TSTP Solution File: GRP485-1 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP485-1 : TPTP v8.1.2. Released v2.6.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n016.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 01:15:07 EDT 2023
% Result : Unsatisfiable 1.94s 2.26s
% Output : CNFRefutation 1.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.13 % Problem : GRP485-1 : TPTP v8.1.2. Released v2.6.0.
% 0.03/0.14 % Command : toma --casc %s
% 0.13/0.35 % Computer : n016.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Mon Aug 28 23:54:00 EDT 2023
% 0.13/0.36 % CPUTime :
% 1.94/2.26 % SZS status Unsatisfiable
% 1.94/2.26 % SZS output start Proof
% 1.94/2.26 original problem:
% 1.94/2.26 axioms:
% 1.94/2.26 double_divide(double_divide(A, double_divide(double_divide(double_divide(A, B), C), double_divide(B, identity()))), double_divide(identity(), identity())) = C
% 1.94/2.26 multiply(A, B) = double_divide(double_divide(B, A), identity())
% 1.94/2.26 inverse(A) = double_divide(A, identity())
% 1.94/2.26 identity() = double_divide(A, inverse(A))
% 1.94/2.26 goal:
% 1.94/2.26 multiply(identity(), a2()) != a2()
% 1.94/2.26 To show the unsatisfiability of the original goal,
% 1.94/2.26 it suffices to show that multiply(identity(), a2()) = a2() (skolemized goal) is valid under the axioms.
% 1.94/2.26 Here is an equational proof:
% 1.94/2.26 0: double_divide(double_divide(X0, double_divide(double_divide(double_divide(X0, X1), X2), double_divide(X1, identity()))), double_divide(identity(), identity())) = X2.
% 1.94/2.26 Proof: Axiom.
% 1.94/2.26
% 1.94/2.26 1: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.94/2.26 Proof: Axiom.
% 1.94/2.26
% 1.94/2.26 2: inverse(X0) = double_divide(X0, identity()).
% 1.94/2.26 Proof: Axiom.
% 1.94/2.26
% 1.94/2.26 3: identity() = double_divide(X0, inverse(X0)).
% 1.94/2.26 Proof: Axiom.
% 1.94/2.26
% 1.94/2.26 4: double_divide(double_divide(X0, double_divide(double_divide(double_divide(X0, X1), X2), inverse(X1))), inverse(identity())) = X2.
% 1.94/2.26 Proof: Rewrite equation 0,
% 1.94/2.26 lhs with equations [2,2]
% 1.94/2.26 rhs with equations [].
% 1.94/2.26
% 1.94/2.26 5: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.94/2.26 Proof: Rewrite equation 1,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 6: X2 = double_divide(double_divide(X3, double_divide(double_divide(inverse(X3), X2), inverse(identity()))), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 4 and 2.
% 1.94/2.26
% 1.94/2.26 8: X2 = double_divide(double_divide(X3, double_divide(double_divide(identity(), X2), inverse(inverse(X3)))), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 4 and 3.
% 1.94/2.26
% 1.94/2.26 9: inverse(double_divide(X0, X1)) = double_divide(double_divide(X0, double_divide(identity(), inverse(X1))), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 4 and 3.
% 1.94/2.26
% 1.94/2.26 10: double_divide(double_divide(double_divide(double_divide(X0, identity()), X4), X5), inverse(X4)) = double_divide(double_divide(X0, X5), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 4 and 4.
% 1.94/2.26
% 1.94/2.26 15: double_divide(double_divide(X0, X1), identity()) = double_divide(double_divide(X0, double_divide(identity(), double_divide(X1, identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 9,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 16: X2 = double_divide(double_divide(X3, double_divide(double_divide(identity(), X2), double_divide(double_divide(X3, identity()), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 8,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 18: X2 = double_divide(double_divide(X3, double_divide(double_divide(double_divide(X3, identity()), X2), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 6,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 19: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.94/2.26 Proof: Rewrite equation 3,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 20: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.94/2.26 Proof: Rewrite equation 5,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 21: double_divide(double_divide(X0, double_divide(double_divide(double_divide(X0, X1), X2), double_divide(X1, identity()))), double_divide(identity(), identity())) = X2.
