0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.11/0.33	% Computer   : n025.cluster.edu
0.11/0.33	% Model      : x86_64 x86_64
0.11/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.33	% Memory     : 8042.1875MB
0.11/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.33	% CPULimit   : 180
0.11/0.33	% DateTime   : Thu Aug 29 11:01:00 EDT 2019
0.11/0.33	% CPUTime    : 
0.17/0.43	% SZS status Unsatisfiable
0.17/0.43	
0.17/0.43	% SZS output start Proof
0.17/0.43	Take the following subset of the input axioms:
0.17/0.45	  fof(identity, axiom, ![A]: double_divide(A, inverse(A))=identity).
0.17/0.45	  fof(inverse, axiom, ![A]: double_divide(A, identity)=inverse(A)).
0.17/0.45	  fof(multiply, axiom, ![A, B]: multiply(A, B)=double_divide(double_divide(B, A), identity)).
0.17/0.45	  fof(prove_these_axioms_3, negated_conjecture, multiply(a3, multiply(b3, c3))!=multiply(multiply(a3, b3), c3)).
0.17/0.45	  fof(single_axiom, axiom, ![A, B, C]: double_divide(double_divide(A, double_divide(double_divide(B, double_divide(C, A)), double_divide(C, identity))), double_divide(identity, identity))=B).
0.17/0.45	
0.17/0.45	Now clausify the problem and encode Horn clauses using encoding 3 of
0.17/0.45	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.17/0.45	We repeatedly replace C & s=t => u=v by the two clauses:
0.17/0.45	  fresh(y, y, x1...xn) = u
0.17/0.45	  C => fresh(s, t, x1...xn) = v
0.17/0.45	where fresh is a fresh function symbol and x1..xn are the free
0.17/0.45	variables of u and v.
0.17/0.45	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.17/0.45	input problem has no model of domain size 1).
0.17/0.45	
0.17/0.45	The encoding turns the above axioms into the following unit equations and goals:
0.17/0.45	
0.17/0.45	Axiom 1 (single_axiom): double_divide(double_divide(X, double_divide(double_divide(Y, double_divide(Z, X)), double_divide(Z, identity))), double_divide(identity, identity)) = Y.
0.17/0.45	Axiom 2 (inverse): double_divide(X, identity) = inverse(X).
0.17/0.45	Axiom 3 (multiply): multiply(X, Y) = double_divide(double_divide(Y, X), identity).
0.17/0.46	Axiom 4 (identity): double_divide(X, inverse(X)) = identity.
0.17/0.46	
0.17/0.46	Lemma 5: inverse(double_divide(Y, X)) = multiply(X, Y).
0.17/0.46	Proof:
0.17/0.46	  inverse(double_divide(Y, X))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(Y, X), identity)
0.17/0.46	= { by axiom 3 (multiply) }
0.17/0.46	  multiply(X, Y)
0.17/0.46	
0.17/0.46	Lemma 6: double_divide(double_divide(identity, double_divide(double_divide(X, inverse(Y)), inverse(Y))), inverse(identity)) = X.
0.17/0.46	Proof:
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(X, inverse(Y)), inverse(Y))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(X, double_divide(Y, identity)), inverse(Y))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(X, double_divide(Y, identity)), double_divide(Y, identity))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(X, double_divide(Y, identity)), double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.46	= { by axiom 1 (single_axiom) }
0.17/0.46	  X
0.17/0.46	
0.17/0.46	Lemma 7: double_divide(double_divide(identity, double_divide(identity, inverse(X))), inverse(identity)) = X.
0.17/0.46	Proof:
0.17/0.46	  double_divide(double_divide(identity, double_divide(identity, inverse(X))), inverse(identity))
0.17/0.46	= { by axiom 4 (identity) }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(X, inverse(X)), inverse(X))), inverse(identity))
0.17/0.46	= { by lemma 6 }
0.17/0.46	  X
0.17/0.46	
0.17/0.46	Lemma 8: inverse(identity) = identity.
0.17/0.46	Proof:
0.17/0.46	  inverse(identity)
0.17/0.46	= { by lemma 6 }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(inverse(identity), inverse(identity)), inverse(identity))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(double_divide(identity, identity), inverse(identity)), inverse(identity))), inverse(identity))
0.17/0.46	= { by axiom 4 (identity) }
0.17/0.46	  double_divide(double_divide(identity, double_divide(double_divide(double_divide(identity, double_divide(identity, inverse(identity))), inverse(identity)), inverse(identity))), inverse(identity))
0.17/0.46	= { by lemma 7 }
0.17/0.46	  double_divide(double_divide(identity, double_divide(identity, inverse(identity))), inverse(identity))
0.17/0.46	= { by lemma 7 }
0.17/0.46	  identity
0.17/0.46	
0.17/0.46	Lemma 9: inverse(inverse(X)) = multiply(identity, X).
