TSTP Solution File: GRP584-1 by Toma---0.4
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Toma---0.4
% Problem : GRP584-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% Transfm : none
% Format : tptp:raw
% Command : toma --casc %s
% Computer : n024.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 01:15:26 EDT 2023
% Result : Unsatisfiable 3.83s 4.09s
% Output : CNFRefutation 3.83s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GRP584-1 : TPTP v8.1.2. Bugfixed v2.7.0.
% 0.07/0.13 % Command : toma --casc %s
% 0.13/0.34 % Computer : n024.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 22:12:53 EDT 2023
% 0.13/0.34 % CPUTime :
% 3.83/4.09 % SZS status Unsatisfiable
% 3.83/4.09 % SZS output start Proof
% 3.83/4.09 original problem:
% 3.83/4.09 axioms:
% 3.83/4.09 double_divide(double_divide(A, double_divide(double_divide(identity(), B), double_divide(C, double_divide(B, A)))), double_divide(identity(), identity())) = C
% 3.83/4.09 multiply(A, B) = double_divide(double_divide(B, A), identity())
% 3.83/4.09 inverse(A) = double_divide(A, identity())
% 3.83/4.09 identity() = double_divide(A, inverse(A))
% 3.83/4.09 goal:
% 3.83/4.09 multiply(a(), b()) != multiply(b(), a())
% 3.83/4.09 To show the unsatisfiability of the original goal,
% 3.83/4.09 it suffices to show that multiply(a(), b()) = multiply(b(), a()) (skolemized goal) is valid under the axioms.
% 3.83/4.09 Here is an equational proof:
% 3.83/4.09 0: double_divide(double_divide(X0, double_divide(double_divide(identity(), X1), double_divide(X2, double_divide(X1, X0)))), double_divide(identity(), identity())) = X2.
% 3.83/4.09 Proof: Axiom.
% 3.83/4.09
% 3.83/4.09 1: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 3.83/4.09 Proof: Axiom.
% 3.83/4.09
% 3.83/4.09 2: inverse(X0) = double_divide(X0, identity()).
% 3.83/4.09 Proof: Axiom.
% 3.83/4.09
% 3.83/4.09 3: identity() = double_divide(X0, inverse(X0)).
% 3.83/4.09 Proof: Axiom.
% 3.83/4.09
% 3.83/4.09 4: double_divide(double_divide(X0, double_divide(double_divide(identity(), X1), double_divide(X2, double_divide(X1, X0)))), inverse(identity())) = X2.
% 3.83/4.09 Proof: Rewrite equation 0,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 5: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 3.83/4.09 Proof: Rewrite equation 1,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 6: X2 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X3), double_divide(X2, inverse(X3)))), inverse(identity())).
% 3.83/4.09 Proof: A critical pair between equations 4 and 2.
% 3.83/4.09
% 3.83/4.09 7: X2 = double_divide(double_divide(X0, double_divide(inverse(identity()), double_divide(X2, double_divide(identity(), X0)))), inverse(identity())).
% 3.83/4.09 Proof: A critical pair between equations 4 and 2.
% 3.83/4.09
% 3.83/4.09 8: X2 = double_divide(double_divide(inverse(X3), double_divide(double_divide(identity(), X3), double_divide(X2, identity()))), inverse(identity())).
% 3.83/4.09 Proof: A critical pair between equations 4 and 3.
% 3.83/4.09
% 3.83/4.09 9: X2 = double_divide(double_divide(X0, double_divide(identity(), double_divide(X2, double_divide(inverse(identity()), X0)))), inverse(identity())).
% 3.83/4.09 Proof: A critical pair between equations 4 and 3.
% 3.83/4.09
% 3.83/4.09 10: X2 = double_divide(double_divide(inverse(identity()), double_divide(double_divide(identity(), double_divide(X3, double_divide(double_divide(identity(), X4), double_divide(X5, double_divide(X4, X3))))), double_divide(X2, X5))), inverse(identity())).
% 3.83/4.09 Proof: A critical pair between equations 4 and 4.
