%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV029^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n116.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 13:33:36 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  : SEV029^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.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 07:36:46 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 0x197b7e8>, <kernel.Type object at 0x197bfc8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xr:(a->(a->Prop))) (Xs:(a->Prop)), (((and ((and (forall (Xx:a), ((Xs Xx)->((Xr Xx) Xx)))) (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xr Xx) Xy)) ((Xr Xy) Xz))->((Xr Xx) Xz))))->((and ((and (forall (Xa:(a->Prop)), (((ex a) (fun (Xx:a)=> (forall (Xx_1:a), ((iff (Xa Xx_1)) ((Xr Xx) Xx_1)))))->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (Xs Xx)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> (forall (Xx_2:a), ((iff (S Xx_2)) ((Xr Xx0) Xx_2)))))) (S Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))) of role conjecture named cTHM260A_pme
% Conjecture to prove = (forall (Xr:(a->(a->Prop))) (Xs:(a->Prop)), (((and ((and (forall (Xx:a), ((Xs Xx)->((Xr Xx) Xx)))) (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xr Xx) Xy)) ((Xr Xy) Xz))->((Xr Xx) Xz))))->((and ((and (forall (Xa:(a->Prop)), (((ex a) (fun (Xx:a)=> (forall (Xx_1:a), ((iff (Xa Xx_1)) ((Xr Xx) Xx_1)))))->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (Xs Xx)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> (forall (Xx_2:a), ((iff (S Xx_2)) ((Xr Xx0) Xx_2)))))) (S Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xr:(a->(a->Prop))) (Xs:(a->Prop)), (((and ((and (forall (Xx:a), ((Xs Xx)->((Xr Xx) Xx)))) (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xr Xx) Xy)) ((Xr Xy) Xz))->((Xr Xx) Xz))))->((and ((and (forall (Xa:(a->Prop)), (((ex a) (fun (Xx:a)=> (forall (Xx_1:a), ((iff (Xa Xx_1)) ((Xr Xx) Xx_1)))))->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (Xs Xx)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> (forall (Xx_2:a), ((iff (S Xx_2)) ((Xr Xx0) Xx_2)))))) (S Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->Prop))) (Xs:(a->Prop)), (((and ((and (forall (Xx:a), ((Xs Xx)->((Xr Xx) Xx)))) (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xr Xx) Xy)) ((Xr Xy) Xz))->((Xr Xx) Xz))))->((and ((and (forall (Xa:(a->Prop)), (((ex a) (fun (Xx:a)=> (forall (Xx_1:a), ((iff (Xa Xx_1)) ((Xr Xx) Xx_1)))))->((ex a) (fun (Xx:a)=> (Xa Xx)))))) (forall (Xx:a), ((iff (Xs Xx)) ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and ((ex a) (fun (Xx0:a)=> (forall (Xx_2:a), ((iff (S Xx_2)) ((Xr Xx0) Xx_2)))))) (S Xx)))))))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xb):(((eq (a->Prop)) Xb) (fun (x:a)=> (Xb x)))
% Found (eta_expansion_dep00 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))):(((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))
% Found (eq_ref0 (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found x10:(P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P0 Xb)
% Found ex_ind:(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P)))
% Instantiate: b:=(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P))):Prop
% Found ex_ind as proof of b
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))):(((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))
% Found (eq_ref0 (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found x10:(P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P0 Xb)
% Found x10:(P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P0 Xb)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))):(((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc))))
% Found (eq_ref0 (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found ((eq_ref Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) as proof of (((eq Prop) (forall (Xb:(a->Prop)) (Xc:(a->Prop)), (((and ((and ((ex a) (fun (Xx:a)=> (forall (Xx_3:a), ((iff (Xb Xx_3)) ((Xr Xx) Xx_3)))))) ((ex a) (fun (Xx:a)=> (forall (Xx_4:a), ((iff (Xc Xx_4)) ((Xr Xx) Xx_4))))))) ((ex a) (fun (Xx:a)=> ((and (Xb Xx)) (Xc Xx)))))->(((eq (a->Prop)) Xb) Xc)))) b)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion000:=(eta_expansion00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion0 Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion a) Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion a) Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion a) Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 (Xb x1)):(((eq Prop) (Xb x1)) (Xb x1))
