TSTP Solution File: GRP570-1 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP570-1 : TPTP v8.1.2. Released v2.6.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n007.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 1.11s 1.57s
% Output : CNFRefutation 1.11s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP570-1 : TPTP v8.1.2. Released v2.6.0.
% 0.07/0.13 % Command : toma --casc %s
% 0.13/0.34 % Computer : n007.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Mon Aug 28 23:00:58 EDT 2023
% 0.13/0.35 % CPUTime :
% 1.11/1.57 % SZS status Unsatisfiable
% 1.11/1.57 % SZS output start Proof
% 1.11/1.57 original problem:
% 1.11/1.57 axioms:
% 1.11/1.57 double_divide(double_divide(A, double_divide(double_divide(B, double_divide(A, C)), double_divide(C, identity()))), double_divide(identity(), identity())) = B
% 1.11/1.57 multiply(A, B) = double_divide(double_divide(B, A), identity())
% 1.11/1.57 inverse(A) = double_divide(A, identity())
% 1.11/1.57 identity() = double_divide(A, inverse(A))
% 1.11/1.57 goal:
% 1.11/1.57 multiply(identity(), a2()) != a2()
% 1.11/1.57 To show the unsatisfiability of the original goal,
% 1.11/1.57 it suffices to show that multiply(identity(), a2()) = a2() (skolemized goal) is valid under the axioms.
% 1.11/1.57 Here is an equational proof:
% 1.11/1.57 0: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.11/1.57 Proof: Axiom.
% 1.11/1.57
% 1.11/1.57 1: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.11/1.57 Proof: Axiom.
% 1.11/1.57
% 1.11/1.57 2: inverse(X0) = double_divide(X0, identity()).
% 1.11/1.57 Proof: Axiom.
% 1.11/1.57
% 1.11/1.57 3: identity() = double_divide(X0, inverse(X0)).
% 1.11/1.57 Proof: Axiom.
% 1.11/1.57
% 1.11/1.57 4: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), inverse(X2))), inverse(identity())) = X1.
% 1.11/1.57 Proof: Rewrite equation 0,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 5: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.11/1.57 Proof: Rewrite equation 1,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 6: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, inverse(X3)), inverse(identity()))), inverse(identity())).
% 1.11/1.57 Proof: A critical pair between equations 4 and 2.
% 1.11/1.57
% 1.11/1.57 7: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, identity()), inverse(inverse(X3)))), inverse(identity())).
% 1.11/1.57 Proof: A critical pair between equations 4 and 3.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: A critical pair between equations 4 and 4.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 8,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2,2].
% 1.11/1.57
% 1.11/1.57 10: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, identity()), double_divide(double_divide(X3, identity()), identity()))), double_divide(identity(), identity())).
% 1.11/1.57 Proof: Rewrite equation 7,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2].
% 1.11/1.57
% 1.11/1.57 11: X1 = double_divide(double_divide(X3, double_divide(double_divide(X1, double_divide(X3, identity())), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.11/1.57 Proof: Rewrite equation 6,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2].
% 1.11/1.57
% 1.11/1.57 12: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.11/1.57 Proof: Rewrite equation 3,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 13: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.11/1.57 Proof: Rewrite equation 5,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 14: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.11/1.57 Proof: Rewrite equation 4,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 15: X1 = double_divide(double_divide(X1, identity()), double_divide(identity(), identity())).
% 1.11/1.57 Proof: A critical pair between equations 10 and 12.
% 1.11/1.57
% 1.11/1.57 16: X4 = double_divide(double_divide(X4, double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.11/1.57 Proof: A critical pair between equations 11 and 12.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: A critical pair between equations 11 and 10.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: A critical pair between equations 11 and 11.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: A critical pair between equations 11 and 14.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: A critical pair between equations 9 and 11.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 25,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2,2,2,2].
% 1.11/1.57
% 1.11/1.57 34: X4 = double_divide(double_divide(X4, double_divide(identity(), inverse(identity()))), inverse(identity())).
