TSTP Solution File: GRP103-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP103-1 : 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 : n005.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 0.19s 0.53s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP103-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n005.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 02:15:22 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.53 Command-line arguments: --no-flatten-goal
% 0.19/0.53
% 0.19/0.53 % SZS status Unsatisfiable
% 0.19/0.53
% 0.19/0.59 % SZS output start Proof
% 0.19/0.59 Take the following subset of the input axioms:
% 0.19/0.59 fof(identity, axiom, ![X]: identity=double_divide(X, inverse(X))).
% 0.19/0.59 fof(inverse, axiom, ![X2]: inverse(X2)=double_divide(X2, identity)).
% 0.19/0.59 fof(multiply, axiom, ![Y, X2]: multiply(X2, Y)=double_divide(double_divide(Y, X2), identity)).
% 0.19/0.59 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)))).
% 0.19/0.59 fof(single_axiom, axiom, ![Z, X2, Y2]: double_divide(double_divide(X2, double_divide(double_divide(identity, Y2), double_divide(Z, double_divide(Y2, X2)))), double_divide(identity, identity))=Z).
% 0.19/0.59
% 0.19/0.59 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.59 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.59 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.59 fresh(y, y, x1...xn) = u
% 0.19/0.59 C => fresh(s, t, x1...xn) = v
% 0.19/0.59 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.59 variables of u and v.
% 0.19/0.59 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.59 input problem has no model of domain size 1).
% 0.19/0.59
% 0.19/0.59 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.59
% 0.19/0.59 Axiom 1 (inverse): inverse(X) = double_divide(X, identity).
% 0.19/0.59 Axiom 2 (identity): identity = double_divide(X, inverse(X)).
% 0.19/0.59 Axiom 3 (multiply): multiply(X, Y) = double_divide(double_divide(Y, X), identity).
% 0.19/0.59 Axiom 4 (single_axiom): double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), double_divide(identity, identity)) = Z.
% 0.19/0.59
% 0.19/0.59 Lemma 5: double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), inverse(identity)) = Z.
% 0.19/0.59 Proof:
% 0.19/0.60 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), double_divide(identity, identity))
% 0.19/0.60 = { by axiom 4 (single_axiom) }
% 0.19/0.60 Z
% 0.19/0.60
% 0.19/0.60 Lemma 6: double_divide(double_divide(X, double_divide(inverse(identity), double_divide(Y, double_divide(identity, X)))), inverse(identity)) = Y.
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(Y, double_divide(identity, X)))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(X, double_divide(double_divide(identity, identity), double_divide(Y, double_divide(identity, X)))), inverse(identity))
% 0.19/0.60 = { by lemma 5 }
% 0.19/0.60 Y
% 0.19/0.60
% 0.19/0.60 Lemma 7: double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))), inverse(identity)) = X.
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), double_divide(X, identity))), inverse(identity))
% 0.19/0.60 = { by axiom 2 (identity) }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), double_divide(X, double_divide(identity, inverse(identity))))), inverse(identity))
% 0.19/0.60 = { by lemma 6 }
% 0.19/0.60 X
% 0.19/0.60
% 0.19/0.60 Lemma 8: double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, inverse(identity)))), inverse(identity)) = X.
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 6 }
% 0.19/0.60 X
% 0.19/0.60
% 0.19/0.60 Lemma 9: double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity)) = inverse(inverse(identity)).
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity))
% 0.19/0.60 = { by lemma 7 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(identity)))), inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by axiom 2 (identity) R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(double_divide(inverse(identity), identity), inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(inverse(inverse(identity)), inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 8 }
% 0.19/0.60 inverse(inverse(identity))
% 0.19/0.60
% 0.19/0.60 Lemma 10: inverse(double_divide(X, Y)) = multiply(Y, X).
% 0.19/0.60 Proof:
% 0.19/0.60 inverse(double_divide(X, Y))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(X, Y), identity)
% 0.19/0.60 = { by axiom 3 (multiply) R->L }
% 0.19/0.60 multiply(Y, X)
% 0.19/0.60
% 0.19/0.60 Lemma 11: multiply(inverse(X), X) = inverse(identity).
% 0.19/0.60 Proof:
% 0.19/0.60 multiply(inverse(X), X)
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 inverse(double_divide(X, inverse(X)))
% 0.19/0.60 = { by axiom 2 (identity) R->L }
% 0.19/0.60 inverse(identity)
% 0.19/0.60
% 0.19/0.60 Lemma 12: double_divide(double_divide(identity, double_divide(inverse(identity), X)), inverse(identity)) = double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))).
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), X)), inverse(identity))
% 0.19/0.60 = { by lemma 7 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))), inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 8 }
% 0.19/0.60 double_divide(inverse(identity), double_divide(inverse(identity), inverse(X)))
% 0.19/0.60
% 0.19/0.60 Lemma 13: inverse(inverse(identity)) = inverse(identity).
% 0.19/0.60 Proof:
% 0.19/0.60 inverse(inverse(identity))
% 0.19/0.60 = { by lemma 9 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity))
% 0.19/0.60 = { by lemma 8 R->L }
% 0.19/0.60 double_divide(double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity)))), inverse(identity)), inverse(identity))
% 0.19/0.60 = { by lemma 9 }
% 0.19/0.60 double_divide(double_divide(double_divide(identity, double_divide(inverse(identity), inverse(inverse(identity)))), inverse(identity)), inverse(identity))
% 0.19/0.60 = { by axiom 2 (identity) R->L }
% 0.19/0.60 double_divide(double_divide(double_divide(identity, identity), inverse(identity)), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) R->L }
% 0.19/0.60 double_divide(double_divide(inverse(identity), inverse(identity)), inverse(identity))
% 0.19/0.60 = { by lemma 11 R->L }
% 0.19/0.60 double_divide(double_divide(inverse(identity), multiply(inverse(identity), identity)), inverse(identity))
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 double_divide(double_divide(inverse(identity), inverse(double_divide(identity, inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(double_divide(identity, inverse(identity)), identity)), inverse(identity))
% 0.19/0.60 = { by lemma 8 R->L }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(double_divide(identity, inverse(identity)), double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(identity, inverse(identity)))), inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by axiom 2 (identity) R->L }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(double_divide(identity, inverse(identity)), double_divide(double_divide(identity, double_divide(inverse(identity), identity)), inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 12 }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(double_divide(identity, inverse(identity)), double_divide(inverse(identity), double_divide(inverse(identity), inverse(identity))))), inverse(identity))
% 0.19/0.60 = { by lemma 5 }
% 0.19/0.60 inverse(identity)
% 0.19/0.60
% 0.19/0.60 Lemma 14: inverse(identity) = identity.
