TSTP Solution File: GRP102-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP102-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 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 : n017.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 01:16:59 EDT 2023
% Result : Unsatisfiable 0.22s 0.50s
% Output : Proof 0.22s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.16/0.16 % Problem : GRP102-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.16/0.17 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.37 % Computer : n017.cluster.edu
% 0.17/0.37 % Model : x86_64 x86_64
% 0.17/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.37 % Memory : 8042.1875MB
% 0.17/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.37 % CPULimit : 300
% 0.17/0.37 % WCLimit : 300
% 0.17/0.37 % DateTime : Tue Aug 29 00:25:13 EDT 2023
% 0.17/0.37 % CPUTime :
% 0.22/0.50 Command-line arguments: --no-flatten-goal
% 0.22/0.50
% 0.22/0.50 % SZS status Unsatisfiable
% 0.22/0.50
% 0.22/0.56 % SZS output start Proof
% 0.22/0.56 Take the following subset of the input axioms:
% 0.22/0.56 fof(identity, axiom, ![X]: identity=double_divide(X, inverse(X))).
% 0.22/0.56 fof(inverse, axiom, ![X2]: inverse(X2)=double_divide(X2, identity)).
% 0.22/0.56 fof(multiply, axiom, ![Y, X2]: multiply(X2, Y)=double_divide(double_divide(Y, X2), identity)).
% 0.22/0.56 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)) | multiply(a4, b4)!=multiply(b4, a4)))).
% 0.22/0.56 fof(single_axiom, axiom, ![Z, X2, Y2]: double_divide(double_divide(X2, double_divide(double_divide(double_divide(Y2, X2), Z), double_divide(Y2, identity))), double_divide(identity, identity))=Z).
% 0.22/0.56
% 0.22/0.56 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.22/0.56 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.22/0.56 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.22/0.56 fresh(y, y, x1...xn) = u
% 0.22/0.56 C => fresh(s, t, x1...xn) = v
% 0.22/0.56 where fresh is a fresh function symbol and x1..xn are the free
% 0.22/0.56 variables of u and v.
% 0.22/0.56 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.22/0.56 input problem has no model of domain size 1).
% 0.22/0.56
% 0.22/0.56 The encoding turns the above axioms into the following unit equations and goals:
% 0.22/0.56
% 0.22/0.56 Axiom 1 (inverse): inverse(X) = double_divide(X, identity).
% 0.22/0.56 Axiom 2 (identity): identity = double_divide(X, inverse(X)).
% 0.22/0.57 Axiom 3 (multiply): multiply(X, Y) = double_divide(double_divide(Y, X), identity).
% 0.22/0.57 Axiom 4 (single_axiom): double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), Z), double_divide(Y, identity))), double_divide(identity, identity)) = Z.
% 0.22/0.57
% 0.22/0.57 Lemma 5: double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), Z), inverse(Y))), inverse(identity)) = Z.
% 0.22/0.57 Proof:
% 0.22/0.57 double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), Z), inverse(Y))), inverse(identity))
% 0.22/0.57 = { by axiom 1 (inverse) }
% 0.22/0.57 double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), Z), inverse(Y))), double_divide(identity, identity))
% 0.22/0.57 = { by axiom 1 (inverse) }
% 0.22/0.57 double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), Z), double_divide(Y, identity))), double_divide(identity, identity))
% 0.22/0.57 = { by axiom 4 (single_axiom) }
% 0.22/0.57 Z
% 0.22/0.57
% 0.22/0.57 Lemma 6: inverse(double_divide(X, Y)) = multiply(Y, X).
% 0.22/0.57 Proof:
% 0.22/0.57 inverse(double_divide(X, Y))
% 0.22/0.57 = { by axiom 1 (inverse) }
% 0.22/0.57 double_divide(double_divide(X, Y), identity)
% 0.22/0.57 = { by axiom 3 (multiply) R->L }
% 0.22/0.57 multiply(Y, X)
% 0.22/0.57
% 0.22/0.57 Lemma 7: double_divide(double_divide(X, double_divide(identity, inverse(Y))), inverse(identity)) = multiply(X, Y).
% 0.22/0.57 Proof:
% 0.22/0.57 double_divide(double_divide(X, double_divide(identity, inverse(Y))), inverse(identity))
% 0.22/0.57 = { by axiom 2 (identity) }
% 0.22/0.57 double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), inverse(double_divide(Y, X))), inverse(Y))), inverse(identity))
% 0.22/0.57 = { by lemma 5 }
% 0.22/0.57 inverse(double_divide(Y, X))
% 0.22/0.57 = { by lemma 6 }
% 0.22/0.57 multiply(X, Y)
% 0.22/0.57
% 0.22/0.57 Lemma 8: double_divide(inverse(X), inverse(identity)) = multiply(X, identity).
% 0.22/0.57 Proof:
% 0.22/0.57 double_divide(inverse(X), inverse(identity))
% 0.22/0.57 = { by axiom 1 (inverse) }
% 0.22/0.57 double_divide(double_divide(X, identity), inverse(identity))
% 0.22/0.57 = { by axiom 2 (identity) }
% 0.22/0.57 double_divide(double_divide(X, double_divide(identity, inverse(identity))), inverse(identity))
% 0.22/0.57 = { by lemma 7 }
% 0.22/0.57 multiply(X, identity)
% 0.22/0.57
% 0.22/0.57 Lemma 9: multiply(identity, X) = inverse(inverse(X)).
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(identity, X)
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 inverse(double_divide(X, identity))
% 0.22/0.57 = { by axiom 1 (inverse) R->L }
% 0.22/0.57 inverse(inverse(X))
% 0.22/0.57
% 0.22/0.57 Lemma 10: multiply(inverse(X), X) = inverse(identity).
