TSTP Solution File: GRP578-1 by Toma---0.4
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%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP578-1 : TPTP v8.1.2. Released v2.6.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n018.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 01:15:25 EDT 2023
% Result : Unsatisfiable 9.52s 9.89s
% Output : CNFRefutation 9.52s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP578-1 : TPTP v8.1.2. Released v2.6.0.
% 0.11/0.13 % Command : toma --casc %s
% 0.13/0.34 % Computer : n018.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 23:48:32 EDT 2023
% 0.13/0.34 % CPUTime :
% 9.52/9.89 % SZS status Unsatisfiable
% 9.52/9.89 % SZS output start Proof
% 9.52/9.89 original problem:
% 9.52/9.89 axioms:
% 9.52/9.89 double_divide(double_divide(A, double_divide(double_divide(double_divide(B, A), C), double_divide(B, identity()))), double_divide(identity(), identity())) = C
% 9.52/9.89 multiply(A, B) = double_divide(double_divide(B, A), identity())
% 9.52/9.89 inverse(A) = double_divide(A, identity())
% 9.52/9.89 identity() = double_divide(A, inverse(A))
% 9.52/9.89 goal:
% 9.52/9.89 multiply(identity(), a2()) != a2()
% 9.52/9.89 To show the unsatisfiability of the original goal,
% 9.52/9.89 it suffices to show that multiply(identity(), a2()) = a2() (skolemized goal) is valid under the axioms.
% 9.52/9.89 Here is an equational proof:
% 9.52/9.89 0: double_divide(double_divide(X0, double_divide(double_divide(double_divide(X1, X0), X2), double_divide(X1, identity()))), double_divide(identity(), identity())) = X2.
% 9.52/9.89 Proof: Axiom.
% 9.52/9.89
% 9.52/9.89 1: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 9.52/9.89 Proof: Axiom.
% 9.52/9.89
% 9.52/9.89 2: inverse(X0) = double_divide(X0, identity()).
% 9.52/9.89 Proof: Axiom.
% 9.52/9.89
% 9.52/9.89 3: identity() = double_divide(X0, inverse(X0)).
% 9.52/9.89 Proof: Axiom.
% 9.52/9.89
% 9.52/9.89 4: double_divide(double_divide(X0, double_divide(double_divide(double_divide(X1, X0), X2), inverse(X1))), inverse(identity())) = X2.
% 9.52/9.89 Proof: Rewrite equation 0,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [].
% 9.52/9.89
% 9.52/9.89 5: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 9.52/9.89 Proof: Rewrite equation 1,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 6: X2 = double_divide(double_divide(identity(), double_divide(double_divide(inverse(X3), X2), inverse(X3))), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 4 and 2.
% 9.52/9.89
% 9.52/9.89 7: identity() = double_divide(double_divide(X0, double_divide(inverse(double_divide(X1, X0)), inverse(X1))), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 4 and 2.
% 9.52/9.89
% 9.52/9.89 8: X2 = double_divide(double_divide(inverse(X3), double_divide(double_divide(identity(), X2), inverse(X3))), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 4 and 3.
% 9.52/9.89
% 9.52/9.89 9: inverse(double_divide(X1, X0)) = double_divide(double_divide(X0, double_divide(identity(), inverse(X1))), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 4 and 3.
% 9.52/9.89
% 9.52/9.89 10: double_divide(double_divide(double_divide(X4, double_divide(identity(), X0)), X5), inverse(X4)) = double_divide(double_divide(X0, X5), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 4 and 4.
% 9.52/9.89
% 9.52/9.89 15: double_divide(double_divide(X1, X0), identity()) = double_divide(double_divide(X0, double_divide(identity(), double_divide(X1, identity()))), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 9,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 16: X2 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(identity(), X2), double_divide(X3, identity()))), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 8,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 17: identity() = double_divide(double_divide(X0, double_divide(double_divide(double_divide(X1, X0), identity()), double_divide(X1, identity()))), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 7,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 18: X2 = double_divide(double_divide(identity(), double_divide(double_divide(double_divide(X3, identity()), X2), double_divide(X3, identity()))), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 6,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 19: identity() = double_divide(X0, double_divide(X0, identity())).
