50	  double_divide(X, Y)
0.21/0.50	
0.21/0.50	Lemma 18: multiply(X, Y) = multiply(Y, X).
0.21/0.50	Proof:
0.21/0.50	  multiply(X, Y)
0.21/0.50	= { by lemma 16 }
0.21/0.50	  multiply(inverse(inverse(X)), Y)
0.21/0.50	= { by lemma 15 }
0.21/0.50	  double_divide(inverse(X), inverse(Y))
0.21/0.50	= { by lemma 8 }
0.21/0.50	  double_divide(multiply(inverse(double_divide(inverse(X), inverse(Y))), X), inverse(X))
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(multiply(multiply(inverse(Y), inverse(X)), X), inverse(X))
0.21/0.50	= { by lemma 15 }
0.21/0.50	  multiply(inverse(multiply(multiply(inverse(Y), inverse(X)), X)), X)
0.21/0.50	= { by lemma 17 }
0.21/0.50	  multiply(double_divide(X, multiply(inverse(Y), inverse(X))), X)
0.21/0.50	= { by lemma 14 }
0.21/0.50	  multiply(Y, X)
0.21/0.50	
0.21/0.50	Lemma 19: double_divide(multiply(multiply(multiply(Y, X), Z), double_divide(Z, W)), W) = double_divide(X, Y).
0.21/0.50	Proof:
0.21/0.50	  double_divide(multiply(multiply(multiply(Y, X), Z), double_divide(Z, W)), W)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(multiply(inverse(double_divide(Z, multiply(Y, X))), double_divide(Z, W)), W)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(inverse(double_divide(double_divide(Z, W), inverse(double_divide(Z, multiply(Y, X))))), W)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(inverse(double_divide(double_divide(Z, W), inverse(double_divide(Z, inverse(double_divide(X, Y)))))), W)
0.21/0.50	= { by axiom 1 (single_axiom) }
0.21/0.50	  double_divide(X, Y)
0.21/0.50	
0.21/0.50	Lemma 20: double_divide(X, Y) = double_divide(Y, X).
0.21/0.50	Proof:
0.21/0.50	  double_divide(X, Y)
0.21/0.50	= { by lemma 19 }
0.21/0.50	  double_divide(multiply(multiply(multiply(Y, X), ?), double_divide(?, ?)), ?)
0.21/0.50	= { by lemma 18 }
0.21/0.50	  double_divide(multiply(multiply(multiply(X, Y), ?), double_divide(?, ?)), ?)
0.21/0.50	= { by lemma 19 }
0.21/0.50	  double_divide(Y, X)
0.21/0.50	
0.21/0.50	Lemma 21: double_divide(multiply(inverse(Y), double_divide(multiply(inverse(Y), X), Z)), Z) = X.
0.21/0.50	Proof:
0.21/0.50	  double_divide(multiply(inverse(Y), double_divide(multiply(inverse(Y), X), Z)), Z)
0.21/0.50	= { by lemma 5 }
0.21/0.50	  double_divide(multiply(inverse(Y), double_divide(multiply(inverse(Y), double_divide(multiply(inverse(?), ?), inverse(X))), Z)), Z)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(inverse(double_divide(double_divide(multiply(inverse(Y), double_divide(multiply(inverse(?), ?), inverse(X))), Z), inverse(Y))), Z)
0.21/0.50	= { by axiom 1 (single_axiom) }
0.21/0.50	  double_divide(inverse(double_divide(double_divide(multiply(inverse(Y), double_divide(multiply(inverse(?), ?), inverse(X))), Z), inverse(double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), inverse(X)), inverse(double_divide(multiply(inverse(?), ?), inverse(Y))))), inverse(X))))), Z)
0.21/0.50	= { by lemma 5 }
0.21/0.50	  double_divide(inverse(double_divide(double_divide(multiply(inverse(Y), double_divide(multiply(inverse(?), ?), inverse(X))), Z), inverse(double_divide(inverse(double_divide(double_divide(multiply(inverse(?), ?), inverse(X)), inverse(Y))), inverse(X))))), Z)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(inverse(double_divide(double_divide(multiply(inverse(Y), double_divide(multiply(inverse(?), ?), inverse(X))), Z), inverse(double_divide(multiply(inverse(Y), double_divide(multiply(inverse(?), ?), inverse(X))), inverse(X))))), Z)
0.21/0.50	= { by axiom 1 (single_axiom) }
0.21/0.50	  X
0.21/0.50	
0.21/0.50	Lemma 22: double_divide(multiply(inverse(Y), double_divide(inverse(Y), X)), X) = multiply(inverse(?), ?).