0.17/0.46	Proof:
0.17/0.46	  inverse(inverse(X))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  inverse(double_divide(X, identity))
0.17/0.46	= { by lemma 5 }
0.17/0.46	  multiply(identity, X)
0.17/0.46	
0.17/0.46	Lemma 10: double_divide(double_divide(inverse(Y), double_divide(inverse(X), inverse(Y))), inverse(identity)) = X.
0.17/0.46	Proof:
0.17/0.46	  double_divide(double_divide(inverse(Y), double_divide(inverse(X), inverse(Y))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(inverse(Y), double_divide(double_divide(X, identity), inverse(Y))), inverse(identity))
0.17/0.46	= { by axiom 4 (identity) }
0.17/0.46	  double_divide(double_divide(inverse(Y), double_divide(double_divide(X, double_divide(Y, inverse(Y))), inverse(Y))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(inverse(Y), double_divide(double_divide(X, double_divide(Y, inverse(Y))), double_divide(Y, identity))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(inverse(Y), double_divide(double_divide(X, double_divide(Y, inverse(Y))), double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.46	= { by axiom 1 (single_axiom) }
0.17/0.46	  X
0.17/0.46	
0.17/0.46	Lemma 11: double_divide(multiply(identity, inverse(X)), inverse(identity)) = X.
0.17/0.46	Proof:
0.17/0.46	  double_divide(multiply(identity, inverse(X)), inverse(identity))
0.17/0.46	= { by lemma 9 }
0.17/0.46	  double_divide(inverse(inverse(inverse(X))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(double_divide(inverse(inverse(X)), identity), inverse(identity))
0.17/0.46	= { by axiom 4 (identity) }
0.17/0.46	  double_divide(double_divide(inverse(inverse(X)), double_divide(inverse(X), inverse(inverse(X)))), inverse(identity))
0.17/0.46	= { by lemma 10 }
0.17/0.46	  X
0.17/0.46	
0.17/0.46	Lemma 12: double_divide(multiply(identity, multiply(Y, X)), inverse(identity)) = double_divide(X, Y).
0.17/0.46	Proof:
0.17/0.46	  double_divide(multiply(identity, multiply(Y, X)), inverse(identity))
0.17/0.46	= { by lemma 5 }
0.17/0.46	  double_divide(multiply(identity, inverse(double_divide(X, Y))), inverse(identity))
0.17/0.46	= { by lemma 11 }
0.17/0.46	  double_divide(X, Y)
0.17/0.46	
0.17/0.46	Lemma 13: inverse(multiply(identity, X)) = multiply(identity, inverse(X)).
0.17/0.46	Proof:
0.17/0.46	  inverse(multiply(identity, X))
0.17/0.46	= { by lemma 9 }
0.17/0.46	  inverse(inverse(inverse(X)))
0.17/0.46	= { by lemma 9 }
0.17/0.46	  multiply(identity, inverse(X))
0.17/0.46	
0.17/0.46	Lemma 14: multiply(identity, multiply(identity, X)) = X.
0.17/0.46	Proof:
0.17/0.46	  multiply(identity, multiply(identity, X))
0.17/0.46	= { by lemma 9 }
0.17/0.46	  multiply(identity, inverse(inverse(X)))
0.17/0.46	= { by lemma 13 }
0.17/0.46	  inverse(multiply(identity, inverse(X)))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(multiply(identity, inverse(X)), identity)
0.17/0.46	= { by lemma 8 }
0.17/0.46	  double_divide(multiply(identity, inverse(X)), inverse(identity))
0.17/0.46	= { by lemma 11 }
0.17/0.46	  X
0.17/0.46	
0.17/0.46	Lemma 15: double_divide(identity, X) = inverse(X).
