%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP103-1 : TPTP v9.3.1. Bugfixed v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:09:16 AM UTC 2026
% Result : Unsatisfiable 0.57s 0.54s
% Output : Proof 1.57s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : GRP103-1 : TPTP v9.3.1. Bugfixed v2.7.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.36 % Computer : n015.cluster.edu
% 0.11/0.36 % Model : x86_64 x86_64
% 0.11/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.36 % Memory : 8046.5625MB
% 0.11/0.36 % OS : Linux 6.8.0-71-generic
% 0.11/0.36 % CPULimit : 300
% 0.11/0.36 % WCLimit : 300
% 0.11/0.36 % DateTime : Sun Sep 27 09:22:29 UTC 2026
% 0.11/0.37 % CPUTime :
% 0.11/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.57/0.54 Command-line arguments: --no-flatten-goal
% 0.57/0.54
% 0.57/0.54 % SZS status Unsatisfiable
% 0.57/0.54
% 0.57/0.61 % SZS output start Proof
% 0.57/0.61 Axiom 1 (inverse): inverse(X) = double_divide(X, identity).
% 0.57/0.61 Axiom 2 (identity): identity = double_divide(X, inverse(X)).
% 0.57/0.61 Axiom 3 (multiply): multiply(X, Y) = double_divide(double_divide(Y, X), identity).
% 0.57/0.61 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.
% 0.57/0.61
% 0.57/0.61 Lemma 5: double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), inverse(identity)) = Z.
% 0.57/0.61 Proof:
% 0.57/0.61 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), inverse(identity))
% 0.57/0.61 = { by axiom 1 (inverse) }
% 0.57/0.61 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(Z, double_divide(Y, X)))), double_divide(identity, identity))
% 0.57/0.61 = { by axiom 4 (single_axiom) }
% 0.57/0.61 Z
% 0.57/0.61
% 0.57/0.61 Lemma 6: double_divide(double_divide(X, double_divide(inverse(identity), double_divide(Y, double_divide(identity, X)))), inverse(identity)) = Y.
% 0.57/0.61 Proof:
% 0.57/0.61 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(Y, double_divide(identity, X)))), inverse(identity))
% 0.57/0.61 = { by axiom 1 (inverse) }
% 0.57/0.61 double_divide(double_divide(X, double_divide(double_divide(identity, identity), double_divide(Y, double_divide(identity, X)))), inverse(identity))
% 0.57/0.61 = { by lemma 5 }
% 0.57/0.61 Y
% 0.57/0.61
% 0.57/0.61 Lemma 7: double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))), inverse(identity)) = X.
% 0.57/0.61 Proof:
% 0.57/0.61 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))), inverse(identity))
% 0.57/0.61 = { by axiom 1 (inverse) }
% 0.57/0.61 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), double_divide(X, identity))), inverse(identity))
% 0.57/0.61 = { by axiom 2 (identity) }
% 0.57/0.61 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), double_divide(X, double_divide(identity, inverse(identity))))), inverse(identity))
% 0.57/0.61 = { by lemma 6 }
% 0.57/0.61 X
% 0.57/0.61
% 0.57/0.61 Lemma 8: double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, inverse(identity)))), inverse(identity)) = X.
% 0.57/0.61 Proof:
% 0.57/0.61 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, inverse(identity)))), inverse(identity))
% 0.57/0.61 = { by axiom 1 (inverse) }
% 0.57/0.61 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))), inverse(identity))
% 0.57/0.61 = { by lemma 6 }
% 0.57/0.61 X
% 0.57/0.61
% 0.57/0.61 Lemma 9: double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity)) = inverse(inverse(identity)).
% 0.57/0.61 Proof:
% 0.57/0.61 double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity))
% 0.57/0.61 = { by lemma 7 R->L }
% 0.57/0.61 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))
% 0.57/0.61 = { by axiom 2 (identity) R->L }
% 0.57/0.61 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(double_divide(inverse(identity), identity), inverse(identity)))), inverse(identity))
% 0.57/0.61 = { by axiom 1 (inverse) R->L }
% 0.57/0.61 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(inverse(inverse(identity)), inverse(identity)))), inverse(identity))
% 0.57/0.61 = { by lemma 8 }
% 0.57/0.61 inverse(inverse(identity))
% 0.57/0.61
% 0.57/0.61 Lemma 10: inverse(double_divide(X, Y)) = multiply(Y, X).
