TSTP Solution File: GRP576-1 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP576-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n003.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 01:15:24 EDT 2023
% Result : Unsatisfiable 1.69s 2.00s
% Output : CNFRefutation 1.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : GRP576-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.12/0.13 % Command : toma --casc %s
% 0.13/0.34 % Computer : n003.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 21:32:25 EDT 2023
% 0.13/0.34 % CPUTime :
% 1.69/2.00 % SZS status Unsatisfiable
% 1.69/2.00 % SZS output start Proof
% 1.69/2.00 original problem:
% 1.69/2.00 axioms:
% 1.69/2.00 double_divide(double_divide(A, double_divide(double_divide(B, double_divide(C, A)), double_divide(C, identity()))), double_divide(identity(), identity())) = B
% 1.69/2.00 multiply(A, B) = double_divide(double_divide(B, A), identity())
% 1.69/2.00 inverse(A) = double_divide(A, identity())
% 1.69/2.00 identity() = double_divide(A, inverse(A))
% 1.69/2.00 goal:
% 1.69/2.00 multiply(a(), b()) != multiply(b(), a())
% 1.69/2.00 To show the unsatisfiability of the original goal,
% 1.69/2.00 it suffices to show that multiply(a(), b()) = multiply(b(), a()) (skolemized goal) is valid under the axioms.
% 1.69/2.00 Here is an equational proof:
% 1.69/2.00 0: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.69/2.00 Proof: Axiom.
% 1.69/2.00
% 1.69/2.00 1: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.69/2.00 Proof: Axiom.
% 1.69/2.00
% 1.69/2.00 2: inverse(X0) = double_divide(X0, identity()).
% 1.69/2.00 Proof: Axiom.
% 1.69/2.00
% 1.69/2.00 3: identity() = double_divide(X0, inverse(X0)).
% 1.69/2.00 Proof: Axiom.
% 1.69/2.00
% 1.69/2.00 4: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), inverse(X2))), inverse(identity())) = X1.
% 1.69/2.00 Proof: Rewrite equation 0,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 5: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.69/2.00 Proof: Rewrite equation 1,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 6: X1 = double_divide(double_divide(identity(), double_divide(double_divide(X1, inverse(X3)), inverse(X3))), inverse(identity())).
% 1.69/2.00 Proof: A critical pair between equations 4 and 2.
% 1.69/2.00
% 1.69/2.00 7: X1 = double_divide(double_divide(inverse(X3), double_divide(double_divide(X1, identity()), inverse(X3))), inverse(identity())).
% 1.69/2.00 Proof: A critical pair between equations 4 and 3.
% 1.69/2.00
% 1.69/2.00 8: X1 = double_divide(double_divide(inverse(identity()), double_divide(double_divide(X1, X4), inverse(double_divide(X3, double_divide(double_divide(X4, double_divide(X5, X3)), inverse(X5)))))), inverse(identity())).
% 1.69/2.00 Proof: A critical pair between equations 4 and 4.
% 1.69/2.00
% 1.69/2.00 9: X1 = double_divide(double_divide(double_divide(identity(), identity()), double_divide(double_divide(X1, X4), double_divide(double_divide(X3, double_divide(double_divide(X4, double_divide(X5, X3)), double_divide(X5, identity()))), identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 8,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 10: X1 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(X1, identity()), double_divide(X3, identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 7,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2].
% 1.69/2.00
% 1.69/2.00 11: X1 = double_divide(double_divide(identity(), double_divide(double_divide(X1, double_divide(X3, identity())), double_divide(X3, identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 6,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2].
% 1.69/2.00
% 1.69/2.00 12: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.69/2.00 Proof: Rewrite equation 3,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 13: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.69/2.00 Proof: Rewrite equation 5,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 14: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.69/2.00 Proof: Rewrite equation 4,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 15: X1 = double_divide(double_divide(double_divide(double_divide(X1, identity()), identity()), identity()), double_divide(identity(), identity())).
% 1.69/2.00 Proof: A critical pair between equations 10 and 12.
% 1.69/2.00
% 1.69/2.00 16: X4 = double_divide(double_divide(identity(), double_divide(identity(), double_divide(X4, identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: A critical pair between equations 11 and 12.
% 1.69/2.00
% 1.69/2.00 18: double_divide(double_divide(X4, identity()), double_divide(double_divide(X5, identity()), double_divide(X4, identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: A critical pair between equations 11 and 10.
% 1.69/2.00
% 1.69/2.00 20: double_divide(X4, double_divide(double_divide(X5, double_divide(X6, X4)), double_divide(X6, identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: A critical pair between equations 11 and 14.
% 1.69/2.00
% 1.69/2.00 33: X4 = double_divide(double_divide(identity(), double_divide(identity(), inverse(X4))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 16,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 34: X1 = double_divide(inverse(inverse(inverse(X1))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 15,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 35: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), inverse(X2))), inverse(identity())) = X1.