% Found (eq_ref0 (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found eq_ref00:=(eq_ref0 (Xb x1)):(((eq Prop) (Xb x1)) (Xb x1))
% Found (eq_ref0 (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x10:(P Xc)
% Found (fun (x10:(P Xc))=> x10) as proof of (P Xc)
% Found (fun (x10:(P Xc))=> x10) as proof of (P0 Xc)
% Found x10:(P Xc)
% Found (fun (x10:(P Xc))=> x10) as proof of (P Xc)
% Found (fun (x10:(P Xc))=> x10) as proof of (P0 Xc)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found x30:(P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P Xb)
% Found (fun (x30:(P Xb))=> x30) as proof of (P0 Xb)
% Found x20:(P (Xb x1))
% Found (fun (x20:(P (Xb x1)))=> x20) as proof of (P (Xb x1))
% Found (fun (x20:(P (Xb x1)))=> x20) as proof of (P0 (Xb x1))
% Found x20:(P (Xb x1))
% Found (fun (x20:(P (Xb x1)))=> x20) as proof of (P (Xb x1))
% Found (fun (x20:(P (Xb x1)))=> x20) as proof of (P0 (Xb x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x1:(P Xb)
% Instantiate: b:=Xb:(a->Prop)
% Found x1 as proof of (P0 b)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eta_expansion000:=(eta_expansion00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion0 Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion a) Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion a) Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion a) Prop) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found eq_ref00:=(eq_ref0 (Xb x3)):(((eq Prop) (Xb x3)) (Xb x3))
% Found (eq_ref0 (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found eq_ref00:=(eq_ref0 (Xb x3)):(((eq Prop) (Xb x3)) (Xb x3))
% Found (eq_ref0 (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found x1:(P Xb)
% Instantiate: f:=Xb:(a->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P Xb)
% Instantiate: f:=Xb:(a->Prop)
% Found x1 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found x30:(P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P0 Xc)
% Found x30:(P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P0 Xc)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eta_expansion000:=(eta_expansion00 Xb):(((eq (a->Prop)) Xb) (fun (x:a)=> (Xb x)))
% Found (eta_expansion00 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eta_expansion0 Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xc x)))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (Xc x2))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) (Xc x)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eta_expansion000:=(eta_expansion00 Xb):(((eq (a->Prop)) Xb) (fun (x:a)=> (Xb x)))
% Found (eta_expansion00 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eta_expansion0 Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found x40:(P (Xb x3))
% Found (fun (x40:(P (Xb x3)))=> x40) as proof of (P (Xb x3))
% Found (fun (x40:(P (Xb x3)))=> x40) as proof of (P0 (Xb x3))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x40:(P (Xb x3))
% Found (fun (x40:(P (Xb x3)))=> x40) as proof of (P (Xb x3))
% Found (fun (x40:(P (Xb x3)))=> x40) as proof of (P0 (Xb x3))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x3:(P Xb)
% Instantiate: b:=Xb:(a->Prop)
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xc):(((eq (a->Prop)) Xc) Xc)
% Found (eq_ref0 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eq_ref (a->Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eq_ref (a->Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eq_ref (a->Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found x10:(P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P Xb)
% Found (fun (x10:(P Xb))=> x10) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eta_expansion000:=(eta_expansion00 Xb):(((eq (a->Prop)) Xb) (fun (x:a)=> (Xb x)))
% Found (eta_expansion00 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eta_expansion0 Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion a) Prop) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eq_ref00:=(eq_ref0 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xb)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found eq_ref00:=(eq_ref0 (Xb x3)):(((eq Prop) (Xb x3)) (Xb x3))