% 3.83/4.09
% 3.83/4.09 11: X2 = double_divide(double_divide(double_divide(identity(), identity()), double_divide(double_divide(identity(), double_divide(X3, double_divide(double_divide(identity(), X4), double_divide(X5, double_divide(X4, X3))))), double_divide(X2, X5))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 10,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 12: X2 = double_divide(double_divide(X0, double_divide(identity(), double_divide(X2, double_divide(double_divide(identity(), identity()), X0)))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 9,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 13: X2 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(identity(), X3), double_divide(X2, identity()))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 8,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 14: X2 = double_divide(double_divide(X0, double_divide(double_divide(identity(), identity()), double_divide(X2, double_divide(identity(), X0)))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 7,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 15: X2 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X3), double_divide(X2, double_divide(X3, identity())))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 6,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 16: identity() = double_divide(X0, double_divide(X0, identity())).
% 3.83/4.09 Proof: Rewrite equation 3,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 17: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 3.83/4.09 Proof: Rewrite equation 5,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 18: double_divide(double_divide(X0, double_divide(double_divide(identity(), X1), double_divide(X2, double_divide(X1, X0)))), double_divide(identity(), identity())) = X2.
% 3.83/4.09 Proof: Rewrite equation 4,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 19: double_divide(identity(), identity()) = double_divide(double_divide(identity(), double_divide(identity(), identity())), double_divide(identity(), identity())).
% 3.83/4.09 Proof: A critical pair between equations 12 and 16.
% 3.83/4.09
% 3.83/4.09 20: double_divide(identity(), X3) = double_divide(double_divide(double_divide(X3, identity()), identity()), double_divide(identity(), identity())).
% 3.83/4.09 Proof: A critical pair between equations 13 and 16.
% 3.83/4.09
% 3.83/4.09 23: X2 = double_divide(double_divide(double_divide(double_divide(identity(), identity()), identity()), double_divide(identity(), double_divide(X2, identity()))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: A critical pair between equations 12 and 16.
% 3.83/4.09
% 3.83/4.09 29: double_divide(X3, double_divide(identity(), double_divide(X4, double_divide(double_divide(identity(), identity()), X3)))) = double_divide(double_divide(identity(), double_divide(double_divide(identity(), identity()), X4)), double_divide(identity(), identity())).
% 3.83/4.09 Proof: A critical pair between equations 14 and 12.
% 3.83/4.09
% 3.83/4.09 30: double_divide(double_divide(X4, identity()), double_divide(double_divide(identity(), X4), double_divide(X5, identity()))) = double_divide(double_divide(identity(), double_divide(double_divide(identity(), identity()), X5)), double_divide(identity(), identity())).
% 3.83/4.09 Proof: A critical pair between equations 14 and 13.
% 3.83/4.09
% 3.83/4.09 35: X2 = double_divide(double_divide(inverse(inverse(identity())), double_divide(identity(), inverse(X2))), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 23,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2].
% 3.83/4.09
% 3.83/4.09 38: double_divide(identity(), X3) = double_divide(inverse(inverse(X3)), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 20,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2].
% 3.83/4.09
% 3.83/4.09 39: inverse(identity()) = double_divide(double_divide(identity(), inverse(identity())), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 19,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 40: double_divide(double_divide(X0, double_divide(double_divide(identity(), X1), double_divide(X2, double_divide(X1, X0)))), inverse(identity())) = X2.
% 3.83/4.09 Proof: Rewrite equation 18,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 41: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 3.83/4.09 Proof: Rewrite equation 17,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 42: identity() = double_divide(X0, inverse(X0)).
% 3.83/4.09 Proof: Rewrite equation 16,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 43: X2 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X3), double_divide(X2, inverse(X3)))), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 15,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 44: X2 = double_divide(double_divide(X0, double_divide(inverse(identity()), double_divide(X2, double_divide(identity(), X0)))), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 14,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 45: X2 = double_divide(double_divide(inverse(X3), double_divide(double_divide(identity(), X3), inverse(X2))), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 13,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2].