% 9.52/9.89 Proof: Rewrite equation 3,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 20: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 9.52/9.89 Proof: Rewrite equation 5,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 22: double_divide(double_divide(double_divide(X4, double_divide(identity(), X0)), X5), double_divide(X4, identity())) = double_divide(double_divide(X0, X5), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 10,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 23: double_divide(double_divide(identity(), X0), identity()) = double_divide(double_divide(X0, identity()), double_divide(identity(), identity())).
% 9.52/9.89 Proof: A critical pair between equations 15 and 19.
% 9.52/9.89
% 9.52/9.89 24: X2 = double_divide(double_divide(double_divide(double_divide(identity(), X2), identity()), identity()), double_divide(identity(), identity())).
% 9.52/9.89 Proof: A critical pair between equations 16 and 19.
% 9.52/9.89
% 9.52/9.89 26: identity() = double_divide(double_divide(double_divide(identity(), X7), double_divide(double_divide(X7, identity()), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 9.52/9.89 Proof: A critical pair between equations 17 and 22.
% 9.52/9.89
% 9.52/9.89 32: double_divide(double_divide(double_divide(X4, identity()), X5), double_divide(X4, identity())) = double_divide(double_divide(double_divide(identity(), identity()), X5), double_divide(identity(), identity())).
% 9.52/9.89 Proof: A critical pair between equations 16 and 18.
% 9.52/9.89
% 9.52/9.89 41: identity() = double_divide(double_divide(double_divide(identity(), X7), double_divide(inverse(X7), inverse(identity()))), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 26,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 43: X2 = double_divide(inverse(inverse(double_divide(identity(), X2))), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 24,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 44: double_divide(double_divide(double_divide(X4, double_divide(identity(), X0)), X5), inverse(X4)) = double_divide(double_divide(X0, X5), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 22,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 46: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 9.52/9.89 Proof: Rewrite equation 20,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 47: identity() = double_divide(X0, inverse(X0)).
% 9.52/9.89 Proof: Rewrite equation 19,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 53: double_divide(double_divide(inverse(X4), X5), inverse(X4)) = double_divide(double_divide(inverse(identity()), X5), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 32,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 55: inverse(double_divide(identity(), X0)) = double_divide(inverse(X0), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 23,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 56: inverse(double_divide(X1, X0)) = double_divide(double_divide(X0, double_divide(identity(), inverse(X1))), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 15,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 57: identity() = double_divide(inverse(inverse(inverse(identity()))), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 43 and 2.
% 9.52/9.89
% 9.52/9.89 68: double_divide(double_divide(inverse(X4), identity()), inverse(X4)) = double_divide(inverse(inverse(identity())), inverse(identity())).
% 9.52/9.89 Proof: A critical pair between equations 53 and 2.
% 9.52/9.89
% 9.52/9.89 78: identity() = double_divide(double_divide(double_divide(double_divide(identity(), identity()), double_divide(identity(), double_divide(double_divide(double_divide(identity(), double_divide(identity(), identity())), double_divide(identity(), double_divide(identity(), identity()))), double_divide(identity(), identity())))), double_divide(identity(), identity())), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 57,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,56,2,2,56,56,2,2,2,2].
% 9.52/9.89
% 9.52/9.89 80: double_divide(double_divide(identity(), X0), identity()) = double_divide(double_divide(X0, identity()), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 55,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 86: identity() = double_divide(X0, double_divide(X0, identity())).
% 9.52/9.89 Proof: Rewrite equation 47,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 87: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 9.52/9.89 Proof: Rewrite equation 46,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 89: X2 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X2), identity())), identity()).
% 9.52/9.89 Proof: Rewrite equation 43,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2,80].
