%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV306^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n189.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 13:34:01 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV306^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n189.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:47:41 CDT 2014
% % CPUTime  : 300.10 
% 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 0x20b84d0>, <kernel.DependentProduct object at 0x2307cb0>) of role type named cK
% Using role type
% Declaring cK:((fofType->Prop)->(fofType->Prop))
% FOF formula ((forall (X:(fofType->Prop)) (Y:(fofType->Prop)), ((forall (Xx:fofType), ((X Xx)->(Y Xx)))->(forall (Xx:fofType), (((cK X) Xx)->((cK Y) Xx)))))->((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U)))) of role conjecture named cTHM2F_pme
% Conjecture to prove = ((forall (X:(fofType->Prop)) (Y:(fofType->Prop)), ((forall (Xx:fofType), ((X Xx)->(Y Xx)))->(forall (Xx:fofType), (((cK X) Xx)->((cK Y) Xx)))))->((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U)))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (X:(fofType->Prop)) (Y:(fofType->Prop)), ((forall (Xx:fofType), ((X Xx)->(Y Xx)))->(forall (Xx:fofType), (((cK X) Xx)->((cK Y) Xx)))))->((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))))']
% Parameter fofType:Type.
% Parameter cK:((fofType->Prop)->(fofType->Prop)).
% Trying to prove ((forall (X:(fofType->Prop)) (Y:(fofType->Prop)), ((forall (Xx:fofType), ((X Xx)->(Y Xx)))->(forall (Xx:fofType), (((cK X) Xx)->((cK Y) Xx)))))->((ex (fofType->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))))
% Found eta_expansion000:=(eta_expansion00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))):(((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) (fun (x:(fofType->Prop))=> (((eq (fofType->Prop)) (cK x)) x)))
% Found (eta_expansion00 (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) b)
% Found ((eta_expansion0 Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U))) b)
% 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)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (cK x)) x)))
% 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)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (((eq (fofType->Prop)) (cK x0)) x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (((eq (fofType->Prop)) (cK x)) x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (fun (x:fofType)=> ((cK x0) x)))
% Found (eta_expansion_dep00 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (fun (x:fofType)=> ((cK x0) x)))
% Found (eta_expansion_dep00 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U)))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (U:(fofType->Prop))=> (((eq (fofType->Prop)) (cK U)) U)))
% Found x1:(P (cK x0))
% Instantiate: b:=(cK x0):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P (cK x0))
% Instantiate: f:=(cK x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found x1:(P (cK x0))
% Instantiate: f:=(cK x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (fun (x:fofType)=> ((cK x0) x)))
% Found (eta_expansion_dep00 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: x0:=(cK b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (cK x0))
% Found (eq_ref0 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eq_ref (fofType->Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eq_ref (fofType->Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eq_ref (fofType->Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((cK x0) x)))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((cK x0) x)))
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: x0:=(cK a):(fofType->Prop)
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x1:(P x0)
% Instantiate: x0:=(cK a):(fofType->Prop)
% Found x1 as proof of (P0 (cK a))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x10:(P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P2 x0)
% Found x10:(P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P1 x0)
% Found (fun (x10:(P1 x0))=> x10) as proof of (P2 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x20:(P1 ((cK x0) x1))
% Found (fun (x20:(P1 ((cK x0) x1)))=> x20) as proof of (P1 ((cK x0) x1))
% Found (fun (x20:(P1 ((cK x0) x1)))=> x20) as proof of (P2 ((cK x0) x1))
% Found x20:(P1 ((cK x0) x1))
% Found (fun (x20:(P1 ((cK x0) x1)))=> x20) as proof of (P1 ((cK x0) x1))
% Found (fun (x20:(P1 ((cK x0) x1)))=> x20) as proof of (P2 ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found x2:(P (x0 x1))
% Instantiate: x0:=(cK b):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x2:(P (x0 x1))
% Instantiate: x0:=(cK b):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P0 b)
% Instantiate: b:=((cK x0) x1):Prop
% Found (fun (x2:(P0 b))=> x2) as proof of (P0 ((cK x0) x1))
% Found (fun (P0:(Prop->Prop)) (x2:(P0 b))=> x2) as proof of ((P0 b)->(P0 ((cK x0) x1)))
% Found (fun (P0:(Prop->Prop)) (x2:(P0 b))=> x2) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P0 b)
% Instantiate: b:=((cK x0) x1):Prop
% Found (fun (x2:(P0 b))=> x2) as proof of (P0 ((cK x0) x1))
% Found (fun (P0:(Prop->Prop)) (x2:(P0 b))=> x2) as proof of ((P0 b)->(P0 ((cK x0) x1)))
% Found (fun (P0:(Prop->Prop)) (x2:(P0 b))=> x2) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found x2:(P (x0 x1))
% Instantiate: b:=(x0 x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found x2:(P (x0 x1))
% Instantiate: b:=(x0 x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found x2:(P (x0 x1))
% Instantiate: b:=(x0 x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found x2:(P (x0 x1))
% Instantiate: b:=(x0 x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (fun (x:fofType)=> ((cK x0) x)))
% Found (eta_expansion_dep00 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b)
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found x20:(P b)
% Found (fun (x20:(P b))=> x20) as proof of (P b)
% Found (fun (x20:(P b))=> x20) as proof of (P0 b)
% Found x20:(P b)
% Found (fun (x20:(P b))=> x20) as proof of (P b)
% Found (fun (x20:(P b))=> x20) as proof of (P0 b)