% 0.57/0.61 Proof:
% 0.57/0.61 inverse(double_divide(X, Y))
% 0.57/0.61 = { by axiom 1 (inverse) }
% 0.57/0.61 double_divide(double_divide(X, Y), identity)
% 0.57/0.61 = { by axiom 3 (multiply) R->L }
% 0.57/0.61 multiply(Y, X)
% 0.57/0.61
% 0.57/0.61 Lemma 11: multiply(inverse(X), X) = inverse(identity).
% 0.57/0.61 Proof:
% 0.57/0.61 multiply(inverse(X), X)
% 0.57/0.61 = { by lemma 10 R->L }
% 0.57/0.61 inverse(double_divide(X, inverse(X)))
% 0.57/0.61 = { by axiom 2 (identity) R->L }
% 0.57/0.61 inverse(identity)
% 0.57/0.61
% 0.57/0.61 Lemma 12: double_divide(double_divide(identity, double_divide(inverse(identity), X)), inverse(identity)) = double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))).
% 0.57/0.61 Proof:
% 0.57/0.61 double_divide(double_divide(identity, double_divide(inverse(identity), X)), inverse(identity))
% 0.57/0.61 = { by lemma 7 R->L }
% 0.57/0.61 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))
% 0.57/0.61 = { by lemma 8 }
% 0.57/0.61 double_divide(inverse(identity), double_divide(inverse(identity), inverse(X)))
% 0.57/0.61
% 0.57/0.61 Lemma 13: inverse(inverse(identity)) = inverse(identity).
% 0.57/0.61 Proof:
% 0.57/0.61 inverse(inverse(identity))
% 0.57/0.62 = { by lemma 9 R->L }
% 0.57/0.62 double_divide(double_divide(identity, double_divide(inverse(identity), inverse(identity))), inverse(identity))
% 0.57/0.62 = { by lemma 8 R->L }
% 0.57/0.62 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))
% 0.57/0.62 = { by lemma 9 }
% 0.57/0.62 double_divide(double_divide(double_divide(identity, double_divide(inverse(identity), inverse(inverse(identity)))), inverse(identity)), inverse(identity))
% 0.57/0.62 = { by axiom 2 (identity) R->L }
% 0.57/0.62 double_divide(double_divide(double_divide(identity, identity), inverse(identity)), inverse(identity))
% 0.57/0.62 = { by axiom 1 (inverse) R->L }
% 0.57/0.62 double_divide(double_divide(inverse(identity), inverse(identity)), inverse(identity))
% 0.57/0.62 = { by lemma 11 R->L }
% 0.57/0.62 double_divide(double_divide(inverse(identity), multiply(inverse(identity), identity)), inverse(identity))
% 1.57/0.62 = { by lemma 10 R->L }
% 1.57/0.62 double_divide(double_divide(inverse(identity), inverse(double_divide(identity, inverse(identity)))), inverse(identity))
% 1.57/0.62 = { by axiom 1 (inverse) }
% 1.57/0.62 double_divide(double_divide(inverse(identity), double_divide(double_divide(identity, inverse(identity)), identity)), inverse(identity))
% 1.57/0.62 = { by lemma 8 R->L }
% 1.57/0.62 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))
% 1.57/0.62 = { by axiom 2 (identity) R->L }
% 1.57/0.62 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))
% 1.57/0.62 = { by lemma 12 }
% 1.57/0.62 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))
% 1.57/0.62 = { by lemma 5 }
% 1.57/0.62 inverse(identity)
% 1.57/0.62
% 1.57/0.62 Lemma 14: inverse(identity) = identity.
% 1.57/0.62 Proof:
% 1.57/0.62 inverse(identity)
% 1.57/0.62 = { by lemma 7 R->L }
% 1.57/0.62 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(identity)))), inverse(identity))
% 1.57/0.62 = { by lemma 13 }
% 1.57/0.62 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(identity))), inverse(identity))
% 1.57/0.62 = { by lemma 7 }
% 1.57/0.62 identity
% 1.57/0.62
% 1.57/0.62 Lemma 15: double_divide(identity, double_divide(identity, inverse(X))) = multiply(double_divide(identity, X), identity).