% 9.52/9.89
% 9.52/9.89 91: identity() = double_divide(double_divide(double_divide(identity(), X7), double_divide(double_divide(X7, identity()), double_divide(identity(), identity()))), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 41,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 98: double_divide(double_divide(double_divide(X4, identity()), identity()), double_divide(X4, identity())) = double_divide(identity(), identity()).
% 9.52/9.89 Proof: Rewrite equation 68,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2,2,2,80,86].
% 9.52/9.89
% 9.52/9.89 99: double_divide(double_divide(double_divide(X4, double_divide(identity(), X0)), X5), double_divide(X4, identity())) = double_divide(double_divide(X0, X5), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 44,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 100: double_divide(double_divide(double_divide(X4, identity()), X5), double_divide(X4, identity())) = double_divide(double_divide(double_divide(identity(), identity()), X5), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 53,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 102: identity() = double_divide(double_divide(identity(), X1), double_divide(double_divide(X1, identity()), double_divide(identity(), identity()))).
% 9.52/9.89 Proof: A critical pair between equations 86 and 80.
% 9.52/9.89
% 9.52/9.89 103: X3 = double_divide(double_divide(identity(), double_divide(double_divide(X3, identity()), double_divide(identity(), identity()))), identity()).
% 9.52/9.89 Proof: A critical pair between equations 89 and 80.
% 9.52/9.89
% 9.52/9.89 108: double_divide(double_divide(identity(), double_divide(identity(), double_divide(double_divide(identity(), X3), identity()))), identity()) = double_divide(X3, double_divide(identity(), identity())).
% 9.52/9.89 Proof: A critical pair between equations 80 and 89.
% 9.52/9.89
% 9.52/9.89 109: double_divide(double_divide(identity(), double_divide(identity(), double_divide(identity(), X7))), identity()) = double_divide(double_divide(X7, identity()), double_divide(identity(), identity())).
% 9.52/9.89 Proof: A critical pair between equations 80 and 99.
% 9.52/9.89
% 9.52/9.89 116: inverse(double_divide(identity(), double_divide(identity(), inverse(double_divide(identity(), X3))))) = double_divide(X3, inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 108,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 121: X3 = inverse(double_divide(identity(), inverse(double_divide(identity(), double_divide(identity(), inverse(double_divide(identity(), inverse(X3)))))))).
% 9.52/9.89 Proof: Rewrite equation 103,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,116,2].
% 9.52/9.89
% 9.52/9.89 122: identity() = double_divide(double_divide(identity(), X1), inverse(double_divide(identity(), double_divide(identity(), inverse(double_divide(identity(), inverse(X1))))))).
% 9.52/9.89 Proof: Rewrite equation 102,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,116].
% 9.52/9.89
% 9.52/9.89 125: double_divide(inverse(inverse(X4)), inverse(X4)) = inverse(identity()).
% 9.52/9.89 Proof: Rewrite equation 98,
% 9.52/9.89 lhs with equations [2,2,2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 132: identity() = double_divide(identity(), inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 91,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,116,122,2].
% 9.52/9.89
% 9.52/9.89 134: X2 = inverse(double_divide(identity(), inverse(double_divide(identity(), X2)))).
% 9.52/9.89 Proof: Rewrite equation 89,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 136: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 9.52/9.89 Proof: Rewrite equation 87,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 137: identity() = double_divide(X0, inverse(X0)).
% 9.52/9.89 Proof: Rewrite equation 86,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 143: identity() = inverse(double_divide(identity(), double_divide(identity(), double_divide(identity(), inverse(double_divide(identity(), double_divide(inverse(identity()), double_divide(identity(), inverse(inverse(identity())))))))))).
% 9.52/9.89 Proof: Rewrite equation 78,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,137,2,137,2,2,116,137,137,2,2,116,2,116,134].
% 9.52/9.89
% 9.52/9.89 151: inverse(double_divide(identity(), double_divide(identity(), double_divide(identity(), X7)))) = inverse(double_divide(identity(), double_divide(identity(), inverse(double_divide(identity(), inverse(X7)))))).