% 3.83/4.09
% 3.83/4.09 47: X2 = double_divide(double_divide(inverse(identity()), double_divide(double_divide(identity(), double_divide(X3, double_divide(double_divide(identity(), X4), double_divide(X5, double_divide(X4, X3))))), double_divide(X2, X5))), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 11,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 48: double_divide(inverse(X4), double_divide(double_divide(identity(), X4), inverse(X5))) = double_divide(double_divide(identity(), double_divide(inverse(identity()), X5)), inverse(identity())).
% 3.83/4.09 Proof: Rewrite equation 30,
% 3.83/4.09 lhs with equations [2,2]
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 49: double_divide(X3, double_divide(identity(), double_divide(X4, double_divide(inverse(identity()), X3)))) = double_divide(inverse(X4), double_divide(double_divide(identity(), X4), inverse(X4))).
% 3.83/4.09 Proof: Rewrite equation 29,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [2,2,48].
% 3.83/4.09
% 3.83/4.09 58: X2 = double_divide(double_divide(inverse(identity()), double_divide(inverse(identity()), inverse(X2))), inverse(identity())).
% 3.83/4.09 Proof: A critical pair between equations 45 and 2.
% 3.83/4.09
% 3.83/4.09 67: X2 = double_divide(double_divide(double_divide(identity(), identity()), double_divide(double_divide(identity(), identity()), double_divide(X2, identity()))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 58,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2].
% 3.83/4.09
% 3.83/4.09 71: X2 = double_divide(double_divide(double_divide(identity(), identity()), double_divide(double_divide(identity(), double_divide(X3, double_divide(double_divide(identity(), X4), double_divide(X5, double_divide(X4, X3))))), double_divide(X2, X5))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 47,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 73: X2 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(identity(), X3), double_divide(X2, identity()))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 45,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2].
% 3.83/4.09
% 3.83/4.09 74: X2 = double_divide(double_divide(X0, double_divide(double_divide(identity(), identity()), double_divide(X2, double_divide(identity(), X0)))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 44,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 75: X2 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X3), double_divide(X2, double_divide(X3, identity())))), double_divide(identity(), identity())).
% 3.83/4.09 Proof: Rewrite equation 43,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 76: identity() = double_divide(X0, double_divide(X0, identity())).
% 3.83/4.09 Proof: Rewrite equation 42,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 77: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 3.83/4.09 Proof: Rewrite equation 41,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 78: double_divide(double_divide(X0, double_divide(double_divide(identity(), X1), double_divide(X2, double_divide(X1, X0)))), double_divide(identity(), identity())) = X2.
% 3.83/4.09 Proof: Rewrite equation 40,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 79: double_divide(identity(), identity()) = identity().
% 3.83/4.09 Proof: Rewrite equation 39,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [2,76,2,76].
% 3.83/4.09
% 3.83/4.09 80: double_divide(identity(), X3) = double_divide(double_divide(double_divide(X3, identity()), identity()), identity()).
% 3.83/4.09 Proof: Rewrite equation 38,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,79].
% 3.83/4.09
% 3.83/4.09 82: X2 = double_divide(double_divide(identity(), double_divide(identity(), double_divide(X2, identity()))), identity()).
% 3.83/4.09 Proof: Rewrite equation 35,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,79,2,79,2,2,79].
% 3.83/4.09
% 3.83/4.09 87: double_divide(X3, double_divide(identity(), double_divide(X4, double_divide(identity(), X3)))) = double_divide(double_divide(X4, identity()), double_divide(double_divide(identity(), X4), double_divide(X4, identity()))).
% 3.83/4.09 Proof: Rewrite equation 49,
% 3.83/4.09 lhs with equations [2,79]
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 88: double_divide(double_divide(X4, identity()), double_divide(double_divide(identity(), X4), double_divide(X5, identity()))) = double_divide(double_divide(identity(), double_divide(identity(), X5)), identity()).
% 3.83/4.09 Proof: Rewrite equation 48,
% 3.83/4.09 lhs with equations [2,2]
% 3.83/4.09 rhs with equations [2,79,2,79].
% 3.83/4.09
% 3.83/4.09 92: identity() = double_divide(double_divide(identity(), double_divide(identity(), double_divide(X3, identity()))), X3).
% 3.83/4.09 Proof: A critical pair between equations 76 and 82.