% 1.57/0.62 Proof:
% 1.57/0.62 double_divide(identity, double_divide(identity, inverse(X)))
% 1.57/0.62 = { by lemma 14 R->L }
% 1.57/0.62 double_divide(inverse(identity), double_divide(identity, inverse(X)))
% 1.57/0.62 = { by lemma 14 R->L }
% 1.57/0.62 double_divide(inverse(identity), double_divide(inverse(identity), inverse(X)))
% 1.57/0.62 = { by lemma 12 R->L }
% 1.57/0.62 double_divide(double_divide(identity, double_divide(inverse(identity), X)), inverse(identity))
% 1.57/0.62 = { by lemma 14 }
% 1.57/0.62 double_divide(double_divide(identity, double_divide(identity, X)), inverse(identity))
% 1.57/0.62 = { by lemma 14 }
% 1.57/0.62 double_divide(double_divide(identity, double_divide(identity, X)), identity)
% 1.57/0.62 = { by axiom 1 (inverse) R->L }
% 1.57/0.62 inverse(double_divide(identity, double_divide(identity, X)))
% 1.57/0.62 = { by lemma 10 }
% 1.57/0.62 multiply(double_divide(identity, X), identity)
% 1.57/0.62
% 1.57/0.62 Lemma 16: multiply(identity, X) = inverse(inverse(X)).
% 1.57/0.62 Proof:
% 1.57/0.62 multiply(identity, X)
% 1.57/0.62 = { by lemma 10 R->L }
% 1.57/0.62 inverse(double_divide(X, identity))
% 1.57/0.62 = { by axiom 1 (inverse) R->L }
% 1.57/0.62 inverse(inverse(X))
% 1.57/0.62
% 1.57/0.62 Lemma 17: double_divide(identity, double_divide(identity, inverse(inverse(X)))) = X.
% 1.57/0.62 Proof:
% 1.57/0.62 double_divide(identity, double_divide(identity, inverse(inverse(X))))
% 1.57/0.62 = { by lemma 16 R->L }
% 1.57/0.62 double_divide(identity, double_divide(identity, multiply(identity, X)))
% 1.57/0.62 = { by lemma 14 R->L }
% 1.57/0.62 double_divide(identity, double_divide(identity, multiply(inverse(identity), X)))
% 1.57/0.62 = { by axiom 1 (inverse) }
% 1.57/0.62 double_divide(identity, double_divide(identity, multiply(double_divide(identity, identity), X)))
% 1.57/0.62 = { by lemma 10 R->L }
% 1.57/0.62 double_divide(identity, double_divide(identity, inverse(double_divide(X, double_divide(identity, identity)))))
% 1.57/0.62 = { by lemma 14 R->L }
% 1.57/0.62 double_divide(inverse(identity), double_divide(identity, inverse(double_divide(X, double_divide(identity, identity)))))
% 1.57/0.62 = { by lemma 14 R->L }
% 1.57/0.62 double_divide(inverse(identity), double_divide(inverse(identity), inverse(double_divide(X, double_divide(identity, identity)))))
% 1.57/0.62 = { by lemma 12 R->L }
% 1.57/0.62 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, double_divide(identity, identity)))), inverse(identity))
% 1.57/0.62 = { by lemma 6 }
% 1.57/0.62 X
% 1.57/0.62
% 1.57/0.62 Lemma 18: multiply(double_divide(identity, multiply(X, Y)), identity) = double_divide(Y, X).
% 1.57/0.62 Proof:
% 1.57/0.62 multiply(double_divide(identity, multiply(X, Y)), identity)
% 1.57/0.62 = { by lemma 15 R->L }
% 1.57/0.62 double_divide(identity, double_divide(identity, inverse(multiply(X, Y))))
% 1.57/0.62 = { by lemma 10 R->L }
% 1.57/0.62 double_divide(identity, double_divide(identity, inverse(inverse(double_divide(Y, X)))))
% 1.57/0.62 = { by lemma 17 }
% 1.57/0.63 double_divide(Y, X)
% 1.57/0.63
% 1.57/0.63 Lemma 19: multiply(double_divide(identity, inverse(X)), identity) = X.
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(double_divide(identity, inverse(X)), identity)
% 1.57/0.63 = { by axiom 1 (inverse) }
% 1.57/0.63 multiply(double_divide(identity, double_divide(X, identity)), identity)
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 inverse(double_divide(identity, double_divide(identity, double_divide(X, identity))))
% 1.57/0.63 = { by axiom 1 (inverse) }
% 1.57/0.63 double_divide(double_divide(identity, double_divide(identity, double_divide(X, identity))), identity)
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 double_divide(double_divide(identity, double_divide(identity, double_divide(X, inverse(identity)))), identity)
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 double_divide(double_divide(identity, double_divide(identity, double_divide(X, inverse(identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 double_divide(double_divide(identity, double_divide(inverse(identity), double_divide(X, inverse(identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 8 }
% 1.57/0.63 X
% 1.57/0.63
% 1.57/0.63 Lemma 20: multiply(double_divide(identity, double_divide(identity, X)), identity) = inverse(inverse(X)).
