TSTP Solution File: GRP077-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP077-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 01:16:53 EDT 2023
% Result : Unsatisfiable 0.21s 0.47s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP077-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n014.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 00:09:45 EDT 2023
% 0.13/0.35 % CPUTime :
% 0.21/0.47 Command-line arguments: --no-flatten-goal
% 0.21/0.47
% 0.21/0.47 % SZS status Unsatisfiable
% 0.21/0.47
% 0.21/0.52 % SZS output start Proof
% 0.21/0.52 Take the following subset of the input axioms:
% 0.21/0.52 fof(identity, axiom, ![X]: identity=double_divide(X, inverse(X))).
% 0.21/0.52 fof(inverse, axiom, ![X2]: inverse(X2)=double_divide(X2, identity)).
% 0.21/0.52 fof(multiply, axiom, ![Y, X2]: multiply(X2, Y)=double_divide(double_divide(Y, X2), identity)).
% 0.21/0.52 fof(prove_these_axioms, negated_conjecture, multiply(inverse(a1), a1)!=identity | (multiply(identity, a2)!=a2 | multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3)))).
% 0.21/0.52 fof(single_axiom, axiom, ![Z, X2, Y2]: double_divide(X2, double_divide(double_divide(double_divide(identity, double_divide(double_divide(X2, identity), double_divide(Y2, Z))), Y2), identity))=Z).
% 0.21/0.52
% 0.21/0.52 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.21/0.52 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.21/0.52 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.21/0.52 fresh(y, y, x1...xn) = u
% 0.21/0.52 C => fresh(s, t, x1...xn) = v
% 0.21/0.52 where fresh is a fresh function symbol and x1..xn are the free
% 0.21/0.52 variables of u and v.
% 0.21/0.52 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.21/0.52 input problem has no model of domain size 1).
% 0.21/0.52
% 0.21/0.52 The encoding turns the above axioms into the following unit equations and goals:
% 0.21/0.52
% 0.21/0.52 Axiom 1 (inverse): inverse(X) = double_divide(X, identity).
% 0.21/0.52 Axiom 2 (identity): identity = double_divide(X, inverse(X)).
% 0.21/0.52 Axiom 3 (multiply): multiply(X, Y) = double_divide(double_divide(Y, X), identity).
% 0.21/0.52 Axiom 4 (single_axiom): double_divide(X, double_divide(double_divide(double_divide(identity, double_divide(double_divide(X, identity), double_divide(Y, Z))), Y), identity)) = Z.
% 0.21/0.52
% 0.21/0.52 Lemma 5: inverse(double_divide(X, Y)) = multiply(Y, X).
% 0.21/0.52 Proof:
% 0.21/0.52 inverse(double_divide(X, Y))
% 0.21/0.52 = { by axiom 1 (inverse) }
% 0.21/0.52 double_divide(double_divide(X, Y), identity)
% 0.21/0.52 = { by axiom 3 (multiply) R->L }
% 0.21/0.52 multiply(Y, X)
% 0.21/0.52
% 0.21/0.52 Lemma 6: double_divide(X, multiply(Y, double_divide(identity, double_divide(inverse(X), double_divide(Y, Z))))) = Z.
% 0.21/0.52 Proof:
% 0.21/0.52 double_divide(X, multiply(Y, double_divide(identity, double_divide(inverse(X), double_divide(Y, Z)))))
% 0.21/0.52 = { by axiom 1 (inverse) }
% 0.21/0.52 double_divide(X, multiply(Y, double_divide(identity, double_divide(double_divide(X, identity), double_divide(Y, Z)))))
% 0.21/0.52 = { by lemma 5 R->L }
% 0.21/0.52 double_divide(X, inverse(double_divide(double_divide(identity, double_divide(double_divide(X, identity), double_divide(Y, Z))), Y)))
% 0.21/0.52 = { by axiom 1 (inverse) }
% 0.21/0.52 double_divide(X, double_divide(double_divide(double_divide(identity, double_divide(double_divide(X, identity), double_divide(Y, Z))), Y), identity))
% 0.21/0.52 = { by axiom 4 (single_axiom) }
% 0.21/0.52 Z
% 0.21/0.52
% 0.21/0.52 Lemma 7: double_divide(X, multiply(Y, double_divide(identity, inverse(inverse(X))))) = inverse(Y).
