TSTP Solution File: GRP572-1 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP572-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% 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 01:15:23 EDT 2023
% Result : Unsatisfiable 1.37s 1.69s
% Output : CNFRefutation 1.37s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.13 % Problem : GRP572-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.10/0.13 % Command : toma --casc %s
% 0.14/0.35 % Computer : n005.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Mon Aug 28 20:46:07 EDT 2023
% 0.14/0.35 % CPUTime :
% 1.37/1.69 % SZS status Unsatisfiable
% 1.37/1.69 % SZS output start Proof
% 1.37/1.69 original problem:
% 1.37/1.69 axioms:
% 1.37/1.69 double_divide(double_divide(A, double_divide(double_divide(B, double_divide(A, C)), double_divide(C, identity()))), double_divide(identity(), identity())) = B
% 1.37/1.69 multiply(A, B) = double_divide(double_divide(B, A), identity())
% 1.37/1.69 inverse(A) = double_divide(A, identity())
% 1.37/1.69 identity() = double_divide(A, inverse(A))
% 1.37/1.69 goal:
% 1.37/1.69 multiply(a(), b()) != multiply(b(), a())
% 1.37/1.69 To show the unsatisfiability of the original goal,
% 1.37/1.69 it suffices to show that multiply(a(), b()) = multiply(b(), a()) (skolemized goal) is valid under the axioms.
% 1.37/1.69 Here is an equational proof:
% 1.37/1.69 0: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.37/1.69 Proof: Axiom.
% 1.37/1.69
% 1.37/1.69 1: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.37/1.69 Proof: Axiom.
% 1.37/1.69
% 1.37/1.69 2: inverse(X0) = double_divide(X0, identity()).
% 1.37/1.69 Proof: Axiom.
% 1.37/1.69
% 1.37/1.69 3: identity() = double_divide(X0, inverse(X0)).
% 1.37/1.69 Proof: Axiom.
% 1.37/1.69
% 1.37/1.69 4: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), inverse(X2))), inverse(identity())) = X1.
% 1.37/1.69 Proof: Rewrite equation 0,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 5: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.37/1.69 Proof: Rewrite equation 1,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 6: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, inverse(X3)), inverse(identity()))), inverse(identity())).
% 1.37/1.69 Proof: A critical pair between equations 4 and 2.
% 1.37/1.69
% 1.37/1.69 7: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, identity()), inverse(inverse(X3)))), inverse(identity())).
% 1.37/1.69 Proof: A critical pair between equations 4 and 3.
% 1.37/1.69
% 1.37/1.69 8: X1 = double_divide(double_divide(double_divide(X3, double_divide(double_divide(X4, double_divide(X3, X5)), inverse(X5))), double_divide(double_divide(X1, X4), inverse(inverse(identity())))), inverse(identity())).
% 1.37/1.69 Proof: A critical pair between equations 4 and 4.
% 1.37/1.69
% 1.37/1.69 9: X1 = double_divide(double_divide(double_divide(X3, double_divide(double_divide(X4, double_divide(X3, X5)), double_divide(X5, identity()))), double_divide(double_divide(X1, X4), double_divide(double_divide(identity(), identity()), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 8,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2,2].
% 1.37/1.69
% 1.37/1.69 10: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, identity()), double_divide(double_divide(X3, identity()), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 7,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2].
% 1.37/1.69
% 1.37/1.69 11: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, double_divide(X3, identity())), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 6,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2].
% 1.37/1.69
% 1.37/1.69 12: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.37/1.69 Proof: Rewrite equation 3,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 13: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.37/1.69 Proof: Rewrite equation 5,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 14: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.37/1.69 Proof: Rewrite equation 4,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 15: X1 = double_divide(double_divide(X1, identity()), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 10 and 12.
% 1.37/1.69
% 1.37/1.69 16: X4 = double_divide(double_divide(X4, double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 11 and 12.
% 1.37/1.69
% 1.37/1.69 19: double_divide(X4, double_divide(double_divide(X5, identity()), double_divide(double_divide(X4, identity()), identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 11 and 10.
% 1.37/1.69
% 1.37/1.69 20: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, identity())), double_divide(identity(), identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 11 and 11.
% 1.37/1.69
% 1.37/1.69 21: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), double_divide(X6, identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 11 and 14.
% 1.37/1.69
% 1.37/1.69 25: X1 = double_divide(double_divide(double_divide(identity(), double_divide(X7, double_divide(identity(), identity()))), double_divide(double_divide(X1, double_divide(X6, double_divide(double_divide(X7, double_divide(X6, identity())), double_divide(identity(), identity())))), double_divide(double_divide(identity(), identity()), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 9 and 11.