% 3.83/4.09
% 3.83/4.09 93: double_divide(identity(), double_divide(X4, identity())) = double_divide(double_divide(identity(), X4), identity()).
% 3.83/4.09 Proof: A critical pair between equations 80 and 80.
% 3.83/4.09
% 3.83/4.09 106: double_divide(identity(), inverse(X4)) = inverse(double_divide(identity(), X4)).
% 3.83/4.09 Proof: Rewrite equation 93,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 107: identity() = double_divide(inverse(double_divide(identity(), double_divide(identity(), X3))), X3).
% 3.83/4.09 Proof: Rewrite equation 92,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,106,106].
% 3.83/4.09
% 3.83/4.09 114: double_divide(identity(), X3) = inverse(inverse(inverse(X3))).
% 3.83/4.09 Proof: Rewrite equation 80,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2].
% 3.83/4.09
% 3.83/4.09 115: inverse(identity()) = identity().
% 3.83/4.09 Proof: Rewrite equation 79,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 116: inverse(double_divide(X0, double_divide(inverse(inverse(inverse(X1))), double_divide(X2, double_divide(X1, X0))))) = X2.
% 3.83/4.09 Proof: Rewrite equation 78,
% 3.83/4.09 lhs with equations [114,2,115,2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 117: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 3.83/4.09 Proof: Rewrite equation 77,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 119: X2 = inverse(inverse(inverse(inverse(double_divide(inverse(inverse(inverse(X3))), double_divide(X2, inverse(X3))))))).
% 3.83/4.09 Proof: Rewrite equation 75,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [114,2,114,2,115,2].
% 3.83/4.09
% 3.83/4.09 120: X2 = inverse(double_divide(X0, inverse(inverse(inverse(double_divide(X2, inverse(inverse(inverse(X0))))))))).
% 3.83/4.09 Proof: Rewrite equation 74,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,115,114,114,2,115,2].
% 3.83/4.09
% 3.83/4.09 121: X2 = inverse(double_divide(inverse(X3), double_divide(inverse(inverse(inverse(X3))), inverse(X2)))).
% 3.83/4.09 Proof: Rewrite equation 73,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,114,2,2,115,2].
% 3.83/4.09
% 3.83/4.09 122: X2 = inverse(inverse(inverse(inverse(double_divide(inverse(inverse(X5)), double_divide(X2, X5)))))).
% 3.83/4.09 Proof: Rewrite equation 71,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,115,114,114,116,114,2,115,2].
% 3.83/4.09
% 3.83/4.09 124: X2 = inverse(inverse(inverse(inverse(inverse(inverse(inverse(inverse(X2)))))))).
% 3.83/4.09 Proof: Rewrite equation 67,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,115,2,115,2,114,114,2,115,2].
% 3.83/4.09
% 3.83/4.09 125: double_divide(inverse(X4), double_divide(inverse(inverse(inverse(X4))), inverse(X5))) = inverse(inverse(inverse(inverse(inverse(inverse(inverse(X5))))))).
% 3.83/4.09 Proof: Rewrite equation 88,
% 3.83/4.09 lhs with equations [2,114,2]
% 3.83/4.09 rhs with equations [114,114,2].
% 3.83/4.09
% 3.83/4.09 126: double_divide(X3, inverse(inverse(inverse(double_divide(X4, inverse(inverse(inverse(X3)))))))) = inverse(inverse(inverse(inverse(inverse(inverse(inverse(X4))))))).
% 3.83/4.09 Proof: Rewrite equation 87,
% 3.83/4.09 lhs with equations [114,114]
% 3.83/4.09 rhs with equations [2,114,2,125].
% 3.83/4.09
% 3.83/4.09 134: identity() = double_divide(inverse(inverse(inverse(inverse(double_divide(identity(), X3))))), X3).
% 3.83/4.09 Proof: A critical pair between equations 107 and 114.
% 3.83/4.09
% 3.83/4.09 136: inverse(inverse(inverse(inverse(inverse(X5))))) = inverse(inverse(inverse(X5))).
% 3.83/4.09 Proof: A critical pair between equations 119 and 121.