% 0.19/0.60 Proof:
% 0.19/0.60 inverse(identity)
% 0.19/0.60 = { by lemma 7 R->L }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 13 }
% 0.19/0.60 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(identity))), inverse(identity))
% 0.19/0.60 = { by lemma 7 }
% 0.19/0.60 identity
% 0.19/0.60
% 0.19/0.60 Lemma 15: double_divide(identity, double_divide(identity, inverse(X))) = multiply(double_divide(identity, X), identity).
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(identity, double_divide(identity, inverse(X)))
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(inverse(identity), double_divide(identity, inverse(X)))
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(inverse(identity), double_divide(inverse(identity), inverse(X)))
% 0.19/0.60 = { by lemma 12 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), X)), inverse(identity))
% 0.19/0.60 = { by lemma 14 }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(identity, X)), inverse(identity))
% 0.19/0.60 = { by lemma 14 }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(identity, X)), identity)
% 0.19/0.60 = { by axiom 1 (inverse) R->L }
% 0.19/0.60 inverse(double_divide(identity, double_divide(identity, X)))
% 0.19/0.60 = { by lemma 10 }
% 0.19/0.60 multiply(double_divide(identity, X), identity)
% 0.19/0.60
% 0.19/0.60 Lemma 16: multiply(identity, X) = inverse(inverse(X)).
% 0.19/0.60 Proof:
% 0.19/0.60 multiply(identity, X)
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 inverse(double_divide(X, identity))
% 0.19/0.60 = { by axiom 1 (inverse) R->L }
% 0.19/0.60 inverse(inverse(X))
% 0.19/0.60
% 0.19/0.60 Lemma 17: double_divide(identity, double_divide(identity, inverse(inverse(X)))) = X.
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(identity, double_divide(identity, inverse(inverse(X))))
% 0.19/0.60 = { by lemma 16 R->L }
% 0.19/0.60 double_divide(identity, double_divide(identity, multiply(identity, X)))
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(identity, double_divide(identity, multiply(inverse(identity), X)))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(identity, double_divide(identity, multiply(double_divide(identity, identity), X)))
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 double_divide(identity, double_divide(identity, inverse(double_divide(X, double_divide(identity, identity)))))
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(inverse(identity), double_divide(identity, inverse(double_divide(X, double_divide(identity, identity)))))
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(inverse(identity), double_divide(inverse(identity), inverse(double_divide(X, double_divide(identity, identity)))))
% 0.19/0.60 = { by lemma 12 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 6 }
% 0.19/0.60 X
% 0.19/0.60
% 0.19/0.60 Lemma 18: multiply(double_divide(identity, multiply(X, Y)), identity) = double_divide(Y, X).
% 0.19/0.60 Proof:
% 0.19/0.60 multiply(double_divide(identity, multiply(X, Y)), identity)
% 0.19/0.60 = { by lemma 15 R->L }
% 0.19/0.60 double_divide(identity, double_divide(identity, inverse(multiply(X, Y))))
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 double_divide(identity, double_divide(identity, inverse(inverse(double_divide(Y, X)))))
% 0.19/0.60 = { by lemma 17 }
% 0.19/0.60 double_divide(Y, X)
% 0.19/0.60
% 0.19/0.60 Lemma 19: inverse(multiply(double_divide(identity, X), identity)) = X.
% 0.19/0.60 Proof:
% 0.19/0.60 inverse(multiply(double_divide(identity, X), identity))
% 0.19/0.60 = { by lemma 15 R->L }
% 0.19/0.60 inverse(double_divide(identity, double_divide(identity, inverse(X))))
% 0.19/0.60 = { by lemma 10 }
% 0.19/0.60 multiply(double_divide(identity, inverse(X)), identity)
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 multiply(double_divide(identity, double_divide(X, identity)), identity)
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 inverse(double_divide(identity, double_divide(identity, double_divide(X, identity))))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(identity, double_divide(X, identity))), identity)
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(identity, double_divide(X, inverse(identity)))), identity)
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(identity, double_divide(X, inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, inverse(identity)))), inverse(identity))
% 0.19/0.60 = { by lemma 8 }
% 0.19/0.60 X
% 0.19/0.60
% 0.19/0.60 Lemma 20: multiply(double_divide(double_divide(identity, X), inverse(Y)), inverse(X)) = Y.
% 0.19/0.60 Proof:
% 0.19/0.60 multiply(double_divide(double_divide(identity, X), inverse(Y)), inverse(X))
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 inverse(double_divide(inverse(X), double_divide(double_divide(identity, X), inverse(Y))))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, X), inverse(Y))), identity)
% 0.19/0.60 = { by lemma 14 R->L }
% 0.19/0.60 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, X), inverse(Y))), inverse(identity))
% 0.19/0.60 = { by axiom 1 (inverse) }
% 0.19/0.60 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, X), double_divide(Y, identity))), inverse(identity))
% 0.19/0.60 = { by axiom 2 (identity) }
% 0.19/0.60 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, X), double_divide(Y, double_divide(X, inverse(X))))), inverse(identity))
% 0.19/0.60 = { by lemma 5 }
% 0.19/0.60 Y
% 0.19/0.60
% 0.19/0.60 Lemma 21: inverse(inverse(inverse(X))) = double_divide(identity, X).