% 1.37/1.69
% 1.37/1.69 28: X1 = double_divide(double_divide(double_divide(identity(), double_divide(X7, inverse(identity()))), double_divide(double_divide(X1, double_divide(X6, double_divide(double_divide(X7, inverse(X6)), inverse(identity())))), inverse(inverse(identity())))), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 25,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2,2,2,2].
% 1.37/1.69
% 1.37/1.69 34: X4 = double_divide(double_divide(X4, double_divide(identity(), inverse(identity()))), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 16,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 35: X1 = double_divide(inverse(X1), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 15,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 36: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), inverse(X2))), inverse(identity())) = X1.
% 1.37/1.69 Proof: Rewrite equation 14,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 37: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.37/1.69 Proof: Rewrite equation 13,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 38: identity() = double_divide(X0, inverse(X0)).
% 1.37/1.69 Proof: Rewrite equation 12,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 42: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), inverse(X6))) = double_divide(double_divide(identity(), double_divide(X5, inverse(identity()))), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 21,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 43: double_divide(X4, double_divide(double_divide(X5, inverse(X4)), inverse(identity()))) = double_divide(double_divide(identity(), double_divide(X5, inverse(identity()))), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 20,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 44: double_divide(X4, double_divide(inverse(X5), inverse(inverse(X4)))) = double_divide(double_divide(identity(), double_divide(X5, inverse(identity()))), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 19,
% 1.37/1.69 lhs with equations [2,2,2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 47: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), inverse(X6))) = identity().
% 1.37/1.69 Proof: A critical pair between equations 42 and 34.
% 1.37/1.69
% 1.37/1.69 50: double_divide(X4, double_divide(inverse(double_divide(X7, double_divide(identity(), identity()))), inverse(inverse(X4)))) = X7.
% 1.37/1.69 Proof: A critical pair between equations 44 and 36.
% 1.37/1.69
% 1.37/1.69 51: X8 = double_divide(double_divide(double_divide(identity(), double_divide(X8, inverse(identity()))), inverse(identity())), inverse(identity())).
% 1.37/1.69 Proof: A critical pair between equations 36 and 42.
% 1.37/1.69
% 1.37/1.69 52: double_divide(X4, double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), inverse(X4)), inverse(identity()))) = X7.
% 1.37/1.69 Proof: A critical pair between equations 43 and 36.
% 1.37/1.69
% 1.37/1.69 53: double_divide(X4, double_divide(inverse(identity()), inverse(inverse(X4)))) = double_divide(double_divide(identity(), identity()), inverse(identity())).
% 1.37/1.69 Proof: A critical pair between equations 44 and 38.
% 1.37/1.69
% 1.37/1.69 56: X1 = double_divide(double_divide(inverse(X3), double_divide(double_divide(X1, X3), inverse(inverse(identity())))), inverse(identity())).
% 1.37/1.69 Proof: A critical pair between equations 36 and 35.
% 1.37/1.69
% 1.37/1.69 57: X1 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(X1, X3), double_divide(double_divide(identity(), identity()), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 56,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2,2].
% 1.37/1.69
% 1.37/1.69 60: double_divide(X4, double_divide(double_divide(identity(), identity()), double_divide(double_divide(X4, identity()), identity()))) = double_divide(double_divide(identity(), identity()), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 53,
% 1.37/1.69 lhs with equations [2,2,2]
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 61: double_divide(X4, double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), double_divide(X4, identity())), double_divide(identity(), identity()))) = X7.
% 1.37/1.69 Proof: Rewrite equation 52,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 62: X8 = double_divide(double_divide(double_divide(identity(), double_divide(X8, double_divide(identity(), identity()))), double_divide(identity(), identity())), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 51,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2].
% 1.37/1.69
% 1.37/1.69 63: double_divide(X4, double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), identity()), double_divide(double_divide(X4, identity()), identity()))) = X7.
% 1.37/1.69 Proof: Rewrite equation 50,
% 1.37/1.69 lhs with equations [2,2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 64: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), double_divide(X6, identity()))) = identity().
% 1.37/1.69 Proof: Rewrite equation 47,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 69: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.37/1.69 Proof: Rewrite equation 38,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 70: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.37/1.69 Proof: Rewrite equation 37,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 71: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.37/1.69 Proof: Rewrite equation 36,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 72: X1 = double_divide(double_divide(X1, identity()), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 35,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 76: X1 = double_divide(double_divide(double_divide(identity(), double_divide(X7, double_divide(identity(), identity()))), double_divide(double_divide(X1, double_divide(X6, double_divide(double_divide(X7, double_divide(X6, identity())), double_divide(identity(), identity())))), double_divide(double_divide(identity(), identity()), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 28,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2,2,2,2].