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(inverse(X), X)
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 inverse(double_divide(X, inverse(X)))
% 0.22/0.57 = { by axiom 2 (identity) R->L }
% 0.22/0.57 inverse(identity)
% 0.22/0.57
% 0.22/0.57 Lemma 11: double_divide(double_divide(X, double_divide(multiply(X, Y), inverse(Y))), inverse(identity)) = identity.
% 0.22/0.57 Proof:
% 0.22/0.57 double_divide(double_divide(X, double_divide(multiply(X, Y), inverse(Y))), inverse(identity))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 double_divide(double_divide(X, double_divide(inverse(double_divide(Y, X)), inverse(Y))), inverse(identity))
% 0.22/0.57 = { by axiom 1 (inverse) }
% 0.22/0.57 double_divide(double_divide(X, double_divide(double_divide(double_divide(Y, X), identity), inverse(Y))), inverse(identity))
% 0.22/0.57 = { by lemma 5 }
% 0.22/0.57 identity
% 0.22/0.57
% 0.22/0.57 Lemma 12: multiply(double_divide(multiply(X, Y), inverse(Y)), X) = inverse(identity).
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(double_divide(multiply(X, Y), inverse(Y)), X)
% 0.22/0.57 = { by lemma 7 R->L }
% 0.22/0.57 double_divide(double_divide(double_divide(multiply(X, Y), inverse(Y)), double_divide(identity, inverse(X))), inverse(identity))
% 0.22/0.57 = { by lemma 11 R->L }
% 0.22/0.57 double_divide(double_divide(double_divide(multiply(X, Y), inverse(Y)), double_divide(double_divide(double_divide(X, double_divide(multiply(X, Y), inverse(Y))), inverse(identity)), inverse(X))), inverse(identity))
% 0.22/0.57 = { by lemma 5 }
% 0.22/0.57 inverse(identity)
% 0.22/0.57
% 0.22/0.57 Lemma 13: inverse(inverse(inverse(inverse(identity)))) = inverse(identity).
% 0.22/0.57 Proof:
% 0.22/0.57 inverse(inverse(inverse(inverse(identity))))
% 0.22/0.57 = { by lemma 9 R->L }
% 0.22/0.57 multiply(identity, inverse(inverse(identity)))
% 0.22/0.57 = { by axiom 2 (identity) }
% 0.22/0.57 multiply(double_divide(inverse(identity), inverse(inverse(identity))), inverse(inverse(identity)))
% 0.22/0.57 = { by lemma 10 R->L }
% 0.22/0.57 multiply(double_divide(multiply(inverse(inverse(identity)), inverse(identity)), inverse(inverse(identity))), inverse(inverse(identity)))
% 0.22/0.57 = { by lemma 12 }
% 0.22/0.57 inverse(identity)
% 0.22/0.57
% 0.22/0.57 Lemma 14: multiply(inverse(inverse(identity)), identity) = identity.
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(inverse(inverse(identity)), identity)
% 0.22/0.57 = { by lemma 8 R->L }
% 0.22/0.57 double_divide(inverse(inverse(inverse(identity))), inverse(identity))
% 0.22/0.57 = { by lemma 13 R->L }
% 0.22/0.57 double_divide(inverse(inverse(inverse(identity))), inverse(inverse(inverse(inverse(identity)))))
% 0.22/0.57 = { by axiom 2 (identity) R->L }
% 0.22/0.57 identity
% 0.22/0.57
% 0.22/0.57 Lemma 15: double_divide(multiply(double_divide(identity, X), Y), inverse(Y)) = multiply(X, identity).
% 0.22/0.57 Proof:
% 0.22/0.57 double_divide(multiply(double_divide(identity, X), Y), inverse(Y))
% 0.22/0.57 = { by lemma 5 R->L }
% 0.22/0.57 double_divide(double_divide(X, double_divide(double_divide(double_divide(identity, X), double_divide(multiply(double_divide(identity, X), Y), inverse(Y))), inverse(identity))), inverse(identity))
% 0.22/0.57 = { by lemma 11 }
% 0.22/0.57 double_divide(double_divide(X, identity), inverse(identity))
% 0.22/0.57 = { by axiom 1 (inverse) R->L }
% 0.22/0.57 double_divide(inverse(X), inverse(identity))
% 0.22/0.57 = { by lemma 8 }
% 0.22/0.57 multiply(X, identity)
% 0.22/0.57
% 0.22/0.57 Lemma 16: inverse(inverse(identity)) = inverse(identity).
% 0.22/0.57 Proof:
% 0.22/0.57 inverse(inverse(identity))
% 0.22/0.57 = { by axiom 1 (inverse) }
% 0.22/0.57 double_divide(inverse(identity), identity)
% 0.22/0.57 = { by lemma 14 R->L }
% 0.22/0.57 double_divide(inverse(identity), multiply(inverse(inverse(identity)), identity))
% 0.22/0.57 = { by lemma 14 R->L }
% 0.22/0.57 double_divide(inverse(multiply(inverse(inverse(identity)), identity)), multiply(inverse(inverse(identity)), identity))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 double_divide(inverse(multiply(inverse(inverse(identity)), identity)), inverse(double_divide(identity, inverse(inverse(identity)))))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 double_divide(inverse(inverse(double_divide(identity, inverse(inverse(identity))))), inverse(double_divide(identity, inverse(inverse(identity)))))
% 0.22/0.57 = { by lemma 9 R->L }
% 0.22/0.57 double_divide(multiply(identity, double_divide(identity, inverse(inverse(identity)))), inverse(double_divide(identity, inverse(inverse(identity)))))
% 0.22/0.57 = { by axiom 2 (identity) }
% 0.22/0.57 double_divide(multiply(double_divide(identity, inverse(identity)), double_divide(identity, inverse(inverse(identity)))), inverse(double_divide(identity, inverse(inverse(identity)))))
% 0.22/0.57 = { by lemma 15 }
% 0.22/0.57 multiply(inverse(identity), identity)
% 0.22/0.57 = { by lemma 10 }
% 0.22/0.57 inverse(identity)
% 0.22/0.57
% 0.22/0.57 Lemma 17: inverse(identity) = identity.