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(double_divide(identity, double_divide(identity, X)), identity)
% 1.57/0.63 = { by lemma 6 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(multiply(double_divide(identity, double_divide(identity, X)), identity), double_divide(identity, X)))), inverse(identity))
% 1.57/0.63 = { by lemma 19 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(multiply(double_divide(identity, double_divide(identity, X)), identity), multiply(double_divide(identity, inverse(double_divide(identity, X))), identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 15 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(double_divide(identity, double_divide(identity, inverse(double_divide(identity, X)))), multiply(double_divide(identity, inverse(double_divide(identity, X))), identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(double_divide(identity, double_divide(identity, inverse(double_divide(identity, X)))), inverse(double_divide(identity, double_divide(identity, inverse(double_divide(identity, X)))))))), inverse(identity))
% 1.57/0.63 = { by axiom 2 (identity) R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(inverse(identity), identity)), inverse(identity))
% 1.57/0.63 = { by axiom 1 (inverse) R->L }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(identity))), inverse(identity))
% 1.57/0.63 = { by lemma 13 }
% 1.57/0.63 double_divide(double_divide(X, inverse(identity)), inverse(identity))
% 1.57/0.63 = { by lemma 14 }
% 1.57/0.63 double_divide(double_divide(X, identity), inverse(identity))
% 1.57/0.63 = { by lemma 14 }
% 1.57/0.63 double_divide(double_divide(X, identity), identity)
% 1.57/0.63 = { by axiom 1 (inverse) R->L }
% 1.57/0.63 inverse(double_divide(X, identity))
% 1.57/0.63 = { by lemma 10 }
% 1.57/0.63 multiply(identity, X)
% 1.57/0.63 = { by lemma 16 }
% 1.57/0.63 inverse(inverse(X))
% 1.57/0.63
% 1.57/0.63 Lemma 21: double_divide(identity, X) = inverse(inverse(inverse(X))).
% 1.57/0.63 Proof:
% 1.57/0.63 double_divide(identity, X)
% 1.57/0.63 = { by lemma 19 R->L }
% 1.57/0.63 multiply(double_divide(identity, inverse(double_divide(identity, X))), identity)
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 inverse(double_divide(identity, double_divide(identity, inverse(double_divide(identity, X)))))
% 1.57/0.63 = { by lemma 15 }
% 1.57/0.63 inverse(multiply(double_divide(identity, double_divide(identity, X)), identity))
% 1.57/0.63 = { by lemma 20 }
% 1.57/0.63 inverse(inverse(inverse(X)))
% 1.57/0.63
% 1.57/0.63 Lemma 22: multiply(double_divide(inverse(inverse(inverse(X))), inverse(Y)), inverse(X)) = Y.
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(double_divide(inverse(inverse(inverse(X))), inverse(Y)), inverse(X))
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 inverse(double_divide(inverse(X), double_divide(inverse(inverse(inverse(X))), inverse(Y))))
% 1.57/0.63 = { by axiom 1 (inverse) }
% 1.57/0.63 double_divide(double_divide(inverse(X), double_divide(inverse(inverse(inverse(X))), inverse(Y))), identity)
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 double_divide(double_divide(inverse(X), double_divide(inverse(inverse(inverse(X))), inverse(Y))), inverse(identity))
% 1.57/0.63 = { by axiom 1 (inverse) }
% 1.57/0.63 double_divide(double_divide(inverse(X), double_divide(inverse(inverse(inverse(X))), double_divide(Y, identity))), inverse(identity))
% 1.57/0.63 = { by lemma 21 R->L }
% 1.57/0.63 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, X), double_divide(Y, identity))), inverse(identity))
% 1.57/0.63 = { by axiom 2 (identity) }
% 1.57/0.63 double_divide(double_divide(inverse(X), double_divide(double_divide(identity, X), double_divide(Y, double_divide(X, inverse(X))))), inverse(identity))
% 1.57/0.63 = { by lemma 5 }
% 1.57/0.63 Y
% 1.57/0.63
% 1.57/0.63 Lemma 23: multiply(inverse(inverse(multiply(inverse(inverse(inverse(X))), Y))), X) = Y.