% 3.83/4.09
% 3.83/4.09 141: double_divide(double_divide(double_divide(double_divide(double_divide(X5, identity()), identity()), identity()), identity()), identity()) = double_divide(double_divide(double_divide(X5, identity()), identity()), identity()).
% 3.83/4.09 Proof: Rewrite equation 136,
% 3.83/4.09 lhs with equations [2,2,2,2,2]
% 3.83/4.09 rhs with equations [2,2,2].
% 3.83/4.09
% 3.83/4.09 143: identity() = double_divide(double_divide(double_divide(double_divide(double_divide(double_divide(identity(), X3), identity()), identity()), identity()), identity()), X3).
% 3.83/4.09 Proof: Rewrite equation 134,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2].
% 3.83/4.09
% 3.83/4.09 150: X2 = double_divide(double_divide(double_divide(double_divide(X2, identity()), identity()), identity()), identity()).
% 3.83/4.09 Proof: Rewrite equation 124,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2,2,141,2,2,141,2].
% 3.83/4.09
% 3.83/4.09 152: X2 = double_divide(double_divide(double_divide(X5, identity()), identity()), double_divide(X2, X5)).
% 3.83/4.09 Proof: Rewrite equation 122,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2,2,2,150].
% 3.83/4.09
% 3.83/4.09 153: X2 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(double_divide(double_divide(X3, identity()), identity()), identity()), double_divide(X2, identity()))), identity()).
% 3.83/4.09 Proof: Rewrite equation 121,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2,2,2].
% 3.83/4.09
% 3.83/4.09 154: X2 = double_divide(double_divide(X0, double_divide(double_divide(double_divide(double_divide(X2, double_divide(double_divide(double_divide(X0, identity()), identity()), identity())), identity()), identity()), identity())), identity()).
% 3.83/4.09 Proof: Rewrite equation 120,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2,2,2,2].
% 3.83/4.09
% 3.83/4.09 156: multiply(X0, X1) = double_divide(double_divide(X1, X0), identity()).
% 3.83/4.09 Proof: Rewrite equation 117,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 158: double_divide(identity(), identity()) = identity().
% 3.83/4.09 Proof: Rewrite equation 115,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 159: double_divide(identity(), X3) = double_divide(double_divide(double_divide(X3, identity()), identity()), identity()).
% 3.83/4.09 Proof: Rewrite equation 114,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2].
% 3.83/4.09
% 3.83/4.09 165: double_divide(identity(), double_divide(X4, identity())) = double_divide(double_divide(identity(), X4), identity()).
% 3.83/4.09 Proof: Rewrite equation 106,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 169: double_divide(X3, double_divide(identity(), double_divide(X4, double_divide(identity(), X3)))) = double_divide(identity(), double_divide(identity(), double_divide(X4, identity()))).
% 3.83/4.09 Proof: Rewrite equation 126,
% 3.83/4.09 lhs with equations [2,2,2,159,2,2,2,159]
% 3.83/4.09 rhs with equations [2,2,2,159,2,165,2,165,2,165,159,2,165,165].
% 3.83/4.09
% 3.83/4.09 170: double_divide(double_divide(X4, identity()), double_divide(double_divide(identity(), X4), double_divide(X5, identity()))) = double_divide(identity(), double_divide(identity(), double_divide(X5, identity()))).
% 3.83/4.09 Proof: Rewrite equation 125,
% 3.83/4.09 lhs with equations [2,2,2,2,159,2]
% 3.83/4.09 rhs with equations [2,2,2,159,2,165,2,165,2,165,159,2,165,165].
% 3.83/4.09
% 3.83/4.09 171: identity() = double_divide(double_divide(identity(), X3), X3).
% 3.83/4.09 Proof: A critical pair between equations 143 and 150.
% 3.83/4.09
% 3.83/4.09 176: X2 = double_divide(double_divide(identity(), identity()), double_divide(X2, identity())).
% 3.83/4.09 Proof: A critical pair between equations 152 and 158.
% 3.83/4.09
% 3.83/4.09 177: double_divide(double_divide(double_divide(X6, identity()), identity()), identity()) = double_divide(X6, identity()).
% 3.83/4.09 Proof: A critical pair between equations 141 and 150.