0.21/0.50	Proof:
0.21/0.50	  double_divide(multiply(inverse(Y), double_divide(inverse(Y), X)), X)
0.21/0.50	= { by lemma 6 }
0.21/0.50	  double_divide(multiply(inverse(Y), double_divide(multiply(inverse(Y), multiply(inverse(?), ?)), X)), X)
0.21/0.50	= { by lemma 21 }
0.21/0.50	  multiply(inverse(?), ?)
0.21/0.50	
0.21/0.50	Lemma 23: multiply(multiply(inverse(?), ?), X) = X.
0.21/0.50	Proof:
0.21/0.50	  multiply(multiply(inverse(?), ?), X)
0.21/0.50	= { by lemma 22 }
0.21/0.50	  multiply(double_divide(multiply(inverse(?), double_divide(inverse(?), ?)), ?), X)
0.21/0.50	= { by lemma 7 }
0.21/0.50	  multiply(double_divide(multiply(inverse(?), ?), multiply(?, multiply(inverse(?), double_divide(inverse(?), ?)))), X)
0.21/0.50	= { by lemma 17 }
0.21/0.50	  multiply(inverse(multiply(multiply(?, multiply(inverse(?), double_divide(inverse(?), ?))), multiply(inverse(?), ?))), X)
0.21/0.50	= { by lemma 15 }
0.21/0.50	  double_divide(multiply(multiply(?, multiply(inverse(?), double_divide(inverse(?), ?))), multiply(inverse(?), ?)), inverse(X))
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(multiply(inverse(double_divide(multiply(inverse(?), double_divide(inverse(?), ?)), ?)), multiply(inverse(?), ?)), inverse(X))
0.21/0.50	= { by lemma 22 }
0.21/0.50	  double_divide(multiply(inverse(double_divide(multiply(inverse(?), double_divide(inverse(?), ?)), ?)), double_divide(multiply(inverse(?), double_divide(inverse(?), ?)), ?)), inverse(X))
0.21/0.50	= { by lemma 5 }
0.21/0.50	  X
0.21/0.50	
0.21/0.50	Lemma 24: multiply(inverse(X), ?) = multiply(?, inverse(X)).
0.21/0.50	Proof:
0.21/0.50	  multiply(inverse(X), ?)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  inverse(double_divide(?, inverse(X)))
0.21/0.50	= { by lemma 10 }
0.21/0.50	  multiply(inverse(multiply(inverse(X), double_divide(?, inverse(X)))), inverse(X))
0.21/0.50	= { by lemma 23 }
0.21/0.50	  multiply(inverse(multiply(inverse(X), multiply(multiply(inverse(?), ?), double_divide(?, inverse(X))))), inverse(X))
0.21/0.50	= { by lemma 3 }
0.21/0.50	  multiply(inverse(inverse(?)), inverse(X))
0.21/0.50	= { by lemma 16 }
0.21/0.50	  multiply(?, inverse(X))
0.21/0.50	
0.21/0.50	Lemma 25: double_divide(Z, double_divide(Z, double_divide(X, Y))) = double_divide(X, Y).