% 0.19/0.60 Proof:
% 0.19/0.60 inverse(inverse(inverse(X)))
% 0.19/0.60 = { by lemma 16 R->L }
% 0.19/0.60 multiply(identity, inverse(X))
% 0.19/0.60 = { by axiom 2 (identity) }
% 0.19/0.60 multiply(double_divide(double_divide(identity, X), inverse(double_divide(identity, X))), inverse(X))
% 0.19/0.60 = { by lemma 20 }
% 0.19/0.60 double_divide(identity, X)
% 0.19/0.60
% 0.19/0.60 Lemma 22: double_divide(identity, inverse(X)) = multiply(X, identity).
% 0.19/0.60 Proof:
% 0.19/0.60 double_divide(identity, inverse(X))
% 0.19/0.60 = { by lemma 21 R->L }
% 0.19/0.60 inverse(inverse(inverse(inverse(X))))
% 0.19/0.60 = { by lemma 21 }
% 0.19/0.60 inverse(double_divide(identity, X))
% 0.19/0.60 = { by lemma 10 }
% 0.19/0.60 multiply(X, identity)
% 0.19/0.60
% 0.19/0.60 Lemma 23: multiply(inverse(X), identity) = inverse(multiply(X, identity)).
% 0.19/0.60 Proof:
% 0.19/0.60 multiply(inverse(X), identity)
% 0.19/0.60 = { by lemma 10 R->L }
% 0.19/0.60 inverse(double_divide(identity, inverse(X)))
% 0.19/0.60 = { by lemma 22 }
% 0.19/0.61 inverse(multiply(X, identity))
% 0.19/0.61
% 0.19/0.61 Lemma 24: double_divide(inverse(multiply(X, identity)), X) = identity.
% 0.19/0.61 Proof:
% 0.19/0.61 double_divide(inverse(multiply(X, identity)), X)
% 0.19/0.61 = { by lemma 23 R->L }
% 0.19/0.61 double_divide(multiply(inverse(X), identity), X)
% 0.19/0.61 = { by lemma 18 R->L }
% 0.19/0.61 multiply(double_divide(identity, multiply(X, multiply(inverse(X), identity))), identity)
% 0.19/0.61 = { by axiom 1 (inverse) }
% 0.19/0.61 multiply(double_divide(identity, multiply(X, multiply(double_divide(X, identity), identity))), identity)
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 multiply(double_divide(identity, multiply(X, multiply(double_divide(X, inverse(identity)), identity))), identity)
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 multiply(double_divide(identity, multiply(X, multiply(double_divide(X, inverse(identity)), inverse(identity)))), identity)
% 0.19/0.61 = { by axiom 1 (inverse) }
% 0.19/0.61 multiply(double_divide(identity, multiply(X, multiply(double_divide(X, double_divide(identity, identity)), inverse(identity)))), identity)
% 0.19/0.61 = { by lemma 10 R->L }
% 0.19/0.61 multiply(double_divide(identity, multiply(X, inverse(double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))))), identity)
% 0.19/0.61 = { by lemma 6 R->L }
% 0.19/0.61 multiply(double_divide(identity, multiply(double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))), inverse(identity)), inverse(double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))))), identity)
% 0.19/0.61 = { by lemma 20 }
% 0.19/0.61 multiply(double_divide(identity, identity), identity)
% 0.19/0.61 = { by axiom 1 (inverse) R->L }
% 0.19/0.61 multiply(inverse(identity), identity)
% 0.19/0.61 = { by lemma 23 }
% 0.19/0.61 inverse(multiply(identity, identity))
% 0.19/0.61 = { by lemma 16 }
% 0.19/0.61 inverse(inverse(inverse(identity)))
% 0.19/0.61 = { by lemma 21 }
% 0.19/0.61 double_divide(identity, identity)
% 0.19/0.61 = { by axiom 1 (inverse) R->L }
% 0.19/0.61 inverse(identity)
% 0.19/0.61 = { by lemma 14 }
% 0.19/0.61 identity
% 0.19/0.61
% 0.19/0.61 Lemma 25: inverse(inverse(X)) = X.
% 0.19/0.61 Proof:
% 0.19/0.61 inverse(inverse(X))
% 0.19/0.61 = { by lemma 16 R->L }
% 0.19/0.61 multiply(identity, X)
% 0.19/0.61 = { by lemma 10 R->L }
% 0.19/0.61 inverse(double_divide(X, identity))
% 0.19/0.61 = { by axiom 1 (inverse) }
% 0.19/0.61 double_divide(double_divide(X, identity), identity)
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 double_divide(double_divide(X, inverse(identity)), identity)
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 double_divide(double_divide(X, inverse(identity)), inverse(identity))
% 0.19/0.61 = { by lemma 13 R->L }
% 0.19/0.61 double_divide(double_divide(X, inverse(inverse(identity))), inverse(identity))
% 0.19/0.61 = { by axiom 1 (inverse) }
% 0.19/0.61 double_divide(double_divide(X, double_divide(inverse(identity), identity)), inverse(identity))
% 0.19/0.61 = { by lemma 24 R->L }
% 0.19/0.61 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(inverse(multiply(double_divide(identity, X), identity)), double_divide(identity, X)))), inverse(identity))
% 0.19/0.61 = { by lemma 6 }
% 0.19/0.61 inverse(multiply(double_divide(identity, X), identity))
% 0.19/0.61 = { by lemma 19 }
% 0.19/0.61 X
% 0.19/0.61
% 0.19/0.61 Lemma 26: double_divide(identity, X) = inverse(X).
% 0.19/0.61 Proof:
% 0.19/0.61 double_divide(identity, X)
% 0.19/0.61 = { by lemma 21 R->L }
% 0.19/0.61 inverse(inverse(inverse(X)))
% 0.19/0.61 = { by lemma 25 }
% 0.19/0.61 inverse(X)
% 0.19/0.61
% 0.19/0.61 Lemma 27: double_divide(identity, double_divide(X, Y)) = inverse(inverse(multiply(Y, X))).