% 0.22/0.57 Proof:
% 0.22/0.57 inverse(identity)
% 0.22/0.57 = { by lemma 16 R->L }
% 0.22/0.57 inverse(inverse(identity))
% 0.22/0.57 = { by lemma 9 R->L }
% 0.22/0.57 multiply(identity, identity)
% 0.22/0.57 = { by lemma 8 R->L }
% 0.22/0.57 double_divide(inverse(identity), inverse(identity))
% 0.22/0.57 = { by lemma 16 R->L }
% 0.22/0.57 double_divide(inverse(identity), inverse(inverse(identity)))
% 0.22/0.57 = { by axiom 2 (identity) R->L }
% 0.22/0.57 identity
% 0.22/0.57
% 0.22/0.57 Lemma 18: multiply(X, identity) = inverse(inverse(X)).
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(X, identity)
% 0.22/0.57 = { by lemma 7 R->L }
% 0.22/0.57 double_divide(double_divide(X, double_divide(identity, inverse(identity))), inverse(identity))
% 0.22/0.57 = { by lemma 17 }
% 0.22/0.57 double_divide(double_divide(X, double_divide(identity, identity)), inverse(identity))
% 0.22/0.57 = { by axiom 1 (inverse) R->L }
% 0.22/0.57 double_divide(double_divide(X, inverse(identity)), inverse(identity))
% 0.22/0.57 = { by lemma 17 }
% 0.22/0.57 double_divide(double_divide(X, identity), inverse(identity))
% 0.22/0.57 = { by lemma 17 }
% 0.22/0.57 double_divide(double_divide(X, identity), identity)
% 0.22/0.57 = { by axiom 1 (inverse) R->L }
% 0.22/0.57 inverse(double_divide(X, identity))
% 0.22/0.57 = { by lemma 6 }
% 0.22/0.57 multiply(identity, X)
% 0.22/0.57 = { by lemma 9 }
% 0.22/0.57 inverse(inverse(X))
% 0.22/0.57
% 0.22/0.57 Lemma 19: double_divide(multiply(X, Y), inverse(identity)) = multiply(double_divide(Y, X), identity).
% 0.22/0.57 Proof:
% 0.22/0.57 double_divide(multiply(X, Y), inverse(identity))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 double_divide(inverse(double_divide(Y, X)), inverse(identity))
% 0.22/0.57 = { by lemma 8 }
% 0.22/0.57 multiply(double_divide(Y, X), identity)
% 0.22/0.57
% 0.22/0.57 Lemma 20: inverse(inverse(inverse(inverse(X)))) = inverse(inverse(X)).
% 0.22/0.57 Proof:
% 0.22/0.57 inverse(inverse(inverse(inverse(X))))
% 0.22/0.57 = { by lemma 18 R->L }
% 0.22/0.57 inverse(inverse(multiply(X, identity)))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 inverse(inverse(inverse(double_divide(identity, X))))
% 0.22/0.57 = { by lemma 18 R->L }
% 0.22/0.57 inverse(multiply(double_divide(identity, X), identity))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 inverse(inverse(double_divide(identity, double_divide(identity, X))))
% 0.22/0.57 = { by lemma 18 R->L }
% 0.22/0.57 multiply(double_divide(identity, double_divide(identity, X)), identity)
% 0.22/0.57 = { by lemma 19 R->L }
% 0.22/0.57 double_divide(multiply(double_divide(identity, X), identity), inverse(identity))
% 0.22/0.57 = { by lemma 15 }
% 0.22/0.57 multiply(X, identity)
% 0.22/0.57 = { by lemma 18 }
% 0.22/0.57 inverse(inverse(X))
% 0.22/0.57
% 0.22/0.57 Lemma 21: multiply(X, inverse(inverse(inverse(Y)))) = multiply(X, inverse(Y)).
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(X, inverse(inverse(inverse(Y))))
% 0.22/0.57 = { by lemma 7 R->L }
% 0.22/0.57 double_divide(double_divide(X, double_divide(identity, inverse(inverse(inverse(inverse(Y)))))), inverse(identity))
% 0.22/0.57 = { by lemma 20 }
% 0.22/0.57 double_divide(double_divide(X, double_divide(identity, inverse(inverse(Y)))), inverse(identity))
% 0.22/0.57 = { by lemma 7 }
% 0.22/0.57 multiply(X, inverse(Y))
% 0.22/0.57
% 0.22/0.57 Lemma 22: multiply(inverse(identity), inverse(X)) = inverse(multiply(X, identity)).
% 0.22/0.57 Proof:
% 0.22/0.57 multiply(inverse(identity), inverse(X))
% 0.22/0.57 = { by lemma 6 R->L }
% 0.22/0.57 inverse(double_divide(inverse(X), inverse(identity)))
% 0.22/0.57 = { by lemma 8 }
% 0.22/0.58 inverse(multiply(X, identity))
% 0.22/0.58
% 0.22/0.58 Lemma 23: double_divide(identity, inverse(inverse(X))) = inverse(inverse(inverse(X))).