% 1.94/2.26 Proof: Rewrite equation 4,
% 1.94/2.26 lhs with equations [2,2]
% 1.94/2.26 rhs with equations [].
% 1.94/2.26
% 1.94/2.26 22: double_divide(double_divide(double_divide(double_divide(X0, identity()), X4), X5), double_divide(X4, identity())) = double_divide(double_divide(X0, X5), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 10,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 23: identity() = double_divide(double_divide(identity(), identity()), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 16 and 19.
% 1.94/2.26
% 1.94/2.26 24: double_divide(double_divide(X0, identity()), identity()) = double_divide(double_divide(X0, identity()), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 15 and 19.
% 1.94/2.26
% 1.94/2.26 27: X8 = double_divide(double_divide(double_divide(X6, identity()), double_divide(double_divide(X6, X8), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 18 and 22.
% 1.94/2.26
% 1.94/2.26 28: double_divide(identity(), double_divide(X5, identity())) = double_divide(double_divide(X3, double_divide(double_divide(double_divide(X3, identity()), X5), identity())), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 18 and 15.
% 1.94/2.26
% 1.94/2.26 29: double_divide(double_divide(identity(), X5), double_divide(double_divide(double_divide(X3, identity()), identity()), identity())) = double_divide(double_divide(X3, X5), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 18 and 16.
% 1.94/2.26
% 1.94/2.26 33: double_divide(identity(), identity()) = double_divide(double_divide(X4, double_divide(X5, double_divide(double_divide(double_divide(identity(), X5), double_divide(double_divide(X4, identity()), identity())), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 21 and 16.
% 1.94/2.26
% 1.94/2.26 36: inverse(identity()) = double_divide(double_divide(X4, double_divide(X5, inverse(double_divide(double_divide(identity(), X5), inverse(inverse(X4)))))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 33,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2,2].
% 1.94/2.26
% 1.94/2.26 39: double_divide(identity(), inverse(X5)) = double_divide(double_divide(X3, inverse(double_divide(inverse(X3), X5))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 28,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 40: X8 = double_divide(double_divide(inverse(X6), double_divide(double_divide(X6, X8), inverse(identity()))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 27,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 43: inverse(inverse(X0)) = double_divide(inverse(X0), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 24,
% 1.94/2.26 lhs with equations [2,2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 44: identity() = inverse(inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 23,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,43].
% 1.94/2.26
% 1.94/2.26 47: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.94/2.26 Proof: Rewrite equation 20,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 48: identity() = double_divide(X0, inverse(X0)).
% 1.94/2.26 Proof: Rewrite equation 19,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 49: X2 = double_divide(double_divide(X3, double_divide(double_divide(inverse(X3), X2), inverse(identity()))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 18,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 52: inverse(double_divide(X0, X1)) = double_divide(double_divide(X0, double_divide(identity(), inverse(X1))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 15,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 56: double_divide(double_divide(identity(), X5), inverse(inverse(inverse(X3)))) = double_divide(double_divide(X3, X5), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 29,
% 1.94/2.26 lhs with equations [2,2,2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 65: inverse(identity()) = inverse(double_divide(X6, double_divide(double_divide(identity(), identity()), inverse(inverse(X6))))).
% 1.94/2.26 Proof: A critical pair between equations 36 and 52.
% 1.94/2.26
% 1.94/2.26 66: inverse(X9) = double_divide(double_divide(inverse(X9), double_divide(identity(), inverse(identity()))), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 40 and 48.
% 1.94/2.26
% 1.94/2.26 71: double_divide(X9, identity()) = double_divide(double_divide(double_divide(X9, identity()), double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 66,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 72: double_divide(identity(), identity()) = double_divide(double_divide(X6, double_divide(identity(), double_divide(double_divide(double_divide(identity(), identity()), double_divide(identity(), double_divide(double_divide(double_divide(X6, double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), double_divide(double_divide(identity(), double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())))), double_divide(identity(), identity())))), double_divide(identity(), identity())))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 65,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,52,2,2,52,52,52,52,2,2,2,2,2].