% 0.19/0.61 Proof:
% 0.19/0.61 double_divide(identity, double_divide(X, Y))
% 0.19/0.61 = { by lemma 21 R->L }
% 0.19/0.61 inverse(inverse(inverse(double_divide(X, Y))))
% 0.19/0.61 = { by lemma 10 }
% 0.19/0.61 inverse(inverse(multiply(Y, X)))
% 0.19/0.61
% 0.19/0.61 Lemma 28: multiply(X, identity) = X.
% 0.19/0.61 Proof:
% 0.19/0.61 multiply(X, identity)
% 0.19/0.61 = { by lemma 25 R->L }
% 0.19/0.61 inverse(inverse(multiply(X, identity)))
% 0.19/0.61 = { by lemma 27 R->L }
% 0.19/0.61 double_divide(identity, double_divide(identity, X))
% 0.19/0.61 = { by lemma 25 R->L }
% 0.19/0.61 inverse(inverse(double_divide(identity, double_divide(identity, X))))
% 0.19/0.61 = { by lemma 16 R->L }
% 0.19/0.61 multiply(identity, double_divide(identity, double_divide(identity, X)))
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 multiply(inverse(identity), double_divide(identity, double_divide(identity, X)))
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 multiply(inverse(identity), double_divide(inverse(identity), double_divide(identity, X)))
% 0.19/0.61 = { by lemma 14 R->L }
% 0.19/0.61 multiply(inverse(identity), double_divide(inverse(identity), double_divide(inverse(identity), X)))
% 0.19/0.61 = { by lemma 25 R->L }
% 0.19/0.61 multiply(inverse(identity), double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(X)))))
% 0.19/0.61 = { by lemma 10 R->L }
% 0.19/0.61 inverse(double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(X)))), inverse(identity)))
% 0.19/0.61 = { by lemma 7 }
% 0.19/0.61 inverse(inverse(X))
% 0.19/0.61 = { by lemma 25 }
% 0.19/0.61 X
% 0.19/0.61
% 0.19/0.61 Lemma 29: multiply(X, Y) = multiply(Y, X).
% 0.19/0.61 Proof:
% 0.19/0.61 multiply(X, Y)
% 0.19/0.61 = { by lemma 10 R->L }
% 0.19/0.61 inverse(double_divide(Y, X))
% 0.19/0.61 = { by lemma 5 R->L }
% 0.19/0.61 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(inverse(double_divide(Y, X)), double_divide(Y, X)))), inverse(identity))
% 0.19/0.61 = { by lemma 19 R->L }
% 0.19/0.61 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(inverse(double_divide(Y, X)), inverse(multiply(double_divide(identity, double_divide(Y, X)), identity))))), inverse(identity))
% 0.19/0.61 = { by axiom 1 (inverse) }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), identity), inverse(multiply(double_divide(identity, double_divide(Y, X)), identity))))), inverse(identity))
% 0.19/0.62 = { by lemma 14 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), inverse(identity)), inverse(multiply(double_divide(identity, double_divide(Y, X)), identity))))), inverse(identity))
% 0.19/0.62 = { by lemma 26 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, identity)), inverse(multiply(double_divide(identity, double_divide(Y, X)), identity))))), inverse(identity))
% 0.19/0.62 = { by lemma 24 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), inverse(multiply(double_divide(identity, double_divide(Y, X)), identity))))), inverse(identity))
% 0.19/0.62 = { by lemma 5 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), double_divide(double_divide(double_divide(Y, X), double_divide(double_divide(identity, inverse(identity)), double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(inverse(identity), double_divide(Y, X))))), inverse(identity))))), inverse(identity))
% 0.19/0.62 = { by axiom 2 (identity) R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(inverse(identity), double_divide(Y, X))))), inverse(identity))))), inverse(identity))
% 0.19/0.62 = { by lemma 14 }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(inverse(identity), double_divide(Y, X))))), identity)))), inverse(identity))
% 0.19/0.62 = { by lemma 14 }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), identity)))), inverse(identity))
% 0.19/0.62 = { by axiom 1 (inverse) R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))), inverse(double_divide(double_divide(Y, X), double_divide(identity, double_divide(inverse(multiply(double_divide(identity, double_divide(Y, X)), identity)), double_divide(identity, double_divide(Y, X))))))))), inverse(identity))
% 0.19/0.62 = { by axiom 2 (identity) R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, Y), identity)), inverse(identity))
% 0.19/0.62 = { by axiom 1 (inverse) R->L }
% 0.19/0.62 double_divide(double_divide(X, inverse(double_divide(identity, Y))), inverse(identity))
% 0.19/0.62 = { by lemma 10 }
% 0.19/0.62 double_divide(double_divide(X, multiply(Y, identity)), inverse(identity))
% 0.19/0.62 = { by lemma 28 }
% 0.19/0.62 double_divide(double_divide(X, Y), inverse(identity))
% 0.19/0.62 = { by lemma 14 }
% 0.19/0.62 double_divide(double_divide(X, Y), identity)
% 0.19/0.62 = { by axiom 1 (inverse) R->L }
% 0.19/0.62 inverse(double_divide(X, Y))
% 0.19/0.62 = { by lemma 10 }
% 0.19/0.62 multiply(Y, X)
% 0.19/0.62
% 0.19/0.62 Lemma 30: double_divide(identity, multiply(X, Y)) = multiply(double_divide(Y, X), identity).
% 0.19/0.62 Proof:
% 0.19/0.62 double_divide(identity, multiply(X, Y))
% 0.19/0.62 = { by lemma 10 R->L }
% 0.19/0.62 double_divide(identity, inverse(double_divide(Y, X)))
% 0.19/0.62 = { by lemma 22 }
% 0.19/0.62 multiply(double_divide(Y, X), identity)
% 0.19/0.62
% 0.19/0.62 Lemma 31: double_divide(X, Y) = double_divide(Y, X).