0.17/0.46	Proof:
0.17/0.46	  double_divide(identity, X)
0.17/0.46	= { by lemma 12 }
0.17/0.46	  double_divide(multiply(identity, multiply(X, identity)), inverse(identity))
0.17/0.46	= { by lemma 14 }
0.17/0.46	  double_divide(multiply(identity, multiply(multiply(identity, multiply(identity, X)), identity)), inverse(identity))
0.17/0.46	= { by lemma 9 }
0.17/0.46	  double_divide(multiply(identity, multiply(inverse(inverse(multiply(identity, X))), identity)), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(multiply(identity, multiply(double_divide(inverse(multiply(identity, X)), identity), identity)), inverse(identity))
0.17/0.46	= { by lemma 5 }
0.17/0.46	  double_divide(multiply(identity, inverse(double_divide(identity, double_divide(inverse(multiply(identity, X)), identity)))), inverse(identity))
0.17/0.46	= { by axiom 2 (inverse) }
0.17/0.46	  double_divide(multiply(identity, double_divide(double_divide(identity, double_divide(inverse(multiply(identity, X)), identity)), identity)), inverse(identity))
0.17/0.46	= { by lemma 8 }
0.17/0.46	  double_divide(multiply(identity, double_divide(double_divide(inverse(identity), double_divide(inverse(multiply(identity, X)), identity)), identity)), inverse(identity))
0.17/0.46	= { by lemma 8 }
0.17/0.46	  double_divide(multiply(identity, double_divide(double_divide(inverse(identity), double_divide(inverse(multiply(identity, X)), inverse(identity))), identity)), inverse(identity))
0.17/0.46	= { by lemma 8 }
0.17/0.46	  double_divide(multiply(identity, double_divide(double_divide(inverse(identity), double_divide(inverse(multiply(identity, X)), inverse(identity))), inverse(identity))), inverse(identity))
0.17/0.46	= { by lemma 10 }
0.17/0.46	  double_divide(multiply(identity, multiply(identity, X)), inverse(identity))
0.17/0.46	= { by lemma 9 }
0.17/0.46	  double_divide(multiply(identity, inverse(inverse(X))), inverse(identity))
0.17/0.46	= { by lemma 11 }
0.17/0.46	  inverse(X)
0.17/0.46	
0.17/0.46	Lemma 16: double_divide(double_divide(X, Y), multiply(Y, X)) = identity.
0.17/0.46	Proof:
0.17/0.46	  double_divide(double_divide(X, Y), multiply(Y, X))
0.17/0.46	= { by lemma 5 }
0.17/0.46	  double_divide(double_divide(X, Y), inverse(double_divide(X, Y)))
0.17/0.46	= { by axiom 4 (identity) }
0.17/0.47	  identity
0.17/0.47	
0.17/0.47	Lemma 17: multiply(multiply(inverse(Y), X), Y) = X.
0.17/0.47	Proof:
0.17/0.47	  multiply(multiply(inverse(Y), X), Y)
0.17/0.47	= { by lemma 15 }
0.17/0.47	  multiply(multiply(double_divide(identity, Y), X), Y)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  multiply(multiply(double_divide(inverse(identity), Y), X), Y)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  multiply(multiply(double_divide(inverse(inverse(identity)), Y), X), Y)
0.17/0.47	= { by lemma 9 }
0.17/0.47	  multiply(multiply(double_divide(multiply(identity, identity), Y), X), Y)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  multiply(multiply(double_divide(multiply(identity, inverse(identity)), Y), X), Y)
0.17/0.47	= { by lemma 5 }
0.17/0.47	  inverse(double_divide(Y, multiply(double_divide(multiply(identity, inverse(identity)), Y), X)))
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(Y, multiply(double_divide(multiply(identity, inverse(identity)), Y), X)), identity)
0.17/0.47	= { by lemma 5 }
0.17/0.47	  double_divide(double_divide(Y, inverse(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)))), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), identity)), identity)
0.17/0.47	= { by lemma 16 }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), double_divide(double_divide(multiply(identity, identity), identity), multiply(identity, multiply(identity, identity))))), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), double_divide(inverse(multiply(identity, identity)), multiply(identity, multiply(identity, identity))))), identity)
0.17/0.47	= { by lemma 13 }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), double_divide(multiply(identity, inverse(identity)), multiply(identity, multiply(identity, identity))))), identity)
0.17/0.47	= { by lemma 14 }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), double_divide(multiply(identity, inverse(identity)), identity))), identity)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), double_divide(multiply(identity, inverse(identity)), identity))), inverse(identity))
0.17/0.47	= { by lemma 15 }
0.17/0.47	  double_divide(double_divide(Y, double_divide(double_divide(X, double_divide(multiply(identity, inverse(identity)), Y)), double_divide(multiply(identity, inverse(identity)), identity))), double_divide(identity, identity))
0.17/0.47	= { by axiom 1 (single_axiom) }
0.17/0.47	  X
0.17/0.47	
0.17/0.47	Lemma 18: multiply(identity, X) = X.
0.17/0.47	Proof:
0.17/0.47	  multiply(identity, X)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  multiply(inverse(identity), X)
0.17/0.47	= { by axiom 4 (identity) }
0.17/0.47	  multiply(inverse(double_divide(X, inverse(X))), X)
0.17/0.47	= { by lemma 5 }
0.17/0.47	  multiply(multiply(inverse(X), X), X)
0.17/0.47	= { by lemma 17 }
0.17/0.47	  X
0.17/0.47	
0.17/0.47	Lemma 19: multiply(X, Y) = multiply(Y, X).