% 1.94/2.26
% 1.94/2.26 84: double_divide(double_divide(X0, X1), identity()) = double_divide(double_divide(X0, double_divide(identity(), double_divide(X1, identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 52,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 87: X2 = double_divide(double_divide(X3, double_divide(double_divide(double_divide(X3, identity()), X2), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 49,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 88: identity() = double_divide(X0, double_divide(X0, identity())).
% 1.94/2.26 Proof: Rewrite equation 48,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 89: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.94/2.26 Proof: Rewrite equation 47,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 91: identity() = double_divide(double_divide(identity(), identity()), identity()).
% 1.94/2.26 Proof: Rewrite equation 44,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 92: double_divide(double_divide(X0, identity()), identity()) = double_divide(double_divide(X0, identity()), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 43,
% 1.94/2.26 lhs with equations [2,2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 94: double_divide(identity(), double_divide(X5, identity())) = double_divide(double_divide(X3, double_divide(double_divide(double_divide(X3, identity()), X5), identity())), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 39,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 97: double_divide(identity(), identity()) = double_divide(double_divide(X4, double_divide(X5, double_divide(double_divide(double_divide(identity(), X5), double_divide(double_divide(X4, identity()), identity())), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 36,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2,2].
% 1.94/2.26
% 1.94/2.26 100: double_divide(double_divide(identity(), X5), double_divide(double_divide(double_divide(X3, identity()), identity()), identity())) = double_divide(double_divide(X3, X5), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 56,
% 1.94/2.26 lhs with equations [2,2,2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 101: double_divide(X9, identity()) = double_divide(double_divide(double_divide(X9, identity()), identity()), identity()).
% 1.94/2.26 Proof: A critical pair between equations 71 and 84.
% 1.94/2.26
% 1.94/2.26 104: double_divide(double_divide(double_divide(identity(), identity()), identity()), identity()) = double_divide(identity(), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 92 and 91.
% 1.94/2.26
% 1.94/2.26 107: double_divide(identity(), identity()) = double_divide(double_divide(identity(), double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 97 and 88.
% 1.94/2.26
% 1.94/2.26 111: double_divide(double_divide(X0, double_divide(identity(), identity())), identity()) = double_divide(double_divide(X0, double_divide(identity(), identity())), double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 84 and 91.
% 1.94/2.26
% 1.94/2.26 112: double_divide(double_divide(X6, double_divide(double_divide(identity(), identity()), double_divide(double_divide(X6, identity()), identity()))), identity()) = double_divide(identity(), identity()).
% 1.94/2.26 Proof: A critical pair between equations 84 and 97.
% 1.94/2.26
% 1.94/2.26 113: inverse(double_divide(X6, double_divide(inverse(identity()), inverse(inverse(X6))))) = inverse(identity()).
% 1.94/2.26 Proof: Rewrite equation 112,
% 1.94/2.26 lhs with equations [2,2,2,2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 114: inverse(double_divide(X0, inverse(identity()))) = double_divide(double_divide(X0, inverse(identity())), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 111,
% 1.94/2.26 lhs with equations [2,2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 118: inverse(identity()) = double_divide(double_divide(identity(), double_divide(identity(), inverse(identity()))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 107,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 121: inverse(inverse(inverse(identity()))) = double_divide(identity(), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 104,
% 1.94/2.26 lhs with equations [2,2,2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 124: inverse(X9) = inverse(inverse(inverse(X9))).
% 1.94/2.26 Proof: Rewrite equation 101,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 130: double_divide(identity(), inverse(X5)) = double_divide(double_divide(X3, inverse(double_divide(inverse(X3), X5))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 94,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 132: inverse(inverse(X0)) = double_divide(inverse(X0), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 92,
% 1.94/2.26 lhs with equations [2,2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 133: identity() = inverse(inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 91,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 135: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 1.94/2.26 Proof: Rewrite equation 89,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 136: identity() = double_divide(X0, inverse(X0)).