% 1.11/1.57 Proof: Rewrite equation 16,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 35: X1 = double_divide(inverse(X1), inverse(identity())).
% 1.11/1.57 Proof: Rewrite equation 15,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 36: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), inverse(X2))), inverse(identity())) = X1.
% 1.11/1.57 Proof: Rewrite equation 14,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 37: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.11/1.57 Proof: Rewrite equation 13,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 38: identity() = double_divide(X0, inverse(X0)).
% 1.11/1.57 Proof: Rewrite equation 12,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 21,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 20,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 44: double_divide(X4, double_divide(inverse(X5), inverse(inverse(X4)))) = double_divide(double_divide(identity(), double_divide(X5, inverse(identity()))), inverse(identity())).
% 1.11/1.57 Proof: Rewrite equation 19,
% 1.11/1.57 lhs with equations [2,2,2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 47: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), inverse(X6))) = identity().
% 1.11/1.57 Proof: A critical pair between equations 42 and 34.
% 1.11/1.57
% 1.11/1.57 50: double_divide(X4, double_divide(inverse(double_divide(X7, double_divide(identity(), identity()))), inverse(inverse(X4)))) = X7.
% 1.11/1.57 Proof: A critical pair between equations 44 and 36.
% 1.11/1.57
% 1.11/1.57 51: X8 = double_divide(double_divide(double_divide(identity(), double_divide(X8, inverse(identity()))), inverse(identity())), inverse(identity())).
% 1.11/1.57 Proof: A critical pair between equations 36 and 42.
% 1.11/1.57
% 1.11/1.57 52: double_divide(X4, double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), inverse(X4)), inverse(identity()))) = X7.
% 1.11/1.57 Proof: A critical pair between equations 43 and 36.
% 1.11/1.57
% 1.11/1.57 53: double_divide(X4, double_divide(inverse(identity()), inverse(inverse(X4)))) = double_divide(double_divide(identity(), identity()), inverse(identity())).
% 1.11/1.57 Proof: A critical pair between equations 44 and 38.
% 1.11/1.57
% 1.11/1.57 56: X1 = double_divide(double_divide(inverse(X3), double_divide(double_divide(X1, X3), inverse(inverse(identity())))), inverse(identity())).
% 1.11/1.57 Proof: A critical pair between equations 36 and 35.
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 56,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2,2].
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 53,
% 1.11/1.57 lhs with equations [2,2,2]
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 61: double_divide(X4, double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), double_divide(X4, identity())), double_divide(identity(), identity()))) = X7.
% 1.11/1.57 Proof: Rewrite equation 52,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 62: X8 = double_divide(double_divide(double_divide(identity(), double_divide(X8, double_divide(identity(), identity()))), double_divide(identity(), identity())), double_divide(identity(), identity())).
% 1.11/1.57 Proof: Rewrite equation 51,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2].
% 1.11/1.57
% 1.11/1.57 63: double_divide(X4, double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), identity()), double_divide(double_divide(X4, identity()), identity()))) = X7.
% 1.11/1.57 Proof: Rewrite equation 50,
% 1.11/1.57 lhs with equations [2,2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 64: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), double_divide(X6, identity()))) = identity().
% 1.11/1.57 Proof: Rewrite equation 47,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 69: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.11/1.57 Proof: Rewrite equation 38,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 70: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.11/1.57 Proof: Rewrite equation 37,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 71: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.11/1.57 Proof: Rewrite equation 36,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 72: X1 = double_divide(double_divide(X1, identity()), double_divide(identity(), identity())).
% 1.11/1.57 Proof: Rewrite equation 35,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 28,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2,2,2,2].
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 43,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 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())).
% 1.11/1.57 Proof: Rewrite equation 42,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 81: X7 = double_divide(double_divide(X7, double_divide(identity(), identity())), identity()).