0.17/0.47	Proof:
0.17/0.47	  multiply(X, Y)
0.17/0.47	= { by lemma 5 }
0.17/0.47	  inverse(double_divide(Y, X))
0.17/0.47	= { by axiom 1 (single_axiom) }
0.17/0.47	  double_divide(double_divide(X, double_divide(double_divide(inverse(double_divide(Y, X)), double_divide(Y, X)), double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(X, double_divide(double_divide(double_divide(double_divide(Y, X), identity), double_divide(Y, X)), double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(double_divide(X, double_divide(double_divide(double_divide(double_divide(Y, X), identity), multiply(identity, double_divide(Y, X))), double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.47	= { by lemma 16 }
0.17/0.47	  double_divide(double_divide(X, double_divide(identity, double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.47	= { by lemma 15 }
0.17/0.47	  double_divide(double_divide(X, inverse(double_divide(Y, identity))), double_divide(identity, identity))
0.17/0.47	= { by lemma 5 }
0.17/0.47	  double_divide(double_divide(X, multiply(identity, Y)), double_divide(identity, identity))
0.17/0.47	= { by lemma 15 }
0.17/0.47	  double_divide(double_divide(X, multiply(identity, Y)), inverse(identity))
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(double_divide(X, multiply(identity, Y)), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  inverse(double_divide(X, multiply(identity, Y)))
0.17/0.47	= { by lemma 5 }
0.17/0.47	  multiply(multiply(identity, Y), X)
0.17/0.47	= { by lemma 18 }
0.17/0.47	  multiply(Y, X)
0.17/0.47	
0.17/0.47	Lemma 20: multiply(identity, double_divide(X, Y)) = inverse(multiply(Y, X)).
0.17/0.47	Proof:
0.17/0.47	  multiply(identity, double_divide(X, Y))
0.17/0.47	= { by lemma 9 }
0.17/0.47	  inverse(inverse(double_divide(X, Y)))
0.17/0.47	= { by lemma 5 }
0.17/0.47	  inverse(multiply(Y, X))
0.17/0.47	
0.17/0.47	Lemma 21: inverse(multiply(X, Y)) = double_divide(X, Y).
0.17/0.47	Proof:
0.17/0.47	  inverse(multiply(X, Y))
0.17/0.47	= { by lemma 19 }
0.17/0.47	  inverse(multiply(Y, X))
0.17/0.47	= { by lemma 20 }
0.17/0.47	  multiply(identity, double_divide(X, Y))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(X, Y)
0.17/0.47	
0.17/0.47	Lemma 22: double_divide(X, Y) = double_divide(Y, X).
0.17/0.47	Proof:
0.17/0.47	  double_divide(X, Y)
0.17/0.47	= { by lemma 21 }
0.17/0.47	  inverse(multiply(X, Y))
0.17/0.47	= { by lemma 20 }
0.17/0.47	  multiply(identity, double_divide(Y, X))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(Y, X)
0.17/0.47	
0.17/0.47	Lemma 23: double_divide(inverse(X), double_divide(inverse(X), inverse(Y))) = inverse(Y).
0.17/0.47	Proof:
0.17/0.47	  double_divide(inverse(X), double_divide(inverse(X), inverse(Y)))
0.17/0.47	= { by lemma 22 }
0.17/0.47	  double_divide(inverse(X), double_divide(inverse(Y), inverse(X)))
0.17/0.47	= { by lemma 6 }
0.17/0.47	  double_divide(double_divide(identity, double_divide(double_divide(double_divide(inverse(X), double_divide(inverse(Y), inverse(X))), inverse(identity)), inverse(identity))), inverse(identity))
0.17/0.47	= { by lemma 10 }
0.17/0.47	  double_divide(double_divide(identity, double_divide(Y, inverse(identity))), inverse(identity))
0.17/0.47	= { by lemma 15 }
0.17/0.47	  double_divide(inverse(double_divide(Y, inverse(identity))), inverse(identity))
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(inverse(double_divide(Y, inverse(identity))), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  inverse(inverse(double_divide(Y, inverse(identity))))
0.17/0.47	= { by lemma 9 }
0.17/0.47	  multiply(identity, double_divide(Y, inverse(identity)))
0.17/0.47	= { by lemma 20 }
0.17/0.47	  inverse(multiply(inverse(identity), Y))
0.17/0.47	= { by lemma 21 }
0.17/0.47	  double_divide(inverse(identity), Y)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(identity, Y)
0.17/0.47	= { by lemma 15 }
0.17/0.47	  inverse(Y)
0.17/0.47	
0.17/0.47	Lemma 24: double_divide(inverse(Y), double_divide(X, inverse(Y))) = X.