% 0.19/0.62 Proof:
% 0.19/0.62 double_divide(X, Y)
% 0.19/0.62 = { by lemma 18 R->L }
% 0.19/0.62 multiply(double_divide(identity, multiply(Y, X)), identity)
% 0.19/0.62 = { by lemma 29 }
% 0.19/0.62 multiply(double_divide(identity, multiply(X, Y)), identity)
% 0.19/0.62 = { by lemma 28 }
% 0.19/0.62 double_divide(identity, multiply(X, Y))
% 0.19/0.62 = { by lemma 30 }
% 0.19/0.62 multiply(double_divide(Y, X), identity)
% 0.19/0.62 = { by lemma 28 }
% 0.19/0.62 double_divide(Y, X)
% 0.19/0.62
% 0.19/0.62 Lemma 32: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.19/0.62 Proof:
% 0.19/0.62 multiply(inverse(X), multiply(X, Y))
% 0.19/0.62 = { by lemma 29 R->L }
% 0.19/0.62 multiply(multiply(X, Y), inverse(X))
% 0.19/0.62 = { by lemma 25 R->L }
% 0.19/0.62 multiply(inverse(inverse(multiply(X, Y))), inverse(X))
% 0.19/0.62 = { by lemma 27 R->L }
% 0.19/0.62 multiply(double_divide(identity, double_divide(Y, X)), inverse(X))
% 0.19/0.62 = { by lemma 10 R->L }
% 0.19/0.62 inverse(double_divide(inverse(X), double_divide(identity, double_divide(Y, X))))
% 0.19/0.62 = { by axiom 1 (inverse) }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(identity, double_divide(Y, X))), identity)
% 0.19/0.62 = { by lemma 14 R->L }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(inverse(identity), double_divide(Y, X))), identity)
% 0.19/0.62 = { by lemma 14 R->L }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(inverse(identity), double_divide(Y, X))), inverse(identity))
% 0.19/0.62 = { by lemma 26 R->L }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, identity), double_divide(Y, X))), inverse(identity))
% 0.19/0.62 = { by lemma 25 R->L }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, identity), double_divide(Y, inverse(inverse(X))))), inverse(identity))
% 0.19/0.62 = { by lemma 25 R->L }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, identity), double_divide(Y, inverse(inverse(inverse(inverse(X))))))), inverse(identity))
% 0.19/0.62 = { by lemma 21 }
% 0.19/0.62 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, identity), double_divide(Y, double_divide(identity, inverse(X))))), inverse(identity))
% 0.19/0.62 = { by lemma 5 }
% 0.19/0.62 Y
% 0.19/0.62
% 0.19/0.62 Lemma 33: double_divide(identity, inverse(inverse(X))) = inverse(multiply(X, identity)).
% 0.19/0.62 Proof:
% 0.19/0.62 double_divide(identity, inverse(inverse(X)))
% 0.19/0.62 = { by lemma 19 R->L }
% 0.19/0.62 inverse(multiply(double_divide(identity, double_divide(identity, inverse(inverse(X)))), identity))
% 0.19/0.62 = { by lemma 17 }
% 0.19/0.62 inverse(multiply(X, identity))
% 0.19/0.62
% 0.19/0.62 Lemma 34: multiply(X, double_divide(X, inverse(Y))) = Y.
% 0.19/0.62 Proof:
% 0.19/0.62 multiply(X, double_divide(X, inverse(Y)))
% 0.19/0.62 = { by lemma 29 R->L }
% 0.19/0.62 multiply(double_divide(X, inverse(Y)), X)
% 0.19/0.62 = { by lemma 10 R->L }
% 0.19/0.62 inverse(double_divide(X, double_divide(X, inverse(Y))))
% 0.19/0.62 = { by axiom 1 (inverse) }
% 0.19/0.62 double_divide(double_divide(X, double_divide(X, inverse(Y))), identity)
% 0.19/0.62 = { by lemma 14 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(X, inverse(Y))), inverse(identity))
% 0.19/0.62 = { by axiom 1 (inverse) }
% 0.19/0.62 double_divide(double_divide(X, double_divide(X, double_divide(Y, identity))), inverse(identity))
% 0.19/0.62 = { by lemma 17 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, double_divide(identity, inverse(inverse(X)))), double_divide(Y, identity))), inverse(identity))
% 0.19/0.62 = { by lemma 33 }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, inverse(multiply(X, identity))), double_divide(Y, identity))), inverse(identity))
% 0.19/0.62 = { by lemma 24 R->L }
% 0.19/0.62 double_divide(double_divide(X, double_divide(double_divide(identity, inverse(multiply(X, identity))), double_divide(Y, double_divide(inverse(multiply(X, identity)), X)))), inverse(identity))
% 0.19/0.62 = { by lemma 5 }
% 0.19/0.62 Y
% 0.19/0.62
% 0.19/0.62 Lemma 35: double_divide(X, inverse(Y)) = multiply(Y, inverse(X)).
% 0.19/0.62 Proof:
% 0.19/0.62 double_divide(X, inverse(Y))
% 0.19/0.62 = { by lemma 32 R->L }
% 0.19/0.62 multiply(inverse(X), multiply(X, double_divide(X, inverse(Y))))
% 0.19/0.62 = { by lemma 34 }
% 0.19/0.62 multiply(inverse(X), Y)
% 0.19/0.62 = { by lemma 29 }
% 0.19/0.62 multiply(Y, inverse(X))
% 0.19/0.62
% 0.19/0.62 Lemma 36: double_divide(inverse(X), Y) = multiply(X, inverse(Y)).
% 0.19/0.62 Proof:
% 0.19/0.62 double_divide(inverse(X), Y)
% 0.19/0.62 = { by lemma 31 R->L }
% 0.19/0.62 double_divide(Y, inverse(X))
% 0.19/0.62 = { by lemma 35 }
% 0.19/0.62 multiply(X, inverse(Y))
% 0.19/0.62
% 0.19/0.62 Lemma 37: inverse(multiply(X, Y)) = double_divide(X, Y).