% 0.22/0.58 Proof:
% 0.22/0.58 double_divide(identity, inverse(inverse(X)))
% 0.22/0.58 = { by lemma 20 R->L }
% 0.22/0.58 double_divide(identity, inverse(inverse(inverse(inverse(X)))))
% 0.22/0.58 = { by lemma 18 R->L }
% 0.22/0.58 double_divide(identity, inverse(inverse(multiply(X, identity))))
% 0.22/0.58 = { by lemma 22 R->L }
% 0.22/0.58 double_divide(identity, inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 double_divide(inverse(identity), inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 12 R->L }
% 0.22/0.58 double_divide(multiply(double_divide(multiply(multiply(inverse(identity), inverse(X)), double_divide(inverse(X), inverse(identity))), inverse(double_divide(inverse(X), inverse(identity)))), multiply(inverse(identity), inverse(X))), inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 double_divide(multiply(double_divide(multiply(inverse(double_divide(inverse(X), inverse(identity))), double_divide(inverse(X), inverse(identity))), inverse(double_divide(inverse(X), inverse(identity)))), multiply(inverse(identity), inverse(X))), inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 10 }
% 0.22/0.58 double_divide(multiply(double_divide(inverse(identity), inverse(double_divide(inverse(X), inverse(identity)))), multiply(inverse(identity), inverse(X))), inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 17 }
% 0.22/0.58 double_divide(multiply(double_divide(identity, inverse(double_divide(inverse(X), inverse(identity)))), multiply(inverse(identity), inverse(X))), inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 6 }
% 0.22/0.58 double_divide(multiply(double_divide(identity, multiply(inverse(identity), inverse(X))), multiply(inverse(identity), inverse(X))), inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 15 }
% 0.22/0.58 multiply(multiply(inverse(identity), inverse(X)), identity)
% 0.22/0.58 = { by lemma 18 }
% 0.22/0.58 inverse(inverse(multiply(inverse(identity), inverse(X))))
% 0.22/0.58 = { by lemma 22 }
% 0.22/0.58 inverse(inverse(inverse(multiply(X, identity))))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 inverse(inverse(inverse(inverse(double_divide(identity, X)))))
% 0.22/0.58 = { by lemma 20 }
% 0.22/0.58 inverse(inverse(double_divide(identity, X)))
% 0.22/0.58 = { by lemma 6 }
% 0.22/0.58 inverse(multiply(X, identity))
% 0.22/0.58 = { by lemma 18 }
% 0.22/0.58 inverse(inverse(inverse(X)))
% 0.22/0.58
% 0.22/0.58 Lemma 24: multiply(inverse(inverse(inverse(X))), Y) = multiply(Y, inverse(X)).
% 0.22/0.58 Proof:
% 0.22/0.58 multiply(inverse(inverse(inverse(X))), Y)
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 inverse(double_divide(Y, inverse(inverse(inverse(X)))))
% 0.22/0.58 = { by axiom 1 (inverse) }
% 0.22/0.58 double_divide(double_divide(Y, inverse(inverse(inverse(X)))), identity)
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 double_divide(double_divide(Y, inverse(inverse(inverse(X)))), inverse(identity))
% 0.22/0.58 = { by lemma 23 R->L }
% 0.22/0.58 double_divide(double_divide(Y, double_divide(identity, inverse(inverse(X)))), inverse(identity))
% 0.22/0.58 = { by lemma 7 }
% 0.22/0.58 multiply(Y, inverse(X))
% 0.22/0.58
% 0.22/0.58 Lemma 25: multiply(inverse(inverse(X)), multiply(Y, inverse(X))) = Y.
% 0.22/0.58 Proof:
% 0.22/0.58 multiply(inverse(inverse(X)), multiply(Y, inverse(X)))
% 0.22/0.58 = { by lemma 20 R->L }
% 0.22/0.58 multiply(inverse(inverse(inverse(inverse(X)))), multiply(Y, inverse(X)))
% 0.22/0.58 = { by lemma 24 }
% 0.22/0.58 multiply(multiply(Y, inverse(X)), inverse(inverse(X)))
% 0.22/0.58 = { by lemma 21 R->L }
% 0.22/0.58 multiply(multiply(Y, inverse(inverse(inverse(X)))), inverse(inverse(X)))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 multiply(inverse(double_divide(inverse(inverse(inverse(X))), Y)), inverse(inverse(X)))
% 0.22/0.58 = { by axiom 1 (inverse) }
% 0.22/0.58 multiply(double_divide(double_divide(inverse(inverse(inverse(X))), Y), identity), inverse(inverse(X)))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 inverse(double_divide(inverse(inverse(X)), double_divide(double_divide(inverse(inverse(inverse(X))), Y), identity)))
% 0.22/0.58 = { by axiom 1 (inverse) }
% 0.22/0.58 double_divide(double_divide(inverse(inverse(X)), double_divide(double_divide(inverse(inverse(inverse(X))), Y), identity)), identity)
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 double_divide(double_divide(inverse(inverse(X)), double_divide(double_divide(inverse(inverse(inverse(X))), Y), inverse(identity))), identity)
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 double_divide(double_divide(inverse(inverse(X)), double_divide(double_divide(inverse(inverse(inverse(X))), Y), inverse(identity))), inverse(identity))
% 0.22/0.58 = { by lemma 23 R->L }
% 0.22/0.58 double_divide(double_divide(inverse(inverse(X)), double_divide(double_divide(double_divide(identity, inverse(inverse(X))), Y), inverse(identity))), inverse(identity))
% 0.22/0.58 = { by lemma 5 }
% 0.22/0.58 Y
% 0.22/0.58
% 0.22/0.58 Lemma 26: inverse(inverse(X)) = X.