% 3.83/4.09
% 3.83/4.09 181: X2 = double_divide(X6, double_divide(X2, double_divide(double_divide(X6, identity()), identity()))).
% 3.83/4.09 Proof: A critical pair between equations 152 and 150.
% 3.83/4.09
% 3.83/4.09 182: X2 = double_divide(double_divide(identity(), X6), double_divide(X2, double_divide(X6, identity()))).
% 3.83/4.09 Proof: A critical pair between equations 152 and 159.
% 3.83/4.09
% 3.83/4.09 183: X2 = double_divide(double_divide(identity(), X6), double_divide(X2, inverse(X6))).
% 3.83/4.09 Proof: Rewrite equation 182,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 184: X2 = double_divide(X6, double_divide(X2, inverse(inverse(X6)))).
% 3.83/4.09 Proof: Rewrite equation 181,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 187: inverse(inverse(inverse(X6))) = inverse(X6).
% 3.83/4.09 Proof: Rewrite equation 177,
% 3.83/4.09 lhs with equations [2,2,2]
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 188: X2 = double_divide(identity(), inverse(X2)).
% 3.83/4.09 Proof: Rewrite equation 176,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [158,2].
% 3.83/4.09
% 3.83/4.09 190: double_divide(inverse(X4), double_divide(double_divide(identity(), X4), inverse(X5))) = double_divide(identity(), X5).
% 3.83/4.09 Proof: Rewrite equation 170,
% 3.83/4.09 lhs with equations [2,2]
% 3.83/4.09 rhs with equations [2,188].
% 3.83/4.09
% 3.83/4.09 191: X4 = inverse(double_divide(identity(), X4)).
% 3.83/4.09 Proof: Rewrite equation 165,
% 3.83/4.09 lhs with equations [2,188]
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 193: double_divide(identity(), X3) = inverse(X3).
% 3.83/4.09 Proof: Rewrite equation 159,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,187].
% 3.83/4.09
% 3.83/4.09 194: inverse(identity()) = identity().
% 3.83/4.09 Proof: Rewrite equation 158,
% 3.83/4.09 lhs with equations [2]
% 3.83/4.09 rhs with equations [].
% 3.83/4.09
% 3.83/4.09 196: multiply(X0, X1) = inverse(double_divide(X1, X0)).
% 3.83/4.09 Proof: Rewrite equation 156,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2].
% 3.83/4.09
% 3.83/4.09 197: X2 = inverse(double_divide(X0, inverse(double_divide(X2, inverse(X0))))).
% 3.83/4.09 Proof: Rewrite equation 154,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,187,2,2,2,187,2].
% 3.83/4.09
% 3.83/4.09 198: X2 = inverse(double_divide(inverse(X3), double_divide(inverse(X3), inverse(X2)))).
% 3.83/4.09 Proof: Rewrite equation 153,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2,2,2,187,2,2].
% 3.83/4.09
% 3.83/4.09 199: X2 = double_divide(inverse(inverse(X5)), double_divide(X2, X5)).
% 3.83/4.09 Proof: Rewrite equation 152,
% 3.83/4.09 lhs with equations []
% 3.83/4.09 rhs with equations [2,2].
% 3.83/4.09
% 3.83/4.09 200: X2 = inverse(inverse(X2)).
% 3.83/4.09 Proof: Rewrite equation 150,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [2,2,2,187,2].
% 3.83/4.10
% 3.83/4.10 201: double_divide(X3, inverse(double_divide(X4, inverse(X3)))) = inverse(X4).
% 3.83/4.10 Proof: Rewrite equation 169,
% 3.83/4.10 lhs with equations [193,193]
% 3.83/4.10 rhs with equations [2,193,200,193].
% 3.83/4.10
% 3.83/4.10 202: X2 = double_divide(X7, double_divide(X2, X7)).
% 3.83/4.10 Proof: A critical pair between equations 184 and 200.
% 3.83/4.10
% 3.83/4.10 209: X2 = inverse(double_divide(inverse(X3), inverse(double_divide(X2, X3)))).
% 3.83/4.10 Proof: A critical pair between equations 197 and 200.