% 0.21/0.52 Proof:
% 0.21/0.52 double_divide(X, multiply(Y, double_divide(identity, inverse(inverse(X)))))
% 0.21/0.52 = { by axiom 1 (inverse) }
% 0.21/0.52 double_divide(X, multiply(Y, double_divide(identity, double_divide(inverse(X), identity))))
% 0.21/0.52 = { by axiom 2 (identity) }
% 0.21/0.52 double_divide(X, multiply(Y, double_divide(identity, double_divide(inverse(X), double_divide(Y, inverse(Y))))))
% 0.21/0.52 = { by lemma 6 }
% 0.21/0.52 inverse(Y)
% 0.21/0.52
% 0.21/0.52 Lemma 8: multiply(identity, X) = inverse(inverse(X)).
% 0.21/0.52 Proof:
% 0.21/0.52 multiply(identity, X)
% 0.21/0.52 = { by lemma 5 R->L }
% 0.21/0.52 inverse(double_divide(X, identity))
% 0.21/0.52 = { by axiom 1 (inverse) R->L }
% 0.21/0.52 inverse(inverse(X))
% 0.21/0.52
% 0.21/0.52 Lemma 9: multiply(inverse(X), X) = inverse(identity).
% 0.21/0.52 Proof:
% 0.21/0.52 multiply(inverse(X), X)
% 0.21/0.52 = { by lemma 5 R->L }
% 0.21/0.52 inverse(double_divide(X, inverse(X)))
% 0.21/0.52 = { by axiom 2 (identity) R->L }
% 0.21/0.52 inverse(identity)
% 0.21/0.52
% 0.21/0.52 Lemma 10: inverse(multiply(inverse(inverse(X)), identity)) = double_divide(X, inverse(identity)).
% 0.21/0.52 Proof:
% 0.21/0.52 inverse(multiply(inverse(inverse(X)), identity))
% 0.21/0.52 = { by lemma 5 R->L }
% 0.21/0.52 inverse(inverse(double_divide(identity, inverse(inverse(X)))))
% 0.21/0.52 = { by lemma 7 R->L }
% 0.21/0.52 double_divide(X, multiply(inverse(double_divide(identity, inverse(inverse(X)))), double_divide(identity, inverse(inverse(X)))))
% 0.21/0.52 = { by lemma 9 }
% 0.21/0.52 double_divide(X, inverse(identity))
% 0.21/0.52
% 0.21/0.52 Lemma 11: double_divide(X, multiply(inverse(X), identity)) = inverse(identity).
% 0.21/0.52 Proof:
% 0.21/0.52 double_divide(X, multiply(inverse(X), identity))
% 0.21/0.52 = { by axiom 2 (identity) }
% 0.21/0.52 double_divide(X, multiply(inverse(X), double_divide(identity, inverse(identity))))
% 0.21/0.52 = { by lemma 7 R->L }
% 0.21/0.52 double_divide(X, multiply(inverse(X), double_divide(identity, double_divide(inverse(X), multiply(identity, double_divide(identity, inverse(inverse(inverse(X)))))))))
% 0.21/0.52 = { by lemma 8 }
% 0.21/0.52 double_divide(X, multiply(inverse(X), double_divide(identity, double_divide(inverse(X), inverse(inverse(double_divide(identity, inverse(inverse(inverse(X))))))))))
% 0.21/0.52 = { by lemma 5 }
% 0.21/0.52 double_divide(X, multiply(inverse(X), double_divide(identity, double_divide(inverse(X), inverse(multiply(inverse(inverse(inverse(X))), identity))))))
% 0.21/0.52 = { by lemma 10 }
% 0.21/0.52 double_divide(X, multiply(inverse(X), double_divide(identity, double_divide(inverse(X), double_divide(inverse(X), inverse(identity))))))
% 0.21/0.52 = { by lemma 6 }
% 0.21/0.52 inverse(identity)
% 0.21/0.52
% 0.21/0.52 Lemma 12: inverse(identity) = identity.
% 0.21/0.52 Proof:
% 0.21/0.52 inverse(identity)
% 0.21/0.52 = { by lemma 11 R->L }
% 0.21/0.52 double_divide(identity, multiply(inverse(identity), identity))
% 0.21/0.52 = { by lemma 9 }
% 0.21/0.52 double_divide(identity, inverse(identity))
% 0.21/0.52 = { by axiom 2 (identity) R->L }
% 0.21/0.52 identity
% 0.21/0.52
% 0.21/0.52 Lemma 13: inverse(multiply(inverse(multiply(X, Y)), identity)) = multiply(X, Y).