% 1.37/1.69
% 1.37/1.69 79: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, identity())), double_divide(identity(), identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 43,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 80: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), double_divide(X6, identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.37/1.69 Proof: Rewrite equation 42,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 81: X7 = double_divide(double_divide(X7, double_divide(identity(), identity())), identity()).
% 1.37/1.69 Proof: A critical pair between equations 63 and 69.
% 1.37/1.69
% 1.37/1.69 88: double_divide(X8, identity()) = double_divide(X4, double_divide(double_divide(X8, double_divide(X4, identity())), double_divide(identity(), identity()))).
% 1.37/1.69 Proof: A critical pair between equations 61 and 72.
% 1.37/1.69
% 1.37/1.69 90: double_divide(X9, identity()) = double_divide(double_divide(double_divide(identity(), X9), double_divide(identity(), identity())), double_divide(identity(), identity())).
% 1.37/1.69 Proof: A critical pair between equations 62 and 72.
% 1.37/1.69
% 1.37/1.69 94: inverse(X9) = double_divide(double_divide(double_divide(identity(), X9), inverse(identity())), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 90,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 96: inverse(X8) = double_divide(X4, double_divide(double_divide(X8, inverse(X4)), inverse(identity()))).
% 1.37/1.69 Proof: Rewrite equation 88,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 102: X7 = inverse(double_divide(X7, inverse(identity()))).
% 1.37/1.69 Proof: Rewrite equation 81,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 104: X1 = double_divide(double_divide(double_divide(identity(), double_divide(X7, inverse(identity()))), double_divide(double_divide(X1, inverse(X7)), inverse(inverse(identity())))), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 76,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,2,96,2,2,2].
% 1.37/1.69
% 1.37/1.69 107: X1 = double_divide(inverse(X1), inverse(identity())).
% 1.37/1.69 Proof: Rewrite equation 72,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2].
% 1.37/1.69
% 1.37/1.69 108: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), inverse(X2))), inverse(identity())) = X1.
% 1.37/1.69 Proof: Rewrite equation 71,
% 1.37/1.69 lhs with equations [2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 109: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.37/1.69 Proof: Rewrite equation 70,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 112: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), inverse(X6))) = identity().
% 1.37/1.69 Proof: Rewrite equation 64,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 113: double_divide(X4, double_divide(X7, inverse(inverse(X4)))) = X7.
% 1.37/1.69 Proof: Rewrite equation 63,
% 1.37/1.69 lhs with equations [2,2,102,2,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 114: inverse(identity()) = identity().
% 1.37/1.69 Proof: Rewrite equation 60,
% 1.37/1.69 lhs with equations [2,2,2,113]
% 1.37/1.69 rhs with equations [2,2,107].
% 1.37/1.69
% 1.37/1.69 116: X1 = inverse(double_divide(inverse(X3), inverse(double_divide(X1, X3)))).
% 1.37/1.69 Proof: Rewrite equation 57,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,2,114,2,114,2,2,114,2].
% 1.37/1.69
% 1.37/1.69 117: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), inverse(X6))) = inverse(double_divide(identity(), inverse(X5))).
% 1.37/1.69 Proof: Rewrite equation 80,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2,114,2,2,114,2].
% 1.37/1.69
% 1.37/1.69 118: double_divide(X4, inverse(double_divide(X5, inverse(X4)))) = inverse(double_divide(identity(), inverse(X5))).
% 1.37/1.69 Proof: Rewrite equation 79,
% 1.37/1.69 lhs with equations [2,2,114,2]
% 1.37/1.69 rhs with equations [2,114,2,2,114,2].
% 1.37/1.69
% 1.37/1.69 120: X7 = inverse(double_divide(X7, identity())).
% 1.37/1.69 Proof: A critical pair between equations 102 and 114.
% 1.37/1.69
% 1.37/1.69 126: inverse(double_divide(identity(), inverse(X8))) = double_divide(identity(), X8).
% 1.37/1.69 Proof: A critical pair between equations 118 and 102.
% 1.37/1.69
% 1.37/1.69 127: inverse(double_divide(identity(), X10)) = double_divide(identity(), inverse(X10)).
% 1.37/1.69 Proof: A critical pair between equations 96 and 94.
% 1.37/1.69
% 1.37/1.69 131: X1 = inverse(double_divide(identity(), inverse(double_divide(X1, identity())))).
% 1.37/1.69 Proof: A critical pair between equations 116 and 114.
% 1.37/1.69
% 1.37/1.69 132: X1 = double_divide(identity(), double_divide(X1, identity())).
% 1.37/1.69 Proof: Rewrite equation 131,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [120,127,2].
