%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : ALG280^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n108.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:18:22 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : ALG280^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n108.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 09:06:16 CDT 2014
% % CPUTime  : 300.09 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x119fea8>, <kernel.Type object at 0x119fb48>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x157c998>, <kernel.Constant object at 0x119f9e0>) of role type named cE
% Using role type
% Declaring cE:a
% FOF formula (<kernel.Constant object at 0x119fcf8>, <kernel.DependentProduct object at 0x119fb90>) of role type named cP
% Using role type
% Declaring cP:(a->(a->a))
% FOF formula (<kernel.Constant object at 0x119ff80>, <kernel.DependentProduct object at 0x119f878>) of role type named cJ
% Using role type
% Declaring cJ:(a->a)
% FOF formula (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a), (((eq a) ((cP X) cE)) X))) of role conjecture named cTHM17_pme
% Conjecture to prove = (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a), (((eq a) ((cP X) cE)) X))):Prop
% We need to prove ['(((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a), (((eq a) ((cP X) cE)) X)))']
% Parameter a:Type.
% Parameter cE:a.
% Parameter cP:(a->(a->a)).
% Parameter cJ:(a->a).
% Trying to prove (((and ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))->(forall (X:a), (((eq a) ((cP X) cE)) X)))
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x0:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x2:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x0:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x0:(P X)
% Instantiate: b:=X:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x4:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x2:(P X)
% Instantiate: b:=X:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found x0:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x2:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) ((cP (cJ Xy)) Xy))):(((eq a) ((cP X) ((cP (cJ Xy)) Xy))) ((cP X) ((cP (cJ Xy)) Xy)))
% Found (eq_ref0 ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x2:(P X)
% Instantiate: b:=X:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found x0:(P X)
% Instantiate: b:=X:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found x4:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x4 as proof of (P0 b)
% Found x30:=(x3 X):(((eq a) ((cP cE) X)) X)
% Instantiate: b:=X:a
% Found x30 as proof of (((eq a) ((cP cE) X)) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x0:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x0 as proof of (P0 b)
% Found x40:=(x4 X):(((eq a) ((cP cE) X)) X)
% Instantiate: b:=X:a
% Found x40 as proof of (((eq a) ((cP cE) X)) b)
% Found x2:(P ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x2 as proof of (P0 b)
% Found x40:=(x4 X):(((eq a) ((cP cE) X)) X)
% Instantiate: b:=X:a
% Found x40 as proof of (((eq a) ((cP cE) X)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) ((cP (cJ Xy)) Xy))):(((eq a) ((cP X) ((cP (cJ Xy)) Xy))) ((cP X) ((cP (cJ Xy)) Xy)))
% Found (eq_ref0 ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found x4:(P X)
% Instantiate: b:=X:a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) ((cP (cJ Xy)) Xy))):(((eq a) ((cP X) ((cP (cJ Xy)) Xy))) ((cP X) ((cP (cJ Xy)) Xy)))
% Found (eq_ref0 ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found ((eq_ref a) ((cP X) ((cP (cJ Xy)) Xy))) as proof of (((eq a) ((cP X) ((cP (cJ Xy)) Xy))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x4:(P X)
% Instantiate: b:=X:a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found x4:(P X)
% Instantiate: b:=X:a
% Found x4 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found x0:(P X)
% Instantiate: b:=X:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found x2:(P X)
% Instantiate: b:=X:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found x2:(P X)
% Instantiate: b:=X:a
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) ((cP (cJ Xy)) Xy)))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found x30:=(x3 cE):(((eq a) ((cP cE) cE)) cE)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (eq_sym010 (x3 cE)) as proof of (P b)
% Found ((eq_sym01 b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x30:=(x3 cE):(((eq a) ((cP cE) cE)) cE)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (eq_sym010 (x3 cE)) as proof of (P b)
% Found ((eq_sym01 b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found x30:=(x3 cE):(((eq a) ((cP cE) cE)) cE)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (eq_sym010 (x3 cE)) as proof of (P b)
% Found ((eq_sym01 b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found x30:=(x3 ((cP X) cE)):(((eq a) ((cP cE) ((cP X) cE))) ((cP X) cE))
% Instantiate: b:=((cP X) cE):a
% Found x30 as proof of (((eq a) ((cP cE) ((cP X) cE))) b)
% Found x4:(P X)
% Instantiate: b:=X:a