% 0.22/0.58 Proof:
% 0.22/0.58 inverse(inverse(X))
% 0.22/0.58 = { by lemma 20 R->L }
% 0.22/0.58 inverse(inverse(inverse(inverse(X))))
% 0.22/0.58 = { by lemma 18 R->L }
% 0.22/0.58 inverse(inverse(multiply(X, identity)))
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 inverse(inverse(multiply(X, inverse(identity))))
% 0.22/0.58 = { by lemma 21 R->L }
% 0.22/0.58 inverse(inverse(multiply(X, inverse(inverse(inverse(identity))))))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 inverse(inverse(inverse(double_divide(inverse(inverse(inverse(identity))), X))))
% 0.22/0.58 = { by lemma 18 R->L }
% 0.22/0.58 inverse(multiply(double_divide(inverse(inverse(inverse(identity))), X), identity))
% 0.22/0.58 = { by lemma 22 R->L }
% 0.22/0.58 multiply(inverse(identity), inverse(double_divide(inverse(inverse(inverse(identity))), X)))
% 0.22/0.58 = { by lemma 6 }
% 0.22/0.58 multiply(inverse(identity), multiply(X, inverse(inverse(inverse(identity)))))
% 0.22/0.58 = { by lemma 13 R->L }
% 0.22/0.58 multiply(inverse(inverse(inverse(inverse(identity)))), multiply(X, inverse(inverse(inverse(identity)))))
% 0.22/0.58 = { by lemma 25 }
% 0.22/0.58 X
% 0.22/0.58
% 0.22/0.58 Lemma 27: multiply(Y, X) = multiply(X, Y).
% 0.22/0.58 Proof:
% 0.22/0.58 multiply(Y, X)
% 0.22/0.58 = { by lemma 26 R->L }
% 0.22/0.58 multiply(inverse(inverse(Y)), X)
% 0.22/0.58 = { by lemma 26 R->L }
% 0.22/0.58 multiply(inverse(inverse(inverse(inverse(Y)))), X)
% 0.22/0.58 = { by lemma 24 }
% 0.22/0.58 multiply(X, inverse(inverse(Y)))
% 0.22/0.58 = { by lemma 26 }
% 0.22/0.58 multiply(X, Y)
% 0.22/0.58
% 0.22/0.58 Lemma 28: inverse(multiply(X, Y)) = double_divide(X, Y).
% 0.22/0.58 Proof:
% 0.22/0.58 inverse(multiply(X, Y))
% 0.22/0.58 = { by axiom 1 (inverse) }
% 0.22/0.58 double_divide(multiply(X, Y), identity)
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 double_divide(multiply(X, Y), inverse(identity))
% 0.22/0.58 = { by lemma 27 }
% 0.22/0.58 double_divide(multiply(Y, X), inverse(identity))
% 0.22/0.58 = { by lemma 19 }
% 0.22/0.58 multiply(double_divide(X, Y), identity)
% 0.22/0.58 = { by lemma 18 }
% 0.22/0.58 inverse(inverse(double_divide(X, Y)))
% 0.22/0.58 = { by lemma 26 }
% 0.22/0.58 double_divide(X, Y)
% 0.22/0.58
% 0.22/0.58 Lemma 29: double_divide(Y, X) = double_divide(X, Y).
% 0.22/0.58 Proof:
% 0.22/0.58 double_divide(Y, X)
% 0.22/0.58 = { by lemma 28 R->L }
% 0.22/0.58 inverse(multiply(Y, X))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 inverse(inverse(double_divide(X, Y)))
% 0.22/0.58 = { by lemma 26 }
% 0.22/0.58 double_divide(X, Y)
% 0.22/0.58
% 0.22/0.58 Lemma 30: double_divide(identity, X) = inverse(X).
% 0.22/0.58 Proof:
% 0.22/0.58 double_divide(identity, X)
% 0.22/0.58 = { by lemma 26 R->L }
% 0.22/0.58 double_divide(identity, inverse(inverse(X)))
% 0.22/0.58 = { by lemma 23 }
% 0.22/0.58 inverse(inverse(inverse(X)))
% 0.22/0.58 = { by lemma 26 }
% 0.22/0.58 inverse(X)
% 0.22/0.58
% 0.22/0.58 Lemma 31: multiply(X, double_divide(inverse(Y), multiply(X, Y))) = identity.