0.21/0.50	Proof:
0.21/0.50	  double_divide(Z, double_divide(Z, double_divide(X, Y)))
0.21/0.50	= { by lemma 20 }
0.21/0.50	  double_divide(Z, double_divide(Z, double_divide(Y, X)))
0.21/0.50	= { by lemma 20 }
0.21/0.50	  double_divide(Z, double_divide(double_divide(Y, X), Z))
0.21/0.50	= { by lemma 20 }
0.21/0.50	  double_divide(double_divide(double_divide(Y, X), Z), Z)
0.21/0.50	= { by lemma 5 }
0.21/0.50	  double_divide(double_divide(multiply(inverse(multiply(?, inverse(?))), multiply(?, inverse(?))), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by lemma 23 }
0.21/0.50	  double_divide(double_divide(multiply(multiply(multiply(inverse(?), ?), inverse(multiply(?, inverse(?)))), multiply(?, inverse(?))), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(double_divide(multiply(inverse(double_divide(inverse(multiply(?, inverse(?))), multiply(inverse(?), ?))), multiply(?, inverse(?))), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by lemma 24 }
0.21/0.50	  double_divide(double_divide(multiply(inverse(double_divide(inverse(multiply(?, inverse(?))), multiply(inverse(?), ?))), multiply(inverse(?), ?)), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by lemma 6 }
0.21/0.50	  double_divide(double_divide(inverse(double_divide(inverse(multiply(?, inverse(?))), multiply(inverse(?), ?))), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.50	  double_divide(double_divide(multiply(multiply(inverse(?), ?), inverse(multiply(?, inverse(?)))), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by lemma 23 }
0.21/0.50	  double_divide(double_divide(inverse(multiply(?, inverse(?))), inverse(double_divide(double_divide(Y, X), Z))), Z)
0.21/0.50	= { by lemma 15 }
0.21/0.50	  double_divide(multiply(inverse(inverse(multiply(?, inverse(?)))), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.50	= { by lemma 16 }
0.21/0.50	  double_divide(multiply(multiply(?, inverse(?)), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.50	= { by lemma 24 }
0.21/0.50	  double_divide(multiply(multiply(inverse(?), ?), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.50	= { by lemma 22 }
0.21/0.50	  double_divide(multiply(double_divide(multiply(inverse(?), double_divide(inverse(?), ?)), ?), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.50	= { by lemma 6 }
0.21/0.50	  double_divide(multiply(double_divide(multiply(inverse(?), double_divide(multiply(inverse(?), multiply(inverse(double_divide(X, Y)), double_divide(X, Y))), ?)), ?), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.50	= { by axiom 2 (multiply) }
0.21/0.51	  double_divide(multiply(double_divide(multiply(inverse(?), double_divide(multiply(inverse(?), multiply(multiply(Y, X), double_divide(X, Y))), ?)), ?), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.51	= { by lemma 20 }
0.21/0.51	  double_divide(multiply(double_divide(multiply(inverse(?), double_divide(multiply(inverse(?), multiply(multiply(Y, X), double_divide(Y, X))), ?)), ?), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.51	= { by lemma 21 }
0.21/0.51	  double_divide(multiply(multiply(multiply(Y, X), double_divide(Y, X)), double_divide(double_divide(Y, X), Z)), Z)
0.21/0.51	= { by lemma 19 }
0.21/0.51	  double_divide(X, Y)
0.21/0.51	
0.21/0.51	Lemma 26: double_divide(X, double_divide(X, Y)) = Y.
0.21/0.51	Proof:
0.21/0.51	  double_divide(X, double_divide(X, Y))
0.21/0.51	= { by axiom 1 (single_axiom) }
0.21/0.51	  double_divide(X, double_divide(X, double_divide(inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))), ?)))
0.21/0.51	= { by lemma 25 }
0.21/0.51	  double_divide(inverse(double_divide(double_divide(?, ?), inverse(double_divide(?, inverse(Y))))), ?)
0.21/0.51	= { by axiom 1 (single_axiom) }
0.21/0.53	  Y
0.21/0.53	
0.21/0.53	Goal 1 (prove_these_axioms_3): multiply(a3, multiply(b3, c3)) = multiply(multiply(a3, b3), c3).