% 1.11/1.57 Proof: A critical pair between equations 63 and 69.
% 1.11/1.57
% 1.11/1.57 88: double_divide(X8, identity()) = double_divide(X4, double_divide(double_divide(X8, double_divide(X4, identity())), double_divide(identity(), identity()))).
% 1.11/1.57 Proof: A critical pair between equations 61 and 72.
% 1.11/1.57
% 1.11/1.57 90: double_divide(X9, identity()) = double_divide(double_divide(double_divide(identity(), X9), double_divide(identity(), identity())), double_divide(identity(), identity())).
% 1.11/1.57 Proof: A critical pair between equations 62 and 72.
% 1.11/1.57
% 1.11/1.57 94: inverse(X9) = double_divide(double_divide(double_divide(identity(), X9), inverse(identity())), inverse(identity())).
% 1.11/1.57 Proof: Rewrite equation 90,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 96: inverse(X8) = double_divide(X4, double_divide(double_divide(X8, inverse(X4)), inverse(identity()))).
% 1.11/1.57 Proof: Rewrite equation 88,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 102: X7 = inverse(double_divide(X7, inverse(identity()))).
% 1.11/1.57 Proof: Rewrite equation 81,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 104: 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())).
% 1.11/1.57 Proof: Rewrite equation 76,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,2,96,2,2,2].
% 1.11/1.57
% 1.11/1.57 107: X1 = double_divide(inverse(X1), inverse(identity())).
% 1.11/1.57 Proof: Rewrite equation 72,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2].
% 1.11/1.57
% 1.11/1.57 108: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), inverse(X2))), inverse(identity())) = X1.
% 1.11/1.57 Proof: Rewrite equation 71,
% 1.11/1.57 lhs with equations [2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 109: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.11/1.57 Proof: Rewrite equation 70,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 112: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), inverse(X6))) = identity().
% 1.11/1.57 Proof: Rewrite equation 64,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 113: double_divide(X4, double_divide(X7, inverse(inverse(X4)))) = X7.
% 1.11/1.57 Proof: Rewrite equation 63,
% 1.11/1.57 lhs with equations [2,2,102,2,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 114: inverse(identity()) = identity().
% 1.11/1.57 Proof: Rewrite equation 60,
% 1.11/1.57 lhs with equations [2,2,2,113]
% 1.11/1.57 rhs with equations [2,2,107].
% 1.11/1.57
% 1.11/1.57 116: X1 = inverse(double_divide(inverse(X3), inverse(double_divide(X1, X3)))).
% 1.11/1.57 Proof: Rewrite equation 57,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,2,114,2,114,2,2,114,2].
% 1.11/1.57
% 1.11/1.57 117: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), inverse(X6))) = inverse(double_divide(identity(), inverse(X5))).
% 1.11/1.57 Proof: Rewrite equation 80,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,114,2,2,114,2].
% 1.11/1.57
% 1.11/1.57 118: double_divide(X4, inverse(double_divide(X5, inverse(X4)))) = inverse(double_divide(identity(), inverse(X5))).
% 1.11/1.57 Proof: Rewrite equation 79,
% 1.11/1.57 lhs with equations [2,2,114,2]
% 1.11/1.57 rhs with equations [2,114,2,2,114,2].
% 1.11/1.57
% 1.11/1.57 120: X7 = inverse(double_divide(X7, identity())).
% 1.11/1.57 Proof: A critical pair between equations 102 and 114.
% 1.11/1.57
% 1.11/1.57 126: inverse(double_divide(identity(), inverse(X8))) = double_divide(identity(), X8).
% 1.11/1.57 Proof: A critical pair between equations 118 and 102.
% 1.11/1.57
% 1.11/1.57 127: inverse(double_divide(identity(), X10)) = double_divide(identity(), inverse(X10)).
% 1.11/1.57 Proof: A critical pair between equations 96 and 94.