% 0.21/0.52 Proof:
% 0.21/0.52 inverse(multiply(inverse(multiply(X, Y)), identity))
% 0.21/0.52 = { by lemma 5 R->L }
% 0.21/0.52 inverse(multiply(inverse(inverse(double_divide(Y, X))), identity))
% 0.21/0.52 = { by lemma 10 }
% 0.21/0.52 double_divide(double_divide(Y, X), inverse(identity))
% 0.21/0.52 = { by lemma 12 }
% 0.21/0.52 double_divide(double_divide(Y, X), identity)
% 0.21/0.52 = { by axiom 1 (inverse) R->L }
% 0.21/0.52 inverse(double_divide(Y, X))
% 0.21/0.52 = { by lemma 5 }
% 0.21/0.52 multiply(X, Y)
% 0.21/0.52
% 0.21/0.52 Lemma 14: double_divide(identity, multiply(X, identity)) = inverse(X).
% 0.21/0.52 Proof:
% 0.21/0.52 double_divide(identity, multiply(X, identity))
% 0.21/0.52 = { by axiom 2 (identity) }
% 0.21/0.52 double_divide(identity, multiply(X, double_divide(identity, inverse(identity))))
% 0.21/0.52 = { by lemma 12 R->L }
% 0.21/0.52 double_divide(identity, multiply(X, double_divide(identity, inverse(inverse(identity)))))
% 0.21/0.52 = { by lemma 7 }
% 0.21/0.52 inverse(X)
% 0.21/0.52
% 0.21/0.52 Lemma 15: double_divide(identity, inverse(inverse(X))) = inverse(multiply(X, identity)).
% 0.21/0.52 Proof:
% 0.21/0.52 double_divide(identity, inverse(inverse(X)))
% 0.21/0.52 = { by lemma 14 R->L }
% 0.21/0.52 double_divide(identity, inverse(double_divide(identity, multiply(X, identity))))
% 0.21/0.52 = { by lemma 5 }
% 0.21/0.52 double_divide(identity, multiply(multiply(X, identity), identity))
% 0.21/0.52 = { by lemma 14 }
% 0.21/0.52 inverse(multiply(X, identity))
% 0.21/0.52
% 0.21/0.52 Lemma 16: multiply(inverse(inverse(X)), identity) = inverse(multiply(inverse(X), identity)).
% 0.21/0.52 Proof:
% 0.21/0.52 multiply(inverse(inverse(X)), identity)
% 0.21/0.52 = { by lemma 13 R->L }
% 0.21/0.52 inverse(multiply(inverse(multiply(inverse(inverse(X)), identity)), identity))
% 0.21/0.52 = { by lemma 10 }
% 0.21/0.52 inverse(multiply(double_divide(X, inverse(identity)), identity))
% 0.21/0.52 = { by lemma 12 }
% 0.21/0.52 inverse(multiply(double_divide(X, identity), identity))
% 0.21/0.52 = { by axiom 1 (inverse) R->L }
% 0.21/0.52 inverse(multiply(inverse(X), identity))
% 0.21/0.52
% 0.21/0.52 Lemma 17: inverse(inverse(multiply(inverse(X), identity))) = inverse(X).
% 0.21/0.52 Proof:
% 0.21/0.52 inverse(inverse(multiply(inverse(X), identity)))
% 0.21/0.52 = { by lemma 16 R->L }
% 0.21/0.52 inverse(multiply(inverse(inverse(X)), identity))
% 0.21/0.52 = { by lemma 10 }
% 0.21/0.52 double_divide(X, inverse(identity))
% 0.21/0.52 = { by lemma 12 }
% 0.21/0.52 double_divide(X, identity)
% 0.21/0.52 = { by axiom 1 (inverse) R->L }
% 0.21/0.53 inverse(X)
% 0.21/0.53
% 0.21/0.53 Lemma 18: multiply(inverse(X), identity) = inverse(multiply(X, identity)).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(inverse(X), identity)
% 0.21/0.53 = { by lemma 13 R->L }
% 0.21/0.53 inverse(multiply(inverse(multiply(inverse(X), identity)), identity))
% 0.21/0.53 = { by lemma 15 R->L }
% 0.21/0.53 double_divide(identity, inverse(inverse(inverse(multiply(inverse(X), identity)))))
% 0.21/0.53 = { by lemma 17 }
% 0.21/0.53 double_divide(identity, inverse(inverse(X)))
% 0.21/0.53 = { by lemma 15 }
% 0.21/0.53 inverse(multiply(X, identity))
% 0.21/0.53
% 0.21/0.53 Lemma 19: inverse(inverse(inverse(X))) = inverse(X).