% 1.94/2.26 Proof: Rewrite equation 88,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 137: X2 = double_divide(double_divide(X3, double_divide(double_divide(inverse(X3), X2), inverse(identity()))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 87,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,2].
% 1.94/2.26
% 1.94/2.26 139: inverse(double_divide(X0, X1)) = double_divide(double_divide(X0, double_divide(identity(), inverse(X1))), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 84,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2].
% 1.94/2.26
% 1.94/2.26 146: inverse(identity()) = inverse(double_divide(X6, double_divide(inverse(identity()), X6))).
% 1.94/2.26 Proof: Rewrite equation 72,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,136,2,2,136,2,2,132,133,2,132,2,132,124,2,139,2,139].
% 1.94/2.26
% 1.94/2.26 148: double_divide(double_divide(identity(), X5), inverse(X3)) = double_divide(double_divide(X3, X5), inverse(identity())).
% 1.94/2.26 Proof: Rewrite equation 100,
% 1.94/2.26 lhs with equations [2,2,2,124]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 149: inverse(inverse(inverse(identity()))) = identity().
% 1.94/2.26 Proof: A critical pair between equations 121 and 136.
% 1.94/2.26
% 1.94/2.26 151: inverse(identity()) = inverse(double_divide(identity(), identity())).
% 1.94/2.26 Proof: A critical pair between equations 113 and 136.
% 1.94/2.26
% 1.94/2.26 156: inverse(identity()) = double_divide(double_divide(identity(), identity()), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 118 and 136.
% 1.94/2.26
% 1.94/2.26 157: inverse(identity()) = inverse(double_divide(identity(), inverse(inverse(identity())))).
% 1.94/2.26 Proof: A critical pair between equations 146 and 2.
% 1.94/2.26
% 1.94/2.26 160: inverse(double_divide(identity(), inverse(identity()))) = double_divide(identity(), inverse(identity())).
% 1.94/2.26 Proof: A critical pair between equations 114 and 136.
% 1.94/2.26
% 1.94/2.26 161: double_divide(double_divide(identity(), double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())) = double_divide(identity(), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 160,
% 1.94/2.26 lhs with equations [2,139,151,2,2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 164: double_divide(identity(), identity()) = double_divide(identity(), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 157,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,151,2,139,151,2,2,161].
% 1.94/2.26
% 1.94/2.26 165: double_divide(identity(), identity()) = double_divide(double_divide(identity(), identity()), double_divide(identity(), identity())).
% 1.94/2.26 Proof: Rewrite equation 156,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 167: double_divide(identity(), identity()) = identity().
% 1.94/2.26 Proof: Rewrite equation 149,
% 1.94/2.26 lhs with equations [2,139,2,164,164,2,165,139,2,164,164,2,165]
% 1.94/2.26 rhs with equations [].
% 1.94/2.26
% 1.94/2.26 177: X2 = double_divide(double_divide(X3, double_divide(double_divide(double_divide(X3, identity()), X2), identity())), identity()).
% 1.94/2.26 Proof: Rewrite equation 137,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2,2,167,2,167].
% 1.94/2.26
% 1.94/2.26 179: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 1.94/2.26 Proof: Rewrite equation 135,
% 1.94/2.26 lhs with equations []
% 1.94/2.26 rhs with equations [2].
% 1.94/2.26
% 1.94/2.26 182: double_divide(identity(), double_divide(X5, identity())) = X5.
% 1.94/2.26 Proof: Rewrite equation 130,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,2,2,167,177].
% 1.94/2.26
% 1.94/2.26 188: double_divide(double_divide(identity(), X5), double_divide(X3, identity())) = double_divide(double_divide(X3, X5), identity()).
% 1.94/2.26 Proof: Rewrite equation 148,
% 1.94/2.26 lhs with equations [2]
% 1.94/2.26 rhs with equations [2,167].
% 1.94/2.26
% 1.94/2.26 189: multiply(identity(), a2()) = a2().
% 1.94/2.26 Proof: Rewrite lhs with equations [179,188,167,182]
% 1.94/2.26 rhs with equations [].
% 1.94/2.26
% 1.94/2.26 % SZS output end Proof
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