% 0.19/0.62 Proof:
% 0.19/0.62 inverse(multiply(X, Y))
% 0.19/0.62 = { by lemma 29 R->L }
% 0.19/0.62 inverse(multiply(Y, X))
% 0.19/0.62 = { by lemma 26 R->L }
% 0.19/0.62 double_divide(identity, multiply(Y, X))
% 0.19/0.62 = { by lemma 29 R->L }
% 0.19/0.62 double_divide(identity, multiply(X, Y))
% 0.19/0.62 = { by lemma 30 }
% 0.19/0.62 multiply(double_divide(Y, X), identity)
% 0.19/0.62 = { by lemma 28 }
% 0.19/0.62 double_divide(Y, X)
% 0.19/0.62 = { by lemma 31 }
% 0.19/0.62 double_divide(X, Y)
% 0.19/0.63
% 0.19/0.63 Lemma 38: multiply(X, double_divide(X, Y)) = inverse(Y).
% 0.19/0.63 Proof:
% 0.19/0.63 multiply(X, double_divide(X, Y))
% 0.19/0.63 = { by lemma 25 R->L }
% 0.19/0.63 multiply(X, double_divide(X, inverse(inverse(Y))))
% 0.19/0.63 = { by lemma 34 }
% 0.19/0.63 inverse(Y)
% 0.19/0.63
% 0.19/0.63 Lemma 39: multiply(multiply(X, Y), multiply(Z, double_divide(X, Y))) = Z.
% 0.19/0.63 Proof:
% 0.19/0.63 multiply(multiply(X, Y), multiply(Z, double_divide(X, Y)))
% 0.19/0.63 = { by lemma 29 R->L }
% 0.19/0.63 multiply(multiply(Z, double_divide(X, Y)), multiply(X, Y))
% 0.19/0.63 = { by lemma 10 R->L }
% 0.19/0.63 multiply(multiply(Z, double_divide(X, Y)), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 37 R->L }
% 0.19/0.63 multiply(multiply(Z, inverse(multiply(X, Y))), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 35 R->L }
% 0.19/0.63 multiply(double_divide(multiply(X, Y), inverse(Z)), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 31 R->L }
% 0.19/0.63 multiply(double_divide(inverse(Z), multiply(X, Y)), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 32 R->L }
% 0.19/0.63 multiply(multiply(inverse(inverse(Z)), multiply(inverse(Z), double_divide(inverse(Z), multiply(X, Y)))), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 38 }
% 0.19/0.63 multiply(multiply(inverse(inverse(Z)), inverse(multiply(X, Y))), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 29 }
% 0.19/0.63 multiply(multiply(inverse(multiply(X, Y)), inverse(inverse(Z))), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 36 R->L }
% 0.19/0.63 multiply(double_divide(inverse(inverse(multiply(X, Y))), inverse(Z)), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 27 R->L }
% 0.19/0.63 multiply(double_divide(double_divide(identity, double_divide(Y, X)), inverse(Z)), inverse(double_divide(Y, X)))
% 0.19/0.63 = { by lemma 20 }
% 0.19/0.63 Z
% 0.19/0.63
% 0.19/0.63 Lemma 40: double_divide(double_divide(X, Y), multiply(Z, multiply(X, Y))) = inverse(Z).
% 0.19/0.63 Proof:
% 0.19/0.63 double_divide(double_divide(X, Y), multiply(Z, multiply(X, Y)))
% 0.19/0.63 = { by lemma 31 R->L }
% 0.19/0.63 double_divide(multiply(Z, multiply(X, Y)), double_divide(X, Y))
% 0.19/0.63 = { by lemma 18 R->L }
% 0.19/0.63 multiply(double_divide(identity, multiply(double_divide(X, Y), multiply(Z, multiply(X, Y)))), identity)
% 0.19/0.63 = { by lemma 31 R->L }
% 0.19/0.63 multiply(double_divide(identity, multiply(double_divide(Y, X), multiply(Z, multiply(X, Y)))), identity)
% 0.19/0.63 = { by lemma 25 R->L }
% 0.19/0.63 multiply(double_divide(identity, multiply(double_divide(Y, X), multiply(Z, inverse(inverse(multiply(X, Y)))))), identity)
% 0.19/0.63 = { by lemma 25 R->L }
% 0.19/0.63 multiply(double_divide(identity, multiply(inverse(inverse(double_divide(Y, X))), multiply(Z, inverse(inverse(multiply(X, Y)))))), identity)
% 0.19/0.63 = { by lemma 16 R->L }
% 0.19/0.63 multiply(double_divide(identity, multiply(multiply(identity, double_divide(Y, X)), multiply(Z, inverse(inverse(multiply(X, Y)))))), identity)
% 0.19/0.63 = { by lemma 27 R->L }
% 0.19/0.63 multiply(double_divide(identity, multiply(multiply(identity, double_divide(Y, X)), multiply(Z, double_divide(identity, double_divide(Y, X))))), identity)
% 0.19/0.63 = { by lemma 39 }
% 0.19/0.63 multiply(double_divide(identity, Z), identity)
% 0.19/0.63 = { by lemma 28 }
% 0.19/0.63 double_divide(identity, Z)
% 0.19/0.63 = { by lemma 26 }
% 0.19/0.63 inverse(Z)
% 0.19/0.63
% 0.19/0.63 Lemma 41: multiply(inverse(X), double_divide(Y, Z)) = double_divide(X, multiply(Y, Z)).
% 0.19/0.63 Proof:
% 0.19/0.63 multiply(inverse(X), double_divide(Y, Z))
% 0.19/0.63 = { by lemma 29 R->L }
% 0.19/0.63 multiply(double_divide(Y, Z), inverse(X))
% 0.19/0.63 = { by lemma 40 R->L }
% 0.19/0.63 multiply(double_divide(Y, Z), double_divide(double_divide(Y, Z), multiply(X, multiply(Y, Z))))
% 0.19/0.63 = { by lemma 38 }
% 0.19/0.63 inverse(multiply(X, multiply(Y, Z)))
% 0.19/0.63 = { by lemma 37 }
% 0.19/0.63 double_divide(X, multiply(Y, Z))
% 0.19/0.63
% 0.19/0.63 Lemma 42: multiply(inverse(X), multiply(Y, Z)) = double_divide(X, double_divide(Y, Z)).