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(inverse(inverse(multiply(inverse(inverse(inverse(X))), Y))), X)
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 inverse(double_divide(X, inverse(inverse(multiply(inverse(inverse(inverse(X))), Y)))))
% 1.57/0.63 = { by axiom 1 (inverse) }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(multiply(inverse(inverse(inverse(X))), Y)))), identity)
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(multiply(inverse(inverse(inverse(X))), Y)))), inverse(identity))
% 1.57/0.63 = { by lemma 21 R->L }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(multiply(double_divide(identity, X), Y)))), inverse(identity))
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(inverse(double_divide(Y, double_divide(identity, X)))))), inverse(identity))
% 1.57/0.63 = { by lemma 21 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(identity, double_divide(Y, double_divide(identity, X)))), inverse(identity))
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(inverse(identity), double_divide(Y, double_divide(identity, X)))), inverse(identity))
% 1.57/0.63 = { by lemma 6 }
% 1.57/0.63 Y
% 1.57/0.63
% 1.57/0.63 Lemma 24: inverse(inverse(inverse(X))) = inverse(X).
% 1.57/0.63 Proof:
% 1.57/0.63 inverse(inverse(inverse(X)))
% 1.57/0.63 = { by lemma 16 R->L }
% 1.57/0.63 multiply(identity, inverse(X))
% 1.57/0.63 = { by lemma 14 R->L }
% 1.57/0.63 multiply(inverse(identity), inverse(X))
% 1.57/0.63 = { by lemma 13 R->L }
% 1.57/0.63 multiply(inverse(inverse(identity)), inverse(X))
% 1.57/0.63 = { by lemma 22 R->L }
% 1.57/0.63 multiply(inverse(inverse(multiply(double_divide(inverse(inverse(inverse(X))), inverse(identity)), inverse(X)))), inverse(X))
% 1.57/0.63 = { by lemma 14 }
% 1.57/0.63 multiply(inverse(inverse(multiply(double_divide(inverse(inverse(inverse(X))), identity), inverse(X)))), inverse(X))
% 1.57/0.63 = { by axiom 1 (inverse) R->L }
% 1.57/0.63 multiply(inverse(inverse(multiply(inverse(inverse(inverse(inverse(X)))), inverse(X)))), inverse(X))
% 1.57/0.63 = { by lemma 23 }
% 1.57/0.63 inverse(X)
% 1.57/0.63
% 1.57/0.63 Lemma 25: inverse(inverse(X)) = X.
% 1.57/0.63 Proof:
% 1.57/0.63 inverse(inverse(X))
% 1.57/0.63 = { by lemma 7 R->L }
% 1.57/0.63 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(inverse(inverse(X))))), inverse(identity))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 double_divide(double_divide(inverse(identity), double_divide(inverse(identity), inverse(X))), inverse(identity))
% 1.57/0.63 = { by lemma 7 }
% 1.57/0.63 X
% 1.57/0.63
% 1.57/0.63 Lemma 26: multiply(X, identity) = inverse(inverse(inverse(inverse(X)))).
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(X, identity)
% 1.57/0.63 = { by lemma 17 R->L }
% 1.57/0.63 multiply(double_divide(identity, double_divide(identity, inverse(inverse(X)))), identity)
% 1.57/0.63 = { by lemma 20 }
% 1.57/0.63 inverse(inverse(inverse(inverse(X))))
% 1.57/0.63
% 1.57/0.63 Lemma 27: multiply(X, Y) = multiply(Y, X).
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(X, Y)
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 inverse(double_divide(Y, X))
% 1.57/0.63 = { by lemma 5 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(inverse(double_divide(Y, X)), double_divide(Y, X)))), inverse(identity))
% 1.57/0.63 = { by lemma 25 R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(double_divide(identity, Y), double_divide(inverse(double_divide(Y, X)), inverse(inverse(double_divide(Y, X)))))), inverse(identity))
% 1.57/0.63 = { by axiom 2 (identity) R->L }
% 1.57/0.63 double_divide(double_divide(X, double_divide(double_divide(identity, Y), identity)), inverse(identity))
% 1.57/0.63 = { by axiom 1 (inverse) R->L }
% 1.57/0.63 double_divide(double_divide(X, inverse(double_divide(identity, Y))), inverse(identity))
% 1.57/0.63 = { by lemma 10 }
% 1.57/0.63 double_divide(double_divide(X, multiply(Y, identity)), inverse(identity))
% 1.57/0.63 = { by lemma 26 }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(inverse(inverse(Y))))), inverse(identity))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 double_divide(double_divide(X, inverse(inverse(Y))), inverse(identity))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 double_divide(double_divide(X, Y), inverse(identity))
% 1.57/0.63 = { by lemma 14 }
% 1.57/0.63 double_divide(double_divide(X, Y), identity)
% 1.57/0.63 = { by axiom 1 (inverse) R->L }
% 1.57/0.63 inverse(double_divide(X, Y))
% 1.57/0.63 = { by lemma 10 }
% 1.57/0.63 multiply(Y, X)
% 1.57/0.63
% 1.57/0.63 Lemma 28: inverse(multiply(X, Y)) = double_divide(X, Y).