0.17/0.47	Proof:
0.17/0.47	  double_divide(inverse(Y), double_divide(X, inverse(Y)))
0.17/0.47	= { by lemma 22 }
0.17/0.47	  double_divide(inverse(Y), double_divide(inverse(Y), X))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(inverse(Y), double_divide(inverse(Y), multiply(identity, X)))
0.17/0.47	= { by lemma 9 }
0.17/0.47	  double_divide(inverse(Y), double_divide(inverse(Y), inverse(inverse(X))))
0.17/0.47	= { by lemma 23 }
0.17/0.47	  inverse(inverse(X))
0.17/0.47	= { by lemma 9 }
0.17/0.47	  multiply(identity, X)
0.17/0.47	= { by lemma 18 }
0.17/0.47	  X
0.17/0.47	
0.17/0.47	Lemma 25: multiply(Y, inverse(X)) = double_divide(X, inverse(Y)).
0.17/0.47	Proof:
0.17/0.47	  multiply(Y, inverse(X))
0.17/0.47	= { by lemma 19 }
0.17/0.47	  multiply(inverse(X), Y)
0.17/0.47	= { by lemma 24 }
0.17/0.47	  multiply(inverse(double_divide(inverse(inverse(inverse(Y))), double_divide(X, inverse(inverse(inverse(Y)))))), Y)
0.17/0.47	= { by lemma 9 }
0.17/0.47	  multiply(inverse(double_divide(multiply(identity, inverse(Y)), double_divide(X, inverse(inverse(inverse(Y)))))), Y)
0.17/0.47	= { by lemma 18 }
0.17/0.47	  multiply(inverse(double_divide(inverse(Y), double_divide(X, inverse(inverse(inverse(Y)))))), Y)
0.17/0.47	= { by lemma 9 }
0.17/0.47	  multiply(inverse(double_divide(inverse(Y), double_divide(X, multiply(identity, inverse(Y))))), Y)
0.17/0.47	= { by lemma 18 }
0.17/0.47	  multiply(inverse(double_divide(inverse(Y), double_divide(X, inverse(Y)))), Y)
0.17/0.47	= { by lemma 5 }
0.17/0.47	  multiply(multiply(double_divide(X, inverse(Y)), inverse(Y)), Y)
0.17/0.47	= { by lemma 19 }
0.17/0.47	  multiply(multiply(inverse(Y), double_divide(X, inverse(Y))), Y)
0.17/0.47	= { by lemma 22 }
0.17/0.47	  multiply(multiply(inverse(Y), double_divide(inverse(Y), X)), Y)
0.17/0.47	= { by lemma 17 }
0.17/0.47	  double_divide(inverse(Y), X)
0.17/0.47	= { by lemma 22 }
0.17/0.47	  double_divide(X, inverse(Y))
0.17/0.47	
0.17/0.47	Lemma 26: double_divide(Y, double_divide(Y, inverse(X))) = inverse(X).
0.17/0.47	Proof:
0.17/0.47	  double_divide(Y, double_divide(Y, inverse(X)))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(multiply(identity, Y), double_divide(Y, inverse(X)))
0.17/0.47	= { by lemma 9 }
0.17/0.47	  double_divide(inverse(inverse(Y)), double_divide(Y, inverse(X)))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(inverse(inverse(Y)), double_divide(multiply(identity, Y), inverse(X)))
0.17/0.47	= { by lemma 9 }
0.17/0.47	  double_divide(inverse(inverse(Y)), double_divide(inverse(inverse(Y)), inverse(X)))
0.17/0.47	= { by lemma 23 }
0.17/0.47	  inverse(X)
0.17/0.47	
0.17/0.47	Lemma 27: double_divide(inverse(X), inverse(Y)) = multiply(X, Y).
0.17/0.47	Proof:
0.17/0.47	  double_divide(inverse(X), inverse(Y))
0.17/0.47	= { by lemma 17 }
0.17/0.47	  multiply(multiply(inverse(X), double_divide(inverse(X), inverse(Y))), X)
0.17/0.47	= { by lemma 19 }
0.17/0.47	  multiply(multiply(double_divide(inverse(X), inverse(Y)), inverse(X)), X)
0.17/0.47	= { by lemma 5 }
0.17/0.47	  multiply(inverse(double_divide(inverse(X), double_divide(inverse(X), inverse(Y)))), X)
0.17/0.47	= { by lemma 26 }
0.17/0.47	  multiply(inverse(inverse(Y)), X)
0.17/0.47	= { by lemma 9 }
0.17/0.47	  multiply(multiply(identity, Y), X)
0.17/0.47	= { by lemma 18 }
0.17/0.47	  multiply(Y, X)
0.17/0.47	= { by lemma 19 }
0.17/0.47	  multiply(X, Y)
0.17/0.47	
0.17/0.47	Lemma 28: multiply(multiply(Y, Z), multiply(X, double_divide(Y, Z))) = X.