% 0.21/0.53 Proof:
% 0.21/0.53 inverse(inverse(inverse(X)))
% 0.21/0.53 = { by lemma 7 R->L }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, inverse(inverse(Y)))))
% 0.21/0.53 = { by axiom 1 (inverse) }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, double_divide(inverse(Y), identity))))
% 0.21/0.53 = { by lemma 12 R->L }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, double_divide(inverse(Y), inverse(identity)))))
% 0.21/0.53 = { by lemma 11 R->L }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(inverse(X)), multiply(inverse(inverse(inverse(X))), identity))))))
% 0.21/0.53 = { by lemma 16 }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(inverse(X)), inverse(multiply(inverse(inverse(X)), identity)))))))
% 0.21/0.53 = { by lemma 18 }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(inverse(X)), inverse(inverse(multiply(inverse(X), identity))))))))
% 0.21/0.53 = { by lemma 17 }
% 0.21/0.53 double_divide(Y, multiply(inverse(inverse(X)), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(inverse(X)), inverse(X))))))
% 0.21/0.53 = { by lemma 6 }
% 0.21/0.53 inverse(X)
% 0.21/0.53
% 0.21/0.53 Lemma 20: inverse(multiply(X, identity)) = inverse(X).
% 0.21/0.53 Proof:
% 0.21/0.53 inverse(multiply(X, identity))
% 0.21/0.53 = { by lemma 15 R->L }
% 0.21/0.53 double_divide(identity, inverse(inverse(X)))
% 0.21/0.53 = { by lemma 19 R->L }
% 0.21/0.53 double_divide(identity, inverse(inverse(inverse(inverse(X)))))
% 0.21/0.53 = { by lemma 15 }
% 0.21/0.53 inverse(multiply(inverse(inverse(X)), identity))
% 0.21/0.53 = { by lemma 10 }
% 0.21/0.53 double_divide(X, inverse(identity))
% 0.21/0.53 = { by lemma 12 }
% 0.21/0.53 double_divide(X, identity)
% 0.21/0.53 = { by axiom 1 (inverse) R->L }
% 0.21/0.53 inverse(X)
% 0.21/0.53
% 0.21/0.53 Lemma 21: multiply(X, identity) = inverse(inverse(X)).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(X, identity)
% 0.21/0.53 = { by lemma 13 R->L }
% 0.21/0.53 inverse(multiply(inverse(multiply(X, identity)), identity))
% 0.21/0.53 = { by lemma 20 }
% 0.21/0.53 inverse(multiply(inverse(X), identity))
% 0.21/0.53 = { by lemma 20 }
% 0.21/0.53 inverse(inverse(X))
% 0.21/0.53
% 0.21/0.53 Lemma 22: multiply(multiply(X, inverse(Y)), Y) = inverse(inverse(X)).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(multiply(X, inverse(Y)), Y)
% 0.21/0.53 = { by lemma 20 R->L }
% 0.21/0.53 multiply(multiply(X, inverse(multiply(Y, identity))), Y)
% 0.21/0.53 = { by lemma 15 R->L }
% 0.21/0.53 multiply(multiply(X, double_divide(identity, inverse(inverse(Y)))), Y)
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 inverse(double_divide(Y, multiply(X, double_divide(identity, inverse(inverse(Y))))))
% 0.21/0.53 = { by lemma 7 }
% 0.21/0.53 inverse(inverse(X))
% 0.21/0.53
% 0.21/0.53 Lemma 23: inverse(inverse(multiply(X, Y))) = multiply(X, Y).
% 0.21/0.53 Proof:
% 0.21/0.53 inverse(inverse(multiply(X, Y)))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 inverse(inverse(inverse(double_divide(Y, X))))
% 0.21/0.53 = { by lemma 19 }
% 0.21/0.53 inverse(double_divide(Y, X))
% 0.21/0.53 = { by lemma 5 }
% 0.21/0.53 multiply(X, Y)
% 0.21/0.53
% 0.21/0.53 Lemma 24: multiply(inverse(inverse(X)), Y) = multiply(X, inverse(inverse(Y))).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(inverse(inverse(X)), Y)
% 0.21/0.53 = { by lemma 22 R->L }
% 0.21/0.53 multiply(multiply(multiply(X, inverse(inverse(Y))), inverse(Y)), Y)
% 0.21/0.53 = { by lemma 22 }
% 0.21/0.53 inverse(inverse(multiply(X, inverse(inverse(Y)))))
% 0.21/0.53 = { by lemma 23 }
% 0.21/0.53 multiply(X, inverse(inverse(Y)))
% 0.21/0.53
% 0.21/0.53 Lemma 25: multiply(inverse(X), multiply(X, inverse(Y))) = inverse(Y).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(inverse(X), multiply(X, inverse(Y)))
% 0.21/0.53 = { by lemma 19 R->L }
% 0.21/0.53 multiply(inverse(X), multiply(X, inverse(inverse(inverse(Y)))))