% 3.83/4.10
% 3.83/4.10 210: X2 = double_divide(inverse(X6), double_divide(X2, double_divide(identity(), X6))).
% 3.83/4.10 Proof: A critical pair between equations 199 and 191.
% 3.83/4.10
% 3.83/4.10 211: inverse(X4) = double_divide(inverse(X5), inverse(double_divide(X4, X5))).
% 3.83/4.10 Proof: A critical pair between equations 201 and 200.
% 3.83/4.10
% 3.83/4.10 212: X7 = double_divide(double_divide(identity(), double_divide(X8, inverse(X7))), inverse(X8)).
% 3.83/4.10 Proof: A critical pair between equations 183 and 201.
% 3.83/4.10
% 3.83/4.10 214: X7 = double_divide(double_divide(identity(), double_divide(X8, double_divide(X7, identity()))), double_divide(X8, identity())).
% 3.83/4.10 Proof: Rewrite equation 212,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [2,2].
% 3.83/4.10
% 3.83/4.10 215: double_divide(X4, identity()) = double_divide(double_divide(X5, identity()), double_divide(double_divide(X4, X5), identity())).
% 3.83/4.10 Proof: Rewrite equation 211,
% 3.83/4.10 lhs with equations [2]
% 3.83/4.10 rhs with equations [2,2].
% 3.83/4.10
% 3.83/4.10 216: X2 = double_divide(double_divide(X6, identity()), double_divide(X2, double_divide(identity(), X6))).
% 3.83/4.10 Proof: Rewrite equation 210,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [2].
% 3.83/4.10
% 3.83/4.10 217: X2 = double_divide(double_divide(X2, identity()), identity()).
% 3.83/4.10 Proof: Rewrite equation 209,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [2,2,215,2].
% 3.83/4.10
% 3.83/4.10 221: X2 = double_divide(double_divide(double_divide(X3, identity()), double_divide(double_divide(X3, identity()), double_divide(X2, identity()))), identity()).
% 3.83/4.10 Proof: Rewrite equation 198,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [2,2,2,2].
% 3.83/4.10
% 3.83/4.10 223: multiply(X0, X1) = double_divide(identity(), double_divide(X1, X0)).
% 3.83/4.10 Proof: Rewrite equation 196,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [193].
% 3.83/4.10
% 3.83/4.10 225: double_divide(identity(), identity()) = identity().
% 3.83/4.10 Proof: Rewrite equation 194,
% 3.83/4.10 lhs with equations [193]
% 3.83/4.10 rhs with equations [].
% 3.83/4.10
% 3.83/4.10 226: X4 = double_divide(identity(), double_divide(identity(), X4)).
% 3.83/4.10 Proof: Rewrite equation 191,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [193].
% 3.83/4.10
% 3.83/4.10 227: double_divide(double_divide(identity(), X4), double_divide(double_divide(identity(), X4), double_divide(identity(), X5))) = double_divide(identity(), X5).
% 3.83/4.10 Proof: Rewrite equation 190,
% 3.83/4.10 lhs with equations [193,193]
% 3.83/4.10 rhs with equations [].
% 3.83/4.10
% 3.83/4.10 230: X8 = double_divide(double_divide(X9, X8), X9).
% 3.83/4.10 Proof: A critical pair between equations 202 and 202.
% 3.83/4.10
% 3.83/4.10 231: double_divide(X6, identity()) = double_divide(double_divide(identity(), identity()), X6).
% 3.83/4.10 Proof: A critical pair between equations 215 and 217.
% 3.83/4.10
% 3.83/4.10 242: X2 = double_divide(identity(), double_divide(double_divide(X3, identity()), double_divide(double_divide(X3, identity()), double_divide(X2, identity())))).
% 3.83/4.10 Proof: Rewrite equation 221,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [231,225].
% 3.83/4.10
% 3.83/4.10 249: double_divide(X11, double_divide(identity(), X10)) = double_divide(X11, double_divide(X10, identity())).
% 3.83/4.10 Proof: A critical pair between equations 230 and 216.
% 3.83/4.10
% 3.83/4.10 254: double_divide(double_divide(identity(), X6), X11) = double_divide(double_divide(X6, identity()), X11).
