TSTP Solution File: GRP078-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP078-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n008.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 01:16:54 EDT 2023
% Result : Unsatisfiable 0.19s 0.41s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP078-1 : TPTP v8.1.2. Bugfixed v2.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.34 % Computer : n008.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Mon Aug 28 22:09:47 EDT 2023
% 0.17/0.34 % CPUTime :
% 0.19/0.41 Command-line arguments: --no-flatten-goal
% 0.19/0.41
% 0.19/0.41 % SZS status Unsatisfiable
% 0.19/0.41
% 0.19/0.45 % SZS output start Proof
% 0.19/0.45 Take the following subset of the input axioms:
% 0.19/0.45 fof(identity, axiom, ![X]: identity=double_divide(X, inverse(X))).
% 0.19/0.45 fof(inverse, axiom, ![X2]: inverse(X2)=double_divide(X2, identity)).
% 0.19/0.45 fof(multiply, axiom, ![Y, X2]: multiply(X2, Y)=double_divide(double_divide(Y, X2), identity)).
% 0.19/0.45 fof(prove_these_axioms, negated_conjecture, multiply(inverse(a1), a1)!=identity | (multiply(identity, a2)!=a2 | multiply(multiply(a3, b3), c3)!=multiply(a3, multiply(b3, c3)))).
% 0.19/0.45 fof(single_axiom, axiom, ![Z, X2, Y2]: double_divide(double_divide(identity, X2), double_divide(identity, double_divide(double_divide(double_divide(X2, Y2), identity), double_divide(Z, Y2))))=Z).
% 0.19/0.45
% 0.19/0.45 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.45 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.45 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.45 fresh(y, y, x1...xn) = u
% 0.19/0.45 C => fresh(s, t, x1...xn) = v
% 0.19/0.45 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.45 variables of u and v.
% 0.19/0.45 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.45 input problem has no model of domain size 1).
% 0.19/0.45
% 0.19/0.45 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.45
% 0.19/0.45 Axiom 1 (inverse): inverse(X) = double_divide(X, identity).
% 0.19/0.45 Axiom 2 (identity): identity = double_divide(X, inverse(X)).
% 0.19/0.45 Axiom 3 (multiply): multiply(X, Y) = double_divide(double_divide(Y, X), identity).
% 0.19/0.45 Axiom 4 (single_axiom): double_divide(double_divide(identity, X), double_divide(identity, double_divide(double_divide(double_divide(X, Y), identity), double_divide(Z, Y)))) = Z.
% 0.19/0.45
% 0.19/0.45 Lemma 5: inverse(double_divide(X, Y)) = multiply(Y, X).
% 0.19/0.45 Proof:
% 0.19/0.45 inverse(double_divide(X, Y))
% 0.19/0.45 = { by axiom 1 (inverse) }
% 0.19/0.45 double_divide(double_divide(X, Y), identity)
% 0.19/0.45 = { by axiom 3 (multiply) R->L }
% 0.19/0.45 multiply(Y, X)
% 0.19/0.45
% 0.19/0.45 Lemma 6: double_divide(double_divide(identity, X), double_divide(identity, double_divide(multiply(Y, X), double_divide(Z, Y)))) = Z.
% 0.19/0.45 Proof:
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, double_divide(multiply(Y, X), double_divide(Z, Y))))
% 0.19/0.45 = { by lemma 5 R->L }
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, double_divide(inverse(double_divide(X, Y)), double_divide(Z, Y))))
% 0.19/0.45 = { by axiom 1 (inverse) }
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, double_divide(double_divide(double_divide(X, Y), identity), double_divide(Z, Y))))
% 0.19/0.45 = { by axiom 4 (single_axiom) }
% 0.19/0.45 Z
% 0.19/0.45
% 0.19/0.45 Lemma 7: double_divide(double_divide(identity, X), double_divide(identity, inverse(multiply(inverse(Y), X)))) = Y.
% 0.19/0.45 Proof:
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, inverse(multiply(inverse(Y), X))))
% 0.19/0.45 = { by axiom 1 (inverse) }
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, double_divide(multiply(inverse(Y), X), identity)))
% 0.19/0.45 = { by axiom 2 (identity) }
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, double_divide(multiply(inverse(Y), X), double_divide(Y, inverse(Y)))))
% 0.19/0.45 = { by lemma 6 }
% 0.19/0.45 Y
% 0.19/0.45
% 0.19/0.45 Lemma 8: multiply(inverse(X), X) = inverse(identity).