% 0.22/0.58 Proof:
% 0.22/0.58 multiply(X, double_divide(inverse(Y), multiply(X, Y)))
% 0.22/0.58 = { by lemma 29 }
% 0.22/0.58 multiply(X, double_divide(multiply(X, Y), inverse(Y)))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 inverse(double_divide(double_divide(multiply(X, Y), inverse(Y)), X))
% 0.22/0.58 = { by lemma 30 R->L }
% 0.22/0.58 double_divide(identity, double_divide(double_divide(multiply(X, Y), inverse(Y)), X))
% 0.22/0.58 = { by lemma 17 R->L }
% 0.22/0.58 double_divide(inverse(identity), double_divide(double_divide(multiply(X, Y), inverse(Y)), X))
% 0.22/0.58 = { by lemma 12 R->L }
% 0.22/0.58 double_divide(multiply(double_divide(multiply(X, Y), inverse(Y)), X), double_divide(double_divide(multiply(X, Y), inverse(Y)), X))
% 0.22/0.58 = { by lemma 29 }
% 0.22/0.58 double_divide(double_divide(double_divide(multiply(X, Y), inverse(Y)), X), multiply(double_divide(multiply(X, Y), inverse(Y)), X))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 double_divide(double_divide(double_divide(multiply(X, Y), inverse(Y)), X), inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y)))))
% 0.22/0.58 = { by lemma 28 R->L }
% 0.22/0.58 double_divide(inverse(multiply(double_divide(multiply(X, Y), inverse(Y)), X)), inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y)))))
% 0.22/0.58 = { by lemma 26 R->L }
% 0.22/0.58 inverse(inverse(double_divide(inverse(multiply(double_divide(multiply(X, Y), inverse(Y)), X)), inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y)))))))
% 0.22/0.58 = { by lemma 18 R->L }
% 0.22/0.58 multiply(double_divide(inverse(multiply(double_divide(multiply(X, Y), inverse(Y)), X)), inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y))))), identity)
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 multiply(double_divide(inverse(inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y))))), inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y))))), identity)
% 0.22/0.58 = { by lemma 9 R->L }
% 0.22/0.58 multiply(double_divide(multiply(identity, double_divide(X, double_divide(multiply(X, Y), inverse(Y)))), inverse(double_divide(X, double_divide(multiply(X, Y), inverse(Y))))), identity)
% 0.22/0.58 = { by lemma 12 }
% 0.22/0.58 inverse(identity)
% 0.22/0.58 = { by lemma 17 }
% 0.22/0.58 identity
% 0.22/0.58
% 0.22/0.58 Lemma 32: double_divide(X, inverse(Y)) = multiply(Y, inverse(X)).
% 0.22/0.58 Proof:
% 0.22/0.58 double_divide(X, inverse(Y))
% 0.22/0.58 = { by lemma 29 }
% 0.22/0.58 double_divide(inverse(Y), X)
% 0.22/0.58 = { by lemma 26 R->L }
% 0.22/0.58 double_divide(inverse(Y), inverse(inverse(X)))
% 0.22/0.58 = { by lemma 18 R->L }
% 0.22/0.58 double_divide(inverse(Y), multiply(X, identity))
% 0.22/0.58 = { by lemma 15 R->L }
% 0.22/0.58 double_divide(inverse(Y), double_divide(multiply(double_divide(identity, X), double_divide(inverse(Y), multiply(double_divide(identity, X), Y))), inverse(double_divide(inverse(Y), multiply(double_divide(identity, X), Y)))))
% 0.22/0.58 = { by lemma 31 }
% 0.22/0.58 double_divide(inverse(Y), double_divide(identity, inverse(double_divide(inverse(Y), multiply(double_divide(identity, X), Y)))))
% 0.22/0.58 = { by lemma 30 }
% 0.22/0.58 double_divide(inverse(Y), inverse(inverse(double_divide(inverse(Y), multiply(double_divide(identity, X), Y)))))
% 0.22/0.58 = { by lemma 26 }
% 0.22/0.58 double_divide(inverse(Y), double_divide(inverse(Y), multiply(double_divide(identity, X), Y)))
% 0.22/0.58 = { by lemma 30 }
% 0.22/0.58 double_divide(inverse(Y), double_divide(inverse(Y), multiply(inverse(X), Y)))
% 0.22/0.58 = { by lemma 27 R->L }
% 0.22/0.58 double_divide(inverse(Y), double_divide(inverse(Y), multiply(Y, inverse(X))))
% 0.22/0.58 = { by lemma 26 R->L }
% 0.22/0.58 double_divide(inverse(Y), double_divide(inverse(Y), inverse(inverse(multiply(Y, inverse(X))))))
% 0.22/0.58 = { by lemma 18 R->L }
% 0.22/0.58 double_divide(inverse(Y), double_divide(inverse(Y), multiply(multiply(Y, inverse(X)), identity)))
% 0.22/0.58 = { by lemma 6 R->L }
% 0.22/0.58 double_divide(inverse(Y), double_divide(inverse(Y), inverse(double_divide(identity, multiply(Y, inverse(X))))))
% 0.22/0.58 = { by lemma 28 R->L }
% 0.22/0.58 double_divide(inverse(Y), inverse(multiply(inverse(Y), inverse(double_divide(identity, multiply(Y, inverse(X)))))))
% 0.22/0.58 = { by lemma 25 R->L }
% 0.22/0.58 double_divide(multiply(inverse(inverse(double_divide(identity, multiply(Y, inverse(X))))), multiply(inverse(Y), inverse(double_divide(identity, multiply(Y, inverse(X)))))), inverse(multiply(inverse(Y), inverse(double_divide(identity, multiply(Y, inverse(X)))))))
% 0.22/0.59 = { by lemma 26 }
% 0.22/0.59 double_divide(multiply(double_divide(identity, multiply(Y, inverse(X))), multiply(inverse(Y), inverse(double_divide(identity, multiply(Y, inverse(X)))))), inverse(multiply(inverse(Y), inverse(double_divide(identity, multiply(Y, inverse(X)))))))
% 0.22/0.59 = { by lemma 15 }
% 0.22/0.59 multiply(multiply(Y, inverse(X)), identity)
% 0.22/0.59 = { by lemma 18 }
% 0.22/0.59 inverse(inverse(multiply(Y, inverse(X))))
% 0.22/0.59 = { by lemma 26 }
% 0.22/0.59 multiply(Y, inverse(X))
% 0.22/0.59
% 0.22/0.59 Lemma 33: double_divide(inverse(X), Y) = multiply(X, inverse(Y)).