0.17/0.47	Proof:
0.17/0.47	  multiply(multiply(Y, Z), multiply(X, double_divide(Y, Z)))
0.17/0.47	= { by lemma 22 }
0.17/0.47	  multiply(multiply(Y, Z), multiply(X, double_divide(Z, Y)))
0.17/0.47	= { by lemma 19 }
0.17/0.47	  multiply(multiply(X, double_divide(Z, Y)), multiply(Y, Z))
0.17/0.47	= { by lemma 5 }
0.17/0.47	  inverse(double_divide(multiply(Y, Z), multiply(X, double_divide(Z, Y))))
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), multiply(X, double_divide(Z, Y))), identity)
0.17/0.47	= { by lemma 27 }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), double_divide(inverse(X), inverse(double_divide(Z, Y)))), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), double_divide(double_divide(X, identity), inverse(double_divide(Z, Y)))), identity)
0.17/0.47	= { by lemma 16 }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), double_divide(double_divide(X, double_divide(double_divide(Z, Y), multiply(Y, Z))), inverse(double_divide(Z, Y)))), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), double_divide(double_divide(X, double_divide(double_divide(Z, Y), multiply(Y, Z))), double_divide(double_divide(Z, Y), identity))), identity)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), double_divide(double_divide(X, double_divide(double_divide(Z, Y), multiply(Y, Z))), double_divide(double_divide(Z, Y), identity))), inverse(identity))
0.17/0.47	= { by lemma 15 }
0.17/0.47	  double_divide(double_divide(multiply(Y, Z), double_divide(double_divide(X, double_divide(double_divide(Z, Y), multiply(Y, Z))), double_divide(double_divide(Z, Y), identity))), double_divide(identity, identity))
0.17/0.47	= { by axiom 1 (single_axiom) }
0.17/0.47	  X
0.17/0.47	
0.17/0.47	Lemma 29: double_divide(multiply(X, Y), multiply(Z, double_divide(X, Y))) = inverse(Z).
0.17/0.47	Proof:
0.17/0.47	  double_divide(multiply(X, Y), multiply(Z, double_divide(X, Y)))
0.17/0.47	= { by lemma 19 }
0.17/0.47	  double_divide(multiply(Y, X), multiply(Z, double_divide(X, Y)))
0.17/0.47	= { by lemma 22 }
0.17/0.47	  double_divide(multiply(Y, X), multiply(Z, double_divide(Y, X)))
0.17/0.47	= { by lemma 22 }
0.17/0.47	  double_divide(multiply(Z, double_divide(Y, X)), multiply(Y, X))
0.17/0.47	= { by lemma 12 }
0.17/0.47	  double_divide(multiply(identity, multiply(multiply(Y, X), multiply(Z, double_divide(Y, X)))), inverse(identity))
0.17/0.47	= { by lemma 28 }
0.17/0.47	  double_divide(multiply(identity, Z), inverse(identity))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  double_divide(Z, inverse(identity))
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(Z, identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  inverse(Z)
0.17/0.47	
0.17/0.47	Lemma 30: multiply(double_divide(Y, inverse(Z)), double_divide(Z, double_divide(X, Y))) = X.
0.17/0.47	Proof:
0.17/0.47	  multiply(double_divide(Y, inverse(Z)), double_divide(Z, double_divide(X, Y)))
0.17/0.47	= { by lemma 22 }
0.17/0.47	  multiply(double_divide(Y, inverse(Z)), double_divide(double_divide(X, Y), Z))
0.17/0.47	= { by lemma 19 }
0.17/0.47	  multiply(double_divide(double_divide(X, Y), Z), double_divide(Y, inverse(Z)))
0.17/0.47	= { by lemma 18 }
0.17/0.47	  multiply(double_divide(double_divide(X, Y), multiply(identity, Z)), double_divide(Y, inverse(Z)))
0.17/0.47	= { by lemma 5 }
0.17/0.47	  inverse(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, Y), multiply(identity, Z))))
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, Y), multiply(identity, Z))), identity)
0.17/0.47	= { by lemma 24 }
0.17/0.47	  double_divide(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, double_divide(inverse(Z), double_divide(Y, inverse(Z)))), multiply(identity, Z))), identity)
0.17/0.47	= { by lemma 9 }
0.17/0.47	  double_divide(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, double_divide(inverse(Z), double_divide(Y, inverse(Z)))), inverse(inverse(Z)))), identity)
0.17/0.47	= { by axiom 2 (inverse) }
0.17/0.47	  double_divide(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, double_divide(inverse(Z), double_divide(Y, inverse(Z)))), double_divide(inverse(Z), identity))), identity)
0.17/0.47	= { by lemma 8 }
0.17/0.47	  double_divide(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, double_divide(inverse(Z), double_divide(Y, inverse(Z)))), double_divide(inverse(Z), identity))), inverse(identity))
0.17/0.47	= { by lemma 15 }
0.17/0.47	  double_divide(double_divide(double_divide(Y, inverse(Z)), double_divide(double_divide(X, double_divide(inverse(Z), double_divide(Y, inverse(Z)))), double_divide(inverse(Z), identity))), double_divide(identity, identity))
0.17/0.47	= { by axiom 1 (single_axiom) }
0.17/0.48	  X
0.17/0.48	
0.17/0.48	Lemma 31: double_divide(inverse(X), multiply(X, Y)) = inverse(Y).