% 0.19/0.45 Proof:
% 0.19/0.45 multiply(inverse(X), X)
% 0.19/0.45 = { by lemma 5 R->L }
% 0.19/0.45 inverse(double_divide(X, inverse(X)))
% 0.19/0.45 = { by axiom 2 (identity) R->L }
% 0.19/0.45 inverse(identity)
% 0.19/0.45
% 0.19/0.45 Lemma 9: double_divide(double_divide(identity, X), double_divide(identity, inverse(inverse(identity)))) = X.
% 0.19/0.45 Proof:
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.45 = { by lemma 8 R->L }
% 0.19/0.45 double_divide(double_divide(identity, X), double_divide(identity, inverse(multiply(inverse(X), X))))
% 0.19/0.45 = { by lemma 7 }
% 0.19/0.45 X
% 0.19/0.45
% 0.19/0.45 Lemma 10: double_divide(identity, inverse(inverse(identity))) = identity.
% 0.19/0.45 Proof:
% 0.19/0.45 double_divide(identity, inverse(inverse(identity)))
% 0.19/0.45 = { by lemma 9 R->L }
% 0.19/0.45 double_divide(double_divide(identity, double_divide(identity, inverse(inverse(identity)))), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.45 = { by axiom 2 (identity) }
% 0.19/0.45 double_divide(double_divide(double_divide(identity, inverse(identity)), double_divide(identity, inverse(inverse(identity)))), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.45 = { by lemma 9 }
% 0.19/0.45 double_divide(inverse(identity), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.45 = { by axiom 1 (inverse) }
% 0.19/0.45 double_divide(double_divide(identity, identity), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.45 = { by lemma 9 }
% 0.19/0.45 identity
% 0.19/0.45
% 0.19/0.46 Lemma 11: inverse(identity) = identity.
% 0.19/0.46 Proof:
% 0.19/0.46 inverse(identity)
% 0.19/0.46 = { by lemma 7 R->L }
% 0.19/0.46 double_divide(double_divide(identity, X), double_divide(identity, inverse(multiply(inverse(inverse(identity)), X))))
% 0.19/0.46 = { by axiom 1 (inverse) }
% 0.19/0.46 double_divide(double_divide(identity, X), double_divide(identity, double_divide(multiply(inverse(inverse(identity)), X), identity)))
% 0.19/0.46 = { by lemma 10 R->L }
% 0.19/0.46 double_divide(double_divide(identity, X), double_divide(identity, double_divide(multiply(inverse(inverse(identity)), X), double_divide(identity, inverse(inverse(identity))))))
% 0.19/0.46 = { by lemma 6 }
% 0.19/0.46 identity
% 0.19/0.46
% 0.19/0.46 Lemma 12: multiply(X, identity) = X.
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(X, identity)
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 inverse(double_divide(identity, X))
% 0.19/0.46 = { by axiom 1 (inverse) }
% 0.19/0.46 double_divide(double_divide(identity, X), identity)
% 0.19/0.46 = { by lemma 10 R->L }
% 0.19/0.46 double_divide(double_divide(identity, X), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.46 = { by lemma 9 }
% 0.19/0.46 X
% 0.19/0.46
% 0.19/0.46 Lemma 13: double_divide(identity, inverse(X)) = X.