% 1.69/2.00 Proof: Rewrite equation 14,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 36: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.69/2.00 Proof: Rewrite equation 13,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 37: identity() = double_divide(X0, inverse(X0)).
% 1.69/2.00 Proof: Rewrite equation 12,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 38: X1 = double_divide(double_divide(identity(), double_divide(double_divide(X1, inverse(X3)), inverse(X3))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 11,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2].
% 1.69/2.00
% 1.69/2.00 39: X1 = double_divide(double_divide(inverse(X3), double_divide(inverse(X1), inverse(X3))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 10,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 40: X1 = double_divide(double_divide(inverse(identity()), double_divide(double_divide(X1, X4), inverse(double_divide(X3, double_divide(double_divide(X4, double_divide(X5, X3)), inverse(X5)))))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 9,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 42: double_divide(X4, double_divide(double_divide(X5, double_divide(X6, X4)), inverse(X6))) = double_divide(double_divide(identity(), double_divide(X5, inverse(identity()))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 20,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 44: double_divide(inverse(X4), double_divide(inverse(X5), inverse(X4))) = double_divide(double_divide(identity(), double_divide(X5, inverse(identity()))), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 18,
% 1.69/2.00 lhs with equations [2,2,2]
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 45: identity() = double_divide(double_divide(identity(), identity()), inverse(identity())).
% 1.69/2.00 Proof: A critical pair between equations 33 and 37.
% 1.69/2.00
% 1.69/2.00 47: double_divide(inverse(X4), double_divide(inverse(identity()), inverse(X4))) = identity().
% 1.69/2.00 Proof: A critical pair between equations 44 and 33.
% 1.69/2.00
% 1.69/2.00 49: inverse(inverse(X4)) = double_divide(double_divide(inverse(identity()), X4), inverse(identity())).
% 1.69/2.00 Proof: A critical pair between equations 39 and 34.
% 1.69/2.00
% 1.69/2.00 50: double_divide(X4, double_divide(double_divide(identity(), double_divide(X6, X4)), inverse(X6))) = identity().
% 1.69/2.00 Proof: A critical pair between equations 42 and 33.
% 1.69/2.00
% 1.69/2.00 52: double_divide(double_divide(identity(), double_divide(identity(), inverse(identity()))), inverse(identity())) = double_divide(inverse(X7), X7).
% 1.69/2.00 Proof: A critical pair between equations 42 and 33.
% 1.69/2.00
% 1.69/2.00 54: double_divide(inverse(X4), double_divide(inverse(double_divide(X7, double_divide(identity(), identity()))), inverse(X4))) = X7.
% 1.69/2.00 Proof: A critical pair between equations 44 and 35.
% 1.69/2.00
% 1.69/2.00 56: inverse(inverse(inverse(X4))) = double_divide(double_divide(identity(), double_divide(X4, inverse(identity()))), inverse(identity())).
% 1.69/2.00 Proof: A critical pair between equations 38 and 34.
% 1.69/2.00
% 1.69/2.00 57: double_divide(double_divide(double_divide(X4, identity()), identity()), identity()) = double_divide(double_divide(identity(), double_divide(X4, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 56,
% 1.69/2.00 lhs with equations [2,2,2]
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 59: double_divide(double_divide(X4, identity()), double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), identity()), double_divide(X4, identity()))) = X7.
% 1.69/2.00 Proof: Rewrite equation 54,
% 1.69/2.00 lhs with equations [2,2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 61: double_divide(X4, double_divide(double_divide(identity(), double_divide(X6, X4)), double_divide(X6, identity()))) = identity().
% 1.69/2.00 Proof: Rewrite equation 50,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 62: double_divide(double_divide(X4, identity()), identity()) = double_divide(double_divide(double_divide(identity(), identity()), X4), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 49,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 63: double_divide(double_divide(X4, identity()), double_divide(double_divide(identity(), identity()), double_divide(X4, identity()))) = identity().
% 1.69/2.00 Proof: Rewrite equation 47,
% 1.69/2.00 lhs with equations [2,2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 65: identity() = double_divide(double_divide(identity(), identity()), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 45,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 66: X1 = double_divide(double_divide(double_divide(identity(), identity()), double_divide(double_divide(X1, X4), double_divide(double_divide(X3, double_divide(double_divide(X4, double_divide(X5, X3)), double_divide(X5, identity()))), identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 40,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 67: X1 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(X1, identity()), double_divide(X3, identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 39,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 69: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.69/2.00 Proof: Rewrite equation 37,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 70: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.69/2.00 Proof: Rewrite equation 36,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 71: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), double_divide(X2, identity()))), double_divide(identity(), identity())) = X1.
% 1.69/2.00 Proof: Rewrite equation 35,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 79: identity() = double_divide(double_divide(X7, identity()), X7).