% 1.37/1.69
% 1.37/1.69 135: double_divide(X8, identity()) = double_divide(identity(), X8).
% 1.37/1.69 Proof: Rewrite equation 126,
% 1.37/1.69 lhs with equations [2,132,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 138: double_divide(identity(), identity()) = identity().
% 1.37/1.69 Proof: Rewrite equation 114,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 139: double_divide(X4, double_divide(X7, X4)) = X7.
% 1.37/1.69 Proof: Rewrite equation 113,
% 1.37/1.69 lhs with equations [2,2,135,132]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 140: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), double_divide(X6, identity()))) = identity().
% 1.37/1.69 Proof: Rewrite equation 112,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 142: multiply(X0, X1) = double_divide(identity(), double_divide(X1, X0)).
% 1.37/1.69 Proof: Rewrite equation 109,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,135].
% 1.37/1.69
% 1.37/1.69 143: double_divide(identity(), double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity())))) = X1.
% 1.37/1.69 Proof: Rewrite equation 108,
% 1.37/1.69 lhs with equations [2,2,138,135]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 144: X1 = double_divide(identity(), double_divide(X7, double_divide(identity(), double_divide(X1, double_divide(X7, identity()))))).
% 1.37/1.69 Proof: Rewrite equation 104,
% 1.37/1.69 lhs with equations []
% 1.37/1.69 rhs with equations [2,138,139,2,2,138,2,138,135,2,138,135].
% 1.37/1.69
% 1.37/1.69 146: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), double_divide(X6, identity()))) = double_divide(X5, identity()).
% 1.37/1.69 Proof: Rewrite equation 117,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2,139,2].
% 1.37/1.69
% 1.37/1.69 150: X8 = double_divide(double_divide(X9, X8), X9).
% 1.37/1.69 Proof: A critical pair between equations 139 and 139.
% 1.37/1.69
% 1.37/1.69 152: double_divide(X9, double_divide(identity(), identity())) = double_divide(identity(), X9).
% 1.37/1.69 Proof: A critical pair between equations 144 and 143.
% 1.37/1.69
% 1.37/1.69 157: double_divide(X8, identity()) = double_divide(X9, double_divide(X9, double_divide(X8, identity()))).
% 1.37/1.69 Proof: A critical pair between equations 146 and 139.
% 1.37/1.69
% 1.37/1.69 160: inverse(X8) = double_divide(X9, double_divide(X9, inverse(X8))).
% 1.37/1.69 Proof: Rewrite equation 157,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 164: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), inverse(X6))) = inverse(X5).
% 1.37/1.69 Proof: Rewrite equation 146,
% 1.37/1.69 lhs with equations [2]
% 1.37/1.69 rhs with equations [2].
% 1.37/1.69
% 1.37/1.69 166: inverse(identity()) = identity().
% 1.37/1.69 Proof: Rewrite equation 140,
% 1.37/1.69 lhs with equations [2,164]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 170: inverse(X9) = double_divide(identity(), X9).
% 1.37/1.69 Proof: Rewrite equation 152,
% 1.37/1.69 lhs with equations [2,166,2]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 180: inverse(X8) = double_divide(double_divide(inverse(X8), X11), X11).
% 1.37/1.69 Proof: A critical pair between equations 160 and 150.
% 1.37/1.69
% 1.37/1.69 184: double_divide(identity(), X8) = double_divide(double_divide(double_divide(identity(), X8), X11), X11).
% 1.37/1.69 Proof: Rewrite equation 180,
% 1.37/1.69 lhs with equations [170]
% 1.37/1.69 rhs with equations [170].
% 1.37/1.69
% 1.37/1.69 199: double_divide(identity(), double_divide(X13, identity())) = double_divide(double_divide(X13, X11), X11).
% 1.37/1.69 Proof: A critical pair between equations 184 and 139.
% 1.37/1.69
% 1.37/1.69 207: X13 = double_divide(double_divide(X13, X11), X11).
% 1.37/1.69 Proof: Rewrite equation 199,
% 1.37/1.69 lhs with equations [139]
% 1.37/1.69 rhs with equations [].
% 1.37/1.69
% 1.37/1.69 212: double_divide(X14, X15) = double_divide(X15, X14).
% 1.37/1.69 Proof: A critical pair between equations 207 and 150.
% 1.37/1.69
% 1.37/1.69 232: multiply(a(), b()) = multiply(b(), a()).
% 1.37/1.69 Proof: Rewrite lhs with equations [142,212]
% 1.37/1.69 rhs with equations [142].
% 1.37/1.69
% 1.37/1.69 % SZS output end Proof
%------------------------------------------------------------------------------