TSTP Solution File: SEV173^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV173^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n102.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 13:33:50 EDT 2014

% Result   : Timeout 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV173^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n102.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 08:20:26 CDT 2014
% % CPUTime  : 300.04 
% 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 0x1c2a3b0>, <kernel.Type object at 0x1c2a6c8>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1c2add0>, <kernel.DependentProduct object at 0x1c2b830>) of role type named cZ
% Using role type
% Declaring cZ:(b->Prop)
% FOF formula (<kernel.Constant object at 0x1c2af80>, <kernel.DependentProduct object at 0x1c2b710>) of role type named cR
% Using role type
% Declaring cR:(b->Prop)
% FOF formula (<kernel.Constant object at 0x1c2a3b0>, <kernel.DependentProduct object at 0x1c2b7a0>) of role type named cS
% Using role type
% Declaring cS:(b->Prop)
% FOF formula (forall (Xx:((b->(b->b))->b)), ((iff ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))) ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))))) of role conjecture named cTHM32_pme
% Conjecture to prove = (forall (Xx:((b->(b->b))->b)), ((iff ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))) ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))))):Prop
% Parameter b_DUMMY:b.
% We need to prove ['(forall (Xx:((b->(b->b))->b)), ((iff ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))) ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))))']
% Parameter b:Type.
% Parameter cZ:(b->Prop).
% Parameter cR:(b->Prop).
% Parameter cS:(b->Prop).
% Trying to prove (forall (Xx:((b->(b->b))->b)), ((iff ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))) ((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR x)) (cS x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion0 Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR x)) (cZ x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion0 Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR x)) (cS x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion0 Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eq_ref0 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eq_ref0 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion0000:=(eta_expansion000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR x)) (cZ x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion0 Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion b) Prop) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eq_ref0 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eq_ref0 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eq_ref (b->Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR x)) (cZ x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion_dep00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion_dep0 (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR x)) (cS x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion_dep00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion_dep0 (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR x)) (cZ x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion_dep00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion_dep0 (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex b)):(((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))->((ex b) (fun (x:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR x)) (cS x)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) x)))))))
% Found (eta_expansion_dep000 (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((eta_expansion_dep00 (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found (((eta_expansion_dep0 (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found ((((eta_expansion_dep b) (fun (x2:b)=> Prop)) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))) (ex b)) as proof of (P (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: b0:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: b0:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref00:=(eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: b0:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P b0)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: b0:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref00:=(eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y)))))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref00:=(eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref000:=(eq_ref00 (ex b)):(((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))))
% Found (eq_ref00 (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (ex b)) as proof of (P (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eq_ref00:=(eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (eq_ref0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eq_ref (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found x1:((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Instantiate: b0:=(fun (Y:b)=> ((and ((and ((or (cR x0)) (cS x0))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))):(b->Prop)
% Found x1 as proof of (P b0)
% Found x1:((ex b) (fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))))
% Instantiate: b0:=(fun (Y:b)=> ((and ((and ((or (cR x0)) (cZ x0))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x0) Y))))):(b->Prop)
% Found x1 as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion_dep00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))):(((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y))))))))
% Found (eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found (((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) as proof of (((eq (b->Prop)) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) b0)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found x:((ex b) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Instantiate: f:=(fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))):(b->Prop)
% Found x as proof of (P f)
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->(P0 (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->(P0 (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->(P0 (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cZ x))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((eta_expansion00 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found (((eta_expansion0 Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found ((((eta_expansion b) Prop) (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y)))))))) P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cZ X))) ((or (cR Y)) (cZ Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR X)) (cS X))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G X) Y))))))))->(P0 (fun (x:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (cR x)) (cS x))) ((or (cR Y)) (cS Y)))) (((eq ((b->(b->b))->b)) Xx) (fun (G:(b->(b->b)))=> ((G x) Y)))))))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (X:b)=> ((ex b) (fun (Y:b)=> ((and ((and ((or (
% EOF
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