% 1.57/0.63 Proof:
% 1.57/0.63 inverse(multiply(X, Y))
% 1.57/0.63 = { by lemma 27 R->L }
% 1.57/0.63 inverse(multiply(Y, X))
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 inverse(inverse(double_divide(X, Y)))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 double_divide(X, Y)
% 1.57/0.63
% 1.57/0.63 Lemma 29: double_divide(X, Y) = double_divide(Y, X).
% 1.57/0.63 Proof:
% 1.57/0.63 double_divide(X, Y)
% 1.57/0.63 = { by lemma 18 R->L }
% 1.57/0.63 multiply(double_divide(identity, multiply(Y, X)), identity)
% 1.57/0.63 = { by lemma 15 R->L }
% 1.57/0.63 double_divide(identity, double_divide(identity, inverse(multiply(Y, X))))
% 1.57/0.63 = { by lemma 28 }
% 1.57/0.63 double_divide(identity, double_divide(identity, double_divide(Y, X)))
% 1.57/0.63 = { by lemma 21 }
% 1.57/0.63 inverse(inverse(inverse(double_divide(identity, double_divide(Y, X)))))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 inverse(double_divide(identity, double_divide(Y, X)))
% 1.57/0.63 = { by lemma 10 }
% 1.57/0.63 multiply(double_divide(Y, X), identity)
% 1.57/0.63 = { by lemma 26 }
% 1.57/0.63 inverse(inverse(inverse(inverse(double_divide(Y, X)))))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 inverse(inverse(double_divide(Y, X)))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 double_divide(Y, X)
% 1.57/0.63
% 1.57/0.63 Lemma 30: inverse(double_divide(X, Y)) = multiply(X, Y).
% 1.57/0.63 Proof:
% 1.57/0.63 inverse(double_divide(X, Y))
% 1.57/0.63 = { by lemma 29 R->L }
% 1.57/0.63 inverse(double_divide(Y, X))
% 1.57/0.63 = { by lemma 28 R->L }
% 1.57/0.63 inverse(inverse(multiply(Y, X)))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 multiply(Y, X)
% 1.57/0.63 = { by lemma 27 }
% 1.57/0.63 multiply(X, Y)
% 1.57/0.63
% 1.57/0.63 Lemma 31: multiply(X, double_divide(X, inverse(Y))) = Y.
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(X, double_divide(X, inverse(Y)))
% 1.57/0.63 = { by lemma 29 R->L }
% 1.57/0.63 multiply(X, double_divide(inverse(Y), X))
% 1.57/0.63 = { by lemma 25 R->L }
% 1.57/0.63 multiply(X, double_divide(inverse(Y), inverse(inverse(X))))
% 1.57/0.63 = { by lemma 16 R->L }
% 1.57/0.63 multiply(X, double_divide(inverse(Y), multiply(identity, X)))
% 1.57/0.63 = { by lemma 25 R->L }
% 1.57/0.63 multiply(inverse(inverse(X)), double_divide(inverse(Y), multiply(identity, X)))
% 1.57/0.63 = { by lemma 16 R->L }
% 1.57/0.63 multiply(multiply(identity, X), double_divide(inverse(Y), multiply(identity, X)))
% 1.57/0.63 = { by lemma 27 R->L }
% 1.57/0.63 multiply(multiply(identity, X), double_divide(inverse(Y), multiply(X, identity)))
% 1.57/0.63 = { by lemma 29 R->L }
% 1.57/0.63 multiply(multiply(identity, X), double_divide(multiply(X, identity), inverse(Y)))
% 1.57/0.63 = { by lemma 27 R->L }
% 1.57/0.63 multiply(double_divide(multiply(X, identity), inverse(Y)), multiply(identity, X))
% 1.57/0.63 = { by lemma 30 R->L }
% 1.57/0.63 multiply(double_divide(multiply(X, identity), inverse(Y)), inverse(double_divide(identity, X)))
% 1.57/0.63 = { by lemma 25 R->L }
% 1.57/0.63 multiply(double_divide(inverse(inverse(multiply(X, identity))), inverse(Y)), inverse(double_divide(identity, X)))
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 multiply(double_divide(inverse(inverse(inverse(double_divide(identity, X)))), inverse(Y)), inverse(double_divide(identity, X)))
% 1.57/0.63 = { by lemma 22 }
% 1.57/0.63 Y
% 1.57/0.63
% 1.57/0.63 Lemma 32: double_divide(X, double_divide(X, inverse(Y))) = inverse(Y).