0.17/0.48	Proof:
0.17/0.48	  double_divide(inverse(X), multiply(X, Y))
0.17/0.48	= { by lemma 19 }
0.17/0.48	  double_divide(inverse(X), multiply(Y, X))
0.17/0.48	= { by lemma 22 }
0.17/0.48	  double_divide(multiply(Y, X), inverse(X))
0.17/0.48	= { by lemma 12 }
0.17/0.48	  double_divide(multiply(identity, multiply(inverse(X), multiply(Y, X))), inverse(identity))
0.17/0.48	= { by lemma 14 }
0.17/0.48	  double_divide(multiply(identity, multiply(multiply(identity, multiply(identity, inverse(X))), multiply(Y, X))), inverse(identity))
0.17/0.48	= { by lemma 9 }
0.17/0.48	  double_divide(multiply(identity, multiply(inverse(inverse(multiply(identity, inverse(X)))), multiply(Y, X))), inverse(identity))
0.17/0.48	= { by lemma 15 }
0.17/0.48	  double_divide(multiply(identity, multiply(double_divide(identity, inverse(multiply(identity, inverse(X)))), multiply(Y, X))), inverse(identity))
0.17/0.48	= { by lemma 25 }
0.17/0.48	  double_divide(multiply(identity, multiply(multiply(multiply(identity, inverse(X)), inverse(identity)), multiply(Y, X))), inverse(identity))
0.17/0.48	= { by lemma 11 }
0.17/0.48	  double_divide(multiply(identity, multiply(multiply(multiply(identity, inverse(X)), inverse(identity)), multiply(Y, double_divide(multiply(identity, inverse(X)), inverse(identity))))), inverse(identity))
0.17/0.48	= { by lemma 28 }
0.17/0.48	  double_divide(multiply(identity, Y), inverse(identity))
0.17/0.48	= { by lemma 18 }
0.17/0.48	  double_divide(Y, inverse(identity))
0.17/0.48	= { by lemma 8 }
0.17/0.48	  double_divide(Y, identity)
0.17/0.48	= { by axiom 2 (inverse) }
0.17/0.49	  inverse(Y)
0.17/0.49	
0.17/0.49	Goal 1 (prove_these_axioms_3): multiply(a3, multiply(b3, c3)) = multiply(multiply(a3, b3), c3).
0.17/0.49	Proof:
0.17/0.49	  multiply(a3, multiply(b3, c3))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  multiply(multiply(b3, c3), a3)
0.17/0.49	= { by lemma 5 }
0.17/0.49	  inverse(double_divide(a3, multiply(b3, c3)))
0.17/0.49	= { by lemma 31 }
0.17/0.49	  double_divide(inverse(c3), multiply(c3, double_divide(a3, multiply(b3, c3))))
0.17/0.49	= { by lemma 22 }
0.17/0.49	  double_divide(inverse(c3), multiply(c3, double_divide(multiply(b3, c3), a3)))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  double_divide(inverse(c3), multiply(double_divide(multiply(b3, c3), a3), c3))
0.17/0.49	= { by lemma 5 }
0.17/0.49	  double_divide(inverse(c3), inverse(double_divide(c3, double_divide(multiply(b3, c3), a3))))
0.17/0.49	= { by lemma 31 }
0.17/0.49	  double_divide(inverse(c3), double_divide(inverse(double_divide(a3, inverse(c3))), multiply(double_divide(a3, inverse(c3)), double_divide(c3, double_divide(multiply(b3, c3), a3)))))
0.17/0.49	= { by lemma 5 }
0.17/0.49	  double_divide(inverse(c3), double_divide(multiply(inverse(c3), a3), multiply(double_divide(a3, inverse(c3)), double_divide(c3, double_divide(multiply(b3, c3), a3)))))
0.17/0.49	= { by lemma 30 }
0.17/0.49	  double_divide(inverse(c3), double_divide(multiply(inverse(c3), a3), multiply(b3, c3)))
0.17/0.49	= { by lemma 22 }
0.17/0.49	  double_divide(inverse(c3), double_divide(multiply(b3, c3), multiply(inverse(c3), a3)))
0.17/0.49	= { by lemma 27 }
0.17/0.49	  double_divide(inverse(c3), double_divide(double_divide(inverse(b3), inverse(c3)), multiply(inverse(c3), a3)))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  double_divide(inverse(c3), double_divide(double_divide(inverse(b3), inverse(c3)), multiply(a3, inverse(c3))))