% Found x4 as proof of (P0 b)
% Found x30:=(x3 X):(((eq a) ((cP cE) X)) X)
% Instantiate: a0:=((cP cE) X):a
% Found x30 as proof of (((eq a) a0) X)
% Found x30:=(x3 X):(((eq a) ((cP cE) X)) X)
% Instantiate: a0:=((cP cE) X):a
% Found x30 as proof of (((eq a) a0) X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found x40:=(x4 X):(((eq a) ((cP cE) X)) X)
% Instantiate: a0:=((cP cE) X):a
% Found x40 as proof of (((eq a) a0) X)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 ((cP X) cE)):(((eq a) ((cP X) cE)) ((cP X) cE))
% Found (eq_ref0 ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found ((eq_ref a) ((cP X) cE)) as proof of (((eq a) ((cP X) cE)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found x40:=(x4 X):(((eq a) ((cP cE) X)) X)
% Instantiate: a0:=((cP cE) X):a
% Found x40 as proof of (((eq a) a0) X)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found x40:=(x4 X):(((eq a) ((cP cE) X)) X)
% Instantiate: a0:=((cP cE) X):a
% Found x40 as proof of (((eq a) a0) X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found x40:=(x4 X):(((eq a) ((cP cE) X)) X)
% Instantiate: a0:=((cP cE) X):a
% Found x40 as proof of (((eq a) a0) X)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found x30:=(x3 cE):(((eq a) ((cP cE) cE)) cE)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (eq_sym010 (x3 cE)) as proof of (P b)
% Found ((eq_sym01 b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found (fun (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of (P b)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of ((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->(P b))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->(P b)))
% Found (and_rect10 (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found x30:=(x3 cE):(((eq a) ((cP cE) cE)) cE)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (eq_sym010 (x3 cE)) as proof of (P b)
% Found ((eq_sym01 b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found (fun (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of (P b)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of ((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->(P b))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->(P b)))
% Found (and_rect10 (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found (fun (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))))) as proof of (P b)
% Found (fun (x0:((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))))) as proof of ((forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE))->(P b))
% Found (fun (x0:((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))))) as proof of (((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))->((forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE))->(P b)))
% Found (and_rect00 (fun (x0:((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))))) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x0:((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))))) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))->((forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE))->P0)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE))) P0) x0) x)) (P b)) (fun (x0:((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))))) as proof of (P b)
% Found (((fun (P0:Type) (x0:(((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))->((forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE))->P0)))=> (((((and_rect ((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE))) P0) x0) x)) (P b)) (fun (x0:((and (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))) (x1:(forall (Xy:a), (((eq a) ((cP (cJ Xy)) Xy)) cE)))=> (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))))) as proof of (P b)
% Found x30:=(x3 cE):(((eq a) ((cP cE) cE)) cE)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (x3 cE) as proof of (((eq a) ((cP b) cE)) b)
% Found (eq_sym010 (x3 cE)) as proof of (P b)
% Found ((eq_sym01 b) (x3 cE)) as proof of (P b)
% Found (((eq_sym0 ((cP b) cE)) b) (x3 cE)) as proof of (P b)
% Found (fun (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of (P b)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of ((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->(P b))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->(P b)))
% Found (and_rect10 (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found (((fun (P0:Type) (x2:((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))->((forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))->P0)))=> (((((and_rect (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx))) P0) x2) x0)) (P b)) (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((cP ((cP Xx) Xy)) Xz)) ((cP Xx) ((cP Xy) Xz))))) (x3:(forall (Xx:a), (((eq a) ((cP cE) Xx)) Xx)))=> (((eq_sym0 ((cP b) cE)) b) (x3 cE)))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((cP X) cE))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) cE)
% Found ((eq_ref a) a0) as proof of (((eq a) a0
% EOF
%------------------------------------------------------------------------------