% 0.21/0.53 = { by lemma 24 R->L }
% 0.21/0.53 multiply(inverse(X), multiply(inverse(inverse(X)), inverse(Y)))
% 0.21/0.53 = { by axiom 1 (inverse) }
% 0.21/0.53 multiply(inverse(X), multiply(double_divide(inverse(X), identity), inverse(Y)))
% 0.21/0.53 = { by lemma 12 R->L }
% 0.21/0.53 multiply(inverse(X), multiply(double_divide(inverse(X), inverse(identity)), inverse(Y)))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 multiply(inverse(X), inverse(double_divide(inverse(Y), double_divide(inverse(X), inverse(identity)))))
% 0.21/0.53 = { by lemma 20 R->L }
% 0.21/0.53 multiply(inverse(X), inverse(multiply(double_divide(inverse(Y), double_divide(inverse(X), inverse(identity))), identity)))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 multiply(inverse(X), inverse(inverse(double_divide(identity, double_divide(inverse(Y), double_divide(inverse(X), inverse(identity)))))))
% 0.21/0.53 = { by lemma 24 R->L }
% 0.21/0.53 multiply(inverse(inverse(inverse(X))), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(X), inverse(identity)))))
% 0.21/0.53 = { by lemma 20 R->L }
% 0.21/0.53 multiply(inverse(multiply(inverse(inverse(X)), identity)), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(X), inverse(identity)))))
% 0.21/0.53 = { by lemma 16 R->L }
% 0.21/0.53 multiply(multiply(inverse(inverse(inverse(X))), identity), double_divide(identity, double_divide(inverse(Y), double_divide(inverse(X), inverse(identity)))))
% 0.21/0.53 = { by lemma 10 R->L }
% 0.21/0.53 multiply(multiply(inverse(inverse(inverse(X))), identity), double_divide(identity, double_divide(inverse(Y), inverse(multiply(inverse(inverse(inverse(X))), identity)))))
% 0.21/0.53 = { by lemma 6 R->L }
% 0.21/0.53 double_divide(Z, multiply(Y, double_divide(identity, double_divide(inverse(Z), double_divide(Y, multiply(multiply(inverse(inverse(inverse(X))), identity), double_divide(identity, double_divide(inverse(Y), inverse(multiply(inverse(inverse(inverse(X))), identity))))))))))
% 0.21/0.53 = { by axiom 1 (inverse) }
% 0.21/0.53 double_divide(Z, multiply(Y, double_divide(identity, double_divide(inverse(Z), double_divide(Y, multiply(multiply(inverse(inverse(inverse(X))), identity), double_divide(identity, double_divide(inverse(Y), double_divide(multiply(inverse(inverse(inverse(X))), identity), identity)))))))))
% 0.21/0.53 = { by lemma 6 }
% 0.21/0.53 double_divide(Z, multiply(Y, double_divide(identity, double_divide(inverse(Z), identity))))
% 0.21/0.53 = { by axiom 1 (inverse) R->L }
% 0.21/0.53 double_divide(Z, multiply(Y, double_divide(identity, inverse(inverse(Z)))))
% 0.21/0.53 = { by lemma 7 }
% 0.21/0.53 inverse(Y)
% 0.21/0.53
% 0.21/0.53 Lemma 26: multiply(inverse(X), double_divide(inverse(X), Y)) = inverse(Y).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(inverse(X), double_divide(inverse(X), Y))
% 0.21/0.53 = { by lemma 25 R->L }
% 0.21/0.53 multiply(multiply(inverse(Y), multiply(Y, inverse(X))), double_divide(inverse(X), Y))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 multiply(multiply(inverse(Y), inverse(double_divide(inverse(X), Y))), double_divide(inverse(X), Y))
% 0.21/0.53 = { by lemma 22 }
% 0.21/0.53 inverse(inverse(inverse(Y)))
% 0.21/0.53 = { by lemma 19 }
% 0.21/0.53 inverse(Y)
% 0.21/0.53
% 0.21/0.53 Lemma 27: multiply(inverse(X), inverse(inverse(Y))) = multiply(inverse(X), Y).
% 0.21/0.53 Proof:
% 0.21/0.53 multiply(inverse(X), inverse(inverse(Y)))
% 0.21/0.53 = { by lemma 20 R->L }
% 0.21/0.53 multiply(inverse(multiply(X, identity)), inverse(inverse(Y)))
% 0.21/0.53 = { by lemma 18 R->L }
% 0.21/0.53 multiply(multiply(inverse(X), identity), inverse(inverse(Y)))
% 0.21/0.53 = { by lemma 24 R->L }
% 0.21/0.53 multiply(inverse(inverse(multiply(inverse(X), identity))), Y)
% 0.21/0.53 = { by lemma 17 }
% 0.21/0.53 multiply(inverse(X), Y)
% 0.21/0.53
% 0.21/0.53 Lemma 28: inverse(multiply(inverse(X), Y)) = double_divide(Y, inverse(X)).