% 1.57/0.63 Proof:
% 1.57/0.63 double_divide(X, double_divide(X, inverse(Y)))
% 1.57/0.63 = { by lemma 29 R->L }
% 1.57/0.63 double_divide(double_divide(X, inverse(Y)), X)
% 1.57/0.63 = { by lemma 18 R->L }
% 1.57/0.63 multiply(double_divide(identity, multiply(X, double_divide(X, inverse(Y)))), identity)
% 1.57/0.63 = { by lemma 31 }
% 1.57/0.63 multiply(double_divide(identity, Y), identity)
% 1.57/0.63 = { by lemma 26 }
% 1.57/0.63 inverse(inverse(inverse(inverse(double_divide(identity, Y)))))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 inverse(inverse(double_divide(identity, Y)))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 double_divide(identity, Y)
% 1.57/0.63 = { by lemma 21 }
% 1.57/0.63 inverse(inverse(inverse(Y)))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 inverse(Y)
% 1.57/0.63
% 1.57/0.63 Lemma 33: multiply(X, double_divide(X, Y)) = inverse(Y).
% 1.57/0.63 Proof:
% 1.57/0.63 multiply(X, double_divide(X, Y))
% 1.57/0.63 = { by lemma 31 R->L }
% 1.57/0.63 multiply(X, double_divide(X, multiply(Z, double_divide(Z, inverse(Y)))))
% 1.57/0.63 = { by lemma 10 R->L }
% 1.57/0.63 multiply(X, double_divide(X, inverse(double_divide(double_divide(Z, inverse(Y)), Z))))
% 1.57/0.63 = { by lemma 31 }
% 1.57/0.63 double_divide(double_divide(Z, inverse(Y)), Z)
% 1.57/0.63 = { by lemma 29 }
% 1.57/0.63 double_divide(Z, double_divide(Z, inverse(Y)))
% 1.57/0.63 = { by lemma 32 }
% 1.57/0.63 inverse(Y)
% 1.57/0.63
% 1.57/0.63 Lemma 34: double_divide(X, inverse(Y)) = multiply(Y, inverse(X)).
% 1.57/0.63 Proof:
% 1.57/0.63 double_divide(X, inverse(Y))
% 1.57/0.63 = { by lemma 29 R->L }
% 1.57/0.63 double_divide(inverse(Y), X)
% 1.57/0.63 = { by lemma 24 R->L }
% 1.57/0.63 double_divide(inverse(inverse(inverse(Y))), X)
% 1.57/0.63 = { by lemma 23 R->L }
% 1.57/0.63 multiply(inverse(inverse(multiply(inverse(inverse(inverse(Y))), double_divide(inverse(inverse(inverse(Y))), X)))), Y)
% 1.57/0.63 = { by lemma 33 }
% 1.57/0.63 multiply(inverse(inverse(inverse(X))), Y)
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 multiply(inverse(X), Y)
% 1.57/0.63 = { by lemma 27 }
% 1.57/0.63 multiply(Y, inverse(X))
% 1.57/0.63
% 1.57/0.63 Lemma 35: double_divide(X, Y) = multiply(inverse(X), inverse(Y)).
% 1.57/0.63 Proof:
% 1.57/0.63 double_divide(X, Y)
% 1.57/0.63 = { by lemma 6 R->L }
% 1.57/0.63 double_divide(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), double_divide(identity, double_divide(identity, X))))), inverse(identity))
% 1.57/0.63 = { by lemma 29 R->L }
% 1.57/0.63 double_divide(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), double_divide(double_divide(identity, X), identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 18 R->L }
% 1.57/0.63 double_divide(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), multiply(double_divide(identity, multiply(identity, double_divide(identity, X))), identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 33 }
% 1.57/0.63 double_divide(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), multiply(double_divide(identity, inverse(X)), identity)))), inverse(identity))
% 1.57/0.63 = { by lemma 19 }
% 1.57/0.63 double_divide(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), X))), inverse(identity))
% 1.57/0.63 = { by lemma 34 }
% 1.57/0.63 multiply(identity, inverse(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), X)))))
% 1.57/0.63 = { by lemma 16 }
% 1.57/0.63 inverse(inverse(inverse(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), X))))))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 inverse(double_divide(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), X))))
% 1.57/0.63 = { by lemma 30 }
% 1.57/0.63 multiply(double_divide(identity, X), double_divide(inverse(identity), double_divide(double_divide(X, Y), X)))
% 1.57/0.63 = { by lemma 21 }
% 1.57/0.63 multiply(inverse(inverse(inverse(X))), double_divide(inverse(identity), double_divide(double_divide(X, Y), X)))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 multiply(inverse(X), double_divide(inverse(identity), double_divide(double_divide(X, Y), X)))
% 1.57/0.63 = { by lemma 14 }
% 1.57/0.63 multiply(inverse(X), double_divide(identity, double_divide(double_divide(X, Y), X)))
% 1.57/0.63 = { by lemma 21 }
% 1.57/0.63 multiply(inverse(X), inverse(inverse(inverse(double_divide(double_divide(X, Y), X)))))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 multiply(inverse(X), inverse(double_divide(double_divide(X, Y), X)))
% 1.57/0.63 = { by lemma 30 }
% 1.57/0.63 multiply(inverse(X), multiply(double_divide(X, Y), X))
% 1.57/0.63 = { by lemma 27 }
% 1.57/0.63 multiply(inverse(X), multiply(X, double_divide(X, Y)))
% 1.57/0.63 = { by lemma 33 }
% 1.57/0.63 multiply(inverse(X), inverse(Y))
% 1.57/0.63
% 1.57/0.63 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)).
