%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU847^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n107.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 13:33:16 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU847^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n107.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 11:36:16 CDT 2014
% % CPUTime  : 300.02 
% 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 0x8129e0>, <kernel.Type object at 0x812998>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) S)) of role conjecture named cGAZING_THM41_pme
% Conjecture to prove = (forall (S:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) S)):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) S))']
% Parameter a:Type.
% Trying to prove (forall (S:(a->Prop)), (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) S))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) (fun (x:a)=> ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not False))) ((and False) (not (S x))))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))):(a->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found x:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))):(a->Prop)
% Found x as proof of (P0 f)
% 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)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))):(a->Prop)
% Found x as proof of (P0 f)
% 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)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Instantiate: b:=(fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))):(a->Prop)
% Found x as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) (fun (x:a)=> ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found x:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))):(a->Prop)
% Found x as proof of (P0 f)
% 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)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found x:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Instantiate: f:=(fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))):(a->Prop)
% Found x as proof of (P0 f)
% 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)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (S x0))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) (S x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found x:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) (fun (x:a)=> ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) (fun (x:a)=> ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x as proof of (P0 f)
% Found x:(P S)
% Instantiate: b:=S:(a->Prop)
% Found x as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not False))) ((and False) (not (S x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found x0:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Instantiate: b:=((or ((and (S x)) (False->False))) ((and False) ((S x)->False))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x0:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Instantiate: b:=((or ((and (S x)) (False->False))) ((and False) ((S x)->False))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (eq_ref0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found x:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x as proof of (P0 f)
% 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)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((and (S x)) (not False))) ((and False) (not (S x))))))
% Found x:(P S)
% Instantiate: f:=S:(a->Prop)
% Found x as proof of (P0 f)
% 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)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((or ((and (S x0)) (not False))) ((and False) (not (S x0)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((and (S x)) (not False))) ((and False) (not (S x))))))
% Found x0:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Instantiate: b:=((or ((and (S x)) (not False))) ((and False) (not (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x0:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Instantiate: b:=((or ((and (S x)) (not False))) ((and False) (not (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x2:(P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P2 S)
% Found x2:(P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P2 S)
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found x3:(P S)
% Found (fun (x3:(P S))=> x3) as proof of (P S)
% Found (fun (x3:(P S))=> x3) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b0)
% Found x2:(P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P2 S)
% Found x2:(P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P1 S)
% Found (fun (x2:(P1 S))=> x2) as proof of (P2 S)
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found x3:(P S)
% Found (fun (x3:(P S))=> x3) as proof of (P S)
% Found (fun (x3:(P S))=> x3) as proof of (P0 S)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) (fun (x:a)=> ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))
% Found (eta_expansion00 (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False))))) b)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) S)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not False))) ((and False) (not (S x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found x0:(P (S x))
% Instantiate: b:=(S x):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) x)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (fun (x01:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))=> x01) as proof of (P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (fun (x01:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))=> x01) as proof of (P0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found x01:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (fun (x01:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))=> x01) as proof of (P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (fun (x01:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))))=> x01) as proof of (P0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) S)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) S)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) S)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) S)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) b0)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found x01:(P b)
% Found (fun (x01:(P b))=> x01) as proof of (P b)
% Found (fun (x01:(P b))=> x01) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x01:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (fun (x01:(P ((or ((and (S x)) (not False))) ((and False) (not (S x))))))=> x01) as proof of (P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (fun (x01:(P ((or ((and (S x)) (not False))) ((and False) (not (S x))))))=> x01) as proof of (P0 ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found x01:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (fun (x01:(P ((or ((and (S x)) (not False))) ((and False) (not (S x))))))=> x01) as proof of (P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (fun (x01:(P ((or ((and (S x)) (not False))) ((and False) (not (S x))))))=> x01) as proof of (P0 ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found x0:(P0 b)
% Instantiate: b:=(S x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (S x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (S x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found x0:(P0 b)
% Instantiate: b:=(S x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (S x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (S x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (False->False))) ((and False) ((S Xz)->False)))))
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found x0:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Instantiate: b:=((or ((and (S x)) (False->False))) ((and False) ((S x)->False))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x0:(P ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Instantiate: b:=((or ((and (S x)) (False->False))) ((and False) ((S x)->False))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found x01:(P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P1 (S x))
% Found (fun (x01:(P1 (S x)))=> x01) as proof of (P2 (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (S x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (S x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) (fun (x:a)=> ((or ((and (S x)) (not False))) ((and False) (not (S x))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found x02:(P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P (S x))
% Found (fun (x02:(P (S x)))=> x02) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(S x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (S x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (S x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found x0:(P0 b)
% Instantiate: b:=(S x):Prop
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 (S x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 (S x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b)
% Found x0:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Instantiate: b:=((or ((and (S x)) (not False))) ((and False) (not (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found x0:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Instantiate: b:=((or ((and (S x)) (not False))) ((and False) (not (S x)))):Prop
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found x2:(P b)
% Found (fun (x2:(P b))=> x2) as proof of (P b)
% Found (fun (x2:(P b))=> x2) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))):(((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found (eq_ref0 ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) as proof of (((eq Prop) ((or ((and (S x)) (not False))) ((and False) (not (S x))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (S x)):(((eq Prop) (S x)) (S x))
% Found (eq_ref0 (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found ((eq_ref Prop) (S x)) as proof of (((eq Prop) (S x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found x2:(P S)
% Found (fun (x2:(P S))=> x2) as proof of (P S)
% Found (fun (x2:(P S))=> x2) as proof of (P0 S)
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eq_ref00:=(eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))):(((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False))))
% Found (eq_ref0 ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found ((eq_ref Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) as proof of (((eq Prop) ((or ((and (S x)) (False->False))) ((and False) ((S x)->False)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (S x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x0:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Instantiate: b:=(fun (x1:a)=> ((or ((and (S x1)) (not False))) ((and False) (not (S x1))))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion0 Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion a) Prop) S) as proof of (((eq (a->Prop)) S) b)
% Found x0:(P ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Instantiate: b:=(fun (x1:a)=> ((or ((and (S x1)) (not False))) ((and False) (not (S x1))))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 S):(((eq (a->Prop)) S) (fun (x:a)=> (S x)))
% Found (eta_expansion_dep00 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P (b x))
% Found (fun (x01:(P (b x)))=> x01) as proof of (P0 (b x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found (fun (x2:(P (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz)))))))=> x2) as proof of (P0 (fun (Xz:a)=> ((or ((and (S Xz)) (not False))) ((and False) (not (S Xz))))))
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 S):(((eq (a->Prop)) S) S)
% Found (eq_ref0 S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found ((eq_ref (a->Prop)) S) as proof of (((eq (a->Prop)) S) b)
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found eq_ref00:=(eq_ref0 (b x)):(((eq Prop) (b x)) (b x))
% Found (eq_ref0 (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found ((eq_ref Prop) (b x)) as proof of (((eq Prop) (b x)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((or ((and (S x)) (not False))) ((and False) (not (S x)))))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found x01:(P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P (S x))
% Found (fun (x01:(P (S x)))=> x01) as proof of (P0 (S x))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) ((and False) ((S x)->False)))
% Found ((eq_ref Prop) a0) as proo
% EOF
%------------------------------------------------------------------------------