% 0.19/0.46 Proof:
% 0.19/0.46 double_divide(identity, inverse(X))
% 0.19/0.46 = { by lemma 12 R->L }
% 0.19/0.46 multiply(double_divide(identity, inverse(X)), identity)
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 inverse(double_divide(identity, double_divide(identity, inverse(X))))
% 0.19/0.46 = { by axiom 1 (inverse) }
% 0.19/0.46 double_divide(double_divide(identity, double_divide(identity, inverse(X))), identity)
% 0.19/0.46 = { by axiom 2 (identity) }
% 0.19/0.46 double_divide(double_divide(identity, double_divide(identity, inverse(X))), double_divide(identity, inverse(identity)))
% 0.19/0.46 = { by lemma 11 R->L }
% 0.19/0.46 double_divide(double_divide(identity, double_divide(identity, inverse(X))), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.46 = { by lemma 8 R->L }
% 0.19/0.46 double_divide(double_divide(identity, double_divide(identity, inverse(X))), double_divide(identity, inverse(multiply(inverse(double_divide(identity, inverse(X))), double_divide(identity, inverse(X))))))
% 0.19/0.46 = { by lemma 5 }
% 0.19/0.46 double_divide(double_divide(identity, double_divide(identity, inverse(X))), double_divide(identity, inverse(multiply(multiply(inverse(X), identity), double_divide(identity, inverse(X))))))
% 0.19/0.46 = { by lemma 12 }
% 0.19/0.46 double_divide(double_divide(identity, double_divide(identity, inverse(X))), double_divide(identity, inverse(multiply(inverse(X), double_divide(identity, inverse(X))))))
% 0.19/0.46 = { by lemma 7 }
% 0.19/0.46 X
% 0.19/0.46
% 0.19/0.46 Lemma 14: multiply(identity, X) = inverse(inverse(X)).
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(identity, X)
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 inverse(double_divide(X, identity))
% 0.19/0.46 = { by axiom 1 (inverse) R->L }
% 0.19/0.46 inverse(inverse(X))
% 0.19/0.46
% 0.19/0.46 Lemma 15: inverse(inverse(X)) = X.
% 0.19/0.46 Proof:
% 0.19/0.46 inverse(inverse(X))
% 0.19/0.46 = { by lemma 7 R->L }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, inverse(multiply(inverse(inverse(inverse(X))), Y))))
% 0.19/0.46 = { by axiom 1 (inverse) }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, double_divide(multiply(inverse(inverse(inverse(X))), Y), identity)))
% 0.19/0.46 = { by lemma 7 R->L }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, double_divide(multiply(inverse(inverse(inverse(X))), Y), double_divide(double_divide(identity, inverse(X)), double_divide(identity, inverse(multiply(inverse(identity), inverse(X))))))))
% 0.19/0.46 = { by lemma 13 }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, double_divide(multiply(inverse(inverse(inverse(X))), Y), double_divide(X, double_divide(identity, inverse(multiply(inverse(identity), inverse(X))))))))
% 0.19/0.46 = { by lemma 13 }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, double_divide(multiply(inverse(inverse(inverse(X))), Y), double_divide(X, multiply(inverse(identity), inverse(X))))))
% 0.19/0.46 = { by lemma 11 }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, double_divide(multiply(inverse(inverse(inverse(X))), Y), double_divide(X, multiply(identity, inverse(X))))))
% 0.19/0.46 = { by lemma 14 }
% 0.19/0.46 double_divide(double_divide(identity, Y), double_divide(identity, double_divide(multiply(inverse(inverse(inverse(X))), Y), double_divide(X, inverse(inverse(inverse(X)))))))
% 0.19/0.46 = { by lemma 6 }
% 0.19/0.46 X
% 0.19/0.46
% 0.19/0.46 Lemma 16: double_divide(identity, X) = inverse(X).
% 0.19/0.46 Proof:
% 0.19/0.46 double_divide(identity, X)
% 0.19/0.46 = { by lemma 15 R->L }
% 0.19/0.46 double_divide(identity, inverse(inverse(X)))
% 0.19/0.46 = { by lemma 13 }
% 0.19/0.46 inverse(X)
% 0.19/0.46
% 0.19/0.46 Lemma 17: multiply(multiply(inverse(X), Y), inverse(Y)) = inverse(X).
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(multiply(inverse(X), Y), inverse(Y))
% 0.19/0.46 = { by lemma 16 R->L }
% 0.19/0.46 multiply(multiply(inverse(X), Y), double_divide(identity, Y))
% 0.19/0.46 = { by lemma 13 R->L }
% 0.19/0.46 multiply(double_divide(identity, inverse(multiply(inverse(X), Y))), double_divide(identity, Y))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 inverse(double_divide(double_divide(identity, Y), double_divide(identity, inverse(multiply(inverse(X), Y)))))
% 0.19/0.46 = { by lemma 7 }
% 0.19/0.46 inverse(X)
% 0.19/0.46
% 0.19/0.46 Lemma 18: multiply(multiply(X, Y), inverse(Y)) = X.