% 0.22/0.59 Proof:
% 0.22/0.59 double_divide(inverse(X), Y)
% 0.22/0.59 = { by lemma 29 }
% 0.22/0.59 double_divide(Y, inverse(X))
% 0.22/0.59 = { by lemma 32 }
% 0.22/0.59 multiply(X, inverse(Y))
% 0.22/0.59
% 0.22/0.59 Lemma 34: multiply(X, multiply(Y, multiply(Z, inverse(X)))) = multiply(Z, Y).
% 0.22/0.59 Proof:
% 0.22/0.59 multiply(X, multiply(Y, multiply(Z, inverse(X))))
% 0.22/0.59 = { by lemma 32 R->L }
% 0.22/0.59 multiply(X, multiply(Y, double_divide(X, inverse(Z))))
% 0.22/0.59 = { by lemma 30 R->L }
% 0.22/0.59 multiply(X, multiply(Y, double_divide(X, double_divide(identity, Z))))
% 0.22/0.59 = { by lemma 6 R->L }
% 0.22/0.59 multiply(X, inverse(double_divide(double_divide(X, double_divide(identity, Z)), Y)))
% 0.22/0.59 = { by lemma 32 R->L }
% 0.22/0.59 double_divide(double_divide(double_divide(X, double_divide(identity, Z)), Y), inverse(X))
% 0.22/0.59 = { by lemma 5 R->L }
% 0.22/0.59 double_divide(double_divide(Z, double_divide(double_divide(double_divide(identity, Z), double_divide(double_divide(double_divide(X, double_divide(identity, Z)), Y), inverse(X))), inverse(identity))), inverse(identity))
% 0.22/0.59 = { by lemma 5 }
% 0.22/0.59 double_divide(double_divide(Z, Y), inverse(identity))
% 0.22/0.59 = { by lemma 32 }
% 0.22/0.59 multiply(identity, inverse(double_divide(Z, Y)))
% 0.22/0.59 = { by lemma 9 }
% 0.22/0.59 inverse(inverse(inverse(double_divide(Z, Y))))
% 0.22/0.59 = { by lemma 26 }
% 0.22/0.59 inverse(double_divide(Z, Y))
% 0.22/0.59 = { by lemma 6 }
% 0.22/0.59 multiply(Y, Z)
% 0.22/0.59 = { by lemma 27 R->L }
% 0.22/0.59 multiply(Z, Y)
% 0.22/0.59
% 0.22/0.59 Lemma 35: multiply(inverse(X), multiply(double_divide(identity, Y), X)) = inverse(inverse(inverse(Y))).
% 0.22/0.59 Proof:
% 0.22/0.59 multiply(inverse(X), multiply(double_divide(identity, Y), X))
% 0.22/0.59 = { by lemma 6 R->L }
% 0.22/0.59 inverse(double_divide(multiply(double_divide(identity, Y), X), inverse(X)))
% 0.22/0.59 = { by lemma 15 }
% 0.22/0.59 inverse(multiply(Y, identity))
% 0.22/0.59 = { by lemma 18 }
% 0.22/0.59 inverse(inverse(inverse(Y)))
% 0.22/0.59
% 0.22/0.59 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3), multiply(a4, b4)) = tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4)).
% 0.22/0.59 Proof:
% 0.22/0.59 tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.22/0.59 = { by lemma 9 }
% 0.22/0.59 tuple(multiply(inverse(a1), a1), inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.22/0.59 = { by lemma 10 }
% 0.22/0.59 tuple(inverse(identity), inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.22/0.59 = { by lemma 17 }
% 0.22/0.59 tuple(identity, inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 }
% 0.22/0.59 tuple(identity, a2, multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.22/0.59 = { by lemma 27 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(c3, multiply(a3, inverse(inverse(b3)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 21 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(c3, multiply(a3, inverse(inverse(inverse(inverse(b3)))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 6 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(c3, inverse(double_divide(inverse(inverse(inverse(inverse(b3)))), a3))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 33 R->L }
% 0.22/0.59 tuple(identity, a2, double_divide(inverse(c3), double_divide(inverse(inverse(inverse(inverse(b3)))), a3)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 29 }
% 0.22/0.59 tuple(identity, a2, double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(c3)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 28 R->L }
% 0.22/0.59 tuple(identity, a2, inverse(multiply(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(c3))), multiply(a4, b4))
% 0.22/0.59 = { by axiom 1 (inverse) }
% 0.22/0.59 tuple(identity, a2, double_divide(multiply(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(c3)), identity), multiply(a4, b4))
% 0.22/0.59 = { by lemma 28 R->L }
% 0.22/0.59 tuple(identity, a2, inverse(multiply(multiply(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(c3)), identity)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 22 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(identity), inverse(multiply(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(c3)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 16 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(inverse(identity)), inverse(multiply(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(c3)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 7 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(inverse(identity)), inverse(double_divide(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))), inverse(identity)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 6 }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(inverse(identity)), multiply(inverse(identity), double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 27 }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(inverse(identity)), multiply(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))), inverse(identity))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(inverse(identity)), multiply(inverse(inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 27 }
% 0.22/0.59 tuple(identity, a2, multiply(multiply(inverse(inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity)), inverse(inverse(identity))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 30 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity)), inverse(inverse(identity))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 6 R->L }
% 0.22/0.59 tuple(identity, a2, inverse(double_divide(inverse(inverse(identity)), multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 R->L }
% 0.22/0.59 tuple(identity, a2, inverse(inverse(inverse(double_divide(inverse(inverse(identity)), multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity)))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 18 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(double_divide(inverse(inverse(identity)), multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity)))), identity), multiply(a4, b4))
% 0.22/0.59 = { by lemma 31 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(double_divide(inverse(inverse(identity)), multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity)))), multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), double_divide(inverse(inverse(identity)), multiply(double_divide(identity, inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), inverse(identity))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 35 }
% 0.22/0.59 tuple(identity, a2, inverse(inverse(inverse(inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 }
% 0.22/0.59 tuple(identity, a2, inverse(inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 }
% 0.22/0.59 tuple(identity, a2, double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), double_divide(identity, inverse(inverse(c3)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 30 }
% 0.22/0.59 tuple(identity, a2, double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), inverse(inverse(inverse(c3)))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 32 }