% 9.52/9.89 Proof: Rewrite equation 109,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2,116].
% 9.52/9.89
% 9.52/9.89 152: double_divide(double_divide(inverse(X4), X5), inverse(X4)) = inverse(double_divide(identity(), double_divide(identity(), inverse(double_divide(identity(), double_divide(inverse(identity()), X5)))))).
% 9.52/9.89 Proof: Rewrite equation 100,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2,2,116].
% 9.52/9.89
% 9.52/9.89 153: inverse(double_divide(identity(), X0)) = inverse(double_divide(identity(), double_divide(identity(), double_divide(identity(), X0)))).
% 9.52/9.89 Proof: Rewrite equation 80,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2,116,151].
% 9.52/9.89
% 9.52/9.89 156: X3 = double_divide(identity(), inverse(double_divide(identity(), inverse(X3)))).
% 9.52/9.89 Proof: A critical pair between equations 121 and 134.
% 9.52/9.89
% 9.52/9.89 160: identity() = inverse(double_divide(identity(), inverse(double_divide(identity(), double_divide(identity(), inverse(identity())))))).
% 9.52/9.89 Proof: A critical pair between equations 121 and 132.
% 9.52/9.89
% 9.52/9.89 166: inverse(identity()) = double_divide(inverse(inverse(double_divide(identity(), inverse(double_divide(identity(), X5))))), X5).
% 9.52/9.89 Proof: A critical pair between equations 125 and 134.
% 9.52/9.89
% 9.52/9.89 167: double_divide(identity(), identity()) = double_divide(double_divide(double_divide(double_divide(identity(), double_divide(double_divide(identity(), X5), identity())), identity()), identity()), X5).
% 9.52/9.89 Proof: Rewrite equation 166,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2,2].
% 9.52/9.89
% 9.52/9.89 173: identity() = double_divide(double_divide(identity(), double_divide(double_divide(identity(), identity()), identity())), identity()).
% 9.52/9.89 Proof: Rewrite equation 160,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,153,2,2].
% 9.52/9.89
% 9.52/9.89 175: X3 = double_divide(identity(), double_divide(double_divide(identity(), double_divide(X3, identity())), identity())).
% 9.52/9.89 Proof: Rewrite equation 156,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 176: double_divide(double_divide(identity(), X0), identity()) = double_divide(double_divide(identity(), double_divide(identity(), double_divide(identity(), X0))), identity()).
% 9.52/9.89 Proof: Rewrite equation 153,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 184: identity() = double_divide(double_divide(identity(), double_divide(double_divide(identity(), double_divide(double_divide(identity(), identity()), double_divide(identity(), double_divide(double_divide(identity(), identity()), identity())))), identity())), identity()).
% 9.52/9.89 Proof: Rewrite equation 143,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2,2,2,176].
% 9.52/9.89
% 9.52/9.89 190: identity() = double_divide(X0, double_divide(X0, identity())).
% 9.52/9.89 Proof: Rewrite equation 137,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 191: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 9.52/9.89 Proof: Rewrite equation 136,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 193: X2 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X2), identity())), identity()).
% 9.52/9.89 Proof: Rewrite equation 134,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 208: double_divide(double_divide(identity(), double_divide(identity(), double_divide(double_divide(identity(), X3), identity()))), identity()) = double_divide(X3, double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 116,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 209: double_divide(double_divide(double_divide(X4, identity()), X5), double_divide(X4, identity())) = double_divide(double_divide(double_divide(identity(), identity()), X5), double_divide(identity(), identity())).
% 9.52/9.89 Proof: Rewrite equation 152,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2,2,2,208].
% 9.52/9.89
% 9.52/9.89 211: double_divide(identity(), identity()) = double_divide(double_divide(identity(), identity()), identity()).
% 9.52/9.89 Proof: A critical pair between equations 167 and 173.
% 9.52/9.89
% 9.52/9.89 212: double_divide(identity(), identity()) = double_divide(double_divide(X6, identity()), X6).