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(multiply(X, Y), inverse(Y))
% 0.19/0.46 = { by lemma 12 R->L }
% 0.19/0.46 multiply(multiply(multiply(X, identity), Y), inverse(Y))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 multiply(multiply(inverse(double_divide(identity, X)), Y), inverse(Y))
% 0.19/0.46 = { by lemma 17 }
% 0.19/0.46 inverse(double_divide(identity, X))
% 0.19/0.46 = { by lemma 5 }
% 0.19/0.46 multiply(X, identity)
% 0.19/0.46 = { by lemma 12 }
% 0.19/0.46 X
% 0.19/0.46
% 0.19/0.46 Lemma 19: double_divide(X, multiply(double_divide(Y, multiply(inverse(Z), X)), inverse(Z))) = Y.
% 0.19/0.46 Proof:
% 0.19/0.46 double_divide(X, multiply(double_divide(Y, multiply(inverse(Z), X)), inverse(Z)))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 double_divide(X, inverse(double_divide(inverse(Z), double_divide(Y, multiply(inverse(Z), X)))))
% 0.19/0.46 = { by lemma 16 R->L }
% 0.19/0.46 double_divide(X, double_divide(identity, double_divide(inverse(Z), double_divide(Y, multiply(inverse(Z), X)))))
% 0.19/0.46 = { by lemma 13 R->L }
% 0.19/0.46 double_divide(double_divide(identity, inverse(X)), double_divide(identity, double_divide(inverse(Z), double_divide(Y, multiply(inverse(Z), X)))))
% 0.19/0.46 = { by lemma 17 R->L }
% 0.19/0.46 double_divide(double_divide(identity, inverse(X)), double_divide(identity, double_divide(multiply(multiply(inverse(Z), X), inverse(X)), double_divide(Y, multiply(inverse(Z), X)))))
% 0.19/0.46 = { by lemma 6 }
% 0.19/0.46 Y
% 0.19/0.46
% 0.19/0.46 Lemma 20: multiply(double_divide(X, inverse(Y)), inverse(Y)) = inverse(X).
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(double_divide(X, inverse(Y)), inverse(Y))
% 0.19/0.46 = { by lemma 12 R->L }
% 0.19/0.46 multiply(double_divide(X, multiply(inverse(Y), identity)), inverse(Y))
% 0.19/0.46 = { by lemma 9 R->L }
% 0.19/0.46 double_divide(double_divide(identity, multiply(double_divide(X, multiply(inverse(Y), identity)), inverse(Y))), double_divide(identity, inverse(inverse(identity))))
% 0.19/0.46 = { by lemma 19 }
% 0.19/0.46 double_divide(X, double_divide(identity, inverse(inverse(identity))))
% 0.19/0.46 = { by lemma 10 }
% 0.19/0.46 double_divide(X, identity)
% 0.19/0.46 = { by axiom 1 (inverse) R->L }
% 0.19/0.46 inverse(X)
% 0.19/0.46
% 0.19/0.46 Lemma 21: double_divide(X, inverse(Y)) = multiply(inverse(X), Y).
% 0.19/0.46 Proof:
% 0.19/0.46 double_divide(X, inverse(Y))
% 0.19/0.46 = { by lemma 18 R->L }
% 0.19/0.46 multiply(multiply(double_divide(X, inverse(Y)), inverse(Y)), inverse(inverse(Y)))
% 0.19/0.46 = { by lemma 20 }
% 0.19/0.46 multiply(inverse(X), inverse(inverse(Y)))
% 0.19/0.46 = { by lemma 15 }
% 0.19/0.46 multiply(inverse(X), Y)
% 0.19/0.46
% 0.19/0.46 Lemma 22: double_divide(double_divide(inverse(X), multiply(inverse(Y), X)), inverse(Y)) = identity.
% 0.19/0.46 Proof:
% 0.19/0.46 double_divide(double_divide(inverse(X), multiply(inverse(Y), X)), inverse(Y))
% 0.19/0.46 = { by lemma 17 R->L }
% 0.19/0.46 double_divide(double_divide(inverse(X), multiply(inverse(Y), X)), multiply(multiply(inverse(Y), X), inverse(X)))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 double_divide(double_divide(inverse(X), multiply(inverse(Y), X)), inverse(double_divide(inverse(X), multiply(inverse(Y), X))))
% 0.19/0.46 = { by axiom 2 (identity) R->L }
% 0.19/0.46 identity
% 0.19/0.46
% 0.19/0.46 Lemma 23: multiply(double_divide(X, multiply(Y, Z)), multiply(Y, Z)) = inverse(X).