0.17/0.49	= { by lemma 23 }
0.17/0.49	  double_divide(inverse(c3), double_divide(double_divide(inverse(b3), inverse(c3)), multiply(a3, double_divide(inverse(b3), double_divide(inverse(b3), inverse(c3))))))
0.17/0.49	= { by lemma 22 }
0.17/0.49	  double_divide(inverse(c3), double_divide(double_divide(inverse(b3), inverse(c3)), multiply(a3, double_divide(double_divide(inverse(b3), inverse(c3)), inverse(b3)))))
0.17/0.49	= { by lemma 22 }
0.17/0.49	  double_divide(inverse(c3), double_divide(multiply(a3, double_divide(double_divide(inverse(b3), inverse(c3)), inverse(b3))), double_divide(inverse(b3), inverse(c3))))
0.17/0.49	= { by lemma 30 }
0.17/0.49	  double_divide(inverse(c3), double_divide(multiply(a3, double_divide(double_divide(inverse(b3), inverse(c3)), inverse(b3))), multiply(double_divide(inverse(b3), inverse(a3)), double_divide(a3, double_divide(double_divide(inverse(b3), inverse(c3)), inverse(b3))))))
0.17/0.49	= { by lemma 29 }
0.17/0.49	  double_divide(inverse(c3), inverse(double_divide(inverse(b3), inverse(a3))))
0.17/0.49	= { by lemma 5 }
0.17/0.49	  double_divide(inverse(c3), multiply(inverse(a3), inverse(b3)))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  double_divide(inverse(c3), multiply(inverse(b3), inverse(a3)))
0.17/0.49	= { by lemma 25 }
0.17/0.49	  double_divide(inverse(c3), double_divide(a3, inverse(inverse(b3))))
0.17/0.49	= { by lemma 9 }
0.17/0.49	  double_divide(inverse(c3), double_divide(a3, multiply(identity, b3)))
0.17/0.49	= { by lemma 18 }
0.17/0.49	  double_divide(inverse(c3), double_divide(a3, b3))
0.17/0.49	= { by lemma 22 }
0.17/0.49	  double_divide(double_divide(a3, b3), inverse(c3))
0.17/0.49	= { by lemma 25 }
0.17/0.49	  multiply(c3, inverse(double_divide(a3, b3)))
0.17/0.49	= { by lemma 15 }
0.17/0.49	  multiply(c3, double_divide(identity, double_divide(a3, b3)))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  multiply(double_divide(identity, double_divide(a3, b3)), c3)
0.17/0.49	= { by lemma 5 }
0.17/0.49	  inverse(double_divide(c3, double_divide(identity, double_divide(a3, b3))))
0.17/0.49	= { by lemma 26 }
0.17/0.49	  double_divide(multiply(identity, double_divide(a3, b3)), double_divide(multiply(identity, double_divide(a3, b3)), inverse(double_divide(c3, double_divide(identity, double_divide(a3, b3))))))
0.17/0.49	= { by lemma 5 }
0.17/0.49	  double_divide(multiply(identity, double_divide(a3, b3)), double_divide(multiply(identity, double_divide(a3, b3)), multiply(double_divide(identity, double_divide(a3, b3)), c3)))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  double_divide(multiply(identity, double_divide(a3, b3)), double_divide(multiply(identity, double_divide(a3, b3)), multiply(c3, double_divide(identity, double_divide(a3, b3)))))
0.17/0.49	= { by lemma 29 }
0.17/0.49	  double_divide(multiply(identity, double_divide(a3, b3)), inverse(c3))
0.17/0.49	= { by lemma 22 }
0.17/0.49	  double_divide(inverse(c3), multiply(identity, double_divide(a3, b3)))
0.17/0.49	= { by lemma 20 }
0.17/0.49	  double_divide(inverse(c3), inverse(multiply(b3, a3)))
0.17/0.49	= { by lemma 27 }
0.17/0.49	  multiply(c3, multiply(b3, a3))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  multiply(c3, multiply(a3, b3))
0.17/0.49	= { by lemma 19 }
0.17/0.49	  multiply(multiply(a3, b3), c3)
0.17/0.49	% SZS output end Proof
0.17/0.49	
0.17/0.49	RESULT: Unsatisfiable (the axioms are contradictory).
0.17/0.49	EOF