% 1.69/2.00 Proof: Rewrite equation 52,
% 1.69/2.00 lhs with equations [2,69,2,65]
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 82: double_divide(X4, double_divide(double_divide(X5, double_divide(X6, X4)), double_divide(X6, identity()))) = double_divide(double_divide(identity(), double_divide(X5, double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.69/2.00 Proof: Rewrite equation 42,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 84: identity() = double_divide(identity(), double_divide(identity(), double_divide(identity(), identity()))).
% 1.69/2.00 Proof: A critical pair between equations 61 and 69.
% 1.69/2.00
% 1.69/2.00 85: identity() = double_divide(double_divide(double_divide(identity(), identity()), identity()), identity()).
% 1.69/2.00 Proof: A critical pair between equations 63 and 69.
% 1.69/2.00
% 1.69/2.00 86: double_divide(double_divide(double_divide(identity(), identity()), identity()), identity()) = double_divide(identity(), double_divide(identity(), identity())).
% 1.69/2.00 Proof: A critical pair between equations 62 and 65.
% 1.69/2.00
% 1.69/2.00 87: X7 = double_divide(double_divide(double_divide(double_divide(X7, double_divide(identity(), identity())), identity()), identity()), identity()).
% 1.69/2.00 Proof: A critical pair between equations 59 and 69.
% 1.69/2.00
% 1.69/2.00 95: double_divide(X3, identity()) = double_divide(double_divide(double_divide(X3, identity()), identity()), double_divide(identity(), identity())).
% 1.69/2.00 Proof: A critical pair between equations 67 and 79.
% 1.69/2.00
% 1.69/2.00 96: inverse(X3) = double_divide(inverse(inverse(X3)), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 95,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2,2,2].
% 1.69/2.00
% 1.69/2.00 103: X7 = inverse(inverse(inverse(double_divide(X7, inverse(identity()))))).
% 1.69/2.00 Proof: Rewrite equation 87,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 104: inverse(inverse(inverse(identity()))) = double_divide(identity(), inverse(identity())).
% 1.69/2.00 Proof: Rewrite equation 86,
% 1.69/2.00 lhs with equations [2,2,2]
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 105: identity() = inverse(inverse(inverse(identity()))).
% 1.69/2.00 Proof: Rewrite equation 85,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2].
% 1.69/2.00
% 1.69/2.00 106: identity() = inverse(identity()).
% 1.69/2.00 Proof: Rewrite equation 84,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,104,105,2].
% 1.69/2.00
% 1.69/2.00 116: inverse(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), inverse(X2)))) = X1.
% 1.69/2.00 Proof: Rewrite equation 71,
% 1.69/2.00 lhs with equations [2,2,106,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 117: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.69/2.00 Proof: Rewrite equation 70,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 118: identity() = double_divide(X0, inverse(X0)).
% 1.69/2.00 Proof: Rewrite equation 69,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 121: X1 = inverse(double_divide(identity(), double_divide(double_divide(X1, X4), X4))).
% 1.69/2.00 Proof: Rewrite equation 66,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,106,2,2,116,2,106,2].
% 1.69/2.00
% 1.69/2.00 125: double_divide(inverse(X4), double_divide(inverse(inverse(X7)), inverse(X4))) = X7.
% 1.69/2.00 Proof: Rewrite equation 59,
% 1.69/2.00 lhs with equations [2,2,106,2,2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 127: inverse(inverse(inverse(X4))) = inverse(double_divide(identity(), inverse(X4))).
% 1.69/2.00 Proof: Rewrite equation 57,
% 1.69/2.00 lhs with equations [2,2,2]
% 1.69/2.00 rhs with equations [2,106,2,2,106,2].
% 1.69/2.00
% 1.69/2.00 129: double_divide(X4, double_divide(double_divide(X5, double_divide(X6, X4)), inverse(X6))) = inverse(inverse(inverse(X5))).
% 1.69/2.00 Proof: Rewrite equation 82,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2,106,2,2,106,2,127].
% 1.69/2.00
% 1.69/2.00 131: inverse(inverse(X4)) = inverse(double_divide(identity(), X4)).
% 1.69/2.00 Proof: Rewrite equation 62,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [2,106,2,106,2].
% 1.69/2.00
% 1.69/2.00 134: X8 = double_divide(inverse(identity()), inverse(X8)).
% 1.69/2.00 Proof: A critical pair between equations 125 and 96.
% 1.69/2.00
% 1.69/2.00 139: X7 = double_divide(inverse(inverse(inverse(X7))), identity()).
% 1.69/2.00 Proof: A critical pair between equations 125 and 118.
% 1.69/2.00
% 1.69/2.00 141: inverse(inverse(X8)) = inverse(inverse(inverse(inverse(X8)))).