% 3.83/4.10 Proof: A critical pair between equations 216 and 230.
% 3.83/4.10
% 3.83/4.10 300: double_divide(double_divide(identity(), double_divide(X12, double_divide(X13, identity()))), double_divide(identity(), X12)) = X13.
% 3.83/4.10 Proof: A critical pair between equations 249 and 214.
% 3.83/4.10
% 3.83/4.10 320: double_divide(identity(), X5) = double_divide(double_divide(identity(), double_divide(identity(), X5)), identity()).
% 3.83/4.10 Proof: A critical pair between equations 227 and 171.
% 3.83/4.10
% 3.83/4.10 323: X13 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X14), double_divide(X13, identity()))), X14).
% 3.83/4.10 Proof: A critical pair between equations 300 and 226.
% 3.83/4.10
% 3.83/4.10 324: double_divide(identity(), X5) = double_divide(X5, identity()).
% 3.83/4.10 Proof: Rewrite equation 320,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [226].
% 3.83/4.10
% 3.83/4.10 344: X15 = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X14), double_divide(identity(), X15))), X14).
% 3.83/4.10 Proof: A critical pair between equations 323 and 324.
% 3.83/4.10
% 3.83/4.10 364: double_divide(identity(), X5) = double_divide(double_divide(identity(), double_divide(identity(), X5)), identity()).
% 3.83/4.10 Proof: A critical pair between equations 227 and 171.
% 3.83/4.10
% 3.83/4.10 366: double_divide(X17, identity()) = double_divide(double_divide(identity(), double_divide(double_divide(identity(), X14), X17)), X14).
% 3.83/4.10 Proof: A critical pair between equations 344 and 202.
% 3.83/4.10
% 3.83/4.10 367: double_divide(identity(), X5) = double_divide(X5, identity()).
% 3.83/4.10 Proof: Rewrite equation 364,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [226].
% 3.83/4.10
% 3.83/4.10 388: double_divide(X17, identity()) = double_divide(double_divide(identity(), double_divide(double_divide(X18, identity()), X17)), X18).
% 3.83/4.10 Proof: A critical pair between equations 366 and 367.
% 3.83/4.10
% 3.83/4.10 408: double_divide(X13, identity()) = double_divide(identity(), double_divide(identity(), double_divide(identity(), X13))).
% 3.83/4.10 Proof: A critical pair between equations 226 and 249.
% 3.83/4.10
% 3.83/4.10 411: double_divide(double_divide(X20, double_divide(identity(), X19)), identity()) = double_divide(double_divide(identity(), X20), X19).
% 3.83/4.10 Proof: A critical pair between equations 388 and 216.
% 3.83/4.10
% 3.83/4.10 420: double_divide(identity(), double_divide(X20, double_divide(identity(), X19))) = double_divide(double_divide(identity(), X20), X19).
% 3.83/4.10 Proof: Rewrite equation 411,
% 3.83/4.10 lhs with equations [408,226]
% 3.83/4.10 rhs with equations [].
% 3.83/4.10
% 3.83/4.10 424: double_divide(identity(), double_divide(X6, double_divide(identity(), X11))) = double_divide(double_divide(X6, identity()), X11).
% 3.83/4.10 Proof: Rewrite equation 254,
% 3.83/4.10 lhs with equations [420]
% 3.83/4.10 rhs with equations [].
% 3.83/4.10
% 3.83/4.10 447: X2 = double_divide(X3, double_divide(X3, X2)).
% 3.83/4.10 Proof: Rewrite equation 242,
% 3.83/4.10 lhs with equations []
% 3.83/4.10 rhs with equations [424,202,424,226,226].
% 3.83/4.10
% 3.83/4.10 454: double_divide(X9, X8) = double_divide(X8, X9).
% 3.83/4.10 Proof: A critical pair between equations 447 and 202.
% 3.83/4.10
% 3.83/4.10 475: multiply(a(), b()) = multiply(b(), a()).
% 3.83/4.10 Proof: Rewrite lhs with equations [223,454]
% 3.83/4.10 rhs with equations [223].
% 3.83/4.10
% 3.83/4.10 % SZS output end Proof
%------------------------------------------------------------------------------