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(double_divide(X, multiply(Y, Z)), multiply(Y, Z))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 multiply(double_divide(X, multiply(Y, Z)), inverse(double_divide(Z, Y)))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 multiply(double_divide(X, inverse(double_divide(Z, Y))), inverse(double_divide(Z, Y)))
% 0.19/0.46 = { by lemma 20 }
% 0.19/0.46 inverse(X)
% 0.19/0.46
% 0.19/0.46 Lemma 24: multiply(X, multiply(inverse(X), Y)) = Y.
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(X, multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 15 R->L }
% 0.19/0.46 multiply(inverse(inverse(X)), multiply(inverse(X), Y))
% 0.19/0.46 = { by axiom 1 (inverse) }
% 0.19/0.46 multiply(double_divide(inverse(X), identity), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 11 R->L }
% 0.19/0.46 multiply(double_divide(inverse(X), inverse(identity)), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 8 R->L }
% 0.19/0.46 multiply(double_divide(inverse(X), multiply(inverse(X), X)), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 6 R->L }
% 0.19/0.46 multiply(double_divide(double_divide(identity, Z), double_divide(identity, double_divide(multiply(inverse(X), Z), double_divide(double_divide(inverse(X), multiply(inverse(X), X)), inverse(X))))), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 22 }
% 0.19/0.46 multiply(double_divide(double_divide(identity, Z), double_divide(identity, double_divide(multiply(inverse(X), Z), identity))), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 22 R->L }
% 0.19/0.46 multiply(double_divide(double_divide(identity, Z), double_divide(identity, double_divide(multiply(inverse(X), Z), double_divide(double_divide(inverse(Y), multiply(inverse(X), Y)), inverse(X))))), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 6 }
% 0.19/0.46 multiply(double_divide(inverse(Y), multiply(inverse(X), Y)), multiply(inverse(X), Y))
% 0.19/0.46 = { by lemma 23 }
% 0.19/0.46 inverse(inverse(Y))
% 0.19/0.46 = { by lemma 15 }
% 0.19/0.46 Y
% 0.19/0.46
% 0.19/0.46 Lemma 25: multiply(inverse(X), inverse(Y)) = double_divide(X, Y).
% 0.19/0.46 Proof:
% 0.19/0.46 multiply(inverse(X), inverse(Y))
% 0.19/0.46 = { by lemma 23 R->L }
% 0.19/0.46 multiply(multiply(double_divide(X, multiply(Y, identity)), multiply(Y, identity)), inverse(Y))
% 0.19/0.46 = { by lemma 12 }
% 0.19/0.46 multiply(multiply(double_divide(X, Y), multiply(Y, identity)), inverse(Y))
% 0.19/0.46 = { by lemma 12 }
% 0.19/0.46 multiply(multiply(double_divide(X, Y), Y), inverse(Y))
% 0.19/0.46 = { by lemma 18 }
% 0.19/0.46 double_divide(X, Y)
% 0.19/0.46
% 0.19/0.46 Lemma 26: double_divide(inverse(X), Y) = multiply(X, inverse(Y)).
% 0.19/0.46 Proof:
% 0.19/0.46 double_divide(inverse(X), Y)
% 0.19/0.46 = { by lemma 24 R->L }
% 0.19/0.46 multiply(X, multiply(inverse(X), double_divide(inverse(X), Y)))
% 0.19/0.46 = { by lemma 25 R->L }
% 0.19/0.46 multiply(X, multiply(inverse(X), multiply(inverse(inverse(X)), inverse(Y))))
% 0.19/0.46 = { by lemma 24 }
% 0.19/0.46 multiply(X, inverse(Y))
% 0.19/0.46
% 0.19/0.46 Lemma 27: inverse(multiply(X, Y)) = double_divide(Y, X).