0.21/0.53	Proof:
0.21/0.53	  multiply(a3, multiply(b3, c3))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  multiply(a3, multiply(c3, b3))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  multiply(a3, multiply(b3, c3))
0.21/0.53	= { by lemma 16 }
0.21/0.53	  multiply(a3, multiply(b3, inverse(inverse(c3))))
0.21/0.53	= { by lemma 26 }
0.21/0.53	  double_divide(inverse(c3), double_divide(inverse(c3), multiply(a3, multiply(b3, inverse(inverse(c3))))))
0.21/0.53	= { by lemma 25 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(a3, double_divide(inverse(c3), multiply(a3, multiply(b3, inverse(inverse(c3))))))))
0.21/0.53	= { by lemma 20 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(double_divide(inverse(c3), multiply(a3, multiply(b3, inverse(inverse(c3))))), a3)))
0.21/0.53	= { by lemma 19 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, double_divide(inverse(c3), multiply(a3, multiply(b3, inverse(inverse(c3)))))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, double_divide(inverse(c3), multiply(a3, multiply(inverse(inverse(c3)), b3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, double_divide(inverse(c3), multiply(multiply(inverse(inverse(c3)), b3), a3))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 14 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, double_divide(inverse(c3), double_divide(inverse(c3), multiply(inverse(multiply(multiply(inverse(inverse(c3)), b3), a3)), inverse(inverse(c3)))))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 26 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(multiply(multiply(inverse(inverse(c3)), b3), a3)), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(multiply(a3, multiply(inverse(inverse(c3)), b3))), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by axiom 2 (multiply) }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(inverse(double_divide(multiply(inverse(inverse(c3)), b3), a3))), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 25 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(inverse(double_divide(inverse(inverse(c3)), double_divide(inverse(inverse(c3)), double_divide(multiply(inverse(inverse(c3)), b3), a3))))), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by axiom 2 (multiply) }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(multiply(double_divide(inverse(inverse(c3)), double_divide(multiply(inverse(inverse(c3)), b3), a3)), inverse(inverse(c3)))), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(multiply(inverse(inverse(c3)), double_divide(inverse(inverse(c3)), double_divide(multiply(inverse(inverse(c3)), b3), a3)))), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 10 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, inverse(double_divide(inverse(inverse(c3)), double_divide(multiply(inverse(inverse(c3)), b3), a3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by axiom 2 (multiply) }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(double_divide(multiply(inverse(inverse(c3)), b3), a3), inverse(inverse(c3)))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(multiply(a3, multiply(inverse(inverse(c3)), double_divide(multiply(inverse(inverse(c3)), b3), a3))), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by axiom 2 (multiply) }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(inverse(double_divide(multiply(inverse(inverse(c3)), double_divide(multiply(inverse(inverse(c3)), b3), a3)), a3)), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 21 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, double_divide(multiply(multiply(inverse(b3), ?), double_divide(?, ?)), ?)))
0.21/0.53	= { by lemma 4 }
0.21/0.53	  double_divide(inverse(c3), double_divide(a3, b3))
0.21/0.53	= { by lemma 20 }
0.21/0.53	  double_divide(double_divide(a3, b3), inverse(c3))
0.21/0.53	= { by lemma 15 }
0.21/0.53	  multiply(inverse(double_divide(a3, b3)), c3)
0.21/0.53	= { by lemma 18 }
0.21/0.53	  multiply(c3, inverse(double_divide(a3, b3)))
0.21/0.53	= { by axiom 2 (multiply) }
0.21/0.53	  multiply(c3, multiply(b3, a3))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  multiply(c3, multiply(a3, b3))
0.21/0.53	= { by lemma 18 }
0.21/0.53	  multiply(multiply(a3, b3), c3)
0.21/0.53	% SZS output end Proof
0.21/0.53	
0.21/0.53	RESULT: Unsatisfiable (the axioms are contradictory).
0.21/0.53	EOF