% 0.21/0.53 Proof:
% 0.21/0.53 inverse(multiply(inverse(X), Y))
% 0.21/0.53 = { by lemma 27 R->L }
% 0.21/0.53 inverse(multiply(inverse(X), inverse(inverse(Y))))
% 0.21/0.53 = { by lemma 20 R->L }
% 0.21/0.53 inverse(multiply(inverse(X), inverse(multiply(inverse(Y), identity))))
% 0.21/0.53 = { by lemma 16 R->L }
% 0.21/0.53 inverse(multiply(inverse(X), multiply(inverse(inverse(Y)), identity)))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 inverse(multiply(inverse(X), inverse(double_divide(identity, inverse(inverse(Y))))))
% 0.21/0.53 = { by lemma 7 R->L }
% 0.21/0.53 double_divide(Y, multiply(multiply(inverse(X), inverse(double_divide(identity, inverse(inverse(Y))))), double_divide(identity, inverse(inverse(Y)))))
% 0.21/0.53 = { by lemma 22 }
% 0.21/0.53 double_divide(Y, inverse(inverse(inverse(X))))
% 0.21/0.53 = { by lemma 19 }
% 0.21/0.53 double_divide(Y, inverse(X))
% 0.21/0.53
% 0.21/0.53 Lemma 29: double_divide(X, multiply(Y, inverse(X))) = inverse(Y).
% 0.21/0.53 Proof:
% 0.21/0.53 double_divide(X, multiply(Y, inverse(X)))
% 0.21/0.53 = { by axiom 1 (inverse) }
% 0.21/0.53 double_divide(X, multiply(Y, double_divide(X, identity)))
% 0.21/0.53 = { by lemma 12 R->L }
% 0.21/0.53 double_divide(X, multiply(Y, double_divide(X, inverse(identity))))
% 0.21/0.53 = { by lemma 10 R->L }
% 0.21/0.53 double_divide(X, multiply(Y, inverse(multiply(inverse(inverse(X)), identity))))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 double_divide(X, multiply(Y, inverse(inverse(double_divide(identity, inverse(inverse(X)))))))
% 0.21/0.53 = { by lemma 24 R->L }
% 0.21/0.53 double_divide(X, multiply(inverse(inverse(Y)), double_divide(identity, inverse(inverse(X)))))
% 0.21/0.53 = { by lemma 7 }
% 0.21/0.53 inverse(inverse(inverse(Y)))
% 0.21/0.53 = { by lemma 19 }
% 0.21/0.53 inverse(Y)
% 0.21/0.53
% 0.21/0.53 Lemma 30: inverse(multiply(X, inverse(Y))) = multiply(Y, inverse(X)).
% 0.21/0.53 Proof:
% 0.21/0.53 inverse(multiply(X, inverse(Y)))
% 0.21/0.53 = { by lemma 26 R->L }
% 0.21/0.53 multiply(inverse(inverse(Y)), double_divide(inverse(inverse(Y)), multiply(X, inverse(Y))))
% 0.21/0.53 = { by axiom 1 (inverse) }
% 0.21/0.53 multiply(inverse(inverse(Y)), double_divide(inverse(inverse(Y)), multiply(X, double_divide(Y, identity))))
% 0.21/0.53 = { by lemma 12 R->L }
% 0.21/0.53 multiply(inverse(inverse(Y)), double_divide(inverse(inverse(Y)), multiply(X, double_divide(Y, inverse(identity)))))
% 0.21/0.53 = { by lemma 20 R->L }
% 0.21/0.53 multiply(inverse(inverse(Y)), double_divide(inverse(multiply(inverse(Y), identity)), multiply(X, double_divide(Y, inverse(identity)))))
% 0.21/0.53 = { by lemma 16 R->L }
% 0.21/0.53 multiply(inverse(inverse(Y)), double_divide(multiply(inverse(inverse(Y)), identity), multiply(X, double_divide(Y, inverse(identity)))))
% 0.21/0.53 = { by lemma 10 R->L }
% 0.21/0.53 multiply(inverse(inverse(Y)), double_divide(multiply(inverse(inverse(Y)), identity), multiply(X, inverse(multiply(inverse(inverse(Y)), identity)))))
% 0.21/0.53 = { by lemma 29 }
% 0.21/0.53 multiply(inverse(inverse(Y)), inverse(X))
% 0.21/0.53 = { by lemma 24 }
% 0.21/0.53 multiply(Y, inverse(inverse(inverse(X))))
% 0.21/0.53 = { by lemma 19 }
% 0.21/0.53 multiply(Y, inverse(X))
% 0.21/0.53
% 0.21/0.53 Lemma 31: double_divide(X, inverse(Y)) = multiply(inverse(X), Y).