% Found x20:(P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P0 (b x1))
% Found x20:(P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P0 (b x1))
% Found x20:(P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P2 (x0 x1))
% Found x20:(P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P2 (x0 x1))
% Found x20:(P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P2 (x0 x1))
% Found x20:(P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P2 (x0 x1))
% Found x20:(P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P2 (x0 x1))
% Found x20:(P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P1 (x0 x1))
% Found (fun (x20:(P1 (x0 x1)))=> x20) as proof of (P2 (x0 x1))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% 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 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) 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 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=(cK x0):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=(cK x0):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) 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) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found x20:(P b)
% Found (fun (x20:(P b))=> x20) as proof of (P b)
% Found (fun (x20:(P b))=> x20) as proof of (P0 b)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x20:(P b)
% Found (fun (x20:(P b))=> x20) as proof of (P b)
% Found (fun (x20:(P b))=> x20) as proof of (P0 b)
% Found x20:(P b)
% Found (fun (x20:(P b))=> x20) as proof of (P b)
% Found (fun (x20:(P b))=> x20) as proof of (P0 b)
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P (x0 x1))
% Found (fun (x20:(P (x0 x1)))=> x20) as proof of (P0 (x0 x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) 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 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% 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 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) 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 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x1)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) 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 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b0)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) 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 x1:(P (cK x0))
% Instantiate: b:=(cK x0):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found x1:(P (cK x0))
% Instantiate: f:=(cK x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found x1:(P (cK x0))
% Instantiate: f:=(cK x0):(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) (x0 x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 (b x1)):(((eq Prop) (b x1)) (b x1))
% Found (eq_ref0 (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found ((eq_ref Prop) (b x1)) as proof of (((eq Prop) (b x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (x0 x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (fun (x:fofType)=> ((cK x0) x)))
% Found (eta_expansion_dep00 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P0 (b x1))
% Found x20:(P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P (b x1))
% Found (fun (x20:(P (b x1)))=> x20) as proof of (P0 (b x1))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: x0:=(cK b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: b:=x0:(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (cK x0)):(((eq (fofType->Prop)) (cK x0)) (cK x0))
% Found (eq_ref0 (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eq_ref (fofType->Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eq_ref (fofType->Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found ((eq_ref (fofType->Prop)) (cK x0)) as proof of (((eq (fofType->Prop)) (cK x0)) b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found x10:(P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P x0)
% Found (fun (x10:(P x0))=> x10) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) x0)
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x20:(P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P ((cK x0) x1))
% Found (fun (x20:(P ((cK x0) x1)))=> x20) as proof of (P0 ((cK x0) x1))
% Found x1:(P x0)
% Instantiate: x0:=(cK b):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((cK x0) x)))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((cK x0) x)))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((cK x0) x)))
% Found x1:(P x0)
% Instantiate: f:=x0:(fofType->Prop)
% Found x1 as proof of (P0 f)
% 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)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((cK x0) x2))
% Found (fun (x2:fofType)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((cK x0) x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((cK x0) x1))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found x2:(P ((cK x0) x1))
% Instantiate: b:=((cK x0) x1):Prop
% Found x2 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (cK x0))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found x1:(P x0)
% Instantiate: x0:=(cK a):(fofType->Prop)
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x1:(P x0)
% Instantiate: x0:=(cK a):(fofType->Prop)
% Found x1 as proof of (P0 (cK a))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x1:(P x0)
% Instantiate: x0:=(cK a):(fofType->Prop)
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x1:(P x0)
% Instantiate: x0:=(cK a):(fofType->Prop)
% Found x1 as proof of (P0 (cK a))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x1))
% Found eq_ref00:=(eq_ref0 (x0 x1)):(((eq Prop) (x0 x1)) (x0 x1))
% Found (eq_ref0 (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found ((eq_ref Prop) (x0 x1)) as proof of (((eq Prop) (x0 x1)) b0)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found eq_ref00:=(eq_ref0 ((cK x0) x1)):(((eq Prop) ((cK x0) x1)) ((cK x0) x1))
% Found (eq_ref0 ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found ((eq_ref Prop) ((cK x0) x1)) as proof of (((eq Prop) ((cK x0) x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b)
% EOF
%------------------------------------------------------------------------------