% 0.22/0.59 tuple(identity, a2, multiply(inverse(inverse(c3)), inverse(double_divide(inverse(inverse(inverse(inverse(b3)))), a3))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 33 R->L }
% 0.22/0.59 tuple(identity, a2, double_divide(inverse(inverse(inverse(c3))), double_divide(inverse(inverse(inverse(inverse(b3)))), a3)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 35 R->L }
% 0.22/0.59 tuple(identity, a2, double_divide(multiply(inverse(inverse(inverse(inverse(b3)))), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3))))), double_divide(inverse(inverse(inverse(inverse(b3)))), a3)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 29 }
% 0.22/0.59 tuple(identity, a2, double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), multiply(inverse(inverse(inverse(inverse(b3)))), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 26 R->L }
% 0.22/0.59 tuple(identity, a2, inverse(inverse(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), multiply(inverse(inverse(inverse(inverse(b3)))), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))))))), multiply(a4, b4))
% 0.22/0.59 = { by lemma 18 R->L }
% 0.22/0.59 tuple(identity, a2, multiply(double_divide(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), multiply(inverse(inverse(inverse(inverse(b3)))), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))))), identity), multiply(a4, b4))
% 0.22/0.59 = { by lemma 19 R->L }
% 0.22/0.59 tuple(identity, a2, double_divide(multiply(multiply(inverse(inverse(inverse(inverse(b3)))), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3))))), double_divide(inverse(inverse(inverse(inverse(b3)))), a3)), inverse(identity)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 29 }
% 0.22/0.59 tuple(identity, a2, double_divide(multiply(multiply(inverse(inverse(inverse(inverse(b3)))), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3))))), double_divide(a3, inverse(inverse(inverse(inverse(b3)))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 27 }
% 0.22/0.59 tuple(identity, a2, double_divide(multiply(multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(inverse(inverse(inverse(b3))))), double_divide(a3, inverse(inverse(inverse(inverse(b3)))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 27 }
% 0.22/0.59 tuple(identity, a2, double_divide(multiply(double_divide(a3, inverse(inverse(inverse(inverse(b3))))), multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(inverse(inverse(inverse(b3)))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.59 = { by lemma 34 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(double_divide(a3, inverse(inverse(inverse(inverse(b3))))), multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 29 }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(double_divide(inverse(inverse(inverse(inverse(b3)))), a3), multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 27 }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))), double_divide(inverse(inverse(inverse(inverse(b3)))), a3)), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 28 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))), inverse(multiply(inverse(inverse(inverse(inverse(b3)))), a3))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 26 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), inverse(inverse(multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))))), inverse(multiply(inverse(inverse(inverse(inverse(b3)))), a3))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 18 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), multiply(multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)), identity))), inverse(multiply(inverse(inverse(inverse(inverse(b3)))), a3))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 6 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), inverse(double_divide(identity, multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))))), inverse(multiply(inverse(inverse(inverse(inverse(b3)))), a3))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 32 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(double_divide(multiply(inverse(inverse(inverse(inverse(b3)))), a3), inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), inverse(double_divide(identity, multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 34 R->L }
% 0.22/0.60 tuple(identity, a2, double_divide(double_divide(multiply(double_divide(identity, multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3))), multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), inverse(double_divide(identity, multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3))))))), inverse(multiply(a3, multiply(inverse(inverse(inverse(inverse(b3)))), inverse(double_divide(identity, multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))))))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 15 }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)), identity), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 18 }
% 0.22/0.60 tuple(identity, a2, double_divide(inverse(inverse(multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)))), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 26 }
% 0.22/0.60 tuple(identity, a2, double_divide(multiply(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))), inverse(a3)), inverse(identity)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 19 }
% 0.22/0.60 tuple(identity, a2, multiply(double_divide(inverse(a3), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3))))), identity), multiply(a4, b4))
% 0.22/0.60 = { by lemma 18 }
% 0.22/0.60 tuple(identity, a2, inverse(inverse(double_divide(inverse(a3), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3))))))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 26 }
% 0.22/0.60 tuple(identity, a2, double_divide(inverse(a3), multiply(double_divide(identity, c3), inverse(inverse(inverse(b3))))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 33 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, inverse(multiply(double_divide(identity, c3), inverse(inverse(inverse(b3)))))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 28 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, double_divide(double_divide(identity, c3), inverse(inverse(inverse(b3))))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 30 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, double_divide(inverse(c3), inverse(inverse(inverse(b3))))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 33 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, multiply(c3, inverse(inverse(inverse(inverse(b3)))))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 20 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, multiply(c3, inverse(inverse(b3)))), multiply(a4, b4))
% 0.22/0.60 = { by lemma 26 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, multiply(c3, b3)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 27 R->L }
% 0.22/0.60 tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.22/0.60 = { by lemma 27 }
% 0.22/0.60 tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.22/0.60 % SZS output end Proof
% 0.22/0.60
% 0.22/0.60 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------