% 0.19/0.46 Proof:
% 0.19/0.46 inverse(multiply(X, Y))
% 0.19/0.46 = { by lemma 5 R->L }
% 0.19/0.46 inverse(inverse(double_divide(Y, X)))
% 0.19/0.46 = { by lemma 15 }
% 0.19/0.46 double_divide(Y, X)
% 0.19/0.46
% 0.19/0.46 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3)) = tuple(identity, a2, multiply(a3, multiply(b3, c3))).
% 0.19/0.47 Proof:
% 0.19/0.47 tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3))
% 0.19/0.47 = { by lemma 14 }
% 0.19/0.47 tuple(multiply(inverse(a1), a1), inverse(inverse(a2)), multiply(multiply(a3, b3), c3))
% 0.19/0.47 = { by lemma 8 }
% 0.19/0.47 tuple(inverse(identity), inverse(inverse(a2)), multiply(multiply(a3, b3), c3))
% 0.19/0.47 = { by lemma 11 }
% 0.19/0.47 tuple(identity, inverse(inverse(a2)), multiply(multiply(a3, b3), c3))
% 0.19/0.47 = { by lemma 15 }
% 0.19/0.47 tuple(identity, a2, multiply(multiply(a3, b3), c3))
% 0.19/0.47 = { by lemma 15 R->L }
% 0.19/0.47 tuple(identity, a2, multiply(multiply(inverse(inverse(a3)), b3), c3))
% 0.19/0.47 = { by lemma 5 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(double_divide(c3, multiply(inverse(inverse(a3)), b3))))
% 0.19/0.47 = { by lemma 21 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(double_divide(c3, double_divide(inverse(a3), inverse(b3)))))
% 0.19/0.47 = { by lemma 25 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(inverse(c3), inverse(double_divide(inverse(a3), inverse(b3))))))
% 0.19/0.47 = { by lemma 5 }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(inverse(c3), multiply(inverse(b3), inverse(a3)))))
% 0.19/0.47 = { by lemma 21 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(double_divide(c3, inverse(multiply(inverse(b3), inverse(a3))))))
% 0.19/0.47 = { by lemma 27 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(inverse(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3))))
% 0.19/0.47 = { by lemma 19 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(inverse(double_divide(inverse(a3), multiply(double_divide(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3), multiply(inverse(b3), inverse(a3))), inverse(b3))))))
% 0.19/0.47 = { by lemma 5 }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(multiply(double_divide(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3), multiply(inverse(b3), inverse(a3))), inverse(b3)), inverse(a3))))
% 0.19/0.47 = { by lemma 25 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(multiply(multiply(inverse(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3)), inverse(multiply(inverse(b3), inverse(a3)))), inverse(b3)), inverse(a3))))
% 0.19/0.47 = { by lemma 5 R->L }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(multiply(inverse(double_divide(inverse(multiply(inverse(b3), inverse(a3))), inverse(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3)))), inverse(b3)), inverse(a3))))
% 0.19/0.47 = { by lemma 25 }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(double_divide(double_divide(inverse(multiply(inverse(b3), inverse(a3))), inverse(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3))), b3), inverse(a3))))
% 0.19/0.47 = { by lemma 26 }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(double_divide(multiply(multiply(inverse(b3), inverse(a3)), inverse(inverse(multiply(inverse(multiply(inverse(b3), inverse(a3))), c3)))), b3), inverse(a3))))
% 0.19/0.47 = { by lemma 15 }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(double_divide(multiply(multiply(inverse(b3), inverse(a3)), multiply(inverse(multiply(inverse(b3), inverse(a3))), c3)), b3), inverse(a3))))
% 0.19/0.47 = { by lemma 24 }
% 0.19/0.47 tuple(identity, a2, inverse(multiply(double_divide(c3, b3), inverse(a3))))
% 0.19/0.47 = { by lemma 27 }
% 0.19/0.47 tuple(identity, a2, double_divide(inverse(a3), double_divide(c3, b3)))
% 0.19/0.47 = { by lemma 26 }
% 0.19/0.47 tuple(identity, a2, multiply(a3, inverse(double_divide(c3, b3))))
% 0.19/0.47 = { by lemma 5 }
% 0.19/0.47 tuple(identity, a2, multiply(a3, multiply(b3, c3)))
% 0.19/0.47 % SZS output end Proof
% 0.19/0.47
% 0.19/0.47 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------