% 9.52/9.89 Proof: A critical pair between equations 167 and 193.
% 9.52/9.89
% 9.52/9.89 215: double_divide(double_divide(X4, identity()), double_divide(identity(), identity())) = double_divide(double_divide(identity(), X4), identity()).
% 9.52/9.89 Proof: A critical pair between equations 208 and 175.
% 9.52/9.89
% 9.52/9.89 218: double_divide(identity(), double_divide(identity(), X3)) = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X3), identity())), identity()).
% 9.52/9.89 Proof: A critical pair between equations 193 and 176.
% 9.52/9.89
% 9.52/9.89 227: double_divide(identity(), double_divide(identity(), X3)) = inverse(double_divide(identity(), inverse(double_divide(identity(), X3)))).
% 9.52/9.89 Proof: Rewrite equation 218,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 232: inverse(identity()) = double_divide(inverse(X6), X6).
% 9.52/9.89 Proof: Rewrite equation 212,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 233: inverse(identity()) = inverse(inverse(identity())).
% 9.52/9.89 Proof: Rewrite equation 211,
% 9.52/9.89 lhs with equations [2]
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 249: X2 = double_divide(identity(), double_divide(identity(), X2)).
% 9.52/9.89 Proof: Rewrite equation 193,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,227].
% 9.52/9.89
% 9.52/9.89 251: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 9.52/9.89 Proof: Rewrite equation 191,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 252: identity() = double_divide(X0, inverse(X0)).
% 9.52/9.89 Proof: Rewrite equation 190,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 258: identity() = inverse(identity()).
% 9.52/9.89 Proof: Rewrite equation 184,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,2,233,252,2,233,252,2,252,2].
% 9.52/9.89
% 9.52/9.89 264: X3 = double_divide(identity(), inverse(double_divide(identity(), inverse(X3)))).
% 9.52/9.89 Proof: Rewrite equation 175,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2].
% 9.52/9.89
% 9.52/9.89 267: inverse(inverse(X4)) = inverse(double_divide(identity(), X4)).
% 9.52/9.89 Proof: Rewrite equation 215,
% 9.52/9.89 lhs with equations [2,2,258,2]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 268: double_divide(double_divide(inverse(X4), X5), inverse(X4)) = inverse(inverse(X5)).
% 9.52/9.89 Proof: Rewrite equation 209,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [2,258,2,258,2,267].
% 9.52/9.89
% 9.52/9.89 274: inverse(inverse(X7)) = double_divide(inverse(identity()), inverse(X7)).
% 9.52/9.89 Proof: A critical pair between equations 268 and 232.
% 9.52/9.89
% 9.52/9.89 287: double_divide(double_divide(X7, identity()), identity()) = double_divide(identity(), double_divide(X7, identity())).
% 9.52/9.89 Proof: Rewrite equation 274,
% 9.52/9.89 lhs with equations [2,2]
% 9.52/9.89 rhs with equations [258,2].
% 9.52/9.89
% 9.52/9.89 292: double_divide(identity(), double_divide(X4, identity())) = double_divide(double_divide(identity(), X4), identity()).
% 9.52/9.89 Proof: Rewrite equation 267,
% 9.52/9.89 lhs with equations [2,2,287]
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 294: X3 = double_divide(identity(), double_divide(X3, identity())).
% 9.52/9.89 Proof: Rewrite equation 264,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2,2,292,287,249].
% 9.52/9.89
% 9.52/9.89 304: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 9.52/9.89 Proof: Rewrite equation 251,
% 9.52/9.89 lhs with equations []
% 9.52/9.89 rhs with equations [2].
% 9.52/9.89
% 9.52/9.89 313: multiply(identity(), a2()) = a2().
% 9.52/9.89 Proof: Rewrite lhs with equations [304,287,294]
% 9.52/9.89 rhs with equations [].
% 9.52/9.89
% 9.52/9.89 % SZS output end Proof
%------------------------------------------------------------------------------