% 1.57/0.63 Proof:
% 1.57/0.63 tuple(multiply(inverse(a1), a1), multiply(identity, a2), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 16 }
% 1.57/0.63 tuple(multiply(inverse(a1), a1), inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 11 }
% 1.57/0.63 tuple(inverse(identity), inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 14 }
% 1.57/0.63 tuple(identity, inverse(inverse(a2)), multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 tuple(identity, a2, multiply(multiply(a3, b3), c3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 27 }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(a3, b3)), multiply(a4, b4))
% 1.57/0.63 = { by lemma 27 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, a3)), multiply(a4, b4))
% 1.57/0.63 = { by lemma 23 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, multiply(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), multiply(inverse(c3), inverse(b3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 27 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, multiply(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), multiply(inverse(b3), inverse(c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 24 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, multiply(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), multiply(inverse(inverse(inverse(b3))), inverse(c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 16 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, multiply(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), multiply(multiply(identity, inverse(b3)), inverse(c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 30 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, multiply(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), multiply(inverse(double_divide(identity, inverse(b3))), inverse(c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 35 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, multiply(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3)))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 30 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(b3, inverse(double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 25 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, multiply(inverse(inverse(b3)), inverse(double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 35 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(c3, double_divide(inverse(b3), double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3)))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 30 R->L }
% 1.57/0.63 tuple(identity, a2, inverse(double_divide(c3, double_divide(inverse(b3), double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 24 R->L }
% 1.57/0.63 tuple(identity, a2, inverse(inverse(inverse(double_divide(c3, double_divide(inverse(b3), double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3))))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 16 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(identity, inverse(double_divide(c3, double_divide(inverse(b3), double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3)))))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 34 R->L }
% 1.57/0.63 tuple(identity, a2, double_divide(double_divide(c3, double_divide(inverse(b3), double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3)))), inverse(identity)), multiply(a4, b4))
% 1.57/0.63 = { by lemma 32 R->L }
% 1.57/0.63 tuple(identity, a2, double_divide(double_divide(c3, double_divide(double_divide(identity, double_divide(identity, inverse(b3))), double_divide(inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), double_divide(double_divide(identity, inverse(b3)), c3)))), inverse(identity)), multiply(a4, b4))
% 1.57/0.63 = { by lemma 5 }
% 1.57/0.63 tuple(identity, a2, inverse(inverse(multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3))), multiply(a4, b4))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 tuple(identity, a2, multiply(inverse(inverse(inverse(multiply(inverse(c3), inverse(b3))))), a3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 24 }
% 1.57/0.63 tuple(identity, a2, multiply(inverse(multiply(inverse(c3), inverse(b3))), a3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 28 }
% 1.57/0.63 tuple(identity, a2, multiply(double_divide(inverse(c3), inverse(b3)), a3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 34 }
% 1.57/0.63 tuple(identity, a2, multiply(multiply(b3, inverse(inverse(c3))), a3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 25 }
% 1.57/0.63 tuple(identity, a2, multiply(multiply(b3, c3), a3), multiply(a4, b4))
% 1.57/0.63 = { by lemma 27 }
% 1.57/0.63 tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(a4, b4))
% 1.57/0.63 = { by lemma 27 R->L }
% 1.57/0.63 tuple(identity, a2, multiply(a3, multiply(b3, c3)), multiply(b4, a4))
% 1.57/0.63 % SZS output end Proof
% 1.57/0.63
% 1.57/0.63 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------