% 1.11/1.57
% 1.11/1.57 131: X1 = inverse(double_divide(identity(), inverse(double_divide(X1, identity())))).
% 1.11/1.57 Proof: A critical pair between equations 116 and 114.
% 1.11/1.57
% 1.11/1.57 132: X1 = double_divide(identity(), double_divide(X1, identity())).
% 1.11/1.57 Proof: Rewrite equation 131,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [120,127,2].
% 1.11/1.57
% 1.11/1.57 134: double_divide(double_divide(identity(), X10), identity()) = X10.
% 1.11/1.57 Proof: Rewrite equation 127,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,132].
% 1.11/1.57
% 1.11/1.57 135: double_divide(X8, identity()) = double_divide(identity(), X8).
% 1.11/1.57 Proof: Rewrite equation 126,
% 1.11/1.57 lhs with equations [2,132,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 138: double_divide(identity(), identity()) = identity().
% 1.11/1.57 Proof: Rewrite equation 114,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 139: double_divide(X4, double_divide(X7, X4)) = X7.
% 1.11/1.57 Proof: Rewrite equation 113,
% 1.11/1.57 lhs with equations [2,2,135,132]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 140: double_divide(X4, double_divide(double_divide(identity(), double_divide(X4, X6)), double_divide(X6, identity()))) = identity().
% 1.11/1.57 Proof: Rewrite equation 112,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 142: multiply(X0, X1) = double_divide(identity(), double_divide(X1, X0)).
% 1.11/1.57 Proof: Rewrite equation 109,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,135].
% 1.11/1.57
% 1.11/1.57 143: double_divide(identity(), double_divide(X0, double_divide(double_divide(X1, double_divide(X0, X2)), double_divide(X2, identity())))) = X1.
% 1.11/1.57 Proof: Rewrite equation 108,
% 1.11/1.57 lhs with equations [2,2,138,135]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 144: X1 = double_divide(identity(), double_divide(X7, double_divide(identity(), double_divide(X1, double_divide(X7, identity()))))).
% 1.11/1.57 Proof: Rewrite equation 104,
% 1.11/1.57 lhs with equations []
% 1.11/1.57 rhs with equations [2,138,139,2,2,138,2,138,135,2,138,135].
% 1.11/1.57
% 1.11/1.57 146: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), double_divide(X6, identity()))) = double_divide(X5, identity()).
% 1.11/1.57 Proof: Rewrite equation 117,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2,139,2].
% 1.11/1.57
% 1.11/1.57 148: double_divide(identity(), double_divide(identity(), X11)) = X11.
% 1.11/1.57 Proof: A critical pair between equations 135 and 134.
% 1.11/1.57
% 1.11/1.57 152: double_divide(X9, double_divide(identity(), identity())) = double_divide(identity(), X9).
% 1.11/1.57 Proof: A critical pair between equations 144 and 143.
% 1.11/1.57
% 1.11/1.57 164: double_divide(X4, double_divide(double_divide(X5, double_divide(X4, X6)), inverse(X6))) = inverse(X5).
% 1.11/1.57 Proof: Rewrite equation 146,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [2].
% 1.11/1.57
% 1.11/1.57 166: inverse(identity()) = identity().
% 1.11/1.57 Proof: Rewrite equation 140,
% 1.11/1.57 lhs with equations [2,164]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 170: inverse(X9) = double_divide(identity(), X9).
% 1.11/1.57 Proof: Rewrite equation 152,
% 1.11/1.57 lhs with equations [2,166,2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 188: double_divide(X9, identity()) = double_divide(identity(), X9).
% 1.11/1.57 Proof: Rewrite equation 170,
% 1.11/1.57 lhs with equations [2]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 195: multiply(identity(), a2()) = a2().
% 1.11/1.57 Proof: Rewrite lhs with equations [142,188,148]
% 1.11/1.57 rhs with equations [].
% 1.11/1.57
% 1.11/1.57 % SZS output end Proof
%------------------------------------------------------------------------------