% 0.21/0.53 Proof:
% 0.21/0.53 double_divide(X, inverse(Y))
% 0.21/0.53 = { by lemma 28 R->L }
% 0.21/0.53 inverse(multiply(inverse(Y), X))
% 0.21/0.53 = { by lemma 27 R->L }
% 0.21/0.53 inverse(multiply(inverse(Y), inverse(inverse(X))))
% 0.21/0.53 = { by lemma 30 }
% 0.21/0.53 multiply(inverse(X), inverse(inverse(Y)))
% 0.21/0.53 = { by lemma 27 }
% 0.21/0.53 multiply(inverse(X), Y)
% 0.21/0.53
% 0.21/0.53 Lemma 32: inverse(inverse(X)) = X.
% 0.21/0.53 Proof:
% 0.21/0.53 inverse(inverse(X))
% 0.21/0.53 = { by lemma 21 R->L }
% 0.21/0.53 multiply(X, identity)
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 inverse(double_divide(identity, X))
% 0.21/0.53 = { by lemma 26 R->L }
% 0.21/0.53 multiply(inverse(Y), double_divide(inverse(Y), double_divide(identity, X)))
% 0.21/0.53 = { by lemma 27 R->L }
% 0.21/0.53 multiply(inverse(Y), inverse(inverse(double_divide(inverse(Y), double_divide(identity, X)))))
% 0.21/0.53 = { by lemma 21 R->L }
% 0.21/0.53 multiply(inverse(Y), multiply(double_divide(inverse(Y), double_divide(identity, X)), identity))
% 0.21/0.53 = { by lemma 5 R->L }
% 0.21/0.53 multiply(inverse(Y), inverse(double_divide(identity, double_divide(inverse(Y), double_divide(identity, X)))))
% 0.21/0.53 = { by lemma 31 R->L }
% 0.21/0.53 double_divide(Y, inverse(inverse(double_divide(identity, double_divide(inverse(Y), double_divide(identity, X))))))
% 0.21/0.53 = { by lemma 8 R->L }
% 0.21/0.53 double_divide(Y, multiply(identity, double_divide(identity, double_divide(inverse(Y), double_divide(identity, X)))))
% 0.21/0.53 = { by lemma 6 }
% 0.21/0.54 X
% 0.21/0.54
% 0.21/0.54 Lemma 33: inverse(multiply(X, Y)) = double_divide(Y, X).
% 0.21/0.54 Proof:
% 0.21/0.54 inverse(multiply(X, Y))
% 0.21/0.54 = { by lemma 5 R->L }
% 0.21/0.54 inverse(inverse(double_divide(Y, X)))
% 0.21/0.54 = { by lemma 32 }
% 0.21/0.54 double_divide(Y, X)
% 0.21/0.54
% 0.21/0.54 Lemma 34: multiply(multiply(X, Y), inverse(Y)) = X.
% 0.21/0.54 Proof:
% 0.21/0.54 multiply(multiply(X, Y), inverse(Y))
% 0.21/0.54 = { by lemma 32 R->L }
% 0.21/0.54 multiply(multiply(X, inverse(inverse(Y))), inverse(Y))
% 0.21/0.54 = { by lemma 22 }
% 0.21/0.54 inverse(inverse(X))
% 0.21/0.54 = { by lemma 32 }
% 0.21/0.54 X
% 0.21/0.54
% 0.21/0.54 Lemma 35: multiply(X, double_divide(X, Y)) = inverse(Y).