% Found (eq_ref0 (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found eq_ref00:=(eq_ref0 (Xb x3)):(((eq Prop) (Xb x3)) (Xb x3))
% Found (eq_ref0 (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found eq_ref00:=(eq_ref0 (Xc x1)):(((eq Prop) (Xc x1)) (Xc x1))
% Found (eq_ref0 (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found eq_ref00:=(eq_ref0 (Xc x1)):(((eq Prop) (Xc x1)) (Xc x1))
% Found (eq_ref0 (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xc):(((eq (a->Prop)) Xc) (fun (x:a)=> (Xc x)))
% Found (eta_expansion_dep00 Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xc) as proof of (((eq (a->Prop)) Xc) b)
% Found x1:(P0 b)
% Instantiate: b:=Xb:(a->Prop)
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 Xb)
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 Xb))
% Found (fun (P0:((a->Prop)->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (Xc x1)):(((eq Prop) (Xc x1)) (Xc x1))
% Found (eq_ref0 (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found eq_ref00:=(eq_ref0 (Xc x1)):(((eq Prop) (Xc x1)) (Xc x1))
% Found (eq_ref0 (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found ((eq_ref Prop) (Xc x1)) as proof of (((eq Prop) (Xc x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xb x1))
% Found x3:(P Xb)
% Instantiate: f:=Xb:(a->Prop)
% Found x3 as proof of (P0 f)
% Found x3:(P Xb)
% Instantiate: f:=Xb:(a->Prop)
% Found x3 as proof of (P0 f)
% Found x70:(P Xb)
% Found (fun (x70:(P Xb))=> x70) as proof of (P Xb)
% Found (fun (x70:(P Xb))=> x70) as proof of (P0 Xb)
% 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 Xb):(((eq (a->Prop)) Xb) Xb)
% Found (eq_ref0 Xb) as proof of (((eq (a->Prop)) Xb) b0)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b0)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b0)
% Found ((eq_ref (a->Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b0)
% Found eq_ref00:=(eq_ref0 (Xb x5)):(((eq Prop) (Xb x5)) (Xb x5))
% Found (eq_ref0 (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found ((eq_ref Prop) (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found ((eq_ref Prop) (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found ((eq_ref Prop) (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x5))
% Found eq_ref00:=(eq_ref0 (Xb x5)):(((eq Prop) (Xb x5)) (Xb x5))
% Found (eq_ref0 (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found ((eq_ref Prop) (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found ((eq_ref Prop) (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found ((eq_ref Prop) (Xb x5)) as proof of (((eq Prop) (Xb x5)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x5))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x5))
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) Xc)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xb):(((eq (a->Prop)) Xb) (fun (x:a)=> (Xb x)))
% Found (eta_expansion_dep00 Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xb) as proof of (((eq (a->Prop)) Xb) b)
% Found x30:(P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P0 Xc)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found x30:(P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P Xc)
% Found (fun (x30:(P Xc))=> x30) as proof of (P0 Xc)
% Found x50:(P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P Xb)
% Found (fun (x50:(P Xb))=> x50) as proof of (P0 Xb)
% Found eq_ref00:=(eq_ref0 (Xb x3)):(((eq Prop) (Xb x3)) (Xb x3))
% Found (eq_ref0 (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found eq_ref00:=(eq_ref0 (Xb x3)):(((eq Prop) (Xb x3)) (Xb x3))
% Found (eq_ref0 (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found ((eq_ref Prop) (Xb x3)) as proof of (((eq Prop) (Xb x3)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x3))
% Found eq_ref00:=(eq_ref0 (Xb x1)):(((eq Prop) (Xb x1)) (Xb x1))
% Found (eq_ref0 (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Xc x1))
% Found eq_ref00:=(eq_ref0 (Xb x1)):(((eq Prop) (Xb x1)) (Xb x1))
% Found (eq_ref0 (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found ((eq_ref Prop) (Xb x1)) as proof of (((eq Prop) (Xb x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Xc x1))
% Found ((eq_ref Prop) b) as proof of (
% EOF
%------------------------------------------------------------------------------