% 0.19/0.63 Proof:
% 0.19/0.63 multiply(inverse(X), multiply(Y, Z))
% 0.19/0.63 = { by lemma 29 R->L }
% 0.19/0.63 multiply(multiply(Y, Z), inverse(X))
% 0.19/0.63 = { by lemma 26 R->L }
% 0.19/0.63 multiply(multiply(Y, Z), double_divide(identity, X))
% 0.19/0.63 = { by lemma 39 R->L }
% 0.19/0.63 multiply(multiply(Y, Z), double_divide(identity, multiply(multiply(Y, Z), multiply(X, double_divide(Y, Z)))))
% 0.19/0.63 = { by lemma 30 }
% 0.19/0.63 multiply(multiply(Y, Z), multiply(double_divide(multiply(X, double_divide(Y, Z)), multiply(Y, Z)), identity))
% 0.19/0.63 = { by lemma 28 }
% 0.19/0.63 multiply(multiply(Y, Z), double_divide(multiply(X, double_divide(Y, Z)), multiply(Y, Z)))
% 0.19/0.63 = { by lemma 31 }
% 0.19/0.63 multiply(multiply(Y, Z), double_divide(multiply(Y, Z), multiply(X, double_divide(Y, Z))))
% 0.19/0.63 = { by lemma 38 }
% 0.19/0.63 inverse(multiply(X, double_divide(Y, Z)))
% 0.19/0.63 = { by lemma 37 }
% 0.19/0.63 double_divide(X, double_divide(Y, Z))
% 0.19/0.63
% 0.19/0.63 Lemma 43: double_divide(multiply(X, Y), double_divide(Z, W)) = multiply(multiply(Z, W), double_divide(X, Y)).
% 0.19/0.63 Proof:
% 0.19/0.63 double_divide(multiply(X, Y), double_divide(Z, W))
% 0.19/0.63 = { by lemma 42 R->L }
% 0.19/0.63 multiply(inverse(multiply(X, Y)), multiply(Z, W))
% 0.19/0.63 = { by lemma 37 }
% 0.19/0.63 multiply(double_divide(X, Y), multiply(Z, W))
% 0.19/0.63 = { by lemma 29 }
% 0.19/0.63 multiply(multiply(Z, W), double_divide(X, Y))
% 0.19/0.63
% 0.19/0.63 Lemma 44: double_divide(multiply(X, Y), multiply(Z, W)) = multiply(double_divide(X, Y), double_divide(Z, W)).
% 0.19/0.63 Proof:
% 0.19/0.63 double_divide(multiply(X, Y), multiply(Z, W))
% 0.19/0.63 = { by lemma 29 R->L }
% 0.19/0.63 double_divide(multiply(X, Y), multiply(W, Z))
% 0.19/0.63 = { by lemma 29 R->L }
% 0.19/0.63 double_divide(multiply(Y, X), multiply(W, Z))
% 0.19/0.63 = { by lemma 31 R->L }
% 0.19/0.63 double_divide(multiply(W, Z), multiply(Y, X))
% 0.19/0.63 = { by lemma 41 R->L }
% 0.19/0.63 multiply(inverse(multiply(W, Z)), double_divide(Y, X))
% 0.19/0.63 = { by lemma 37 }
% 0.19/0.63 multiply(double_divide(W, Z), double_divide(Y, X))
% 0.19/0.63 = { by lemma 29 }
% 0.19/0.63 multiply(double_divide(Y, X), double_divide(W, Z))
% 0.19/0.63 = { by lemma 31 }
% 0.19/0.63 multiply(double_divide(X, Y), double_divide(W, Z))
% 0.19/0.63 = { by lemma 31 }
% 0.19/0.63 multiply(double_divide(X, Y), double_divide(Z, W))
% 0.19/0.63
% 0.19/0.63 Lemma 45: double_divide(X, multiply(Y, double_divide(Z, multiply(Y, X)))) = Z.
% 0.19/0.63 Proof:
% 0.19/0.63 double_divide(X, multiply(Y, double_divide(Z, multiply(Y, X))))
% 0.19/0.63 = { by lemma 25 R->L }
% 0.19/0.63 double_divide(X, multiply(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by lemma 41 R->L }
% 0.19/0.63 multiply(inverse(X), double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by axiom 1 (inverse) }
% 0.19/0.63 multiply(double_divide(X, identity), double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by lemma 44 R->L }
% 0.19/0.63 double_divide(multiply(X, identity), multiply(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by lemma 41 R->L }
% 0.19/0.63 multiply(inverse(multiply(X, identity)), double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by axiom 1 (inverse) }
% 0.19/0.63 multiply(double_divide(multiply(X, identity), identity), double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by lemma 44 R->L }
% 0.19/0.63 double_divide(multiply(multiply(X, identity), identity), multiply(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by lemma 41 R->L }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(X))))))
% 0.19/0.63 = { by lemma 7 R->L }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(inverse(X))))), inverse(identity))))))
% 0.19/0.63 = { by lemma 21 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, double_divide(double_divide(inverse(identity), double_divide(inverse(identity), double_divide(identity, X))), inverse(identity))))))
% 0.19/0.63 = { by lemma 14 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, double_divide(double_divide(inverse(identity), double_divide(inverse(identity), double_divide(identity, X))), identity)))))
% 0.19/0.63 = { by lemma 14 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, double_divide(double_divide(inverse(identity), double_divide(identity, double_divide(identity, X))), identity)))))
% 0.19/0.63 = { by lemma 14 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, double_divide(double_divide(identity, double_divide(identity, double_divide(identity, X))), identity)))))
% 0.19/0.63 = { by axiom 1 (inverse) R->L }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, inverse(double_divide(identity, double_divide(identity, double_divide(identity, X))))))))
% 0.19/0.63 = { by lemma 10 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, multiply(double_divide(identity, double_divide(identity, X)), identity)))))
% 0.19/0.63 = { by lemma 27 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, multiply(inverse(inverse(multiply(X, identity))), identity)))))
% 0.19/0.63 = { by lemma 10 R->L }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, inverse(double_divide(identity, inverse(inverse(multiply(X, identity)))))))))
% 0.19/0.63 = { by lemma 33 }
% 0.19/0.63 multiply(inverse(multiply(multiply(X, identity), identity)), double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(multiply(multiply(X, identity), identity)))))))
% 0.19/0.63 = { by lemma 29 R->L }
% 0.19/0.63 multiply(double_divide(Y, double_divide(Z, multiply(Y, inverse(inverse(multiply(multiply(X, identity), identity)))))), inverse(multiply(multiply(X, identity), identity)))