% 0.21/0.54 Proof:
% 0.21/0.54 multiply(X, double_divide(X, Y))
% 0.21/0.54 = { by lemma 33 R->L }
% 0.21/0.54 multiply(X, inverse(multiply(Y, X)))
% 0.21/0.54 = { by lemma 20 R->L }
% 0.21/0.54 multiply(X, inverse(multiply(multiply(Y, X), identity)))
% 0.21/0.54 = { by lemma 15 R->L }
% 0.21/0.54 multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))
% 0.21/0.54 = { by lemma 23 R->L }
% 0.21/0.54 inverse(inverse(multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))))
% 0.21/0.54 = { by lemma 29 R->L }
% 0.21/0.54 double_divide(double_divide(multiply(Y, X), multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), multiply(inverse(multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), inverse(double_divide(multiply(Y, X), multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))))))
% 0.21/0.54 = { by lemma 5 }
% 0.21/0.54 double_divide(double_divide(multiply(Y, X), multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), multiply(inverse(multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), multiply(multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X))))), multiply(Y, X))))
% 0.21/0.54 = { by lemma 5 R->L }
% 0.21/0.54 double_divide(double_divide(multiply(Y, X), multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), multiply(inverse(multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), multiply(multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X))))), inverse(double_divide(X, Y)))))
% 0.21/0.54 = { by lemma 25 }
% 0.21/0.54 double_divide(double_divide(multiply(Y, X), multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), inverse(double_divide(X, Y)))
% 0.21/0.54 = { by lemma 5 }
% 0.21/0.54 double_divide(double_divide(multiply(Y, X), multiply(X, double_divide(identity, inverse(inverse(multiply(Y, X)))))), multiply(Y, X))
% 0.21/0.54 = { by lemma 7 }
% 0.21/0.54 double_divide(inverse(X), multiply(Y, X))
% 0.21/0.54 = { by lemma 33 R->L }
% 0.21/0.54 inverse(multiply(multiply(Y, X), inverse(X)))
% 0.21/0.54 = { by lemma 34 }
% 0.21/0.54 inverse(Y)
% 0.21/0.54
% 0.21/0.54 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3)) = tuple(identity, a2, multiply(a3, multiply(b3, c3))).
% 0.21/0.54 Proof:
% 0.21/0.54 tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3))
% 0.21/0.54 = { by lemma 8 }
% 0.21/0.54 tuple(multiply(inverse(a1), a1), inverse(inverse(a2)), multiply(multiply(a3, b3), c3))
% 0.21/0.54 = { by lemma 9 }
% 0.21/0.54 tuple(inverse(identity), inverse(inverse(a2)), multiply(multiply(a3, b3), c3))
% 0.21/0.54 = { by lemma 12 }
% 0.21/0.54 tuple(identity, inverse(inverse(a2)), multiply(multiply(a3, b3), c3))
% 0.21/0.54 = { by lemma 32 }
% 0.21/0.54 tuple(identity, a2, multiply(multiply(a3, b3), c3))
% 0.21/0.54 = { by lemma 5 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(double_divide(c3, multiply(a3, b3))))
% 0.21/0.54 = { by lemma 33 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(inverse(multiply(multiply(a3, b3), c3))))
% 0.21/0.54 = { by lemma 5 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(inverse(multiply(inverse(double_divide(b3, a3)), c3))))
% 0.21/0.54 = { by lemma 28 }
% 0.21/0.54 tuple(identity, a2, inverse(double_divide(c3, inverse(double_divide(b3, a3)))))
% 0.21/0.54 = { by lemma 31 }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), double_divide(b3, a3))))
% 0.21/0.54 = { by lemma 34 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), double_divide(multiply(multiply(b3, c3), inverse(c3)), a3))))
% 0.21/0.54 = { by lemma 33 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), inverse(multiply(a3, multiply(multiply(b3, c3), inverse(c3)))))))
% 0.21/0.54 = { by lemma 5 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), inverse(multiply(a3, inverse(double_divide(inverse(c3), multiply(b3, c3))))))))
% 0.21/0.54 = { by lemma 30 }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), multiply(double_divide(inverse(c3), multiply(b3, c3)), inverse(a3)))))
% 0.21/0.54 = { by lemma 5 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), inverse(double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3)))))))
% 0.21/0.54 = { by lemma 35 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), multiply(identity, double_divide(identity, double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3))))))))
% 0.21/0.54 = { by lemma 8 }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), inverse(inverse(double_divide(identity, double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3)))))))))
% 0.21/0.54 = { by lemma 32 }
% 0.21/0.54 tuple(identity, a2, inverse(multiply(inverse(c3), double_divide(identity, double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3)))))))
% 0.21/0.54 = { by lemma 32 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(inverse(inverse(multiply(inverse(c3), double_divide(identity, double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3)))))))))
% 0.21/0.54 = { by lemma 35 R->L }
% 0.21/0.54 tuple(identity, a2, inverse(inverse(multiply(a3, double_divide(a3, multiply(inverse(c3), double_divide(identity, double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3))))))))))
% 0.21/0.54 = { by lemma 33 }
% 0.21/0.54 tuple(identity, a2, inverse(double_divide(double_divide(a3, multiply(inverse(c3), double_divide(identity, double_divide(inverse(a3), double_divide(inverse(c3), multiply(b3, c3)))))), a3)))
% 0.21/0.54 = { by lemma 6 }
% 0.21/0.54 tuple(identity, a2, inverse(double_divide(multiply(b3, c3), a3)))
% 0.21/0.54 = { by lemma 5 }
% 0.21/0.54 tuple(identity, a2, multiply(a3, multiply(b3, c3)))
% 0.21/0.54 % SZS output end Proof
% 0.21/0.54
% 0.21/0.54 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------