% 1.69/2.00 Proof: A critical pair between equations 103 and 96.
% 1.69/2.00
% 1.69/2.00 146: double_divide(double_divide(X8, identity()), identity()) = double_divide(double_divide(double_divide(double_divide(X8, identity()), identity()), identity()), identity()).
% 1.69/2.00 Proof: Rewrite equation 141,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [2,2,2,2].
% 1.69/2.00
% 1.69/2.00 148: X7 = double_divide(double_divide(X7, identity()), identity()).
% 1.69/2.00 Proof: Rewrite equation 139,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2,2,146].
% 1.69/2.00
% 1.69/2.00 149: X8 = double_divide(double_divide(identity(), identity()), double_divide(X8, identity())).
% 1.69/2.00 Proof: Rewrite equation 134,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,2].
% 1.69/2.00
% 1.69/2.00 150: X4 = double_divide(double_divide(identity(), X4), identity()).
% 1.69/2.00 Proof: Rewrite equation 131,
% 1.69/2.00 lhs with equations [2,2,148]
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 151: double_divide(double_divide(X4, identity()), double_divide(X7, double_divide(X4, identity()))) = X7.
% 1.69/2.00 Proof: Rewrite equation 125,
% 1.69/2.00 lhs with equations [2,2,2,148,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 154: X1 = double_divide(double_divide(X1, X4), X4).
% 1.69/2.00 Proof: Rewrite equation 121,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,150].
% 1.69/2.00
% 1.69/2.00 156: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.69/2.00 Proof: Rewrite equation 117,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 157: double_divide(double_divide(X0, double_divide(double_divide(X1, double_divide(X2, X0)), double_divide(X2, identity()))), identity()) = X1.
% 1.69/2.00 Proof: Rewrite equation 116,
% 1.69/2.00 lhs with equations [2,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 161: identity() = double_divide(identity(), identity()).
% 1.69/2.00 Proof: Rewrite equation 106,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 164: double_divide(X4, double_divide(double_divide(X5, double_divide(X6, X4)), double_divide(X6, identity()))) = double_divide(X5, identity()).
% 1.69/2.00 Proof: Rewrite equation 129,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2,2,154,2].
% 1.69/2.00
% 1.69/2.00 167: double_divide(X9, identity()) = double_divide(double_divide(identity(), identity()), X9).
% 1.69/2.00 Proof: A critical pair between equations 149 and 148.
% 1.69/2.00
% 1.69/2.00 174: double_divide(X8, double_divide(X4, identity())) = double_divide(double_divide(X4, identity()), X8).
% 1.69/2.00 Proof: A critical pair between equations 151 and 154.
% 1.69/2.00
% 1.69/2.00 187: double_divide(X4, double_divide(double_divide(X5, double_divide(X6, X4)), inverse(X6))) = inverse(X5).
% 1.69/2.00 Proof: Rewrite equation 164,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 188: identity() = inverse(identity()).
% 1.69/2.00 Proof: Rewrite equation 161,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 191: inverse(inverse(X1)) = X1.
% 1.69/2.00 Proof: Rewrite equation 157,
% 1.69/2.00 lhs with equations [2,187,2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 192: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.69/2.00 Proof: Rewrite equation 156,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 196: double_divide(X8, inverse(X4)) = double_divide(inverse(X4), X8).
% 1.69/2.00 Proof: Rewrite equation 174,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2].
% 1.69/2.00
% 1.69/2.00 197: inverse(X9) = double_divide(identity(), X9).
% 1.69/2.00 Proof: Rewrite equation 167,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [2,188].
% 1.69/2.00
% 1.69/2.00 199: double_divide(X8, inverse(inverse(X9))) = double_divide(X9, X8).
% 1.69/2.00 Proof: A critical pair between equations 196 and 191.
% 1.69/2.00
% 1.69/2.00 214: double_divide(X9, identity()) = double_divide(identity(), X9).
% 1.69/2.00 Proof: Rewrite equation 197,
% 1.69/2.00 lhs with equations [2]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 215: multiply(X0, X1) = double_divide(identity(), double_divide(X1, X0)).
% 1.69/2.00 Proof: Rewrite equation 192,
% 1.69/2.00 lhs with equations []
% 1.69/2.00 rhs with equations [2,214].
% 1.69/2.00
% 1.69/2.00 224: double_divide(X8, X9) = double_divide(X9, X8).
% 1.69/2.00 Proof: Rewrite equation 199,
% 1.69/2.00 lhs with equations [2,2,154]
% 1.69/2.00 rhs with equations [].
% 1.69/2.00
% 1.69/2.00 227: multiply(a(), b()) = multiply(b(), a()).
% 1.69/2.00 Proof: Rewrite lhs with equations [215,224]
% 1.69/2.00 rhs with equations [215].
% 1.69/2.00
% 1.69/2.00 % SZS output end Proof
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