% 0.19/0.63 = { by lemma 42 R->L }
% 0.19/0.63 multiply(multiply(inverse(Y), multiply(Z, multiply(Y, inverse(inverse(multiply(multiply(X, identity), identity)))))), inverse(multiply(multiply(X, identity), identity)))
% 0.19/0.63 = { by lemma 29 }
% 0.19/0.63 multiply(multiply(multiply(Z, multiply(Y, inverse(inverse(multiply(multiply(X, identity), identity))))), inverse(Y)), inverse(multiply(multiply(X, identity), identity)))
% 0.19/0.63 = { by axiom 1 (inverse) }
% 0.19/0.63 multiply(multiply(multiply(Z, multiply(Y, inverse(inverse(multiply(multiply(X, identity), identity))))), double_divide(Y, identity)), inverse(multiply(multiply(X, identity), identity)))
% 0.19/0.63 = { by lemma 36 R->L }
% 0.19/0.63 multiply(multiply(multiply(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity)))), double_divide(Y, identity)), inverse(multiply(multiply(X, identity), identity)))
% 0.19/0.63 = { by lemma 43 R->L }
% 0.19/0.63 multiply(double_divide(multiply(Y, identity), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity))))), inverse(multiply(multiply(X, identity), identity)))
% 0.19/0.63 = { by lemma 25 R->L }
% 0.19/0.63 inverse(inverse(multiply(double_divide(multiply(Y, identity), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity))))), inverse(multiply(multiply(X, identity), identity)))))
% 0.19/0.63 = { by lemma 27 R->L }
% 0.19/0.63 double_divide(identity, double_divide(inverse(multiply(multiply(X, identity), identity)), double_divide(multiply(Y, identity), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity)))))))
% 0.19/0.63 = { by lemma 21 R->L }
% 0.19/0.63 inverse(inverse(inverse(double_divide(inverse(multiply(multiply(X, identity), identity)), double_divide(multiply(Y, identity), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity)))))))))
% 0.19/0.63 = { by lemma 16 R->L }
% 0.19/0.63 multiply(identity, inverse(double_divide(inverse(multiply(multiply(X, identity), identity)), double_divide(multiply(Y, identity), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity))))))))
% 0.19/0.63 = { by lemma 35 R->L }
% 0.19/0.63 double_divide(double_divide(inverse(multiply(multiply(X, identity), identity)), double_divide(multiply(Y, identity), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity)))))), inverse(identity))
% 0.19/0.63 = { by lemma 22 R->L }
% 0.19/0.63 double_divide(double_divide(inverse(multiply(multiply(X, identity), identity)), double_divide(double_divide(identity, inverse(Y)), double_divide(Z, double_divide(inverse(Y), inverse(multiply(multiply(X, identity), identity)))))), inverse(identity))
% 0.19/0.63 = { by lemma 5 }
% 0.19/0.64 Z
% 0.19/0.64
% 0.19/0.64 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)).
% 0.19/0.64 Proof:
% 0.19/0.64 tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.64 = { by lemma 16 }
% 0.19/0.64 tuple(multiply(inverse(a1), a1), inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.64 = { by lemma 11 }
% 0.19/0.64 tuple(inverse(identity), inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.64 = { by lemma 14 }
% 0.19/0.64 tuple(identity, inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.64 = { by lemma 25 }
% 0.19/0.64 tuple(identity, a2, multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 0.19/0.64 = { by lemma 10 R->L }
% 0.19/0.64 tuple(identity, a2, inverse(double_divide(c3, multiply(a3, b3))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 38 R->L }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(a3, double_divide(c3, multiply(a3, b3)))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 29 R->L }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(a3, double_divide(c3, multiply(b3, a3)))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 31 R->L }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(double_divide(c3, multiply(b3, a3)), a3)), multiply(a4, b4))
% 0.19/0.64 = { by lemma 45 R->L }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(double_divide(c3, multiply(b3, a3)), double_divide(double_divide(c3, multiply(b3, a3)), multiply(b3, double_divide(a3, multiply(b3, double_divide(c3, multiply(b3, a3)))))))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 45 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(double_divide(c3, multiply(b3, a3)), double_divide(double_divide(c3, multiply(b3, a3)), multiply(b3, c3)))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 31 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(double_divide(c3, multiply(b3, a3)), double_divide(multiply(b3, c3), double_divide(c3, multiply(b3, a3))))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 43 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(double_divide(c3, multiply(b3, a3)), multiply(multiply(c3, multiply(b3, a3)), double_divide(b3, c3)))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 29 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, double_divide(double_divide(c3, multiply(b3, a3)), multiply(double_divide(b3, c3), multiply(c3, multiply(b3, a3))))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 40 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, inverse(double_divide(b3, c3))), multiply(a4, b4))
% 0.19/0.64 = { by lemma 10 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, multiply(c3, b3)), multiply(a4, b4))
% 0.19/0.64 = { by lemma 29 }
% 0.19/0.64 tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 0.19/0.64 = { by lemma 29 R->L }
% 0.19/0.64 tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 0.19/0.64 % SZS output end Proof
% 0.19/0.64
% 0.19/0.64 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------